Maksym Loginov
email: maksym.loginov@inra.fr

Floriane Doudies

Nicolas Hengl

Frédéric Pignon

Geneviève Gésan-Guiziou

Floriane Doudiès

Geneviève Gésan- Guiziou

Influence of membrane resistance on swelling and removal of colloidal filter cake after filtration pressure release

Keywords: filtration-consolidation theory, gel swelling, membrane rinsing, membrane fouling reversibility, membrane hydraulic resistance. Nomenclature

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In the literature, it is commonly accepted to distinguish two components of the external membrane fouling (fouling resistance): reversible fouling (which is removed from the surface by shear after the filtration pressure release/decrease or after the increasing of the shear rate) and irreversible fouling (that corresponds to the difference between the total fouling resistance and the resistance of reversible fouling). The scientific background for this distinction is DLVO theory applied to the case of a filter cake [START_REF] Bacchin | Model for colloidal fouling of membranes[END_REF][START_REF] Harmant | Coagulation of colloids retained by porous wall[END_REF][START_REF] Bacchin | A unifying model for concentration polarization, gel-layer formation and particle deposition in cross-flow membrane filtration of colloidal suspensions[END_REF][START_REF] Aimar | Slow colloidal aggregation and membrane fouling[END_REF]. Increasing of the local compressive pressure applied to cake particles (solid pressure) in the direction of membrane surface results in the decreasing of local inter-particle distance (increasing of local particle concentration). At the distance to membrane surface, where the local compressive force overcomes the interparticle repulsion force, cake particles undergo an aggregation (sol-gel transition). In contrast to outer part of the cake (sol, concentration polarization layer), the inner part of the cake (rigid gel, deposit) is not removed from the membrane surface by crossflow shear until the shear force overcomes the particle attraction force (Coulomb failure criterion applied to filter cake removal [START_REF] Benkahla | Cake growth mechanism in crossflow microfiltration of mineral suspensions[END_REF]). Therefore, appearance of irreversible external membrane fouling is associated with the gel (or deposit, dense structure) formation on membrane surface.

The extent of irreversible membrane fouling is commonly characterized with the help of cyclic variation of filtration conditions: applied wall shear stress and transmembrane pressure (or imposed filtrate flux) [START_REF] Benkahla | Cake growth mechanism in crossflow microfiltration of mineral suspensions[END_REF][START_REF] Defrance | Reversibility of fouling formed in activated sludge filtration[END_REF][START_REF] Gesan-Guiziou | Critical stability conditions in skimmed milk crossflow microfiltration: Impact on operating modes[END_REF][START_REF] Grandison | Hydrodynamic factors affecting flux and fouling during ultrafiltration of skimmed milk[END_REF][START_REF] Espinasse | On an experimental method to measure critical flux in ultrafiltration[END_REF][START_REF] Youravong | Critical flux in ultrafiltration of skimmed milk[END_REF][START_REF] Gesan-Guiziou | Cake properties in dead-end ultrafiltration of casein micelles: determination of critical operating conditions[END_REF]. The hysteresis of fouling, which is observed in coordinates foulingresistance vs. pressure (or vs. shear stress), is used in order to determine critical fouling conditions (filtration conditions that result in gel formation). It should be noticed that duration of filtration cycles (fouling formation followed by fouling removal) is typically short, and its influence on results of the experiment (extent of irreversible fouling, irreversibility of fouling in general) is out of the scope of the majority of studies.

However, a number of articles demonstrated that both the rate and the extent of fouling removal after a cross-flow and dead-end filtration increases with the increasing of duration of the pressure relaxation step [START_REF] Gesan-Guiziou | Cake properties in dead-end ultrafiltration of casein micelles: determination of critical operating conditions[END_REF][START_REF] Nakanishi | Rinsing behavior of deposited layers formed on membranes in ultrafiltration[END_REF][START_REF] Chua | Controlling fouling in membrane bioreactors operated with a variable throughput[END_REF][START_REF] Van Der Marel | An improved flux-step method to determine the critical flux and the critical flux for irreversibility in a membrane bioreactor[END_REF][START_REF] Ma | An improved step-wise method to determine critical flux of membrane bioreactor by adding relaxation zone[END_REF][START_REF] Jorgensen | On the reversibility of cake buildup and compression in a membrane bioreactor[END_REF][START_REF] Zhang | Membrane cleaning assisted by high shear stress for restoring ultrafiltration membranes fouled by dairy wastewater[END_REF][START_REF] Jorgensen | Modeling approach to describe fouling removal during relaxation[END_REF]. In other words, sufficient pressure relaxation duration can result in the following removal of "irreversible" fouling (at least partial). For example, it was demonstrated that the casein micelles gel, which is considered to cause irreversible fouling during the dead-end filtration-stirring experiment [START_REF] Qu | Dead-end filtration of sponge-like colloids: The case of casein micelle[END_REF], can swell and redisperse in course of time after the osmotic compression experiments [START_REF] Qu | On the cohesive properties of casein micelles in dense systems[END_REF] or in a different filtration experiment [START_REF] Steinhauer | Effect of pH, transmembrane pressure and whey proteins on the properties of casein micelle deposit layers[END_REF].

Articles concerning the modelling of the influence of pressure relaxation duration on the fouling extent and removal rate are rather rare in filtration literature [START_REF] Jorgensen | Modeling approach to describe fouling removal during relaxation[END_REF]. However, significant progress is attained in the modelling of gel dissolution and the hydrodynamic cleaning (i.e., rinsing) of impermeable surface from foulants (deposits), where models consider the timedependent gel swelling (solvent diffusion into a deposit) that precedes the particles/molecules redispersion by applied flow (e.g., [START_REF] Rosner | Lifetime of a highly soluble dense spherical particle[END_REF][START_REF] Lee | Prediction of polymer dissolution in swellable controlled-release systems[END_REF][START_REF] Lee | Mathematical models and experiments for swelling phenomena before dissolution of a polymer film[END_REF][START_REF] Mickaily | Hydrodynamic cleaning of a viscous film from the inside of a long tube[END_REF][START_REF] Devotta | The life time of a dissolving polymeric particle[END_REF][START_REF] Ranade | Convective diffusion from a dissolving polymeric particle[END_REF][START_REF] Gillham | Cleaning-in-place of whey protein fouling deposits: Mechanisms controlling cleaning[END_REF][START_REF] Davey | Mathematical modeling of rice cooking and dissolution in beer production[END_REF][START_REF] Xin | Cleaning rate in the uniform cleaning stage for whey protein gel deposits[END_REF][START_REF] Lu | Crossflow microfiltration of monodispersed deformable particle suspension[END_REF][START_REF] Xin | Removal of a model protein foulant from metal surfaces[END_REF][START_REF] Bouquerand | Swelling and erosion affecting flavor release from glassy particles in water[END_REF][START_REF] Kaunisto | Mechanistic modelling of drug release from polymer-coated and swelling and dissolving polymer matrix systems[END_REF][START_REF] Valois | Understanding the role of molar mass and stirring in polymer dissolution[END_REF][START_REF] Briffaz | Modelling of water transport with convection effects on amylose transfer in a swelling, eroding and gelatinizing starchy matrix[END_REF]). In addition, behavior of filter cakes (and consolidated filter cakes) after the filtration pressure decrease (or release) was modeled in the frame of conventional filtration-consolidation theory and experimentally characterized in a series of papers by Iritani, Iwata, Katagiri, Murase, and Shirato [START_REF] Murase | Expansion of consolidated material after release of load[END_REF][START_REF] Murase | Stress relaxation of expressed cake[END_REF][START_REF] Murase | Expansion of inhomogeneous cake after release of load[END_REF][START_REF] Shirato | Experimental analysis of flux decline mechanism of batch ultrafiltration (filtration characteristics of gel layer)[END_REF][START_REF] Iwata | Expansion and stress relaxation of expressed cake[END_REF][START_REF] Iwata | Theory of compression and expansion of hydrogels[END_REF][START_REF] Iritani | Consolidation and expansion of a granular bed of superabsorbent hydrogels[END_REF][START_REF] Iritani | Compression and expansion properties of filter cake accompanied with step change in applied pressure in membrane filtration[END_REF][START_REF] Iritani | Simplified evaluation of consolidation and expansion behaviour of highly compressible cake[END_REF] and Banks [START_REF] Banks | Theory of constant-rate expression and subsequent relaxation[END_REF]. Particularly, it was experimentally demonstrated that colloidal filter cakes (and hydrogels) can swell and restore lower values of local particle concentration and permeability (at least partially) after the filtration pressure release [START_REF] Shirato | Experimental analysis of flux decline mechanism of batch ultrafiltration (filtration characteristics of gel layer)[END_REF][START_REF] Iwata | Theory of compression and expansion of hydrogels[END_REF][START_REF] Iritani | Consolidation and expansion of a granular bed of superabsorbent hydrogels[END_REF][START_REF] Iritani | Compression and expansion properties of filter cake accompanied with step change in applied pressure in membrane filtration[END_REF][START_REF] Iritani | Simplified evaluation of consolidation and expansion behaviour of highly compressible cake[END_REF] and a new definition of filter cake reversibility, based on filtration resistance recovery, was introduced [START_REF] Iritani | Compression and expansion properties of filter cake accompanied with step change in applied pressure in membrane filtration[END_REF]. It was demonstrated that the response of filter cake properties (accessed from the experimentally measured filter cake resistance) on the variation of applied pressure is not immediate, and kinetics of cake relaxation is related with local cake properties (compressibility and permeability), initial cake thickness and values of compressive pressure applied during the filter cake formation and cake relaxation [START_REF] Murase | Expansion of consolidated material after release of load[END_REF][START_REF] Murase | Stress relaxation of expressed cake[END_REF][START_REF] Murase | Expansion of inhomogeneous cake after release of load[END_REF][START_REF] Shirato | Experimental analysis of flux decline mechanism of batch ultrafiltration (filtration characteristics of gel layer)[END_REF][START_REF] Iwata | Expansion and stress relaxation of expressed cake[END_REF][START_REF] Iwata | Theory of compression and expansion of hydrogels[END_REF][START_REF] Iritani | Consolidation and expansion of a granular bed of superabsorbent hydrogels[END_REF][START_REF] Iritani | Compression and expansion properties of filter cake accompanied with step change in applied pressure in membrane filtration[END_REF][START_REF] Iritani | Simplified evaluation of consolidation and expansion behaviour of highly compressible cake[END_REF]. Unfortunately, previous articles did not take into consideration any possible influence of filtration membrane resistance on the filter cake relaxation and the following membrane rinsing.

The present article is devoted to the modeling of colloidal filter cake swelling after the filtration pressure release in the frame of filtration-consolidation theory. It is assumed that the filter cake compression is reversible, in particular, the local filter cake concentration can decrease below the concentration of sol-gel transition during the swelling. In this case, the sol-gel transition is defined as the transition from liquid-like to solid-like rheological behavior. The present article considers the influence of the membrane resistance to filtrate inflow into the cake on the cake swelling kinetics, and analyzes the influence of parameters (the filter cake thickness, compressibility and permeability, the membrane hydraulic resistance, the swelling duration, flow behavior of swelled material) on the following membrane rinsing by tangential shear. Several possible methods for the experimental verification of model results are proposed and the importance of swelling duration and membrane resistance for the fouling removal is discussed.

Description of a filter cake swelling on membrane surface

When a liquid flows across a porous body under a pressure difference ΔP (e.g., across a filter cake and a membrane during a constant pressure dead-end filtration of particulate suspension, Fig. 1a), the liquid pressure p l in capillaries between particles and in membrane pores decreases in the flow direction. is ω the material coordinate. For colloidal objects, distribution of the local osmotic pressure π in cake and suspension is equivalent to distribution of p s in particulate filter cake, while the liquid pressure p l is not determined.

The liquid pressure decreases because of the friction (viscous drag), which exists between the moving liquid and stationary particles (pores surface). The same force with an opposite sign is exerted on solids by the moving liquid. In the conventional filtration-consolidation theory, the net force applied to particles or porous body is divided by the cross-sectional area of the whole system and represented as an effective (fictive) solid compressive pressure p s (e.g., [START_REF] Olivier | Modelling of cake filtration: An overview[END_REF]). In the case of the pressure filtration, the local force balance (i.e., force balance in a thin cross-sectional layer of the filter cake or filtration membrane) is presented by Eq. (1):

dp s = -dp l (1)
Eq. ( 1) is used in the majority of pressure filtration models [START_REF] Olivier | Modelling of cake filtration: An overview[END_REF][START_REF] Lee | Theories of cake filtration and consolidation and implications to sludge dewatering[END_REF][START_REF] Tiller | The role of porosity in filtration. Part XI: Filtration followed by expression[END_REF]. Given that in suspension p l = ΔP and p s = 0 (there is no relative solid-liquid movement and friction), integration of Eq. (1)

across the system of a filter cake and filtration membrane yields

ΔP = p l + p s (2)
Eqs.

(1) implies that p s increases with decreasing of p l (in the flow direction, Fig. 1a). The solid compressive pressure increases because the compressive force is transmitted and accumulated in the flow direction across the porous body (from particle to particle, from filter cake to membrane, and along the membrane pores). At the given moment of filtration, the solid pressure within the cake increases from zero at the cake-suspension interface to a maximum at the cake-membrane interface (which is denoted herein as p m ). It further increases across the membrane (linearly with the depth, if the membrane is homogeneous and incompressible) and reaches p s = ∆P at the external membrane surface. In filtrate, both p l = 0 and p s = 0 because the applied pressure is transmitted to a membrane support (Fig. 1a).

Eq. ( 1) allows substituting the liquid pressure gradient dp l /dx in the Darcy equation

𝑢𝑢 ∝ d𝑝𝑝 𝑙𝑙 d𝑥𝑥 ⁄ (3)
with the solid pressure gradient dp s /dx and applying the Darcy equation for description of liquid flow in compressible filter cakes (i.e., filter cakes with p s -dependent permeability and particle concentration):

𝑢𝑢 = - 1 𝜇𝜇𝜇𝜇(𝑝𝑝 𝑠𝑠 )𝜌𝜌 𝑠𝑠 d𝑝𝑝 𝑠𝑠 d𝜔𝜔 ( 4
)
where u is the apparent liquid velocity relative to solids, µ is liquid viscosity, α is local specific resistance, ρ s is density of the solids, ω is material coordinate. Therefore, Eq. ( 4) is widely used for description of formation and consolidation of particulate filter cakes [START_REF] Olivier | Modelling of cake filtration: An overview[END_REF][START_REF] Lee | Theories of cake filtration and consolidation and implications to sludge dewatering[END_REF][START_REF] Tiller | The role of porosity in filtration. Part XI: Filtration followed by expression[END_REF].

Usually, membrane filtration of colloidal objects that have a considerable osmotic pressure π is not discussed in terms of local values of p l and p s , which characterize porous bodies and particulate filter cakes; in this case Eq. (4) can not be applied immediately. However, Elimelech

and Bhattacharjee [START_REF] Elimelech | A novel approach for modeling concentration polarization in crossflow membrane filtration based on the equivalence of osmotic pressure model and filtration theory[END_REF] demonstrated that the drag force, which is applied by moving liquid to a layer of colloidal particles or solute molecules retained by the membrane during filtration is equal to a thermodynamic force, which is exerted by solvent molecules on these particles (solute molecules). The thermodynamic force is related with the solvent chemical potential (and not with the solvent pressure alone as the drag force applied to the liquid in conventional filtrationconsolidation theory for particulate and porous systems), and it is directly proportional to a local osmotic pressure gradient dπ/dx [START_REF] Elimelech | A novel approach for modeling concentration polarization in crossflow membrane filtration based on the equivalence of osmotic pressure model and filtration theory[END_REF]. Since the drag force is directly proportional to the apparent liquid velocity u (according to Stokes equation), it follows that

𝑢𝑢 ∝ -d𝜋𝜋 d𝑥𝑥 ⁄ (5)
during the liquid flow across the layer of colloidal particles (solute molecules) [START_REF] Elimelech | A novel approach for modeling concentration polarization in crossflow membrane filtration based on the equivalence of osmotic pressure model and filtration theory[END_REF]. Eq. ( 5) can be transformed into Eq. ( 4), where p s stands for π and local properties of colloidal particle layer are characterized in terms of conventional filtration-consolidation theory (specific cake resistance and solid volume fraction) instead of collective diffusion coefficient and number particle concentration, which are usually employed for description of colloids with considerable osmotic pressure. That is why Eq. ( 4) was applied for description of membrane filtration and consolidation of colloids with considerable osmotic pressure and without a defined capillary structure: protein solutions [START_REF] Iritani | Determination of cake properties in ultrafiltration of nano-colloids based on single step-up pressure filtration test[END_REF] and hydrogel particles (polymer networks) [START_REF] Iwata | Theoretical analysis of centrifugal dewatering of superabsorbent hydrogels using Terzaghi-Voigt combined model[END_REF][START_REF] Tanaka | Kinetics of swelling of gels[END_REF]. Distribution of π during filtration of colloidal objects is equivalent to distribution of p s (which is presented in Fig. 1a), while p l is usually not determined for colloidal filter cakes.

For majority of materials the increasing of solid pressure causes the filter cake compression, i.e., plastic and elastic structure deformation and increasing of solids concentration (because of the decreasing of inter-particle distance together with deformation and squeezing of aggregates and porous particles). For many materials (especially, for colloids with considerable osmotic pressure, e.g., repulsing nanoparticles, polymer molecules, soft particles having elastic polymer networks), the structure deformation under compression can be reversible, at least partially. After release of the applied pressure, which maintained a solid pressure in the reversibly compressible filter cake, the remaining solid pressure tends to dissipate via the structure expansion, as a compressed spring after the load release. Consequently, the liquid flows inside the filter cake, the filter cake swells (expands) (Fig. 1b). In the case of reversibly compressible particulate filter cake, this liquid inflow can be seen as a result of the momentary liquid pressure reduction in the filter cake: after the release of ΔP, p l in the filter cake becomes negative (relative to the atmospheric pressure) in accordance to Eq. ( 2). This decreasing of p l can be seen as result of a tension created in liquid-filled pores and capillaries by expanding elastic solid structure of compressed particulate filter cake. In the case of reversibly compressible colloidal filter cakes with considerable osmotic pressure, the liquid inflow after the pressure release can be seen as osmotic effect. The liquid is absorbed by colloidal filter cake because its excessive osmotic pressure (relative to the osmotic pressure of filtrate and bulk suspensions, which are assumed to be negligible herein) is not compensated by the applied pressure after the pressure release. In both cases, the liquid flows inside the cake, in the direction of higher solid compressive (or osmotic) pressure areas after the release of applied pressure (Fig. 1b).

Basic equations of filtration-consolidation theory (which are normally used to describe cake formation and compression [START_REF] Olivier | Modelling of cake filtration: An overview[END_REF][START_REF] Shirato | Constant pressure expression of solidliquid mixtures with medium resistance[END_REF][START_REF] Tien | Principles of filtration[END_REF]) hold to describe the filter cake swelling as well: a) the continuity equation for the liquid flow in an infinitesimal layer of a filter cake

� 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 � 𝜔𝜔 = -� 𝜕𝜕𝑢𝑢 𝜕𝜕𝜔𝜔 � 𝑡𝑡 (6)
(where t is time, ω is material coordinate, e is local void ratio, which is equal to porosity divided by solids volume fraction (see Eq. (B3) in Appendix B), u is the apparent liquid velocity relative to solids); b) the Darcy equation ( 4); c) and the force balance Eq. ( 1).

Their combination yields the basic consolidation equation [START_REF] Murase | Analysis of expression operations[END_REF] 𝜕𝜕𝑝𝑝

𝑠𝑠 𝜕𝜕𝜕𝜕 = - 𝜕𝜕𝑝𝑝 𝑠𝑠 𝜕𝜕𝜕𝜕 • 𝜕𝜕 𝜕𝜕𝜔𝜔 � 1 𝜇𝜇𝜌𝜌 𝑠𝑠 𝜇𝜇 𝜕𝜕𝑝𝑝 𝑠𝑠 𝜕𝜕𝜔𝜔 � (7)
which can describe the filter cake swelling, if appropriate initial and boundary conditions are used. In the case of colloidal filter cakes with considerable osmotic pressure, p s in Eq. ( 7) and in the following discussion denotes π.

It should be underlined that derivation of Eq. ( 7) from Eq. ( 6) is based on the assumption that the local void fraction explicitly depends on the solid pressure only (and does not depend on time). Therefore, it is assumed that the filter cake behaves a Terzaghi element [START_REF] Iwata | Theoretical analysis of centrifugal dewatering of superabsorbent hydrogels using Terzaghi-Voigt combined model[END_REF]; creep is not considered during the swelling in this model.

Fig. 1b presents the filter cake on the membrane surface at the moment of filtration pressure release. At end of filtration there is a certain solid pressure distribution within the filter cake -the initial solid pressure distribution in the swelling cake p si (ω)

p s = p si (ω) at t = 0 (8)
At the cake-suspension surface the solid pressure is naught (if suspension is diluted and the sample weight is negligible)

p s = 0 at ω = ω 0 (9)
At the moment after filtration pressure release the solid pressure at the cake-membrane surface p m is maximal (as it was during the filter cake formation), while it is absent at the filtrate side of the membrane (Fig. 1b). It is reasonable to assume that during the swelling, the liquid (filtrate) can flow across the membrane into the filter cake under the solid pressure gradient, which is equal to p m (this is also a usual presentation of osmotic flow across a semipermeable membrane). If it is assumed that membrane is a thin semipermeable incompressible continuum, this flow can be described using the Darcy equation as

𝑢𝑢 = 𝑝𝑝 𝑚𝑚 𝜇𝜇𝑟𝑟 𝑖𝑖𝑖𝑖 at ω = 0 ( 10
)
as soon as p s = 0 in the filtrate. In Eq. ( 10), r in is the outside-in membrane hydraulic resistance (resistance to the liquid flow across the membrane from the filtrate side into the cell). The value of r in can differ from the value of the usually discussed inside-out membrane resistance r m (e.g., [START_REF] Qin | Direct osmosis for reverse osmosis fouling control: Principles, applications and recent developments[END_REF]). The liquid flow also penetrates an infinitesimal filter cake layer adjacent to the membrane surface that can be described as

𝑢𝑢 = 1 𝜇𝜇𝜇𝜇 𝑚𝑚 𝜌𝜌 𝑠𝑠 d𝑝𝑝 𝑠𝑠 d𝜔𝜔 at ω = 0 (11)
where α m is specific resistance of this layer. Combination of Eqs. ( 10) and ( 11) yields the second boundary condition required for solution of Eq. ( 7):

d𝑝𝑝 𝑠𝑠 d𝜔𝜔 = 𝜇𝜇 𝑚𝑚 𝜌𝜌 𝑠𝑠 𝑟𝑟 𝑖𝑖𝑖𝑖 𝑝𝑝 𝑚𝑚 at ω = 0 (12)
In Eq. ( 12) p m and α m change with time (decrease) during the swelling.

Eq. ( 12) differs the present paper from important works on the filter cake behavior after the filtration pressure release [37-39, 41, 42, 45, 46, 52], because previous models did not consider the membrane as a hindrance for filtrate inflow into the cake during the swelling (though the value of membrane resistance is usually included in the modeling of cake formation and cake expression [START_REF] Iritani | Compression and expansion properties of filter cake accompanied with step change in applied pressure in membrane filtration[END_REF][START_REF] Iritani | Determination of cake properties in ultrafiltration of nano-colloids based on single step-up pressure filtration test[END_REF][START_REF] Vorobiev | Derivation of filtration equations incorporating the effect of pressure redistribution on the cake-medium interface: A constant pressure filtration[END_REF][START_REF] Iritani | Determination of pressure dependence of permeability characteristics from single constant pressure filtration test[END_REF][START_REF] Iritani | Determination of cake filtration characteristics of dilute suspension of bentonite from various filtration tests[END_REF][START_REF] Iritani | Evaluation of variable pressure dead-end ultrafiltration behaviors of nanocolloids based on filter cake properties[END_REF]).

On the possibility of filtrate inflow and absorption across the membrane

In the previous section it was assumed that the liquid can flow inside the filter cake after the filtration pressure release due to the existence of solid pressure (or osmotic pressure) gradient across the semipermeable membrane. As it follows from Eqs. ( 1) and ( 2), this implies that the liquid pressure in the membrane pores is negative p l < 0 (relative to the atmospheric pressure in filtrate). In the case of particulate filter cakes, the liquid pressure is pores of relaxing cake is negative too (while the liquid pressure in osmotic systems is not determined). Surprisingly, the negative values of liquid pressure in relaxing filter cake or filtration membrane were not pointed out in previous studies on filter cakes expansion during the relaxation [START_REF] Murase | Expansion of consolidated material after release of load[END_REF]. However, this phenomenon is recognized in discussions on osmotic flow across the semipermeable membrane [START_REF] Soodak | Forum on osmosis. IV. More on osmosis and diffusion[END_REF][START_REF] Hirsch | Negative solvent pressure in osmosis: A Gedanken experiment[END_REF][START_REF] Ph | Biological physics: energy, information[END_REF][START_REF] Mauro | Osmotic flow in a rigid porous membrane[END_REF] and clay swelling in soil mechanics [START_REF] Terzaghi | Theoretical soil mechanics[END_REF].

It is necessary to discuss the conditions, which enable the water flow across the membrane (and inside the particular filter cake) under the negative liquid pressure. Though purified and degassed water can exist for a certain time even at pressure below 0 Pa (i.e., under the tension above 1.01•10 5 Pa), it is expected that in a real case application of high tension to the liquid inside porous system will result in formation of cavities or dissolved gas bubbles [START_REF] Soodak | Forum on osmosis. IV. More on osmosis and diffusion[END_REF][START_REF] Mauro | Osmotic flow in a rigid porous membrane[END_REF]. This would reduce the tension and interrupt or slow down the liquid transport across the membrane [START_REF] Soodak | Forum on osmosis. IV. More on osmosis and diffusion[END_REF][START_REF] Mauro | Osmotic flow in a rigid porous membrane[END_REF][START_REF] Gao | Independence of hydraulic pressures on the feed and draw solutions in the osmotically driven membrane processes[END_REF][START_REF] Zhang | Metastable state of water and performance of osmotically driven membrane processes[END_REF][START_REF] Pashley | Unusual properties of water: Effects on desalination processes[END_REF]. The probability of bubbles and cavities formation must be highly dependent on the actual membrane pore structure. We can speculate that it will less likely happen in very small pores (when the pore size is smaller than the critical size of bubble formation) and in membranes with large distant pores (when the actual liquid pressure gradient across the pore is lower that the liquid pressure gradient calculated from the nominal membrane resistance). In any case, the risk of bubbles formation and flow interruption increases with decreasing of the liquid tension (i.e., it increases with the increasing of the pressure applied during the filter cake formation ΔP). For this reason, all calculations in this article were performed for ΔP = 10 5 Pa, which resulted in even lower maximal values of p m and lower absolute values of p l inside the membrane during the filter cake swelling (both depend on assumed filter cake thickness and membrane resistance).

Solution of basic consolidation equation for compressible filter cakes

Analytical solution of Eq. ( 7) for actual compressible filter cakes with pressure-dependent specific filtration resistance and void ratio is difficult to obtain. Therefore, these dependencies are usually substituted with a constant effective parameter (an average modified consolidation coefficient). Following this tradition, analytical solution of Eq. ( 7) with two constant effective parameters was obtained; the solution is presented and the reason of its restricted applicability is discussed in Appendix A.

Eq. ( 7) was also solved numerically for the actual case of compressible filter cakes, as it is discussed in Appendix B. The values of parameters (pressure dependencies of specific filtration resistance and solids volume fraction, local rheological properties, initial filter cake thickness, as well as applied pressure and membrane resistance) were typical for ultrafiltration of colloids that form highly compressible filter cakes with low permeability. Experimentally obtained constitutive equations for permeability, compressibility and shear stress of aqueous dispersions of casein micelles were used for this purpose; they are discussed in Appendix C.

Both solutions were obtained for a constant value of ω 0 : it was assumed that the weight of solids in a filter cake holds during the swelling. This is a weak assumption for a filter cake that swells on horizontal membrane in the absence of agitation; the assumption means that the swelled part of the cake remains at the cake-suspension interface.

Results and discussion

Particle concentration distribution

Fig. 2 presents two examples of solution of Eq. ( 7): filter cake swelling kinetics without (a)

and with (b) filtrate absorption across the membrane. In the case when filtrate inflow into the cake does not happen for any reason (as it would be, for example, for impermeable surface, r in = ∞), the solid pressure decreases continuously from the cake-membrane surface to the cake-bulk interface (Fig. 2a). However, when the membrane surface is permeable for filtrate, the solid pressure distribution in the swelling cake is nonmonotonic (Fig. 2b), with the lower p s values near the cake-membrane interface, which is explained by faster solid pressure relaxation in this area because of filtrate inflow across the membrane. Other observed trends (gradual decreasing of the solid pressure with time, slowdown of the pressure relaxation rate) are common for two cases and typical for relaxation processes. It must be noticed that in best of the authors' knowledge similar bell-like p s (ω,t) behavior was for the first time reported by Murase [START_REF] Murase | Expansion of consolidated material after release of load[END_REF] in the paper devoted to the modeling of expansion of consolidated material after the load release and later discussed in [START_REF] Murase | Expansion of inhomogeneous cake after release of load[END_REF]. However, as soon as the membrane resistance was not considered (which is equal to the case r in = 0), the solid pressure at the drainage (internal cake-membrane) surface attained p s = 0 and the bell-like p s (ω) and φ(ω)

profiles appeared immediately on the beginning of cake expansion. Fig. 3a -c. In the absence of filtrate inflow across the membrane (r in = ∞, Fig. 3a), the profiles φ(x) have no peculiarities: monotonic decrease of solid pressure with x results in the monotonic decrease of particle concentration from the membrane surface to the cake-bulk interface. While in the case of filtrate inflow across the membrane (Figs. 3b,c), appearance of the low p s area near the membrane results in the bell-like distribution of solid concentration, i.e., the maximal concentration is observed at a certain distance from the membrane; this distance increases with swelling duration.

The maximum of the φ(x) curve is more pronounced in the case of initially thick filter cake with ω 0 = 100 µm as compared to thin cake with ω 0 = 10 µm, which is explained by expectedly faster relaxation of thin cakes due to the faster liquid absorption across the cake-bulk interface. It can be noticed that in both cases (r in = ∞ and r in ≠ ∞) the swelling of filter cakes results in rather small increase on the filter cake thickness at the beginning of the swelling (t < 1000 s), which can be difficult to detect by currently available methods for the filter cake surface visualization (e.g., optical coherence tomography [START_REF] Gao | Novel approach to characterizing the growth of a fouling layer during membrane filtration via optical coherence tomography[END_REF], fluid dynamic gauging [START_REF] Gordon | A comparison of local phosphorescence detection and fluid dynamic gauging methods for studying the removal of cohesive fouling layers: Effect of layer roughness[END_REF], electrochemical sensors [START_REF] Gaucher | Determination of cake thickness and porosity during cross-flow ultrafiltration on a plane ceramic membrane surface using an electrochemical method[END_REF],

particle image velocimetry [START_REF] Gaucher | Hydrodynamics study in a plane ultrafiltration module using an electrochemical method and particle image velocimetry visualization[END_REF][START_REF] Rey | Velocity, stress and concentration fields revealed by micro-PIV and SAXS within concentration polarization layers during cross-flow ultrafiltration of colloidal Laponite suspensions[END_REF], x-ray radiography [START_REF] Gholami | Timeresolved 2D concentration maps in flowing suspensions using X-ray[END_REF], magnetic resonance imaging [START_REF] Arndt | In situ MRI of alginate fouling and flow in ceramic hollow fiber membranes[END_REF][START_REF] Arndt | New insights into sodium alginate fouling of ceramic hollow fiber membranes by NMR imaging[END_REF][START_REF] Schork | MRI investigations of filtration: Fouling and cleaning processes[END_REF], ultrasonic [START_REF] Collier | Contribution of the shear wave ultrasonic reflectometry to the stickiness measurements[END_REF], simple optical [START_REF] Tung | In situ measurement of cake thickness distribution by a photointerrupt sensor[END_REF], etc. [START_REF]Monitoring and visualizing membrane-based processes[END_REF][START_REF] Chen | Non-invasive observation of synthetic membrane processes -A review of methods[END_REF]). However, the remarkable difference in local solid concentration distribution near the membrane surface in the case of thick filter cakes with and without filtrate inflow across the membrane could be observed, for example, by the method of in situ SAXS analysis of local solid concentration in filter cakes, described in [START_REF] Rey | Velocity, stress and concentration fields revealed by micro-PIV and SAXS within concentration polarization layers during cross-flow ultrafiltration of colloidal Laponite suspensions[END_REF][START_REF] Jin | Effects of ultrasound on cross-flow ultrafiltration of skim milk: characterization from macro-scale to nano-scale[END_REF][START_REF] David | Spatial and temporal in situ evolution of the concentration profile during casein micelle ultrafiltration probed by small-angle X-ray scattering[END_REF][START_REF] Pignon | Structure and rheological behavior of casein micelle suspensions during ultrafiltration process[END_REF]. In the case of thin filter cakes, the available spatial resolution of SAXS (about 25 µm [START_REF] Doudiès | Structural organization of casein micelles in concentrated layer during cross-flow ultrafiltration[END_REF][START_REF] Doudiès | Understanding of casein micelles concentrated layers properties during cross flow ultrafiltration by in-situ small-angle X-ray scattering (SAXS)[END_REF][START_REF] Doudiès | Small-Angle X-Ray Scattering (SAXS) analyses of build-up, relaxation and erosion of casein micelles fouling layer during cross-flow ultrafiltration[END_REF]) can be insufficient for observation of the bell-like shape of the φ(x) [START_REF] Rey | Velocity, stress and concentration fields revealed by micro-PIV and SAXS within concentration polarization layers during cross-flow ultrafiltration of colloidal Laponite suspensions[END_REF]. Therefore, the general model verification requires an application of indirect methods of the filter cake analysis. For any nonzero value of membrane resistance (including r in = ∞) the value of φ m gradually decreases with time: the lower is r in , the faster is the concentration decrease that is explained by faster filtrate inflow and its absorption by the filter cake part adjacent to the membrane surface. In the limiting case of r in = 0, the solid volume fraction at the membrane surface decreases to the value φ(p s = 0) at the first instance of the swelling [START_REF] Murase | Expansion of consolidated material after release of load[END_REF][START_REF] Murase | Expansion of inhomogeneous cake after release of load[END_REF]. However, in the current model it is assumed that the filter cake compression is reversible (φ → 0 when p s → 0), and the value of φ m can decrease below φ sg , the concentration of sol-gel transition (i.e., transition from liquid-like to solid-like rheological behavior). For any filter cake thickness the value of θ increases with r in and asymptotically approaches the value expected for the case of filter cake swelling in the absence of filtrate inflow across the membrane (cake swelling on impermeable surface with r in = ∞, horizontals in Fig. 5a). The difference in swelling kinetics in the absence and presence of filtrate inflow is more remarkable for the thick filter cakes as compared to thin filter cakes. It is interesting to note that in the case when filtrate inflow is possible (r in ≠ ∞), the required swelling duration θ is far less sensitive to the filter cake thickness increase as compared to the case, when filtrate is not absorbed across the membrane (r in = ∞). For example, when r in = 10 13 m -1 the increasing of θ quickly slows down as ω 0 increases, while θ continuously increases as 𝜔𝜔 0 2 in the case of impermeable membrane (Fig. 5b). This can be explained as follows: in first case the filtrate can be absorbed via two surfaces of the filter cake (Fig. 1), and its flow towards the high p s area is impeded by membrane resistance r in (which is assumed to be constant) and resistance of the filter cake r ci (which is proportional to ω 0 , Fig. 5b). Therefore, the increasing of r ci in the case of a permeable membrane has only limited impact on the swelling kinetics at the membrane-cake interface. While in the case of impermeable surface with r in = ∞, the swelling follows the usual law for swelling and diffusion into a semi-infinite medium 𝜕𝜕 ∝ 𝑙𝑙 2 [START_REF] Tanaka | Kinetics of swelling of gels[END_REF][START_REF] Crank | The mathematics of diffusion[END_REF].

Particle concentration on membrane-cake interface

Membrane rinsing after the filter cake swelling

It must be noticed that for any value of θ, when φ m = φ sg , the φ(x) distribution in the swelling cake is distinct in case of the absence and presence of filtrate inflow across the membrane: in the case of r in = ∞ the highest particle concentration exists at the membrane surface (φ max = φ m = φ sg ), while in the case with filtrate inflow φ max is situated at the certain distance from the membrane surface and a gelled part of the filter cake with φ max > φ sg remains in the system event when t > θ. In principle, this difference can be used for the model verification via the following experiment. When a filter cake with φ m > φ sg is present on the membrane, certain minimal value of tangential force is required in order to sweep it from the surface (e.g., [START_REF] Benkahla | Cake growth mechanism in crossflow microfiltration of mineral suspensions[END_REF]). It can be expected that both attraction between cake particles and adhesion of the cake to the membrane decrease with decreasing of local particle concentration [START_REF] Mattsson | In situ investigation of soft cake fouling layers using fluid dynamic gauging[END_REF][START_REF] Lewis | Investigation of cake fouling and pore blocking phenomena using fluid dynamic gauging and critical flux models[END_REF][START_REF] Mattsson | The use of fluid dynamic gauging in investigating the thickness and cohesive strength of cake fouling layers formed during crossflow microfiltration[END_REF] (for example, due to the swelling [START_REF] Nakanishi | Rinsing behavior of deposited layers formed on membranes in ultrafiltration[END_REF]). Below, it is assumed that the force, required for the sweeping of the filter cake layer with a local particle concentration φ, significantly decreases, when φ ≤ φ sg (i.e., the gel is rigid and strongly attached to membrane surface, while the sol is liquid and can be easily removed). These are reasonable assumptions for a reversibly compressible filter cake, which is in the focus of the current paper. Fig. 6 schematically presents two filter cakes (after the swelling but before the rinsing): cake 1 with φ m > φ sg and cake 2 with φ m < φ sg .

Fig. 6.

Fig. 6. To the experiment on membrane rinsing after the cake swelling: filter cakes with different particle concentration distribution (cake 1 with φ m > φ sg and cake 2 with φ m < φ sg ); dashed horizontal corresponds to the concentration of sol-gel transition, φ sg , shaded parts can be removed from the surface (at left) by small tangential force, because φ ≤ φ sg at their left borders.

According to the previous reasoning, if small tangential force is applied, only shaded part of the cake 1 with φ ≤ φ sg is removed, the quantity of solids remaining on the membrane surface after the swelling-sweeping operation is defined by the solid pressure distribution profile attained during the sweeping and it is denoted as ω r , ω r > 0 (Fig. 6). However, when the value φ m ≤ φ sg is attained during the swelling (cake 2, Fig. 6), the cake is removed entirely because the adhesion of its part adjacent to membrane surface is small. If the hydraulic resistance of the residual part of the filter cake r r is measured by liquid percolation at the same applied pressure ΔP as it was used for the cake formation, the value of r r can be calculated as that of r ci (using Eqs. (B8) -(B9) with the value of ω 0 replaced by ω r ). Fig. 7a compares the values of r r /r ci attained after the swelling of initially thin (ω 0 = 10 µm) filter cakes different in the absence (solid curves) and presence (dashed curves) of filtrate inflow and absorption across the membrane. The observed immediate decrease of r r /r ci at t = 0 for all filter cakes is explained by the fact that the outer part of the cake has φ < φ sg and can be easily removed by sweeping without swelling. In the absence of filtrate inflow and absorption across the membrane (r in = ∞), the difference in r r (t) behavior is explained by initially different filter cake structure, when the cakes are formed at different membrane resistances: the lower is r m , the lower is the initial φ(x) distribution (under constant ΔP and ω 0 ) and the faster is the cake swelling and removal (that corresponds to the small negative slope of horizontals in Fig. 5a). An important common feature for the cakes swelling at r in = ∞ is that the decrease of r r (t) to 0 is gradual. In contrast, when the filtrate inflow and absorption across the membrane are possible, the r r (t)

behavior is different: gradual decreasing is followed by an abrupt drop to 0, because at t = θ the gelled part of the swelling cake is not in a direct contact with the membrane. This difference is less obvious for thin filter cakes at high values of r in (at low ratio of r ci /r in, e.g. for r m = 10 14 m -1 in Fig. 7a) because in this case the swelling behavior at large t approaches to that in the absence of filtrate inflow (Fig. 3c). However, this difference is distinct in the case of low r in (r in = 10 12 and 10 13 m -1 ). For thick cakes (ω 0 = 100 µm), which swell with filtrate inflow across the membrane, an abrupt drop of r r is observed even in the case of high r in (Fig. 7b).

Though an implementation of the experiment on the swelling followed by the sweeping described above can put in evidence the influence of the membrane resistance on the filter cake swelling kinetics, it is based on the assumption that the swelled gel (with φ m < φ sg ) is instantaneously removed from the surface under the application of a small (infinitesimal) tangential shear force. However, in practice, removal of the swelled gel from the surface in a reasonably short time will require an application of certain minimal force. This can also provoke a tangential movement ("sliding") of the non-swelled gel that must be considered in predicting of the r r (t) behavior. It can be assumed that when an external tangential shear force τ ext is applied to the gel after a certain swelling duration, the entire gel can move along the membrane surface in the direction of τ ext , if a part of the swelled gel in vicinity of the membrane attained a concentration, which corresponds to the local yield stress τ 0 < τ ext (at x = 0..h, Fig. 8). If it is assumed that the part of the gel having local yield stress τ 0 (x) > τ ext behaves as a rigid body (shaded area in Fig. 8), its velocity can be estimated from the equation for the plate, which moves relatively a parallel fixed surface (membrane) with the gap filled with non-Newtonian liquid having rheological properties depending on a local solid concentration. If the local rheological properties in flowing sol (Fig. 8) are described by Herschel-Bulkley model (Eq. (C3)), the relative velocity v under the external shear stress τ ext can be calculated as

𝑣𝑣 = � � 𝜏𝜏 𝑒𝑒𝑒𝑒𝑡𝑡 -𝜏𝜏 0 (𝑥𝑥) 𝐾𝐾(𝑥𝑥) � 1 𝑁𝑁(𝑒𝑒) d𝑥𝑥 ℎ 0 ( 13
)
where h is the thickness of the gap between the membrane and the gel (τ 0 = τ ext at x = h, Fig. 8

r in = ∞, φ m = φ(τ 0 = τ ext ).
The lower are r in and ω 0 , the faster is the gel swelling (Fig. 5a), and the faster is the increasing of h and v with t. In the case of initially thin filter cake (ω 0 = 10 µm, Fig. 9a) the values of h attained before the moment of the solid gel disappearance (final points of the curves in Fig. 9) are of the order of tenth of microns. These low values of h are close to the surface roughness of actual membranes that conflicts with the assumption about the parallel membranegel alignment (Fig. 8). It can be speculated that in the case of thin filter cakes, the part of the gel with φ(x) > φ(τ 0 = τ ext ) can not move along the membrane surface because of the membrane roughness. Therefore, in this case ω 0 = 10 µm, the swelled cake is swept from the membrane surface only after sufficient swelling duration, when φ(x) < φ(τ 0 = τ ext ): t ≈ 10 s at r in = 10 12 m -1 and t ≈ 20 s at r in = 10 13 m -1 . This noticeably increases the expected time required for complete gel removal from the membrane in the presence of filtrate inflow, from ≈ 1 s to ≈ 10 s at r in = 10 12 m -1 and from ≈ 15 s to ≈ 18 s at r in = 10 13 m -1 . Nevertheless, these values are lower as compared to the time required for the gel swelling in the absence of filtrate inflow across the membrane (≈ 50 s at r in = ∞, Fig. 9a), which suggests the positive influence of filtrate inflow on the gel removal. At high filter cake thickness ω 0 = 100 µm, the gel swelling is slow, however, the gap thickness h can reach the values h > 100 µm and expected gel tangential velocity can exceed 10 cm•s -1 before the gel disappearance (Fig. 9b). Therefore, in the presence of filtrate inflow and absorption by filter cake, an abrupt decrease of residual resistance r r (Fig. 7a,b) can be also expected in the case of application of non-infinitesimal τ ext for the gel sweeping both for thick and thin filter cakes. This also implies that in the presence of filtrate inflow and absorption across the membrane, the experiment on the swelling-sweeping of thick filter cakes can result in the appearance of gel lumps in the flushing liquid [START_REF] Jimenez-Lopez | Role of milk constituents on critical conditions and deposit structure in skimmilk microfiltration (0.1 μm)[END_REF][START_REF] Diagne | On the actual cleanability of polyethersulfone membrane fouled by proteins at critical or limiting flux[END_REF] (in contrast to the case with r in = ∞, where the membrane is efficiently cleaned, when the gel swelling is completed). Though the swelling duration required for the thick cake sweeping from the surface is too high to have practical application (Fig. 9b), the proposed experiment can be used for the elucidation of the influence of membrane resistance on the filter cake swelling. Tangential movement ("sliding") of the filter cake, which is discussed above, is different from the apparent wall slip (or wall slip) [START_REF] Bonn | Yield stress materials in soft condensed matter[END_REF], which is not included in the current simplified model of the filter cake removal. The discussed "sliding" onsets, when the stress applied to the swelling gel exceeds a minimal local value of the true yield stress in the internal swelling part of the gel (τ ext > min(τ 0 )). The wall slip appears, when τ ext exceeds an apparent local yield stress τ 0a (τ ext > τ 0a ). For homogeneous yield stress materials with a given particle volume fraction φ, the apparent yield stress is lower that the true one, τ 0a (φ) < τ 0 (φ) [START_REF] Seth | Influence of short-range forces on wall-slip in microgel pastes[END_REF]. The value of τ 0a is not a material property, it decreases with the increasing of the smoothness on the fixed wall [START_REF] Seth | Influence of short-range forces on wall-slip in microgel pastes[END_REF]. The information on τ 0a is scarce, however, it was reported that for some materials τ 0a (as well as τ 0 ) strongly decreases with decreasing of φ [START_REF] Seth | Influence of short-range forces on wall-slip in microgel pastes[END_REF]. Hence, the value of τ 0a should also decrease during the filter cake swelling on the membrane surface. If the membrane smoothness is not dependent on membrane resistance, the decreasing of τ 0a should be faster for the membrane with lower resistance to filtrate inflow r in because of the faster decreasing of φ in the vicinity of the membrane surface (and φ m , in particular). Hence, qualitatively, whatever the reason of tangential movement of the swelled filter cake under the external shear stress (wall slip or "sliding"), the decreasing of r in facilitates both (increases the tangential velocity, or decreases τ ext and swelling time required for the onset of tangential movement). The conclusion about easier removal of filter cake in the case of higher membrane permeability and filtrate inflow holds in case of wall slip.

Conclusions and final remarks

The model of one-dimensional swelling of colloidal filter cake after the filtration pressure release is developed in the frame of conventional filtration-consolidation theory. The model considers the filter cake as a Terzaghi element, and it takes into account possible filtrate inflow across the membrane under the solid pressure gradient and its uptake by the cake.

The model predicts that the filter cake swelling mechanism and swelling kinetics are determined by the filter cake properties (compressibility, permeability), the initial filter cake resistance r ci and the membrane resistance to filtrate inflow r in . The rate of the cake swelling on the impermeable surface (r in = ∞) significantly decreases with increasing of r ci (or initial cake thickness). While in the presence of filtrate inflow across the membrane (r in ≠ ∞), the swelling rate is less sensitive to r ci , and it increases with decreasing of the membrane resistance to filtrate inflow r in . This is explained by relatively fast filtrate inflow and absorption across the membrane as compared to slower filtrate flow from the bulk suspension throughout the inner part of the filter cake.

At high values of r ci /r in ratio (relatively thick filter cakes on permeable membrane surface), the swelling results in the non-uniform (bell-like) solid concentration distribution in the swelling cake (in contrast to monotonic decreasing of local solid concentration in the cake swelling on an impermeable surface). At a reasonably high initial cake thickness (ω 0 = 100 µm) this prediction can be verified with the help of in situ SAXS-filtration method [START_REF] Rey | Velocity, stress and concentration fields revealed by micro-PIV and SAXS within concentration polarization layers during cross-flow ultrafiltration of colloidal Laponite suspensions[END_REF][START_REF] Jin | Effects of ultrasound on cross-flow ultrafiltration of skim milk: characterization from macro-scale to nano-scale[END_REF][START_REF] David | Spatial and temporal in situ evolution of the concentration profile during casein micelle ultrafiltration probed by small-angle X-ray scattering[END_REF][START_REF] Pignon | Structure and rheological behavior of casein micelle suspensions during ultrafiltration process[END_REF][START_REF] Doudiès | Structural organization of casein micelles in concentrated layer during cross-flow ultrafiltration[END_REF][START_REF] Doudiès | Understanding of casein micelles concentrated layers properties during cross flow ultrafiltration by in-situ small-angle X-ray scattering (SAXS)[END_REF][START_REF] Doudiès | Small-Angle X-Ray Scattering (SAXS) analyses of build-up, relaxation and erosion of casein micelles fouling layer during cross-flow ultrafiltration[END_REF].

It is interesting to notice that the positive influence of non-uniform deposit swelling on its removal from non-porous surface is widely discussed, but outside of filtration literature [START_REF] Tan | On the origins of sudden adhesion loss at a critical relative humidity: Examination of bulk and interfacial contributions[END_REF][START_REF] Jing | Water-swelling-induced morphological instability of a supported polymethyl methacrylate thin film[END_REF][START_REF] Yang | Harnessing surface wrinkle patterns in soft matter[END_REF][START_REF] Berkelaar | Water-induced blister formation in a thin film polymer[END_REF][START_REF] White | Mechanisms of criticality in environmental adhesion loss[END_REF][START_REF] Wang | A three-dimensional phase diagram of growth-induced surface instabilities[END_REF].

It was demonstrated that formation of more voluminous internal layer under the rigid outer layer of the deposit can result in formation of surface instabilities, which facilitate the deposit detachment. Moreover, an interesting article recently demonstrated that the wrinkles formation during the alginate gel swelling after the reverse osmosis and forward osmosis filtration facilitates the fouling gel detachment and removal from the membrane [START_REF] Tow | In situ visualization of organic fouling and cleaning mechanisms in reverse osmosis and forward osmosis[END_REF].

Further calculations, based on actual rheological properties of colloidal dispersions of casein micelles, show that filtrate inflow across the membrane under the solid pressure gradient can have a noticeable influence on the rate of membrane surface rinsing by external shear force, which is applied after the cake swelling: the decreasing of r in results in the decreasing of the swelling duration required for the filter cake elimination. Because of the bell-like solid concentration distribution in the swelling cake with initially high thickness (ω 0 = 100 µm), it can be swept from the surface under the tangential shear stress, which is lower than the yield stress of the gel. This peculiar dependence of the strength of the filter cake adhesion to the membrane on the swelling duration, the filter cake thickness and the membrane resistance to filtrate inflow can be verified with the help of available methods for analysis of deposit properties [START_REF] Gordon | A comparison of local phosphorescence detection and fluid dynamic gauging methods for studying the removal of cohesive fouling layers: Effect of layer roughness[END_REF][START_REF] Mattsson | In situ investigation of soft cake fouling layers using fluid dynamic gauging[END_REF][START_REF] Lewis | Investigation of cake fouling and pore blocking phenomena using fluid dynamic gauging and critical flux models[END_REF][START_REF] Mattsson | The use of fluid dynamic gauging in investigating the thickness and cohesive strength of cake fouling layers formed during crossflow microfiltration[END_REF][START_REF] Hooper | Comparative studies of fluid dynamic gauging and a micromanipulation probe for strength measurements[END_REF][START_REF] Liu | Direct measurement of the force required to disrupt and remove fouling deposits of whey protein concentrate[END_REF][START_REF] Gordon | Studies into the swelling of gelatine films using a scanning fluid dynamic gauge[END_REF][START_REF] Saikhwan | Swelling and dissolution in cleaning of whey protein gels[END_REF][START_REF] Akhtar | Matching the nano-to the meso-scale: Measuring deposit-surface interactions with atomic force microscopy and micromanipulation[END_REF][START_REF] Ali | Development of a "millimanipulation" device to study the removal of soft solid fouling layers from solid substrates and its application to cooked lard deposits[END_REF].

It should be noted that the positive effect of filtrate inflow into the cell (filtrate back permeation) on the membrane surface cleaning is widely demonstrated and modeled in the literature on osmotic backwash or reverse osmosis and forward osmosis membranes (e.g., [START_REF] Sagiv | Modeling of backwash cleaning methods for RO membranes[END_REF][START_REF] Kim | Application of osmotic backwashing in forward osmosis: Mechanisms and factors involved[END_REF][START_REF] Motsa | Organic fouling in forward osmosis membranes: The role of feed solution chemistry and membrane structural properties[END_REF][START_REF] Blandin | Fouling and cleaning of high permeability forward osmosis membranes[END_REF][START_REF] Motsa | Osmotic backwash of fouled FO membranes: Cleaning mechanisms and membrane surface properties after cleaning[END_REF]) and hydraulic back pulsing of ultrafiltration membranes (e.g., [START_REF] Lalrinsanga | Effect of pressure pulsing on concentration boundary layer over membrane -A numerical investigation[END_REF][START_REF] Vinther | Predicting optimal back-shock times in ultrafiltration hollow fiber modules through path-lines[END_REF][START_REF] Akhondi | Influence of backwashing on the pore size of hollow fiber ultrafiltration membranes[END_REF]). It is discussed that liquid permeation across the membrane (either under the hydraulic pressure application or the osmotic pressure difference between the filtrate and saline bulk solution) reduces the deposit attachment to the membrane and shifts it away from the membrane surface. This facilitates the deposit removal by tangential flow. The present model results suggest that the same effect (filtrate backflow, gel softening, and its lifting from the membrane surface) can be expected without the hydraulic back-pressure application or increasing the bulk osmotic pressure.

A number of assumptions was used in the current model of filter cake swelling: completely reversible cake compression, absence of creep, one-dimensional cake expansion during the swelling, and simplified consideration of cake adhesion to membrane surface and simplified presentation of filtration membrane. Hence, further development of this model can be focused on partial irreversibility of filter cake compression (e.g., as in [START_REF] Iritani | Compression and expansion properties of filter cake accompanied with step change in applied pressure in membrane filtration[END_REF][START_REF] Iritani | Simplified evaluation of consolidation and expansion behaviour of highly compressible cake[END_REF]) and its limited redispersion (as in [START_REF] Qu | On the cohesive properties of casein micelles in dense systems[END_REF]). It also can include more realistic membrane model (porous surface instead of semipermeable continuum) and account for 2D and 3D swelling (in order to be able to describe possible wrinkles formation [START_REF] Tow | In situ visualization of organic fouling and cleaning mechanisms in reverse osmosis and forward osmosis[END_REF]).

Despite of used assumptions and conclusion that the influence of the membrane resistance to filtrate inflow on the filter cake swelling and removal is less important for thin filter cakes (which are met in membrane filtration applications) as compared to thick filter cakes (that are mainly used for laboratory analysis of filter cakes), we hope that conclusions of current work can be useful in different cases: for example, during the interpretation of experimental data on reversibility of cake formation and compression ([18, 44, 45, 118, 119]), filter cake adhesion to membrane [START_REF] Zhou | Investigation of the cohesive strength of membrane fouling layers formed during cross-flow microfiltration: The effects of pH adjustment on the properties and fouling characteristics of microcrystalline cellulose[END_REF], and critical conditions of cake removal from the membrane surface ( [START_REF] Steinhauer | Effect of pH, transmembrane pressure and whey proteins on the properties of casein micelle deposit layers[END_REF][START_REF] Diagne | On the actual cleanability of polyethersulfone membrane fouled by proteins at critical or limiting flux[END_REF]).

𝑝𝑝 𝑚𝑚 (𝜕𝜕) = 𝑝𝑝 𝑚𝑚𝑖𝑖 � 2(sin where S is the membrane surface.

The volume of filtrate absorbed by the cake across the membrane increases with time and asymptotically approaches to a certain value V f∞ that is defined as

𝑉𝑉 𝑓𝑓∞ 𝑆𝑆 = 𝑝𝑝 𝑚𝑚𝑖𝑖 𝜇𝜇𝑟𝑟 𝑖𝑖𝑖𝑖 𝜔𝜔 0 2 𝐶𝐶 𝑒𝑒 � 2 𝜆𝜆 𝑘𝑘 2 • sin 2 𝜆𝜆 𝑘𝑘 -𝜆𝜆 𝑘𝑘 sin 𝜆𝜆 𝑘𝑘 cos 𝜆𝜆 𝑘𝑘 𝜆𝜆 𝑘𝑘 2 -𝜆𝜆 𝑘𝑘 sin 𝜆𝜆 𝑘𝑘 cos 𝜆𝜆 𝑘𝑘 ∞ 𝑘𝑘=1 (A10)
The "completeness" of filtrate absorption by the filter cake can be characterized with the help of

a ratio U 𝑈𝑈 = 𝑉𝑉 𝑓𝑓 (𝜕𝜕) 𝑉𝑉 𝑓𝑓∞ (A11)
which changes from 0 to 1 during the filter cake swelling.

Equations analogous to Eqs. (A9) -(A11) are frequently applied for description of different types of consolidation of moderately compressible materials. Therefore, pressure dependencies of α(p s ) and e(p s ) are generally ignored during the modelling: it is assumed that C e is constant despite the fact that the solid pressure changes with time and across the sample.

The assumption about the constancy of C e can be acceptable for the cases where the average solid pressure, that governs the process, does not change significantly with time: for example, during mechanical compression with small pressure increments.

However, in contrast to different consolidation processes, description of a filter cake swelling on a membrane surface requires an additional parameter, the average specific filtration resistance α e of the filter cake layer adjacent to the membrane surface (parameter α e appears in Eq. (A2) and following equations via Eq. (A5)). It substitutes an actual pressure dependency α(p m ), where p m is the solid pressure on the cake-membrane surface. The latter decreases from p mi to 0 (Eq. (A8)), which implies significant decrease of α during the swelling. In the present paper devoted to the role of membrane resistance in the filter cake swelling, this decrease can not be c = c 0 at ω = ω 0 (B6)

d𝑐𝑐 d𝜔𝜔 = 𝑝𝑝 𝑠𝑠 (𝑐𝑐) 𝑟𝑟 𝑖𝑖𝑖𝑖 • 𝜌𝜌 𝑠𝑠 𝑐𝑐 𝜅𝜅(𝑐𝑐) 𝜕𝜕 𝑝𝑝 𝑠𝑠 (𝑐𝑐) 𝜕𝜕𝑐𝑐 at ω = 0 (B7)
Eq. (B4) can be solved with initial and boundary conditions (B5) -(B7) for known dependencies κ(c) and p s (c) and given values of parameters r in , c mi and ω 0 . The initial solid concentration on the cake-membrane interface of c mi (in Eq. (B5)) is defined by the pressure applied during the cake formation (i.e., dead-end filtration) ΔP and the ratio of inside-out membrane resistance r m to the filter cake resistance attained at the end of filtration r ci as

𝑝𝑝 𝑠𝑠 (𝑐𝑐 𝑚𝑚𝑖𝑖 ) = Δ𝑃𝑃 (1 + 𝑟𝑟 𝑚𝑚 𝑟𝑟 𝑐𝑐𝑖𝑖 ⁄ ) ⁄ (B8)
where r ci is defined from

𝑟𝑟 𝑐𝑐𝑖𝑖 = 𝜌𝜌 𝑠𝑠 𝜔𝜔 0 • 𝑝𝑝 𝑠𝑠 (𝑐𝑐 𝑚𝑚𝑖𝑖 ) ∫ 𝑐𝑐 𝜅𝜅(𝑐𝑐) 𝜕𝜕 𝑝𝑝 𝑠𝑠 (𝑐𝑐) 𝜕𝜕𝑐𝑐 d𝑐𝑐 𝑐𝑐 𝑚𝑚𝑚𝑚 𝑐𝑐 0 (B9)
Eq. ( 7) was solved (dependencies c(ω/ω 0 , t) were obtained) by the Crank-Nicholson method with the help of Maple 18 (Maplesoft). Temporal and spatial step sizes were chosen such that their reduction would have had no observable effect on the result of calculation of local properties (compressibility, permeability, and further tangential velocity) of the swelling cake.

In order to study the influence of the possibility and intensity of filtrate inflow across the membrane on the filter cake swelling, Eq. ( 7) was solved for different values of r in , including the case r in = ∞, which corresponds to the absence of filtrate absorption by the cake. The value of ω 0 , used for solution, was varied from 10 to 200 µm. Particular attention was focused on the cases ω 0 = 10 µm and ω 0 = 100 µm. The former corresponds to "thin" filter cakes, which can be met in membrane filtration applications, while the latter simulates rather "thick" filter cakes (or deposits) that are usually obtained in laboratory studies on filtration and/or surface cleaning [START_REF] Rey | Velocity, stress and concentration fields revealed by micro-PIV and SAXS within concentration polarization layers during cross-flow ultrafiltration of colloidal Laponite suspensions[END_REF][START_REF] Jin | Effects of ultrasound on cross-flow ultrafiltration of skim milk: characterization from macro-scale to nano-scale[END_REF][START_REF] David | Spatial and temporal in situ evolution of the concentration profile during casein micelle ultrafiltration probed by small-angle X-ray scattering[END_REF][START_REF] Pignon | Structure and rheological behavior of casein micelle suspensions during ultrafiltration process[END_REF][START_REF] Doudiès | Structural organization of casein micelles in concentrated layer during cross-flow ultrafiltration[END_REF][START_REF] Doudiès | Understanding of casein micelles concentrated layers properties during cross flow ultrafiltration by in-situ small-angle X-ray scattering (SAXS)[END_REF][START_REF] Doudiès | Small-Angle X-Ray Scattering (SAXS) analyses of build-up, relaxation and erosion of casein micelles fouling layer during cross-flow ultrafiltration[END_REF][START_REF] Mattsson | In situ investigation of soft cake fouling layers using fluid dynamic gauging[END_REF][START_REF] Lewis | Investigation of cake fouling and pore blocking phenomena using fluid dynamic gauging and critical flux models[END_REF][START_REF] Mattsson | The use of fluid dynamic gauging in investigating the thickness and cohesive strength of cake fouling layers formed during crossflow microfiltration[END_REF]. Other parameters were fixed at reasonable values: the pressure applied during the cake formation ΔP = 10 5 Pa, the filtrate viscosity μ = 10 -3 Pa•s, the bulk concentration of casein micelles dispersion φ 0 = 0.02.

Appendix C. Casein micelles dispersions, material properties

Casein micelles, which properties are used in order to describe colloids typically met in filtration, are well characterized natural spherical porous colloidal particles, highly compressible and deformable, stable against the aggregation at neutral pH [START_REF] Broyard | Modifications of structures and functions of caseins: a scientific and technological challenge[END_REF]. Using the methods of osmotic stress and dead-end filtration, Bouchoux et al. obtained dependencies of hydraulic permeability κ and osmotic pressure of casein micelles dispersions on the solids concentration c for a wide range of solid pressure [START_REF] Bouchoux | A general approach for predicting the filtration of soft and permeable colloids: The milk example[END_REF]. These dependencies are presented by Eqs. (B1) -(B2). The osmotic pressure of casein micelles dispersions attained in osmotic stress experiments is equivalent to their compressive pressure (solid pressure). The specific filtration resistance and the solids volume fraction can be obtained from the data of Bouchoux et al. [START_REF] Bouchoux | A general approach for predicting the filtration of soft and permeable colloids: The milk example[END_REF] as

𝜇𝜇 = 1 𝜅𝜅𝑐𝑐 ⁄ (C1) 𝜑𝜑 = 𝑐𝑐 𝜌𝜌 𝑠𝑠 ⁄ (C2)
The density of solids of casein micelles can be calculated from the values of micelles voluminosity and the micelles density reported by de Kruif and Huppertz [START_REF] De Kruif | Casein Micelles: Size distribution in milks from individual cows[END_REF]: ρ s = 1.35 g/ml (which is close to the value provided by Morris et al. [START_REF] Morris | Further observations on the size, shape, and hydration of casein micelles from novel analytical ultracentrifuge and capillary viscometry approaches[END_REF]). The obtained dependencies α(p s ) and φ(p s ) presented in Fig. C1. [121]). Dotted vertical corresponds to the solid pressure in the point of sol-gel transition; dashed vertical corresponds to the solid pressure required for "irreversible" gel formation.

At low particle concentration, casein micelles dispersions are sols with practically linear dependence between solids volume fraction and solid pressure and practically constant and low specific filtration resistance. According to the data of Bouchoux et al. [START_REF] Bouchoux | Casein micelle dispersions under osmotic stress[END_REF][START_REF] Bouchoux | Rheology and phase behavior of dense casein micelle dispersions[END_REF], the point of fluid sol to solid gel transition of casein micelles dispersions corresponds to the solid pressure about p s ≈ 3 kPa. Further, the increase of solid pressure above 30 kPa results into the "irreversible" gel formation; that is determined from the dead-end ultrafiltration experiments [START_REF] Qu | Dead-end filtration of sponge-like colloids: The case of casein micelle[END_REF]). As well as many colloidal filter cakes, at high solid pressure, casein micelles gels have moderate compressibility (𝜑𝜑 ∝ 𝑝𝑝 𝑠𝑠 0.17 ) and high specific filtration resistance that strongly depends on p s (α = 10 16 …10 17 m/kg, 𝜇𝜇 ∝ 𝑝𝑝 𝑠𝑠 0.95 ).

According to Jin et al. [START_REF] Jin | Effects of ultrasound on cross-flow ultrafiltration of skim milk: characterization from macro-scale to nano-scale[END_REF], for a wide range of solid volume fraction φ = 0.02 -0.147 rheological properties of aqueous dispersions of casein micelles can be described by Herschel-Bulkley model

𝜏𝜏 = 𝜏𝜏 0 + 𝐾𝐾𝛾𝛾̇𝑁𝑁 (C3)
where τ is the shear stress, 𝛾𝛾̇ is the shear rate, and K, N and τ 0 (yield stress) are parameters that depend on the solid volume fraction. Figure C2 presents these dependencies obtained via the least-square fitting of experimental 𝜏𝜏(𝛾𝛾) data reported by Jin et al. [START_REF] Jin | Effects of ultrasound on cross-flow ultrafiltration of skim milk: characterization from macro-scale to nano-scale[END_REF] for different casein micelles concentrations.

Figure captions

Fig. 1. Liquid p l and solid p s pressures (dashed and solid curves, respectively) in a system of a particulate filter cake on a membrane surface at the moments (a) before and (b) after the release of filtration pressure ΔP; arrows show the liquid flow direction. The initial solid pressure in the cake psi varies from the membrane-cake (ω = 0) to the cake-suspension interface (ω = ω 0 ), where is ω the material coordinate. For colloidal objects, distribution of the local osmotic pressure π in cake and suspension is equivalent to distribution of p s in particulate filter cake, while the liquid pressure p l is not determined. Fig. 6. To the experiment on membrane rinsing after the cake swelling: filter cakes with different particle concentration distribution (cake 1 with φ m > φ sg and cake 2 with φ m < φ sg ); dashed horizontal corresponds to the concentration of sol-gel transition, φ sg , shaded parts can be removed from the surface (at left) by small tangential force, because φ ≤ φ sg at their left borders.

e

  void ratio (dimensionless) h gap thickness, thickness of a layer on the membrane surface which can flow relatively the membrane and the rigid gel under the applied external shear stress (m) K parameter of Herschel-Bulkley equation (Pa•s N ) l length (m) N parameter of Herschel-Bulkley equation (dimensionless) ΔP applied pressure (Pa) p l liquid pressure (Pa) p m solid pressure at the membrane surface (Pa) p mi initial solid pressure at the membrane surface (at the end of filtration) (Pa) p s compressive solid pressure (Pa) p si initial solid pressure (at the end of filtration) (Pa) r ci initial filter cake resistance (cake resistance at the end of filtration) (m -1 ) r in outside-in membrane hydraulic resistance (m -1 ) r m inside-out membrane hydraulic resistance (m -1 ) r r hydraulic resistance of residual gel (m -1 ) S cross-sectional area of filtration membrane (m 2 ) t time (s) U completeness of filter cake swelling (dimensionless) u apparent liquid velocity relative to solids (m•s -1 ) V f filtrate volume (m 3 ) V f∞ final filtrate volume (m 3 ) v tangential velocity of rigid gel relative to membrane (m•s -1 ) x distance to membrane surface in Cartesian coordinates (m) Greek letters α specific filtration resistance (m•kg -1 ) α e average specific filtration resistance at the membrane surface (m•kg -1 ) α m specific filtration resistance of a layer on a cake-membrane interface (m•kg -1 ) 𝛾𝛾̇ shear rate (s -1 ) θ required swelling duration, duration of swelling required in order to attain φ m = φ sg (s) κ local permeability (m 2 ) λ variable, root of Eq. (A5) (dimensionless) μ filtrate viscosity (Pa•s) ξ variable (dimensionless) ρ s specific weight of solids (kg•m -3 ) τ 0 local yield stress (Pa) τ ext external applied shear stress (wall shear stress) (Pa) φ solid volume fraction (dimensionless) φ m solid volume fraction on cake-membrane interface (dimensionless) φ max maximal instant solid volume fraction in the system (dimensionless) φ sg solid volume fraction of sol-gel transition (dimensionless) ω distance to membrane surface in material coordinates (m) ω 0 position of cake-bulk interface in material coordinates (m) ω r thickness of swelled gel in material coordinates (m)

Fig

  Fig. 1a,b.

Fig. 1 .

 1 Fig. 1. Liquid p l and solid p s pressures (dashed and solid curves, respectively) in a system of a particulate filter cake on a membrane surface at the moments (a) before and (b) after the release of filtration pressure ΔP; arrows show the liquid flow direction. The initial solid pressure in the cake p si varies from the membrane-cake (ω = 0) to the cake-suspension interface (ω = ω 0 ), where

Fig.

  Fig. 2a, b.

Fig. 2 .

 2 Fig. 2. Solid pressure distribution during the swelling of a thick filter cake (ω 0 = 100 µm) formed at the membrane surface with r m = 10 13 m -1 : (a) without filtrate inflow across the membrane (r in =

Fig. 3

 3 Fig. 3 presents the evolution of local solid concentration in the filter cake caused by the solid pressure decrease during the swelling with consideration of membrane resistance.

Fig. 3 .

 3 Fig. 3. Solid volume fraction distribution during the filter cake swelling: (a) thick filter cake without filtrate inflow across the membrane (r in = ∞); (b) thick filter cake with filtrate inflow across the membrane with r in = 10 13 m -1 ; (c) thin filter cake with filtrate inflow across the

Fig. 4

 4 Fig. 4 presents the evolution of particle concentration on the internal membrane surface φ m during the swelling of thick filter cakes at different values of membrane resistance r in .

Fig. 4 .

 4 Fig. 4. Solid volume fraction at the cake-membrane interface versus the swelling duration for the thick filter cake (example for ω 0 = 100 µm) for different values of outside-in membrane resistance r in (shown near the curves). The inside-out membrane resistance is r m = r in (except of r m = 10 13 m -1 for r in = ∞). Dashed horizontal corresponds to the concentration of sol-gel transition φ sg for the studied system.

Fig. 5

 5 Fig. 5 presents the time θ, which is required in order to attain φ m = φ sg at different conditions of swelling.

Fig

  Fig. 5a -b.

Fig. 5 .

 5 Fig. 5. Time required for filter cake swelling θ at different conditions: (a) θ versus the membrane hydraulic resistance r m for thin (ω 0 = 10 µm, dashed curves) and thick (ω 0 = 100 µm, solid curves) filter cakes for the cases with (r in = r m , curves) and without (r in = ∞, horizontals) filtrate inflow across the membrane; (b) θ versus the filter cake thickness ω 0 and initial filter cake resistance r ci for the cases with (example for r in = 10 13 m -1 ) and without (r in = ∞) filtrate inflow across the membrane with r m = 10 13 m -1 .

Fig

  Fig. 7a, b.

Fig. 7 .

 7 Fig. 7. Relative residual cake resistance r r /r ci when the sweeping force is applied after the swelling duration t in the absence (solid curves r in = ∞) and presence (dashed curves, r in = r m ) of filtrate inflow across the membrane: (a) thin filter cakes, ω 0 = 10 µm; (b) thick filter cakes, ω 0 = 100 µm. The values of r m are shown near the curves.

Fig. 8 .Fig. 8 .

 88 Fig. 8.

  ), and the local values of parameters of Herschel-Bulkley equation K(x), N(x) and τ 0 (x) are determined by the local solid concentration φ(x).

Fig. 9

 9 Fig. 9 presents values of the gap thickness h and the velocity v calculated for different swelling durations t. The concentration dependencies of Herschel-Bulkley equation parameters, determined by Jin et al. [83] for casein micelles dispersion and presented in the Appendix C, were used for calculations. The data presented in Fig. 9 are calculated for τ ext = 20 Pa, which is a

Fig. 9 .

 9 Fig. 9. Estimated values of the gap thickness h (dashed curves) and the tangential velocity of the gel v (solid curves) versus the swelling duration for (a) thin (ω 0 = 10 µm) and (b) thick (ω 0 = 100 µm) filter cakes under the external tangential shear τ ext = 20 Pa. The values of the membrane resistance to filtrate inflow r in are indicated near the curves. The curves end, when the solid gel disappears because of the swelling: φ max = φ(τ 0 = τ ext ). Horizontals correspond to the moment of the solid gel disappearance in the absence of filtrate inflow across the membrane: r in = ∞, φ m =

Fig. C1.

  Fig. C1.

Fig. C1 .

 C1 Fig. C1. Pressure dependency of solids volume fraction and specific filtration resistance for aqueous dispersions of casein micelles (calculated from the data reposted by Bouchoux et al.

Fig. C2 .

 C2 Fig. C2.

Fig. 2 .

 2 Fig. 2. Solid pressure distribution during the swelling of a thick filter cake (ω 0 = 100 µm) formed at the membrane surface with r m = 10 13 m -1 : (a) without filtrate inflow across the membrane (r in = ∞); (b) with filtrate inflow across the membrane with r in = 10 13 m -1 . The time after the pressure release (in seconds) is shown near the curves.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Solid volume fraction distribution during the filter cake swelling: (a) thick filter cake without filtrate inflow across the membrane (r in = ∞); (b) thick filter cake with filtrate inflow across the membrane with r in = 10 13 m -1 ; (c) thin filter cake with filtrate inflow across the membrane with r in = 10 13 m -1 . The time after the pressure release (in seconds) is shown near the curves; in each case r m = 10 13 m -1 . Fig. 4. Solid volume fraction at the cake-membrane interface versus the swelling duration for the thick filter cake (example for ω 0 = 100 µm) for different values of outside-in membrane resistance r in (shown near the curves). The inside-out membrane resistance is r m = r in (except of r m = 10 13 m -1 for r in = ∞). Dashed horizontal corresponds to the concentration of sol-gel transition φ sg for the studied system.

Fig. 5 .

 5 Fig. 5. Time required for filter cake swelling θ at different conditions: (a) θ versus the membrane hydraulic resistance r m for thin (ω 0 = 10 µm, dashed curves) and thick (ω 0 = 100 µm, solid curves) filter cakes for the cases with (r in = r m , curves) and without (r in = ∞, horizontals) filtrate inflow across the membrane; (b) θ versus the filter cake thickness ω 0 and initial filter cake resistance r ci for the cases with (example for r in = 10 13 m -1 ) and without (r in = ∞) filtrate inflow across the membrane with r m = 10 13 m -1 .

Fig. 7 .

 7 Fig. 7. Relative residual cake resistance r r /r ci when the sweeping force is applied after the swelling duration t in the absence (solid curves r in = ∞) and presence (dashed curves, r in = r m ) of filtrate inflow across the membrane: (a) thin filter cakes, ω 0 = 10 µm; (b) thick filter cakes, ω 0 = 100 µm. The values of r m are shown near the curves.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Solid concentration distribution φ(x) and corresponding yield stress distribution τ 0 (x) in a swelled gel. In the vicinity of membrane the part of the gel with τ 0 (x) < τ ext can flow under the external shear force, that cause the movement of the denser part of the gel with τ 0 (x) > τ ext . Fig. 9. Estimated values of the gap thickness h (dashed curves) and the tangential velocity of the gel v (solid curves) versus the swelling duration for (a) thin (ω 0 = 10 µm) and (b) thick (ω 0 = 100 µm) filter cakes under the external tangential shear τ ext = 20 Pa. The values of the membrane resistance to filtrate inflow r in are indicated near the curves. The curves end, when the solid gel disappears because of the swelling: φ max = φ(τ 0 = τ ext ). Horizontals correspond to the moment of the solid gel disappearance in the absence of filtrate inflow across the membrane: r in = ∞, φ m = φ(τ 0 = τ ext ).

Fig. C1 .

 C1 Fig. C1. Pressure dependency of solids volume fraction and specific filtration resistance for aqueous dispersions of casein micelles (calculated from the data reposted by Bouchoux et al. [121]). Dotted vertical corresponds to the solid pressure in the point of sol-gel transition; dashed vertical corresponds to the solid pressure required for "irreversible" gel formation. Fig. C2. Values of parameters of Herschel-Bulkley model for casein micelles dispersions at different solid concentrations: symbols -data obtained by least-square fitting of experimental dependencies 𝜏𝜏(𝛾𝛾, 𝜑𝜑) (presented by Jin at al. [83]) with the help of Eq. (C3), curves -result of best curve fitting for these data. Dotted vertical corresponds to the concentration of sol-gel transition.

  2 𝜆𝜆 𝑘𝑘 -𝜆𝜆 𝑘𝑘 sin 𝜆𝜆 𝑘𝑘 cos 𝜆𝜆 𝑘𝑘 ) 𝜆𝜆 𝑘𝑘 2 -𝜆𝜆 𝑘𝑘 sin 𝜆𝜆 𝑘𝑘 cos 𝜆𝜆 𝑘𝑘

				∞ 𝑘𝑘=1		exp �-𝜆𝜆 𝑘𝑘 2 𝐶𝐶 𝑒𝑒 𝜔𝜔 0 2 𝜕𝜕�	(A8)
	The value of p m asymptotically approaches 0 during the filter cake swelling. The volume of
	filtrate absorbed by the swelling filter cake can be obtained with the help of integration of Darcy
	equation (10) over the swelling time:
	𝑉𝑉 𝑓𝑓 (𝜕𝜕) 𝑆𝑆	=	𝑝𝑝 𝑚𝑚𝑖𝑖 𝜇𝜇𝑟𝑟 𝑖𝑖𝑖𝑖	𝜔𝜔 0 2 𝐶𝐶 𝑒𝑒	� ∞ 𝑘𝑘=1	2 𝜆𝜆 𝑘𝑘 2 •	sin 2 𝜆𝜆 𝑘𝑘 -𝜆𝜆 𝑘𝑘 sin 𝜆𝜆 𝑘𝑘 cos 𝜆𝜆 𝑘𝑘 𝜆𝜆 𝑘𝑘 2 -𝜆𝜆 𝑘𝑘 sin 𝜆𝜆 𝑘𝑘 cos 𝜆𝜆 𝑘𝑘	• �1 -exp �-𝜆𝜆 𝑘𝑘 2 𝐶𝐶 𝑒𝑒 𝜔𝜔 0 2 𝜕𝜕��	(A9)

Acknowledgments

ML thanks department CEPIA of INRA for the financial support provided for his work in the frame of the project ANS "Emomilé: étude et modélisation de la microfiltration du lait". The authors thank Dr. Javier Pérez, group leader of the beamline SWING of Synchrotron SOLEIL, for his invaluable help with the project realization. ML equally thanks Dr. Alexander Zaderko for invaluable help with osmotic pressure gedanken experiments.

Appendix A. Analytical solution of basic consolidation equation for cake swelling

Solution of Eq. [START_REF] Gesan-Guiziou | Critical stability conditions in skimmed milk crossflow microfiltration: Impact on operating modes[END_REF] with conditions [START_REF] Grandison | Hydrodynamic factors affecting flux and fouling during ultrafiltration of skimmed milk[END_REF], [START_REF] Espinasse | On an experimental method to measure critical flux in ultrafiltration[END_REF] and [START_REF] Nakanishi | Rinsing behavior of deposited layers formed on membranes in ultrafiltration[END_REF] requires the knowledge of pressure dependencies of void ratio e(p s ) and specific filtration resistance α(p s ). In the filtrationconsolidation literature it became conventional to replace these dependencies with an effective constant, the modified average consolidation coefficient C e

which value can quantitatively characterize the kinetics of entire consolidation (or swelling) process for moderately compressible filter cakes. Following the traditional approach and introducing another effective constant, the average specific filtration resistance on the membrane surface α e , instead of α m (p s ) into Eq. ( 12), Eq. ( 7) can be solved analytically [START_REF] Murase | Expansion of consolidated material after release of load[END_REF][START_REF] Iwata | Expansion and stress relaxation of expressed cake[END_REF][START_REF] Iritani | Consolidation and expansion of a granular bed of superabsorbent hydrogels[END_REF][START_REF] Iritani | Simplified evaluation of consolidation and expansion behaviour of highly compressible cake[END_REF][START_REF] Iwata | Theoretical analysis of centrifugal dewatering of superabsorbent hydrogels using Terzaghi-Voigt combined model[END_REF],

and solid pressure distribution during the swelling can be presented as

where functions 𝑓𝑓 𝑖𝑖 (𝜔𝜔) and 𝑓𝑓 𝑖𝑖 (𝜉𝜉) are

and λ k are roots of equation

in the order of increase of their absolute values.

Eq. (A2) is obtained for an arbitrary initial solid pressure distribution in the filter cake p si (ω). As soon as the current approach is justified only for moderately compressible filter cakes, one can focus only on the linear initial solid pressure distribution:

where p mi is the initial value of solid pressure at the membrane surface. Introducing Eq. (A6) into (A2) yields

Using Eq. (A2) the solid pressure at cake-membrane surface (ω = 0) can be determined as ignored, because, in accordance to Eq. (A5), the value of specific filtration resistance α on the membrane surface is of the same importance for description of cake swelling as the value of the membrane resistance r in .

Therefore, numerical solution of Eq. ( 7) was preferred in the current paper in order to avoid the ambiguities concerning the role of membrane resistance in cake swelling.

Nevertheless, it was observed that same trends are predicted for the solid pressure evolution with the help of numerical solution of Eq. ( 7) and with application of Eqs. (A2) -(A11).

Appendix B. Basic consolidation equation for cake swelling, numerical solution

The dependencies of osmotic pressure (solid pressure) p s (c) and hydraulic permeability κ(c)

and on the mass concentration of solid c reported by Bouchoux et al. [START_REF] Bouchoux | A general approach for predicting the filtration of soft and permeable colloids: The milk example[END_REF] for aqueous dispersions of casein micelles were used in numerical solution of Eq. ( 7):

Here c is expressed in kg/m 3 The boundary conditions ( 9) and ( 12) can be represented as