S Charpentier

L Kosi Ński

Lukasz Kosiński
email: lukasz.kosinski@im.uj.edu.pl

WILD BOUNDARY BEHAVIOUR OF HOLOMORPHIC FUNCTIONS IN DOMAINS OF C N

Keywords: 2010 Mathematics Subject Classification. Primary 32A10 ; Secondary 32A40 Universality, Several complex variables, Boundary behaviour, Labyrinth

Given a domain of holomorphy D in C N , N ≥ 2, we show that the set of holomorphic functions in D whose cluster sets along any finite length paths to the boundary of D is maximal, is residual, densely lineable and spaceable in the space O(D) of holomorphic functions in D. Besides, if D is a strictly pseudoconvex domain in C N , and if a suitable family of smooth curves γ(x, r), x ∈ bD, r ∈ [0, 1), ending at a point of bD is given, then we exhibit a spaceable, densely lineable and residual subset of O(D), every element f of which satisfies the following property: For any measurable function h on bD, there exists a sequence (r n ) n ∈ [0, 1) tending to 1, such that

for almost every x in bD.

In 2000, Boivin, Gauthier and Paramonov proved that, given a domain Ω R N whose boundary has no component consisting of a single point and a homogeneous complex elliptic operator L in R N , there exists an L-analytic function f in Ω -i.e a solution f to Lf = 0 on Ω -satysfing the following property: For any ζ in the boundary bΩ of Ω and any continuous path γ : [0, 1) → Ω with γ(r) → ζ as r → 1, the cluster set of f along γ is maximal (i.e. equal to C ∪ {∞}) [9, Theorem 5]. We recall that the cluster set of f along γ is defined as the set {c ∈ C ∪ {∞} : f • γ(r n ) → c for some (r n ) tending to 1} .

This result was completed and extended in [START_REF] Bernal-González | Maximal cluster sets of L-analytic functions along arbitrary curves[END_REF] where it was proven that the set of such above functions contains a dense vector subspace of the space of L-analytic functions on Ω endowed with the locally uniform convergence topology. In particular, the previous results apply to O(D) where D is a domain in the complex plane (with L = ∂) and to the space of functions harmonic in domains of R N (with L = ∆), and thus improved earlier works, for instance in the unit disc. We refer to the survey [START_REF] Prado-Bassas | Maximal cluster sets on spaces of holomorphic functions[END_REF] for an overview on the topic of (L)-holomorphic functions with maximal cluster sets before 2008. We shall also mention [START_REF] Bernal-González | Universal Taylor series with maximal cluster sets[END_REF] where the authors are interested in the dense lineability and the spaceability (see below for the definition) in O(D) of the set of so-called universal series (see [START_REF] Nestoridis | Universal Taylor series[END_REF]) with maximal cluster sets along any path to the boundary, and [START_REF] Charpentier | Holomorphic functions with universal boundary behaviour[END_REF] where functions holomorphic in D with a universal property implying the maximality of cluster sets along any such paths are exhibited and studied more specifically.

Despite the rather large degree of generality of the previous results, it seems that the natural problem of exhibiting holomorphic functions in domains of C N , N ≥ 2, with maximal cluster sets along continuous paths to the boundary has been left open till now. Actually polynomial and holomorphic approximation in C N makes the problem more involved in this setting. Indeed, the existence of a function f in O(D) with maximal cluster set along every path to the boundary clearly imposes on D to be a domain of holomorphy, hence to be pseudoconvex by Oka's solution to the Levi problem. Moreover it turns out that such a function cannot exist in general: no holomorphic function in B N can be unbounded along every path to the boundary [START_REF] Globevnik | Holomorphic functions with highly noncontinuable boundary behavior[END_REF]Theorem 3].

The first aim of this paper is to address this problem (Theorem A below). Let D be a pseudoconvex domain in C N . From now on, a finite length path to bD refers to as a continuous piecewise C 1 path γ : [0, 1) → D, with finite length, such that γ(r) → ζ ∈ bD as r → 1. We shall also recall that a subset of a topological vector space X is residual if it contains a dense countable intersection of open sets, densely lineable if it contains apart 0 a dense subspace of X and spaceable if it contains apart 0 a closed infinite dimensional subspace of X (see [START_REF] Aron | Lineability: the search for linearity in mathematics[END_REF][START_REF] Bernal-González | Linear subsets of nonlinear sets in topological vector spaces[END_REF]).

Theorem A. Let D be a pseudoconvex domain in C N . There exists a set V residual in O(D) whose every element f satisfies the following property: for any finite length path γ to bD and any complex number c ∈ C, there exists a sequence (r

n ) n ⊂ [0, 1), r n → 1 as n → ∞, such that f • γ(r n ) → c, n → ∞.
Moreover V is densely lineable and spaceable in O(D).

Observe that, since L ∩ B N is the image of a finite length path for any complex line L, it implies that the set of those f ∈ O(B N ) such that f L∩B N is non-extendable for any L is residual in O(D), an answer to a question posed in [4, Page 8].

Let us make some comments on the proofs in [START_REF] Bernal-González | Maximal cluster sets of L-analytic functions along arbitrary curves[END_REF][START_REF] Boivin | Approximation on closed sets by analytic or meromorphic solutions of elliptic equations and applications[END_REF]. The functions with maximal cluster sets along paths are built as (infinite) sums of L-analytic functions in Ω which are simultaneously small on some compact subsets of Ω and arbitrarily close to any given complex number on suitable other compact subsets. Here suitable means that any path to the boundary has to intersect all but finitely many of these sets. To perform this construction, it is built in [START_REF] Boivin | Approximation on closed sets by analytic or meromorphic solutions of elliptic equations and applications[END_REF] a sequence (G n ) of sets in Ω with the property that any path to bΩ intersects all but finitely many G n 's and such that each K n ∪ G n is a Roth-Keldysh-Lavrentiev set -on which a Runge-type theorem for L-analytic function is valid -where (K n ) is some exhaustion of Ω by compact sets (see [START_REF] Bernal-González | Maximal cluster sets of L-analytic functions along arbitrary curves[END_REF]Lemma 3.2] for a precise statement). In 2015, for completely different purpose, Globevnik succeeded in building a labyrinth, that is a sequence (Γ n ) in B N with exactly the same property as the sequence (G n ) above, except that only those paths to bB N with finite length have to intersect all but finitely many Γ n and that K n ∪ Γ n is now polynomially convex [START_REF] Globevnik | A complete complex hypersurface in the ball of C N[END_REF]. This major step allowed him to build a function f ∈ O(B N ) unbounded along any finite length path to bB N , that he used to exhibit a complete, closed complex hypersurface in B N , and thus to give a positive answer to a question posed by P. Yang in 1977 [START_REF] Yang | Curvature of complex submanifolds in C n[END_REF][START_REF] Yang | Curvature of complex submanifolds in C n[END_REF]. He extended this to any pseudoconvex domains of C N [START_REF] Globevnik | Holomorphic functions unbounded on curves of finite length[END_REF]. Later the results of Globevnik from [START_REF] Globevnik | A complete complex hypersurface in the ball of C N[END_REF][START_REF] Globevnik | Holomorphic functions unbounded on curves of finite length[END_REF] were proven by Alarcón [START_REF] Alarcón | Complete complex hypersurfaces in the ball come in foliations[END_REF] for any Stein manifold endowed with a Riemannian metric.

The construction of such labyrinths was much simplified in B N [START_REF] Alarcón | A construction of complete complex hypersurfaces in the ball with control on the topology[END_REF] and also in any pseudoconvex domains [START_REF] Charpentier | Construction of labyrinths in pseudoconvex domains[END_REF]. With this in hand, we will then be able to prove Theorem A. Furthermore, it turns out that holomorphic functions in C N with some erratic boundary behaviour were already exhibited in 2005 by Bayart [5, Theorem 1], who showed the following.

Theorem 1.1. The set of functions in O(B N ) satisfying that, given any measurable function h on bB N , there exists an increasing sequence (r n ) n ⊂ [0, 1), r n → 1, such that f (r n x) → h(x) for almost every x ∈ bB N is residual.

In fact, a more difficult result is proven showing that such f can have an arbitrarily slow radial growth to bB N . This result is in contrast with the existence of holomorphic functions with radial limits equal almost everywhere on bB N to a given measurable function (see [START_REF] Kahane | Sur le comportement radial des fonctions analytiques[END_REF] for the unit disc, [START_REF] Dupain | Extension à la dimension n d'un théorème de Ortel et Schneider[END_REF][START_REF] Iordan | On the Radial Cluster Sets for Holomorphic Functions in the Unit Ball of C n[END_REF] for the unit ball, and [START_REF] Charpentier | Approximation par des fonctions holomorphes à croissance contrôlée[END_REF] for strictly pseudoconvex domains in C N with C ∞ boundary). In [START_REF] Bayart | Universal radial limits of holomorphic functions[END_REF][START_REF] Iordan | On the Radial Cluster Sets for Holomorphic Functions in the Unit Ball of C n[END_REF] the constructions are based on a Mergelyan type theorem for certain star shaped compact subsets of the closed ball, proven in [START_REF] Hakim | Boundary properties of analytic functions in the ball of C n[END_REF].

The second aim of this paper is to prove a simpler and natural polynomial approximation result in strictly convex sets of C N -somewhat inspired by the construction of labyrinths, to extend it to strictly pseudoconvex domains thanks to a deep embedding theorem due to Diederich, Fornaess and Wold [START_REF] Diederich | Exposing Points on the Boundary of a Strictly Pseudoconvex or a Locally Convexifiable Domain of Finite 1-Type[END_REF], and finally to improve and extend, in a rather transparent fashion, Bayart's result to strictly pseudoconvex domains.

Theorem B (Theorem 3.5 below for a precise statement). Let D be a strictly pseudoconvex domain of C N and let γ : bD × [0, 1] → D be a smooth map. Under some assumptions on γ, there is a residual set W in O(D) every element f of which satisfies that given any measurable function h : bD → C, there exists a sequence (r

n ) n ⊂ [0, 1), r n → 1, such that f • γ(x, r n ) → h(x)
for almost every x ∈ bD. Moreover, the set W is densely lineable and spaceable in O(D).

If D = B N our proof can be combined with that of Bayart (or Iordan [START_REF] Iordan | On the Radial Cluster Sets for Holomorphic Functions in the Unit Ball of C n[END_REF]) to get that any functions as in Theorem B can have an arbitrary radial growth.

The paper is organized as follows. The next section introduces the necessary background in complex analysis in several variables, that readers from Operator Theory or one dimensional complex variable may not be familiar with. Section 3.1 is devoted to the proof of Theorem A, and Section 3.2 to that of Theorem B.

Background

In this section, we briefly introduce the necessary background in several complex variables. A domain D in C N is said to be pseudoconvex if there is a continuous plurisubharmonic function u : D → R such for any c ∈ R the set {z ∈ D : u(z) < c} is relatively compact. We refer to [20, Chapter II] for several equivalent definitions. As a solution to the Levi problem, Oka, Bremermann and Norguet established that the class of pseudoconvex domains coincides with that of domains of holomorphy [START_REF] Hörmander | An introduction to complex analysis in several variables[END_REF]Chapter IV].

A strictly (or strongly) pseudoconvex domain D in C N is a bounded domain with C 2 boundary such that the Levi form of some defining function ρ is positive definite on the complex tangent space (see [START_REF] Krantz | Function theory of several complex variables[END_REF]Chapter 3] or [START_REF] Ohsawa | Analysis of several complex variables[END_REF]Chapter 3]). The next theorem will be crucial for the proof of Theorem B. It is a particular case of [START_REF] Diederich | Exposing Points on the Boundary of a Strictly Pseudoconvex or a Locally Convexifiable Domain of Finite 1-Type[END_REF]Theorem 1.1].

Theorem 2.1. Let D ⊂ C N be a strictly pseudoconvex domain. For any x ∈ bD, there is a ball B x centred at x and a holomorphic embedding

k x : D → B N with k x (x) = (1, 0) ∈ C × C N -1 such that each point of k x (B x ∩ bD) is a point of strict convexity of k x (D).
We will constantly make use of polynomial or holomorphic approximation. This is understood in terms of polynomial or holomorphic convexity, that we shall recall now. For this discussion, we refer to the book by Stout [START_REF] Stout | Polynomial convexity[END_REF].

Let D be a domain of C N and K a compact subset of D. The holomorphic hull of K with respect to D is defined by

K D = {z ∈ D : |ϕ(z)| ≤ sup K |ϕ| for every ϕ ∈ O(D)}. It is well known that K D is a compact subset of D provided that D is pseudoconvex. K is said to be holomorphically convex with respect to D -or O(D)-convex -if K = K D . If D = C N , then K = K C N is the polynomial hull of K, and K is polynomially convex whenever K = K.
Any compact convex set is polynomially convex. The C N version of Runge's theorem is the Oka-Weil theorem, that states as follows (see [START_REF] Forstnerič | Stein manifolds and holomorphic mappings[END_REF]Theorem 2.3.1]). A union of polynomially convex sets is not polynomially convex in general. However, a classical result due to Kallin says that, under some conditions, this can hold true (see e.g. [START_REF] Stout | Polynomial convexity[END_REF]Theorem 1.6.19]). In particular, it says that if K and L are compact subsets of C N and there is a polynomial p such that p(K) ∩ p(L) is empty, then K ∪ L = K ∪ L. Thus if K and L are additionally polynomially convex, then K ∪ L is polynomially convex as well. In the next sections we shall need the following analogous result for holomorphic hulls. It can be proven as [START_REF] Stout | Polynomial convexity[END_REF]Theorem 1.6.19] or by embedding D into C 2N +1 ([20, Theorem 5.3.9]), using Cartan's theorem and directly applying [START_REF] Stout | Polynomial convexity[END_REF]Theorem 1.6.19].

Proposition 2.3. Let K, L be compact subsets of a pseudoconvex domain D in C N . Suppose that there is f ∈ O(D) such that f (K) ∩ f (L) is empty. Then K ∪ L D = K D ∪ L D .
This proposition and an easy induction immediately yields the next corollary.

Corollary 2.4. Let D ⊂ C N be a pseudoconvex domain. Let K 0 , . . . , K n be compact O(D)convex sets. Suppose that for each j = 0, . . . , n there is

f ∈ O(D) such that f (K j ) D ∩ f ( i<j K i ) D is empty. Then K 0 ∪ . . . ∪ K n is O(D)-convex.

Proofs of Theorems A and B and some complementary results

Wild holomorphic functions along finite length paths.

Given an open set U ⊂ C N , a map γ : [0, 1) → U is said to be a path to bU -or ending at bU -if γ is piecewise C 1 and there exists x ∈ bU such that lim r→1 γ(r) = x. The length of a path γ is defined as

1 0 |γ (t)|dt.
The main goal of this section is to prove Theorem A. Let us first prove the first part, that we restate as follows.

Theorem 3.1. Let D be a pseudoconvex domain of C N . Let V denote the subset of O(D) consisting in those f satisfying the property that given any finite length path γ to bD and any complex number c, there exists r n tending to

1 such that f • γ(r n ) → c as n → ∞. Then V is residual in O(D).
The proof will be an easy application of the Baire Category Theorem and of the main result of [START_REF] Charpentier | Construction of labyrinths in pseudoconvex domains[END_REF] that we state below under the form we need. Proof of Theorem 3.1. Let (K n ) be given by Theorem 3.2 with M n = 1 for any n, and let (c j ) j be a dense sequence in C. We define

Ṽ := k,j∈N n∈N f ∈ O(D) : sup Γn |f -c j | < 1 k .
We claim that any f ∈ Ṽ belongs to V. Indeed, for a given c ∈ C, there exists a sequence (n k ) k such that sup Γn k |f -c| < 1 k . Let now γ be any continuous path with finite length ending at bD. By the properties of the Γ n 's, γ([0, 1)) must intersect all but finitely many Γ n k . Hence c belongs to the closure of f • γ([0, 1)).

It remains to prove that Ṽ is a dense G δ -subset of O(D). By the Baire Category Theorem, it is enough to prove that for any k, j, the set

n∈N f ∈ O(D) : sup Γn |f -c j | < 1 k is open and dense in O(D). That it is open is clear. To prove it is dense, we fix a compact subset K ⊂ D, h ∈ O(D) and > 0. Then we choose n such that K ⊂ D n and use the O(D)- convexity of D n ∪ Γ n to get f ∈ O(D) such that |f -h| < on D n and |f -c j | < 1/k on Γ n
Let us now complete the proof of Theorem A. Proof. (1) We first prove the dense lineability of V. Let us fix a countable neighbourhood basis (U l ) of O(D). With the notations used in the proof of Theorem 3.1, we easily derive from the latter that, for any increasing sequence (n i ) of integers, the set

Ṽ((n i )) := k,j∈N i∈N f ∈ O(D); sup Γn i |f -c j | < 1 k .
is residual in O(D). Let us build by induction sequences (n l i ) in N and (f l ) in Ṽ as follows: Pick f 0 ∈ Ṽ ∩U 0 and consider an increasing sequence (n 0 i ) such that sup

Γ n 0 i |f 0 | → 0 as i → ∞.
Now assume that f l has been built in Ṽ((n l-1 i )) ∩ U l (with the notation (n -1 i ) := (n i )). There is an increasing subsequence (n l i ) of (n l-1 i ) for which sup

Γ n l i |f l | → 0 as i → ∞. Let us pick any f l+1 ∈ Ṽ((n l i )) ∩ U l+1 . Thus the set E := span(f l : l ≥ 0) is a dense vector subspace of O(D).
To finish, let us check that any f := M l=0 a l f l ∈ E \ {0} belong to Ṽ. We can assume that a M = 0. Fix k, j ∈ N. By definition, there exist infinitely many n M i such that sup

Γ n M i |f M - c j a M | < 1 2k
Moreover, by construction, sup

Γ n M i |f l | → 0 as i → ∞, for any 0 ≤ l ≤ M -1. Thus one can choose i ∈ N such that sup Γ n M i |f -c j | < 1 k , which proves that f ∈ Ṽ((n M i )). Since we clearly have Ṽ((n M i )) ⊂ Ṽ, the proof is complete. (2)
We turn to the proof of the spaceability of V. We denote by ≺ the lexicographical order on N × N. Let (D n ) and (Γ n ) be given as in Theorem 3.2. Up to re-ordering we may assume that the Γ n 's are ordered in a double sequence Γ k,n , k ≥ n ≥ 1, such that Γ k,n ⊂ D j+1 \ D j where j is the rank in which the pair (k, n) appears in the sequence N × N ordered increasingly with respect to ≺ (for instance, if (k, n) = (4, 2) then its rank j is 8). We also fix a basic sequence (e n ) in O(D), generating an infinite dimensional subspace, such that for every n ≥ 1, sup D 1 |e n | = 1 and the sequence (e k ) k≥n is basic in the normed vector space (O(D), sup Dn | • |) with basic constant less than 2. For the existence of such a basic sequence, see [START_REF] Menet | Hypercyclic subspaces and weighted shifts[END_REF]Lemma 1.7]. Let (c k ) be a dense sequence in C and let us fix a sequence ( n ) of positive real numbers such that := n < ∞. As in the proof of [24, Theorem 1.1] and using the O(D)-convexity of Γ n ∪ D n , one can build by induction two families (f i,j ) i≥j≥1 ⊂ O(D) and (n i,j ) i≥j≥1 ⊂ N satisfying the following conditions:

(i) sup

D j |f j,j | < j 2 j+1 ; (ii) sup Γ k,j |f i,j -(c k -i-1 l=j f l,j )| < j 2 i+1 if Γ k,j ⊂ D i+1 \ D i ; (iii) sup D i |f i,j | < j 2 i+1 ; (iv) sup Γ k,m |f i,j + i-1 l=j f l,j | < j 2 i+1 if m = j and Γ k,m ⊂ D i+1 \ D i . By (iii) f j := i≥j f i,j + e j is well-defined in O(D)
and, according to [24, Lemma 1.9], (iii) also implies that (f j ) is a basic sequence in O(D), equivalent to (e j ). Thus the set E := span(f j ; j ≥ 1) is a closed infinite dimensional subspace of O(D). The remaining of the proof consists in showing that any f = n≥1 a n f n ∈ E \ {0} belongs to V. Denoting by N the smallest integer among those n for which a n = 0, we shall assume, without loss of generality, that a N = 1. Moreover, proceeding as in the proof of [24, Theorem 1.10], one can see that there exists a constant K such that |a n | ≤ K for any n ≥ 1.

By the proof of Theorem 3.1, we need only prove that sup

Γ k,N |f -c k | → 0 as k → ∞. For k ≥ N let us denote by j(k) the unique integer such that Γ k,N ⊂ D j(k)+1 \ D j(k) . Note that j(k) > k. Considering the estimate |f (z) -c k | ≤ |f N (z) -c k | + K n =N |f n (z)|, we use first (ii) and (iii) to get, for any z ∈ Γ k,N , |f N (z) -c k | ≤ |f N (z) - j(k) i=N f i,N (z)| + | j(k) i=N f i,N (z) -c k | < N 2 j(k) ,
and secondly (iii) and (iv) to obtain

n =N |f n (z)| ≤ n =N | j(k) i=n f i,n (z)| + n =N |f n (z) - j(k) i=n f i,n (z)| < 2 j(k) .
This concludes the proof. ), and r → γ(x, r) hits bD transversally at x for any x ∈ bD, as r → 1. For r ∈ [0, 1), we denote by bD r := {γ(x, r) : x ∈ bD}. We assume additionally ( †):

: bD × [0, 1] → D such that γ(x, r) ∈ D, r ∈ [0, 1
( †) For any r close enough to 1, bD r bounds a smooth domain D r that is close to D. In particular, this assumption implies that for any compact set K in D and any r close enough to 1, D r contains K.

Note that given a strictly pseudoconvex domain D, there always exists a C 1 map as above satisfying ( †). Indeed, it is the most natural one: For x ∈ bD, let ν x denote the inward unit normal vector and set γ(x, r)

:= x + (1 -r)ν x for r ∈ [0, 1]. It is clear that γ is C 1 on bD × [0, 1]
. Moreover, for r close enough to 1, each point of the set bD r is at distance r from bD. Now, we can use the well-known fact that the signed distance function to the boundary of a smooth domain is C 2 in order to see that D r is a smooth domain, which is close in C 2 -topology to D (see [START_REF] Krantz | Function theory of several complex variables[END_REF]Chapter 3]).

In fact, a classical but a bit more tricky argument shows that there even exist maps γ which are C 2 on bD × [0, 1] and hit transversally bD as r → 1. The following proposition says that, under this stronger assumption, ( †) is automatically satisfied. Proposition 3.4. Keeping the above settings, let us assume additionally that the map γ is C 2 on bD × [0, 1]. Let K be any compact subset of D. There exists r 0 ∈ (0, 1) such that D r is the boundary of a strictly pseudoconvex domain in D that contains K whenever r ∈ (r 0 , 1].

Proof. Fix x ∈ bD.It follows from the implicit function theorem that there exist a neighborhood U of x, an open set U in R 2n-1 and a C 2 function f : U → R such that

U ∩ bD = {(t, f (t)) : t ∈ U }.
Now, using that γ is C 2 , we shall infer that for any r close enough to 1, there are C 2 smooth functions g r such that γ((t, f (t)), r) = (t, g r (t)), t ∈ U , and g r is close to f in C 2 topology. Thus bD r is locally the graph of a smooth function, from which we easily obtain, at first, that bD r bounds a domain, denote it by D r , that contains K for r sufficiently close to 1. Second, we deduce that D r is strictly pseudoconvex, since the Levi form of some defining function for D r , restricted to proper complex tangent spaces, is close to the (positive and definite) Levi form of a defining function of D.

Let us now turn to the proof of Theorem B. The proofs of the dense lineability and the spaceability are omitted as both may be obtained exactly as Proposition 3.3, together with the proof of the following. Here and below, the map γ is defined as above and assumed to be C 1 and satisfy ( †). Theorem 3.5. Let (r j ) be an increasing sequence, 0 < r j < 1, lim j→∞ r j = 1. Then the set W(r j ) of holomorphic functions f on D such that, for any measurable function h : bD → C, there is a subsequence (r j k ) of (r j ) with

lim k→∞ f (γ(x, r j k )) = h(x) for a.e. x ∈ bD is residual in O(D).
The proof of Theorem 3.5 is based on Lemma 3.7 below about holomorphic approximation on D. To obtain this lemma, we will first prove a polynomial approximation result in strictly convex domains (Lemma 3.6), and then use embeddings of D into strictly convex domains (Theorem 2.1).

For a convex set V of C N and H a real hyperplane in C N , we call a cap of V any connected component of the set V ∩ (C N \ H) (which has at most two components). A closed cap of V will be the closure in V of a cap of V . Note that any closed cap is a convex compact subset of C N and, as such, it is polynomially convex. Lemma 3.6. Let V be a strictly convex domain and U a domain of C N intersecting bV . Then for every > 0 there are finitely many disjoint closed caps Γ 1 , . . . ,

Γ n contained entirely in U ∩ V such that m((U ∩ bV ) \ Γ) < , where Γ = Γ 1 ∪ . . . ∪ Γ n . Moreover, given any polynomially convex compact subset K of V with Γ ∩ K = ∅, Γ ∪ K is polynomially convex.
Proof. By compactness and the implicit function theorem, we can assume that there exists an open set

U in R 2N -1 such that bV ∩ U = {(t, f (t)) : t ∈ U },
and that the map F : t → (t, f (t)) is a C 2 diffeomorphism from U onto bV ∩ U . Now it is not difficult to check that the strict convexity of V and the smoothness of F imply the existence of 0 < K 1 ≤ K 2 < ∞ such that for each open cube C contained in U , there exists a closed cap Γ of V such that F (C) ⊃ Γ ∩ bV and K 1 m(C) ≤ m(Γ) ≤ K 2 m(C). Moreover, for C small enough, we shall have Γ ⊂ U . Thus the lemma follows from the choice of finitely many disjoint open cubes C 1 , . . . , C n in U such that the measure of U \ ( n j=1 C j ) is small enough and, again, the fact that F is a diffeomorphism.

The last assertion of the lemma follows from Corollary 2.4.

Γ in U ∩ G such that m((U ∩ bG) \ Γ) < , K ∩ Γ = ∅ and Γ ∪ K is O(D)-convex in D.
Proof. Up to obvious changes, we may and shall assume that U = C N . According to [START_REF] Diederich | Exposing Points on the Boundary of a Strictly Pseudoconvex or a Locally Convexifiable Domain of Finite 1-Type[END_REF] (Theorem 2.1), for each x ∈ bD there is a ball B x centred at x and an embedding k

x : D → B N with k x (x) = (1, 0) ∈ C × C N -1 such that each point of k x (B x ∩ bD) is a point of strict convexity of k x (D). If G is close enough to D, then each point of k x (B x ∩ bG) is a point of strict convexity of k x (G).
Thus one can consider a strictly convex domain

V x containing k x (G) such that k x (G) ∩ k x (B x ) = V x ∩ k x (B x )
. Moreover, we shall assume that bG ⊂ ∪ x∈bD B x , hence one can find finitely many disjoint open sets U 1 , . . . , U n such that each of them lies in some B x , K ∩ U j = ∅, and m(bG \ ∪U j ) < /2. Let us fix j and take x such that U j ⊂ B x and let us apply Lemma 3.6 to V x and the domain k x (U j ). It provides us with a finite union of disjoint closed caps, that we shall denote by Γj , such that m((k x (U j ) ∩ bV x ) \ Γj ) < η, where η will be chosen below. Let us define Γ j := k -1

x ( Γj ) and observe that η can be chosen small enough so that m((U j ∩ bG) \ Γ j ) < /(2n).

Setting Γ = ∪ j Γ j , we get m(bG \ Γ) < . Note also that each Γj is polynomially convex (Lemma 3.6) and is separated from k x (K) and k x (U i ) for i = j by a polynomial. Composing with k x we get that each Γ j is O(D)-convex and separated from K and Γ i , i = j, by a function holomorphic in D. This, according to Corollary 2.4, means that Γ ∪ K is O(D) convex.

We now deduce the following, which is the key-ingredient for Theorem 3.5.

Corollary 3.8. Let > 0, let K be O(D)-convex in D, h ∈ O(D)
and let ϕ be continuous on bD. There exists r 0 ∈ (0, 1) such that for any r ∈ (r 0 , 1), there exist a compact set

E ⊂ bD and f ∈ O(D) such that (1) m(bD \ E) < ; (2) sup x∈E |f • γ(x, r) -ϕ(x)| < ; (3) sup z∈K |f (z) -h(z)| < .
Proof. We fix > 0. By continuity of ϕ on bD, there exists finitely many connected open sets U 1 , . . . , U n ⊂ C N , with U i ∩U j = ∅ if i = j, and c 1 , . . . , c n ∈ C such that |ϕ(x)-c i | < /2 for any x ∈ bD ∩ U i , i = 1, . . . , n, and m(bD

\ (U := ∪ i U i )) < /2.
Since γ is C 1 on bD × [0, 1] and γ(x, 1) = x for any x ∈ bD, there exists r 0 such that for any r 0 ≤ r ≤ 1, the map γ r : x → γ(x, r) is a C 1 diffeomorphism from bD onto bD r . Moreover for any A ⊂ bD, m(A) ≈ m(γ r (A)), where the constants in ≈ do not depend on A or on r ≥ r 0 . We define V r as an open set of

C N such that V r ∩ K = ∅ and V r = ∪ i V i r with V i r = γ r (bD ∩ U i ).
In particular, we thus have m(bD r \ V r ) ≈ /2, and the function φ defined as φ(z) = c i if z ∈ V i r is well-defined and holomorphic in V r . By assumption ( †), we can assume that r 0 is even closer to 1 so that for any r ∈ (r 0 , 1), bD r is the boundary of a strictly pseudoconvex domain in D that contains K, whence, according to Lemma 3.7, there exists a compact set Γ r in V r with m(

V r ∩ bD r \ Γ r ) < /2, such that Γ r ∪ K is O(D)-convex in D.
Thus there exists f ∈ O(D) such that |f (z) -φ(z)| < /2 for any z ∈ bD r ∩ Γ r and |f (z) -h(z)| < for any z ∈ K. Setting E = γ -1 r (bD r ∩ Γ r ) completes the proof, since is arbitrary.

Proof of Theorem 3.5. The rest of the proof is very similar to steps 2 and 3 of that of [START_REF] Bayart | Universal radial limits of holomorphic functions[END_REF]Theorem 1]. Let (K s ) be O(D) convex sets that exhaust D, i.e. K j is contained in the interior of K s+1 and s K s = D. We also fix a dense sequence (h j ) in the set C(bD) of continuous functions on bD. We denote by E k the set of all compact subsets E of bD such that m(bD \ E) > 1 -1/2 k . For positive integers j, k, l, s, we define the set

U (j, k, l, s) = r∈(r j ,1) E∈E k f ∈ O(D) : |f • γ(x, r) -h j (x)| < 1/2 k , x ∈ E .
It is clear that each U (j, k, l, s) is an open subset of O(D) and, by Corollary 3.8, that it is dense. So, by the Baire Category Theorem, we get that U := j,k,l,s U j,k,l,s is a dense G δ -subset of O(D).

Let us take f ∈ U . It remains to prove that f belongs to W(r j ). We thus fix h measurable on bD. Proceeding as in Step 3 of the proof of [5, Theorem 1], we can build by induction an increasing sequence (r j k ), r j k → 1, and a sequence

(E k ) in E k such that |f • γ(x, r j k ) -h(x)| < 1 2 k , x ∈ E k . Setting E = n≥1 k≥n E k , it is easily seen that m(E) = 1 and that f • γ(x, r j k ) → h(x) as k → ∞ for any x ∈ E.
The case of the ball. In the case of the ball, Theorem 3.5 can be strengthened. Assume that we are given a family of maps {γ z : bB N ×[0, 1] → B N ; z ∈ B N } such that each γ z satisfies the assumptions made at the beginning of the section and, additionally, that (z, x, r) → γ z (x, r) is continuous on B N × bB N × [0, 1] and (z, r) → ∂γ z (x, r)/∂r is continuous on B N × [0, 1], for any x ∈ bB N . A typical example of such maps is γ z (x, r) = x + r(x -z).

Let v : [0, +∞[→ [0, +∞[ be a weight, i.e. an increasing function such that v(x) → ∞ as x → ∞. We recall that the space H 0 v is defined as

H 0 v := f ∈ O(B N ) : sup |z|=r |f (z)| v(r) → 0 as r → 1 .
It is well-known that H 0 v is a Banach space endowed with the norm f = sup z∈B N |f (z)|/v(|z|) and that the polynomials are dense in H 0 v . Theorem 3.9. Let v be a weight and let (r j ) be a sequence in (0, 1) with r j → 1. With the notations of Theorem 3.5 and D = B N , the set W(r j ) ∩ H 0 v is residual in H 0 v . This theorem was obtained by Bayart for γ z (x, r) = z + r(x -z) [5, Theorem 1]. Observe that the statement holds true also with O(D) instead of H 0 v and that it is already an improvement of Theorem 3.5 in the case D = B N (the family γ z is indexed by an uncountable set and an uncountable intersection of residual sets may be empty). More surprisingly, the above theorem tells that the functions satisfying the conclusion of Theorem 3.5 can grow arbitrarily slowly. The proof of Theorem 3.9, whose details are left to the reader, follows the same lines as that of [START_REF] Bayart | Universal radial limits of holomorphic functions[END_REF]Theorem 1], upon replacing [5, Lemma 1] and [START_REF] Bayart | Universal radial limits of holomorphic functions[END_REF]Lemma 3] by, respectively, Lemmas 3.10 and 3.11 below.

Let us recall that a continuous function f on B N has K-limit at x ∈ B N if (f (z i )) i converges for any (z i ) ⊂ D α (x) converging to x, where D α (x) is the Korányi approach region defined for α > 1 by

D α (x) = z ∈ B N : |1 -z, x | < α 2 (1 -|z| 2 ) .
Observe that, under the assumptions made on the family {γ z : z ∈ B N }, for any compact subset L of B N , there exists α > 1 and 0 < r < 1 such that

(3.1) {γ z (x, r ) : z ∈ L, r < r < 1} ⊂ D α (B N ).
For L a compact subset of B N and E ⊂ bB N , we define

E * L = {γ z (x, r); x ∈ E, r ∈ [0, 1], z ∈ L} Lemma 3.10.
Let L be a compact subset of B N and h a continuous function on B N with Klimit almost everywhere on bB N . For every > 0, there exist a subset E of bB N , m(bB N \E) <

, and an extension h of h to B N ∪ E such that h is continuous on E * L . Proof. Up to (3.1), it is identical to that of [5, Lemma 1]. Lemma 3.11. Let v be a weight, L a compact subset of B N , and h a continuous function on bB N . For every > 0, there exist a compact subset E of bB N and a function f holomorphic in B N , continuous on E * L , such that:

(1) m(B N \ E) < ; (2) |f (x) -h(x)| < for any x ∈ E; (3) f -g < .
Proof. The proof works the same as that of [START_REF] Bayart | Universal radial limits of holomorphic functions[END_REF]Lemma 3]. However, instead of [21, Lemma 3] (see also [START_REF] Hakim | Boundary properties of analytic functions in the ball of C n[END_REF]), which is the main ingredient of [5, Lemma 3], we observe that one can use Lemma 3.6 (with V = B N and U = C N ), whose proof is rather elementary. Remark 3.12. It would be interesting to know whether Theorem 3.9 extends to any strictly pseudoconvex domain. The most difficult point would consist in controlling the growth of the functions near the boundary. The strategy developed in [START_REF] Charpentier | Approximation par des fonctions holomorphes à croissance contrôlée[END_REF] might be useful.

Proceeding as for V in Proposition 3.3, we can prove the following. Proposition 3.13. Let v be a weight and let (r j ) be a sequence in (0, 1) with r j → 1. W(r j ) ∩ H 0 v is densely lineable and spaceable in H 0 v . A concluding remark. We recall that D stands for the unit disc in the complex plane. In the sequel we say that f ∈ O(D) satisfies property (P) if given any K bD and any h ∈ C(bD), there exists (r j ) ∈ [0, 1) such that f r j → h uniformly on K as j → ∞. It is easily checked that if f has (P), then given any h ∈ C(bD), f r j converges pointwise on bD to h for some sequence (r j ), and that f has maximal cluster sets along any path to bD (with finite or infinite length) γ to bD. It is proved in [START_REF] Charpentier | Holomorphic functions with universal boundary behaviour[END_REF] that the set of functions satisfying (P) is a dense G δ -subset of O(D). In contrast, as we have already mentioned, there do not exist an f ∈ O(B N ), N ≥ 2, such that f • γ has dense range for any path γ to bB N [START_REF] Globevnik | Holomorphic functions with highly noncontinuable boundary behavior[END_REF]. Furthermore, one can show that there cannot exist f ∈ O(B N ), N ≥ 2, such that given any h ∈ C(bB N ), f r j → h pointwise on bB N , for some (r j ) ⊂ [0, 1). Indeed, a standard argument using the Baire Category Theorem yields that if f r j (x) → h(x) for any x ∈ bB N , then it is uniformly bounded on some open subset of bB N . Thus, for some cap Γ ∈ B N , f r j has to be bounded on Γ ∩ bB N . Now, by the maximum modulus principle, f r j |L is also bounded on L ∩ Γ uniformly for any complex affine line L intersecting Γ. Therefore f r j must be bounded on Γ, hence f itself has to be bounded on a non-empty open set B N ∩ B x , for some open ball B x centred at some x ∈ bB N . This of course implies that for some h ∈ C(bB N ), there exists no sequence (r j ) ⊂ [0, 1) such that f r j → h pointwise on bB N , a contradiction. Observe that this argument shows that the pointwise convergence of f r j to any h for some (r j ) cannot even hold on a given open subset of bB N . However, it is possible to prove the following. For x ∈ bB N , we denote by L x the complex line passing through x and the origin, and by bL x the set L x ∩ bB N . Theorem 3.14. Let (x i ) i≥1 be a sequence in bB N . There exists a dense G δ -subset U of O(B N ) such that every f ∈ U satisfies that, given any h i ∈ C(bL x i ) and any compact set K i bL x i , i ≥ 1, there exists a sequence (r j ) in [0, 1) such that for any i ≥ 1, f r j → h i uniformly on K i .

Observe that the sequence (r j ) does not depend on i ≥ 1. For the proof, we will need another particular case of Kallin's lemma (see [START_REF] Stout | Polynomial convexity[END_REF]Page 63]). Lemma 3.15. Let K and L be two compact convex subsets of C N that intersects at one point. If there exists a linear functional ϕ on C N such that ϕ ≤ 0 on K and ϕ ≥ 0 on L, and ϕ -1 (0) meets K only in a point, then K ∪ L is polynomially convex.

From this lemma, it follows that the union of two closed caps in B N which intersect at a single point is polynomially convex.

Proof of Theorem 3.14. By the Baire Category Theorem and a diagonal argument, it is enough to prove the theorem for (x i ) i finite. Let us then fix {x i : i = 1, . . . , m} in bB N . Let us set a sequence (h j 1 , . . . h j m ) dense in C(bL x 1 ) × . . . × C(bL xm ). By the one variable Mergelyan theorem, any continuous function on bL x i can be uniformly approximated on any proper compact subset of bL x i by polynomials on C N , so we can assume that each h j i is a polynomial. Let us also fix a sequence (y s 1 , . . . , y s m ) s dense in bL x 1 × . . . × bL xm and let (K s 1,n , . . . , K s m,n ) n be an exhaustion of compact subsets of (bL For j, n, s, k fixed, the union on the right-hand side is clearly open in O(B N ). To prove that it is dense, we can consider, for any 0 < ρ < 1, finitely many closed caps Γ 1 , . . . , Γ l 0 ⊂ B N such that (1) For any 1 ≤ l ≤ l 0 , Γ l ∩ ρB N = ∅;

(2) For any 1 ≤ l ≤ l 0 , Γ l ∩ l =l Γ l contains at most one point;

(3) K s 1,n ∪ . . . ∪ K s m,n ⊂ l 0 l=1 Γ l . To finish, we use the fact that l 0 l=1 Γ l ∪ ρB N is polynomially convex, that we deduce from Corollary 3.15.

We shall mention that a function in O(D) which satisfies property (P) cannot have an arbitrary slow radial growth ( [START_REF] Charpentier | Holomorphic functions with universal boundary behaviour[END_REF]Corollary 2.11]). As a consequence, and in contrast with Theorem 3.9, a function in the class U cannot grow arbitrarily slowly either.

Finally, using the formalism and the ideas considered in this section, we shall add that Theorem 3.14 can be stated for strictly pseudoconvex domains, up to adequate modifications.

1 .

 1 Introduction Let B N = {z ∈ C N , |z| < 1} denote the open unit ball of C N , N ≥ 2, and D the open unit disc of the complex plane. If D is a domain in C N , N ≥ 1, O(D) will stand for the space of holomorphic functions endowed with the locally uniform convergence topology.

Theorem 2 . 2 .

 22 Let D be a domain of C N . If K ⊂ D is a compact O(D)-convex set, then any function holomorphic in a neighborhood of K can be approximated by functions in O(D) uniformly on K.

Theorem 3 . 2 .

 32 Let D be a pseudoconvex domain in C N and let (D n ) be an exhaustion of D by strictly pseudoconvex domains that are O(D)-convex. Let also (M n ) be a sequence of positive numbers. Then there are compact sets Γ n ⊂ D n+1 \ D n such that D n ∪ m j=n Γ j is O(D)-convex for every m ≥ n, and any continuous path connecting ∂D n to ∂D n+1 and omitting Γ n has length greater than M n .

Proposition 3 . 3 .

 33 With the notations of Theorem 3.1, V is dense lineable and spaceable in O(D).

Lemma 3 . 7 .

 37 Let D be a strictly pseudoconvex domain in C N . For any > 0, any compact O(D)-convex set K ⊂ D, any strictly pseudoconvex domain G ⊂ D sufficiently close to D in C 2 -topology and any open set U intersecting bG, there exists a compact set

x 1 ×

 1 . . . × bL xm ) \ {y s 1 , . . . , y s m }. Now, observe that U = j,n,s,k r∈[1-1 k ,1) m i=1 f ∈ O(B N ) : sup K s i,n |f r (z) -h j i (z)| < 1 k .

The first author is partly supported by the grant ANR-17-CE40-0021 of the French National Research Agency ANR (project Front). The second author is partly supported by the NCN grant SONATA BIS no. 2017/26/E/ST1/00723.

acknowledgement

The authors are grateful to the referee for her/his careful reading of the manuscript.


D E Jessop
email: d.e.jessop@bris.ac.uk

M A Gilbertson

A J Hogg

ON THE MOTION OF FLUIDISED GRANULAR CURRENTS: MOTION ALONG HORIZONTAL SURFACES

The motion of fluidised granular currents over horizontal surfaces is investigated experimentally and by mathematical modelling of solid and fluid phases. Analytical solutions for the bulk motion were tested by experiments involving material being introduced at a constant volume flux, and a reasonable agreement found between theoretical predictions and laboratory measurements.

INTRODUCTION

Aerated granular flows are a feature of large scale and destructive natural hazards that arise from volcanism as pyroclastic flows and phenomena such as avalanches and landslides. They are also utilised in industrial processes where the bulk transport of powders is required in order to produce pharmaceuticals, agricultural materials and foodstuffs. An understanding of the underlying physical processes that control these flows is necessary to avoid or avert disasters and increase efficiency of these industrial processes.

Fluidised beds and the apparatus used in this investigation incorporate a porous base that allows a flow of gas initially perpendicular to the base to permeate through the overlying (granular) material. The presence of this gas flow exerts a drag on particles that constitute the material and supports part, or all, of their effective weight.

Flows of dry granular material propagating over horizontal and inclined surfaces have been treated extensively in the literature (see, for example, Savage & Hutter (1); Balmforth & Kerswell (2); Pouliquen & Forterre (3)). Often the granular material is released from rest behind a lockgate that is rapidly removed to initiate the flow, or the material is introduced continuously at a particular point and flows rapidly away before undergoing a deceleration phase. We consider only the latter case in this paper. As frictional forces between particles as well as basal friction are responsible for this rapid attenuation, disruption of force chains transmitting stress between particles by fluidisation suggests an increased mobility for fluidised granular flows (Roche et al. (4)).

In this paper we aim to determine a model for the flow of fully fluidised granular currents. This will allow us to understand some of the complex physics associated with such phenomena. In order to do this we first discuss the forces and interactions associated with each phase, giving separate mass and momentum conservation equations for both. Next we explore simplifications that can be made for flows along horizontal surfaces and look for analytical solutions to the resulting equation of motion. Then we describe the experiments that were performed and analyse the resulting data with a view to evaluating our assumptions. Finally we give conclusions of the work.

THEORY Background & assumptions

We consider a two-dimensional flow of a granular material over a horizontal surface. The flow is thin so that its horizontal lengthscale, L, far exceeds typical values for its thickness, H. The material is also assumed to be fully fluidised so that the weight of the solid phase is entirely balanced by the drag from the imposed gas flow; there are no residual stresses within the solid phase. Thus, as in figure 1, orientating the x and z coordinate axes to be horizontal and vertical, respectively, and denoting the voidage of the material by , the density of the solid phase by ρ s and gravitational acceleration by -g ẑ, this balance is given by 0

= f f s • ẑ -(1 -) ρ s g, (1)
where f f s is the bulk drag force between the fluid and solid phases. Since the flow is thin, vertical fluid accelerations are negligible and thus the pressure gradient in the fluid phase must also balance the weight of the fluid and the drag between the phases. Thus denoting the fluid pressure and density by p f and

ρ f 0 = - ∂p f ∂z -f f s • ẑ -ρ f g (2)
Thus from ( 1) and ( 2) we deduce that the fluid pressure is hydrostatic and given by

p f = ((1 -)ρ s + ρ f ) g (h -z)
where h is the height of the material surface.

The imposed interstitial gas flow plays a crucial role in fluidised systems and in our model it is represented through the drag term f f s . We model this using the Ergun equation (see Rhodes [START_REF] Rhodes | Introduction to Particle Technology[END_REF]) given by

f f s = 150µ f (1 -) 2 d 2 p 3 (u -v) + 1.75ρ f (1 -) d p 3 |u -v|(u -v). ( 3
)
where µ f is the viscosity of the gas, d p is the particle diameter, u and v are the gas and solid velocities, respectively. In this report we chose to write the z component of the gas velocity as the sum of the imposed gas speed, u gi and the motion induced by the presence of particles u z , or u = (u x , u z + u gi ). Ignoring numerical factors, the ratio of the two terms in the Ergun equation is equivalent to a Reynolds number for the gas flow through the fluidised bed, Re bed ; typical values for physical parameters in the experiments reported later give this to be O(1) so the flow through the bed is laminar and only the left hand term of ( 3) is considered during the analysis. To ease writing, we combine the group of constants multiplying the difference in velocities as

K = 150µ f (1 -) 2 /d 2 p
3 and thus we approximate

f f s = K(u -v).
We now consider the horizontal components of the momentum equations for each phase. In general these could include the inertia of each phase, streamwise pressure gradients, resistive forces and contributions from the inter-phase drag law. In this paper we assume that inertia is negligible. This requires that the reduced Reynolds number is small, given by

a 2 Re 1
where a is the aspect ratio of the flow (a = H/L) and Re = ρ s U H/µ, where U and µ are a velocity scale and the effective viscosity respectively.

Observations made during laboratory experiments suggest that the flows are not inertially driven. For instance, if the flow rate of gas into the apparatus is shut off, the flow of material stops instantaneously. This behaviour is in agreement with Eames & Gilbertson (6) who observed steady flows and concluded that fluidised granular flows propagate in a similar manner to a viscous fluid as described by Huppert [START_REF] Huppert | Propagation of two-dimensional and axisymmetric viscous gravity currents over a horizontal surface[END_REF].

Experiments performed with rotating paddle viscometers such as those performed by Matheson et al. [START_REF] Matheson | Characteristics of fluid-solid systems[END_REF], Furkawa & Ohmae (9) and Schügerl et al (10) using a fluidised bed contained within two concentric cylinders all agree in giving the observed viscosity of fully fluidised beds to be in the range of 0.05 to 2Pas. Caution must be paid to these results, however, as along with 'viscous' losses there is an associated momentum loss due to the acceleration of the particles. Furthermore, the results of such experiments are only valid for low applied shear gradients as the particles have a tendency of migrating to the outside of the cylinder/paddle due to their inertia thus creating a non-homogeneous suspension. This does suggest, however, that the viscosity of a suspension of particles is somehow altered.

On this assumption of negligible inertial effects we find that the horizontal component for the solid phase is given by 0

= K (u x -v x ) (4)
thus implying the horizontal gas and solid velocities are identical. For the fluid phase we find that

0 = - ∂p f ∂x -K (u x -v x ) + ∂τ xz ∂z ( 5
)
where τ xz is the shear stress within the fluid phase. We propose that ∂τ xz /∂z scales as u x /h 2 as it would for a viscous fluid with a constant of proportionality of the same dimensions as viscosity. As shall be discussed later, we assume that is constant. Upon integration of ( 5) and substitution into the depth integrated mass conservation equation for the solid phase, we obtain the governing equation for evolution of the height profile

∂h ∂t = ( 1
-)ρ s + ρ f 3 µ g ∂ ∂x h 3 ∂h ∂x , (6)
which can be solved numerically subject to the boundary conditions h(x f ) = 0 and v x h x=0 = q where x f is the front position and q is the volumetric flux of material. The total volume of material at any time is given by

x f 0 h dx = q t. (7)

Analytic solutions

Huppert [START_REF] Huppert | Propagation of two-dimensional and axisymmetric viscous gravity currents over a horizontal surface[END_REF] obtained an equation of the same form as [START_REF] Eames | The flow of aerated particles over a horizontal rigid surface[END_REF] and was able to construct similarity solutions subject to an identical condition as [START_REF] Huppert | Propagation of two-dimensional and axisymmetric viscous gravity currents over a horizontal surface[END_REF]. We follow the same method (see also Barenblatt (11)) to obtain the following solutions in terms of the similarity variable, ξ and similarity solution H(ξ)

x(t) = Λq 3/4 t 4/5 ξ, (8)
and h(x, t) = q 2 t/Λ 1/5 H(ξ), (9)
where Λ = ((1 -)ρ s + ρ f ) g/3 µ. Normalising (8) with respect to the front position, x f so that y = ξ/ξ f and substituting (9) into the governing equation ( 6) leads, after rescaling, to the following non-linear ODE in H(y)

H 3 H + 3H 2 H 2 + 4/5yH -1/5H = 0. ( 10
)
where a prime denotes differentiation with respect to y. H(y) and ξ f can be found numerically, subject to knowing Λ, allowing us to find x f .

EXPERIMENTAL SETUP

To test the hypotheses made in the previous section a series of experiments was devised wherein material is introduced at a constant flux. These are similar to the experiments performed by Eames & Gilbertson (6) except on a much larger scale so that data series over sufficiently long timescales could be obtained. The apparatus used was designed specifically for this investigation. An image of it is shown in figure 2. To avoid possible electrostatic effects which have arrisen with perspex walled fluidised beds, glass was used for the front and 15mm thick aluminium plate for the back and sides. This has the advantage of being rigid and thus ensuring that the channel in which material flows is straight throughout its length.

A basal distributor plates made from a 'sandwich' of porous Vyon D sheets from Porvair Ltd. surrounding a layer of Geldart A particles ranging from 45-90µm. This causes a large pressure drop so that the gas flow is constant along their length and that the presence of particles above the distributor plates does not locally affect the gas supply.

Experiments were recorded using a digital camera. As the apparatus is quite long a wide camera angle must be used in order to capture the flows in their entirety. It was found to be easier to determine the position of the front by viewing the flows from above; at the same time it is desirable to see how the height profile evolves. To resolve these issues a mirror was positioned above the apparatus as is shown in figure 2b so that both could be recorded simultaneously. Data extraction was done by viewing still images taken from the videos using the UTHSCSA ImageTool programme. Assuming that the front position can be determined from an image to within a few pixels and the 100cm length of apparatus equates to something in the order of 450 pixels, then we estimate the error in our measurements to be O( 1)cm. This is accounted for as error bars of this magnitude in the plots of data in the next section.

The particles used were glass spheres in the range of 250-500µm (Potters Ballotini grade C) corresponding to a Geldart B powder. Before starting any experiments, the powder was conditioned by vigorous fluidisation. Viewing the experiments from above presented the problem that the slightly off-white particles were hard to distinguish from the white Vyon sheet. To ease visualisation they were coloured using very dilute artists acrylic paint and then dried thoroughly to leave only the pigment. Using particles of this size ensured non-cohesive behaviour. Bed expansion using these particles is negligible so we assume that is constant for all values of u gi used in the experiments.

Standard fluidisation curves were constructed using data for the pressure drop through various bed heights from independent experiments (as described in Davidson & Harrison (12); Rhodes (13) etc.) from which the minimum fluidisation velocity, u mf was found to be 10.78cm/s. This is in agreement with theoretical predictions made by equating the first term from the Ergun equation ( 3) with the bed weight per unit volume which gives 5.854cm/s ≤ u mf ≤ 23.416cm/s1 .

RESULTS

We expect from (8) that x f ∼ (q 3 t 4 ) 1/5 . With knowledge of the constant Λ it should be possible to predict accurately the position of the front at any point in time. There is, however, at least one physical variable in Λ that is not well defined, namely the mixture viscosity, µ. This should be some function of , but the exact relationship is not currently clear. The voidage itself may also be hard to measure for any given experiment due to bed expansivity. However, for not too high values of u gi , we assume that is close to that Figure 3: a) (left) Front position, x f as a function of time, t for different volume fluxes of material and gas flow rates: q = 2.544cm 2 /s, u gi /u mf = 1.3 (×); q = 5.2cm 2 /s, u gi /u mf = 1.3 (•); q = 9.515cm 2 /s, u gi /u mf = 1.3 (♦); q = 9.515cm 2 /s, u gi /u mf =1.48 (+); q = 9.515cm 2 /s, u gi /u mf = 1.86 ( ). b) (right) The same data for x f but scaled with q 3/4 t. Also plotted is a best fit curve (-) according to (11) using A = 5.701 and n = 0.634 and the curve predicted by ( 8) (-• -) with A = 2.75 and n = 0.8 measurable when unaerated.

Instead we chose to try to collapse the data onto a single curve by scaling the front position with q 3/4 t so that

x f = A q 3/4 t n , (11)
where A and n are constants to be determined. Comparing this with [START_REF] Matheson | Characteristics of fluid-solid systems[END_REF] shows clearly that if n is in the region of 4/5 then our model is likely to be valid. Also A is related to Λ and we may hence be able to find a value for the mixture viscosity. The front propagation data is shown in figure 3a, and then as a function of q 3/4 t in figure 3b. Figure 3 shows the data points collapse well for all the experiments except when the values of u gi /u mf are particularly high. When at such high gas flow rates, the powder bubbles vigorously which may alter the mechanisms that transport powder within the bulk; otherwise particle movement was as a bulk motion.

Fitting a curve through all the data, we find that A = 5.701 and n = 0.634. We observed a section of the distributor plate that might not be delivering the same flowrate of gas as along the rest of its length. As this only accounted for the last 10cm of the apparatus, it is reasonable to simply ignore the motion over this part. Furthermore, it takes a little time for the flow to become fully developed and enter the intermediate asymptotic regime that gives rise to the similarity solutions. With this in mind, the data points corresponding to the first 30cm and final 10cm of the apparatus were ignored and another curve fitted to the remaining data. This gave A = 5.225 and n = 0.659. Both values are a little short of the 4/5 power we might have expected. [START_REF] Eames | The flow of aerated particles over a horizontal rigid surface[END_REF]. Using [START_REF] Barenblatt | Scaling, self-similarity, and intermediate asymptotics[END_REF] with n = 0.8 seems to best fit their data whilst none of the curves seem to fit our data. On being introduced to the apparatus, the material freefalls 0.5m before encountering the distributor plate2 . Its kinetic energy causes it to scatter forwards which would affect the results at such a short time. The flows were certainly not fully developed and hence not in the intermediate asymptotic regime meaning that similarity solutions do not apply here, so we should not expect any of the curves to fit the new data.

Comparing [START_REF] Furkawa | Liquidlike behaviour of fluidized particles[END_REF] and [START_REF] Barenblatt | Scaling, self-similarity, and intermediate asymptotics[END_REF] gives Λ = (A/ξ f | α=1 ) 1/5 = 3779.56 for A = 5.225. Rearranging Λ = ((1 -)ρ s + ρ f g) 3µ, and using the values stated earlier in this paper gives µ = 0.0129Pas.

CONCLUSIONS

We have presented data on the propagation of fluidised granular currents for several values of material volume fluxes and ratios of u gi /u mf for Geldart B powder in the range of 250-500µm. The minimum fluidisation velocity was determined by independent experiments to be 10.78cm/s. Comparisons have been made to theoretical predictions by use of similarity solutions and a suggestion made for the value of a mixture viscosity by use of the data.

A plot of the front position, x f against rescaled time (q 3/4 t) gives a power-law distribution whose exponent, n = 0.66. According to the theory presented we expect the exponent to be 0.8, although the experimentally determined value is not unreasonable.

Our model is mathematically identical to that determined by Huppert [START_REF] Huppert | Propagation of two-dimensional and axisymmetric viscous gravity currents over a horizontal surface[END_REF] for viscous fluids.

We believe that this is because the flows are driven by streamwise pressure gradients and retarded by stresses that scale as v x /h 2 . These are analogous to viscous stresses found in fluids. This gives rise to viscous-like behaviour in the fluidised granular flow. Using experimental and numerically determined values we find a value for the mixture viscosity to be µ = 0.0129Pas, which not too dissimilar to viscosities found by Matheson et al. (8), Furkawa & Ohmae (9) and Schügerl et al [START_REF] Schügerl | Rheologische Eigenschaften von gasdurchströmten Fliessbettsystemen[END_REF].

Figure 1 :

 1 Figure 1: Orientation of axes, and notation associated with modelling the currents. The image shows a fluidised current approximately 65cm long and 2cm thick.

Figure 2 :

 2 Figure 2: a) (left) Front view of the apparatus. b) (right) Schematic of the experimental set up.

  material, q/[cm 2 /s] Front position after 1s/[cm]

Figure 4 :Figure 4

 44 Figure 4: Front position after 1s as a function of the volume flux of material. Data from Eames & Gilbertson (6) (+) where u gi /u mf = 1.26 is plotted along with our own data for ratios of 1.3 ( ), 1.48 (♦), 1.67 ( ), & 1.86 (×) respectively and lines corresponding to (11) for various values of A and n

physical parameters used were ρs =2500kg/m 3 , g =9.81m/s

, µ f =1.862×10-5 Pas. mf was taken to be 0.4 and the density of air was neglected.

This is compared to about 20cm in the case ofEames & Gilbertson (6)

ACKNOWLEDGEMENTS

D.E.J. would like to thank his co-authors for their support and insightful observations and also R.S.J. Sparks for his many kind suggestions.