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Euler equations (fluid dynamics)

In fluid dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. In particular, they correspond to the Navier–Stokes equations with zero viscosity and zero thermal conductivity.

The Euler equations can be applied to incompressible and compressible flows. The incompressible Euler equations consist of Cauchy equations for conservation of mass and balance of momentum, together with the incompressibility condition that the flow velocity is a solenoidal field. The compressible Euler equations consist of equations for conservation of mass, balance of momentum, and balance of energy, together with a suitable constitutive equation for the specific energy density of the fluid. Historically, only the equations of conservation of mass and balance of momentum were derived by Euler. However, fluid dynamics literature often refers to the full set of the compressible Euler equations – including the energy equation – as "the compressible Euler equations".

The mathematical characters of the incompressible and compressible Euler equations are rather different. For constant fluid density, the incompressible equations can be written as a quasilinear advection equation for the fluid velocity together with an elliptic Poisson's equation for the pressure. On the other hand, the compressible Euler equations form a quasilinear hyperbolic system of conservation equations.

The Euler equations can be formulated in a "convective form" (also called the "Lagrangian form") or a "conservation form" (also called the "Eulerian form"). The convective form emphasizes changes to the state in a frame of reference moving with the fluid. The conservation form emphasizes the mathematical interpretation of the equations as conservation equations for a control volume fixed in space (which is useful from a numerical point of view).

History
The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides", published in Mémoires de l'Académie des Sciences de Berlin in 1757 (although Euler had previously presented his work to the Berlin Academy in 1752). Prior work included contributions from the Bernoulli family as well as from Jean le Rond d'Alembert.

The Euler equations were among the first partial differential equations to be written down, after the wave equation. In Euler's original work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible flow. An additional equation, which was called the adiabatic condition, was supplied by Pierre-Simon Laplace in 1816.

During the second half of the 19th century, it was found that the equation related to the balance of energy must at all times be kept for compressible flows, and the adiabatic condition is a consequence of the fundamental laws in the case of smooth solutions. With the discovery of the special theory of relativity, the concepts of energy density, momentum density, and stress were unified into the concept of the stress–energy tensor, and energy and momentum were likewise unified into a single concept, the energy–momentum vector.

Incompressible Euler equations with constant and uniform density
In convective form (i.e., the form with the convective operator made explicit in the momentum equation), the incompressible Euler equations in case of density constant in time and uniform in space are:

{{Equation box 1 {D\mathbf{u} \over Dt} &= -\nabla w + \mathbf{g} \\ \nabla\cdot \mathbf{u} &= 0 \end{align}\right.$$ }}
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where:
 * $$\mathbf u$$ is the flow velocity vector, with components in an N-dimensional space $$u_1, u_2, \dots, u_N$$,
 * $$\frac{D\boldsymbol\Phi}{Dt} = \frac{\partial\boldsymbol\Phi}{\partial t} + \mathbf v \cdot \nabla \boldsymbol\Phi$$, for a generic function (or field) $$\boldsymbol\Phi$$ denotes its material derivative in time with respect to the advective field $$\mathbf v$$ and
 * $$\nabla w$$ is the gradient of the specific (with the sense of per unit mass) thermodynamic work, the internal source term, and
 * $$\nabla \cdot \mathbf u$$ is the flow velocity divergence.
 * $$\mathbf{g}$$ represents body accelerations (per unit mass) acting on the continuum, for example gravity, inertial accelerations, electric field acceleration, and so on.

The first equation is the Euler momentum equation with uniform density (for this equation it could also not be constant in time). By expanding the material derivative, the equations become: $$\left\{\begin{align} {\partial\mathbf{u} \over \partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} &= -\nabla w + \mathbf{g} \\ \nabla \cdot \mathbf{u} &= 0 \end{align}\right.$$

In fact for a flow with uniform density $$\rho_0$$ the following identity holds: $$\nabla w \equiv \nabla \left(\frac p {\rho_0} \right) = \frac 1 {\rho_0} \nabla p $$ where $$p$$ is the mechanic pressure. The second equation is the incompressible constraint, stating the flow velocity is a solenoidal field (the order of the equations is not causal, but underlines the fact that the incompressible constraint is not a degenerate form of the continuity equation, but rather of the energy equation, as it will become clear in the following). Notably, the continuity equation would be required also in this incompressible case as an additional third equation in case of density varying in time or varying in space. For example, with density uniform but varying in time, the continuity equation to be added to the above set would correspond to: $$\frac{\partial \rho}{\partial t} = 0 $$

So the case of constant and uniform density is the only one not requiring the continuity equation as additional equation regardless of the presence or absence of the incompressible constraint. In fact, the case of incompressible Euler equations with constant and uniform density discussed here is a toy model featuring only two simplified equations, so it is ideal for didactical purposes even if with limited physical relevance.

The equations above thus represent respectively conservation of mass (1 scalar equation) and momentum (1 vector equation containing $$N$$ scalar components, where $$N$$ is the physical dimension of the space of interest). Flow velocity and pressure are the so-called physical variables.

In a coordinate system given by $$\left(x_1, \dots, x_N\right)$$ the velocity and external force vectors $$\mathbf u$$ and $$\mathbf g$$ have components $$(u_1,\dots, u_N)$$ and $$\left(g_1, \dots, g_N\right)$$, respectively. Then the equations may be expressed in subscript notation as:

 * Singularities$$\left\{\begin{align}

{\partial u_i \over \partial t} + \sum_{j=1}^N {\partial \left(u_i u_j + w\delta_{ij}\right) \over \partial x_j} &= g_i \\ \sum_{i=1}^N {\partial u_i \over \partial x_i} &= 0 \end{align}\right.$$

where the $$i$$ and $$j$$ subscripts label the N-dimensional space components, and $$\delta_{ij}$$ is the Kroenecker delta. The use of Einstein notation (where the sum is implied by repeated indices instead of sigma notation) is also frequent.

Properties
Although Euler first presented these equations in 1755, many fundamental questions or concepts about them remain unanswered.

In three space dimensions, in certain simplified scenarios, the Euler equations produce singularities.

Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy: $${\partial \over\partial t} \left(\frac{1}{2} u^2 \right) + \nabla \cdot \left(u^2 \mathbf u + w \mathbf u\right) = 0 $$

In the one-dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid Burgers' equation: $${\partial u \over\partial t}+ u {\partial u \over\partial x} = 0$$

This model equation gives many insights into Euler equations.

Nondimensionalisation
In order to make the equations dimensionless, a characteristic length $$r_0$$, and a characteristic velocity $$u_0$$, need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained: $$\begin{align} u^* & \equiv \frac{u}{u_0}, & r^* & \equiv \frac{r}{r_0}, \\[5pt] t^* & \equiv \frac{u_0}{r_0} t, & p^* & \equiv \frac{w}{u_0^2}, \\[5pt] \nabla^* & \equiv r_0 \nabla \end{align}$$ and of the field unit vector: $$\hat{\mathbf g}\equiv \frac {\mathbf g} g, $$

Substitution of these inversed relations in Euler equations, defining the Froude number, yields (omitting the * at apix): {{Equation box 1 {D\mathbf{u} \over Dt} &= -\nabla w + \frac{1}{\mathrm{Fr}} \hat{\mathbf{g}} \\ \nabla \cdot \mathbf{u} &= 0 \end{align}\right.$$ }}
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Euler equations in the Froude limit (no external field) are named free equations and are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory.

Conservation form
The conservation form emphasizes the mathematical properties of Euler equations, and especially the contracted form is often the most convenient one for computational fluid dynamics simulations. Computationally, there are some advantages in using the conserved variables. This gives rise to a large class of numerical methods called conservative methods.

The free Euler equations are conservative, in the sense they are equivalent to a conservation equation: $$ \frac{\partial \mathbf y}{\partial t}+ \nabla \cdot \mathbf F ={\mathbf 0}, $$ or simply in Einstein notation: $$ \frac{\partial y_j}{\partial t}+ \frac{\partial f_{ij}}{\partial r_i}= 0_i, $$ where the conservation quantity $$\mathbf y$$ in this case is a vector, and $$\mathbf F$$ is a flux matrix. This can be simply proved.

{{hidden $$\nabla \cdot (w \mathbf I) = \mathbf I \cdot \nabla w + w \nabla \cdot \mathbf I = \nabla w $$ $$\mathbf u \cdot \nabla \cdot \mathbf u = \nabla \cdot (\mathbf u \otimes \mathbf u)$$ where $$\otimes$$ denotes the outer product. The same identities expressed in Einstein notation are: $$\partial_i\left(w \delta_{ij}\right) = \delta_{ij} \partial_i w + w \partial_i \delta_{ij} = \delta_{ij} \partial_i w = \partial_j w$$ $$u_j \partial_i u_i = \partial_i \left(u_i u_j\right)$$ where $I$ is the identity matrix with dimension $N$ and $δ_{ij}$ its general element, the Kroenecker delta.
 * Demonstration of the conservation form
 * First, the following identities hold:

Thanks to these vector identities, the incompressible Euler equations with constant and uniform density and without external field can be put in the so-called conservation (or Eulerian) differential form, with vector notation: $$\left\{\begin{align} {\partial\mathbf{u} \over \partial t} + \nabla \cdot \left(\mathbf{u} \otimes \mathbf{u} + w\mathbf{I}\right) &= \mathbf{0} \\ {\partial 0 \over \partial t} + \nabla \cdot \mathbf{u} &= 0, \end{align}\right.$$ or with Einstein notation: $$\left\{\begin{align} \partial_t u_j + \partial_i \left(u_i u_j + w \delta_{ij}\right) &= 0 \\ \partial_t 0 + \partial_j u_j &= 0, \end{align}\right.$$

Then incompressible Euler equations with uniform density have conservation variables: $$ \mathbf y = \begin{pmatrix}\mathbf u \\ 0 \end{pmatrix}; \qquad \mathbf F = \begin{pmatrix}\mathbf u \otimes \mathbf u + w \mathbf I \\ \mathbf u \end{pmatrix}. $$

Note that in the second component u is by itself a vector, with length N, so y has length N+1 and F has size N(N+1). In 3D for example y has length 4, I has size 3×3 and F has size 4×3, so the explicit forms are: $$ {\mathbf y}=\begin{pmatrix} u_1 \\ u_2  \\ u_3 \\0 \end{pmatrix}; \quad {\mathbf F}=\begin{pmatrix} u_1^2 + w & u_1u_2 &  u_1u_3 \\ u_2 u_1 & u_2^2 + w &  u_2 u_3 \\ u_3 u_1 & u_3 u_2 &  u_3^2 + w \\ u_1 & u_2 & u_3 \end{pmatrix}. $$

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At last Euler equations can be recast into the particular equation:

{{Equation box 1 \frac {\partial}{\partial t}\begin{pmatrix} \mathbf u \\ 0 \end{pmatrix} + \nabla \cdot \begin{pmatrix}\mathbf u \otimes \mathbf u + w \mathbf I \\ \mathbf u \end{pmatrix} = \begin{pmatrix}\mathbf g \\ 0\end{pmatrix} $$ }}
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Spatial dimensions
For certain problems, especially when used to analyze compressible flow in a duct or in case the flow is cylindrically or spherically symmetric, the one-dimensional Euler equations are a useful first approximation. Generally, the Euler equations are solved by Riemann's method of characteristics. This involves finding curves in plane of independent variables (i.e., $$x$$ and $$t$$) along which partial differential equations (PDEs) degenerate into ordinary differential equations (ODEs). Numerical solutions of the Euler equations rely heavily on the method of characteristics.

Incompressible Euler equations
In convective form the incompressible Euler equations in case of density variable in space are:

{{Equation box 1 {D\rho \over Dt} &= 0\\ {D\mathbf{u} \over Dt} &= -\frac{\nabla p}{\rho} + \mathbf{g} \\ \nabla \cdot \mathbf{u} &= 0 \end{align}\right.$$ }}
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where the additional variables are:
 * $$\rho$$ is the fluid mass density,
 * $$p$$ is the pressure, $$p = \rho w$$.

The first equation, which is the new one, is the incompressible continuity equation. In fact the general continuity equation would be: $${\partial \rho \over\partial t} + \mathbf u \cdot \nabla \rho + \rho \nabla \cdot \mathbf u  = 0$$

but here the last term is identically zero for the incompressibility constraint.

Conservation form
The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively: $$ \mathbf y = \begin{pmatrix}\rho \\  \rho \mathbf u  \\0\end{pmatrix}; \qquad {\mathbf F} = \begin{pmatrix}\rho \mathbf u \\ \rho \mathbf u \otimes \mathbf u + p \mathbf I\\\mathbf u\end{pmatrix}. $$

Here $$\mathbf y$$ has length $$N+2$$ and $$\mathbf F$$ has size $$(N+2)N$$. In general (not only in the Froude limit) Euler equations are expressible as: $$ \frac {\partial}{\partial t}\begin{pmatrix}\rho \\  \rho \mathbf u  \\0\end{pmatrix}+ \nabla \cdot \begin{pmatrix}\rho \mathbf u\\\rho \mathbf u \otimes \mathbf u + p \mathbf I\\ \mathbf u\end{pmatrix} = \begin{pmatrix}0 \\  \rho \mathbf g \\ 0 \end{pmatrix} $$

Conservation variables
The variables for the equations in conservation form are not yet optimised. In fact we could define: $$ {\mathbf y}=\begin{pmatrix}\rho \\ \mathbf j  \\0\end{pmatrix}; \qquad {\mathbf F}=\begin{pmatrix} \mathbf j \\ \frac {1} \rho \, \mathbf j \otimes \mathbf j+ p \mathbf I\\ \frac \mathbf j \rho \end{pmatrix}. $$ where $$\mathbf j = \rho \mathbf u$$ is the momentum density, a conservation variable.

where $$\mathbf f = \rho \mathbf g$$ is the force density, a conservation variable.

Euler equations
In differential convective form, the compressible (and most general) Euler equations can be written shortly with the material derivative notation:

{{Equation box 1 {D\rho \over Dt} &= -\rho\nabla \cdot \mathbf{u} \\[1.2ex] \frac{D\mathbf{u}}{Dt} &= -\frac{\nabla p}{\rho} + \mathbf{g} \\[1.2ex] {De \over Dt} &= -\frac{p}{\rho}\nabla \cdot \mathbf{u} \end{align}\right.$$ }}
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where the additional variables here is:
 * $$e$$ is the specific internal energy (internal energy per unit mass).

The equations above thus represent conservation of mass, momentum, and energy: the energy equation expressed in the variable internal energy allows to understand the link with the incompressible case, but it is not in the simplest form. Mass density, flow velocity and pressure are the so-called convective variables (or physical variables, or lagrangian variables), while mass density, momentum density and total energy density are the so-called conserved variables (also called eulerian, or mathematical variables).

If one expands the material derivative the equations above are: $$\left\{\begin{align} {\partial\rho \over \partial t} + \mathbf{u} \cdot \nabla\rho + \rho\nabla \cdot \mathbf{u} &= 0 \\[1.2ex] \frac{\partial\mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla\mathbf{u} + \frac{\nabla p}{\rho} &= \mathbf{g} \\[1.2ex] {\partial e \over \partial t} + \mathbf{u} \cdot \nabla e + \frac{p}{\rho}\nabla \cdot \mathbf{u} &= 0 \end{align} \right.$$

Incompressible constraint (revisited)
Coming back to the incompressible case, it now becomes apparent that the incompressible constraint typical of the former cases actually is a particular form valid for incompressible flows of the energy equation, and not of the mass equation. In particular, the incompressible constraint corresponds to the following very simple energy equation: $$\frac{D e}{D t} = 0 $$

Thus for an incompressible inviscid fluid the specific internal energy is constant along the flow lines, also in a time-dependent flow. The pressure in an incompressible flow acts like a Lagrange multiplier, being the multiplier of the incompressible constraint in the energy equation, and consequently in incompressible flows it has no thermodynamic meaning. In fact, thermodynamics is typical of compressible flows and degenerates in incompressible flows.

Basing on the mass conservation equation, one can put this equation in the conservation form: $${\partial \rho e \over \partial t} + \nabla \cdot (\rho e \mathbf u) = 0 $$ meaning that for an incompressible inviscid nonconductive flow a continuity equation holds for the internal energy.

Enthalpy conservation
Since by definition the specific enthalpy is: $$h = e + \frac p \rho $$

The material derivative of the specific internal energy can be expressed as: $${D e \over Dt} = {D h \over Dt} - \frac 1 \rho \left({D p \over Dt} - \frac p \rho {D \rho \over Dt} \right)$$

Then by substituting the momentum equation in this expression, one obtains: $${D e \over Dt}= {D h \over Dt} - \frac 1 \rho \left(p \nabla \cdot \mathbf u + {D p \over Dt} \right)$$

And by substituting the latter in the energy equation, one obtains that the enthalpy expression for the Euler energy equation: $${D h \over Dt} = \frac 1 \rho {D p \over Dt} $$

In a reference frame moving with an inviscid and nonconductive flow, the variation of enthalpy directly corresponds to a variation of pressure.

Thermodynamics of ideal fluids
In thermodynamics the independent variables are the specific volume, and the specific entropy, while the specific energy is a function of state of these two variables.

For a thermodynamic fluid, the compressible Euler equations are consequently best written as:

{{Equation box 1 {Dv \over Dt} &= v \nabla \cdot \mathbf u\\[1.2ex] \frac{D\mathbf{u}}{Dt} &= ve_{vv}\nabla v + ve_{vs}\nabla s + \mathbf{g} \\[1.2ex] {Ds \over Dt} &= 0 \end{align}\right.$$ }}
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where:
 * $$v$$ is the specific volume
 * $$\mathbf u$$ is the flow velocity vector
 * $$s$$ is the specific entropy

In the general case and not only in the incompressible case, the energy equation means that for an inviscid thermodynamic fluid the specific entropy is constant along the flow lines, also in a time-dependent flow. Basing on the mass conservation equation, one can put this equation in the conservation form: $${\partial \rho s \over \partial t} + \nabla \cdot (\rho s \mathbf u) = 0 $$ meaning that for an inviscid nonconductive flow a continuity equation holds for the entropy.

On the other hand, the two second-order partial derivatives of the specific internal energy in the momentum equation require the specification of the fundamental equation of state of the material considered, i.e. of the specific internal energy as function of the two variables specific volume and specific entropy: $$e = e(v, s)$$

The fundamental equation of state contains all the thermodynamic information about the system (Callen, 1985), exactly like the couple of a thermal equation of state together with a caloric equation of state.

Conservation form
The Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively: $$\mathbf y = \begin{pmatrix} \rho \\ \mathbf j \\ E^t \end{pmatrix}; \qquad {\mathbf F} = \begin{pmatrix} \mathbf j \\ \frac 1 \rho \mathbf j \otimes \mathbf j + p \mathbf I \\ \left(E^t + p\right) \frac{1}{\rho}\mathbf{j} \end{pmatrix}. $$

where:
 * $$\mathbf j = \rho \mathbf u$$ is the momentum density, a conservation variable.
 * $E^t = \rho e + \frac{1}{2} \rho u^2$ is the total energy density (total energy per unit volume).

Here $$\mathbf y$$ has length N + 2 and $$\mathbf F$$ has size N(N + 2). In general (not only in the Froude limit) Euler equations are expressible as:

where $$\mathbf f = \rho \mathbf g$$ is the force density, a conservation variable.

We remark that also the Euler equation even when conservative (no external field, Froude limit) have no Riemann invariants in general. Some further assumptions are required

However, we already mentioned that for a thermodynamic fluid the equation for the total energy density is equivalent to the conservation equation: $${\partial \over \partial t} (\rho s) + \nabla \cdot (\rho s \mathbf u) = 0 $$

Then the conservation equations in the case of a thermodynamic fluid are more simply expressed as:

where $$S = \rho s$$ is the entropy density, a thermodynamic conservation variable.

Another possible form for the energy equation, being particularly useful for isobarics, is: $$ \frac{\partial H^t}{\partial t} + \nabla \cdot \left(H^t \mathbf u\right) = \mathbf u \cdot \mathbf f - \frac{\partial p}{\partial t} $$ where $H^t = E^t + p = \rho e + p + \frac{1}{2} \rho u^2$ is the total enthalpy density.

Quasilinear form and characteristic equations
Expanding the fluxes can be an important part of constructing numerical solvers, for example by exploiting (approximate) solutions to the Riemann problem. In regions where the state vector y varies smoothly, the equations in conservative form can be put in quasilinear form: $$ \frac{\partial \mathbf y}{\partial t} + \mathbf A_i \frac{\partial \mathbf y}{\partial r_i} = {\mathbf 0}. $$ where $$\mathbf A_i$$ are called the flux Jacobians defined as the matrices: $$ \mathbf A_i (\mathbf y)=\frac{\partial \mathbf f_i (\mathbf y)}{\partial \mathbf y}. $$

Obviously this Jacobian does not exist in discontinuity regions (e.g. contact discontinuities, shock waves in inviscid nonconductive flows). If the flux Jacobians $$\mathbf A_i$$ are not functions of the state vector $$\mathbf y$$, the equations reveals linear.

Characteristic equations
The compressible Euler equations can be decoupled into a set of N+2 wave equations that describes sound in Eulerian continuum if they are expressed in characteristic variables instead of conserved variables.

In fact the tensor A is always diagonalizable. If the eigenvalues (the case of Euler equations) are all real the system is defined hyperbolic, and physically eigenvalues represent the speeds of propagation of information. If they are all distinguished, the system is defined strictly hyperbolic (it will be proved to be the case of one-dimensional Euler equations). Furthermore, diagonalisation of compressible Euler equation is easier when the energy equation is expressed in the variable entropy (i.e. with equations for thermodynamic fluids) than in other energy variables. This will become clear by considering the 1D case.

If $$\mathbf p_i$$ is the right eigenvector of the matrix $$\mathbf A$$ corresponding to the eigenvalue $$\lambda_i$$, by building the projection matrix: $$\mathbf{P} = \left[\mathbf{p}_1, \mathbf{p}_2, ..., \mathbf{p}_n\right]$$

One can finally find the characteristic variables as: $$\mathbf{w} = \mathbf{P}^{-1} \mathbf{y},$$

Since A is constant, multiplying the original 1-D equation in flux-Jacobian form with P−1 yields the characteristic equations: $$ \frac{\partial w_i}{\partial t} + \lambda_j \frac{\partial w_i}{\partial r_j} = 0_i $$

The original equations have been decoupled into N+2 characteristic equations each describing a simple wave, with the eigenvalues being the wave speeds. The variables wi are called the characteristic variables and are a subset of the conservative variables. The solution of the initial value problem in terms of characteristic variables is finally very simple. In one spatial dimension it is: $$w_i(x, t) = w_i\left(x - \lambda_i t, 0\right)$$

Then the solution in terms of the original conservative variables is obtained by transforming back: $$\mathbf{y} = \mathbf{P} \mathbf{w},$$ this computation can be explicited as the linear combination of the eigenvectors: $$\mathbf{y}(x, t) = \sum_{i=1}^m w_i\left(x - \lambda_i t, 0\right) \mathbf p_i,$$

Now it becomes apparent that the characteristic variables act as weights in the linear combination of the jacobian eigenvectors. The solution can be seen as superposition of waves, each of which is advected independently without change in shape. Each i-th wave has shape wipi and speed of propagation λi. In the following we show a very simple example of this solution procedure.

Waves in 1D inviscid, nonconductive thermodynamic fluid
If one considers Euler equations for a thermodynamic fluid with the two further assumptions of one spatial dimension and free (no external field: g = 0): $$\left\{\begin{align} {\partial v \over \partial t} + u{\partial v \over \partial x} - v {\partial u \over \partial x} &= 0 \\[1.2ex] {\partial u \over \partial t} + u{\partial u \over \partial x} - e_{vv} v {\partial v \over \partial x} - e_{vs}v {\partial s \over \partial x} &= 0 \\[1.2ex] {\partial s \over \partial t} + u{\partial s \over \partial x} &= 0 \end{align}\right.$$

If one defines the vector of variables: $$\mathbf{y} = \begin{pmatrix}v \\ u \\ s\end{pmatrix}$$ recalling that $$v$$ is the specific volume, $$u$$ the flow speed, $$s$$ the specific entropy, the corresponding jacobian matrix is: $${\mathbf A}=\begin{pmatrix}u & -v & 0 \\ - e_{vv} v & u & - e_{vs} v \\ 0 & 0 & u \end{pmatrix}.$$

At first one must find the eigenvalues of this matrix by solving the characteristic equation: $$\det(\mathbf A(\mathbf y) - \lambda(\mathbf y) \mathbf I) = 0$$

that is explicitly: $$\det\begin{bmatrix}u-\lambda & -v & 0 \\ - e_{vv} v & u-\lambda & - e_{vs} v \\ 0 & 0 & u-\lambda \end{bmatrix}=0$$

This determinant is very simple: the fastest computation starts on the last row, since it has the highest number of zero elements. $$(u-\lambda) \det \begin{bmatrix}u-\lambda & -v \\ - e_{vv} v & u -\lambda \end{bmatrix}=0$$

Now by computing the determinant 2×2: $$(u - \lambda)\left((u - \lambda)^2 - e_{vv} v^2\right) = 0$$ by defining the parameter: $$a(v,s) \equiv v \sqrt {e_{vv}} $$ or equivalently in mechanical variables, as: $$a(\rho,p) \equiv \sqrt {\partial p \over \partial \rho} $$

This parameter is always real according to the second law of thermodynamics. In fact the second law of thermodynamics can be expressed by several postulates. The most elementary of them in mathematical terms is the statement of convexity of the fundamental equation of state, i.e. the hessian matrix of the specific energy expressed as function of specific volume and specific entropy: $$ \begin{pmatrix}e_{vv} & e_{vs} \\ e_{vs} & e_{ss} \end{pmatrix}$$ is defined positive. This statement corresponds to the two conditions: $$\left\{\begin{align} e_{vv} &> 0 \\[1.2ex] e_{vv}e_{ss} - e_{vs}^2 &> 0 \end{align}\right.$$

The first condition is the one ensuring the parameter a is defined real.

The characteristic equation finally results: $$(u - \lambda)\left((u - \lambda)^2 - a^2\right) = 0$$

That has three real solutions: $$\lambda_1(v,u,s) = u-a(v,s) \quad \lambda_2(u)= u, \quad \lambda_3(v,u,s) = u+a(v,s)$$

Then the matrix has three real eigenvalues all distinguished: the 1D Euler equations are a strictly hyperbolic system.

At this point one should determine the three eigenvectors: each one is obtained by substituting one eigenvalue in the eigenvalue equation and then solving it. By substituting the first eigenvalue λ1 one obtains: $$\begin{pmatrix}a & -v & 0 \\ - e_{vv} v & a & - e_{vs} v \\ 0 & 0 & a \end{pmatrix} \begin{pmatrix}v_1\\ u_1  \\s_1 \end{pmatrix}=0$$

Basing on the third equation that simply has solution s1=0, the system reduces to: $$\begin{pmatrix}a & -v \\-a^2 /v& a  \end{pmatrix} \begin{pmatrix}v_1\\ u_1 \end{pmatrix}=0$$

The two equations are redundant as usual, then the eigenvector is defined with a multiplying constant. We choose as right eigenvector: $$ \mathbf p_1=\begin{pmatrix}v\\ a \\0\end{pmatrix}$$

The other two eigenvectors can be found with analogous procedure as: $$ \mathbf p_2=\begin{pmatrix} e_{vs} \\ 0\\ - \left(\frac a v \right)^2 \end{pmatrix}, \qquad \mathbf p_3 = \begin{pmatrix}v\\ -a \\0\end{pmatrix}$$

Then the projection matrix can be built: $$ \mathbf P (v,u,s)=( \mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3) =\begin{pmatrix} v & e_{vs} & v\\ a & 0 & -a \\ 0 & - \left(\frac a v \right)^2 & 0 \end{pmatrix}$$

Finally it becomes apparent that the real parameter a previously defined is the speed of propagation of the information characteristic of the hyperbolic system made of Euler equations, i.e. it is the wave speed. It remains to be shown that the sound speed corresponds to the particular case of an isentropic transformation: $$a_s \equiv \sqrt {\left({\partial p \over \partial \rho} \right)_s} $$

Compressibility and sound speed
Sound speed is defined as the wavespeed of an isentropic transformation: $$a_s(\rho,p) \equiv \sqrt {\left({\partial p \over \partial \rho} \right)_s} $$ by the definition of the isoentropic compressibility: $$K_s (\rho,p) \equiv \rho \left({\partial p \over \partial \rho} \right)_s $$ the soundspeed results always the square root of ratio between the isentropic compressibility and the density: $$a_s \equiv \sqrt {\frac {K_s} \rho} $$

Ideal gas
The sound speed in an ideal gas depends only on its temperature: $$a_s (T) = \sqrt {\gamma \frac T m} $$

Since the specific enthalpy in an ideal gas is proportional to its temperature:

$$h = c_p T = \frac {\gamma}{\gamma-1} \frac T m $$

the sound speed in an ideal gas can also be made dependent only on its specific enthalpy:

$$a_s (h) = \sqrt {(\gamma -1) h} $$

Bernoulli's theorem for steady inviscid flow
Bernoulli's theorem is a direct consequence of the Euler equations.

Incompressible case and Lamb's form
The vector calculus identity of the cross product of a curl holds:

$$ \mathbf{v \ \times } \left( \mathbf{ \nabla \times F} \right) = \nabla_F \left( \mathbf{v \cdot F } \right) - \mathbf{v \cdot \nabla } \mathbf{ F} \, $$

where the Feynman subscript notation $$\nabla_F$$ is used, which means the subscripted gradient operates only on the factor $$\mathbf F$$.

Lamb in his famous classical book Hydrodynamics (1895), still in print, used this identity to change the convective term of the flow velocity in rotational form:

$$\mathbf u \cdot \nabla \mathbf u = \frac{1}{2}\nabla\left(u^2\right) + (\nabla \times \mathbf u) \times \mathbf u$$

the Euler momentum equation in Lamb's form becomes:

$$ \frac{\partial\mathbf{u}}{\partial t} + \frac{1}{2}\nabla\left(u^2\right) + (\nabla \times \mathbf{u}) \times \mathbf{u} + \frac{\nabla p}{\rho} = \mathbf{g} = \frac{\partial\mathbf{u}}{\partial t} + \frac{1}{2}\nabla\left(u^2\right) - \mathbf{u} \times (\nabla \times \mathbf{u}) + \frac{\nabla p}{\rho} $$

Now, basing on the other identity:

$$\nabla \left( \frac {p}{\rho} \right) = \frac {\nabla p}{\rho} - \frac{p}{\rho^2} \nabla \rho $$

the Euler momentum equation assumes a form that is optimal to demonstrate Bernoulli's theorem for steady flows:

$$\nabla \left(\frac{1}{2}u^2 + \frac{p}{\rho}\right) - \mathbf g = -\frac{p}{\rho^2} \nabla \rho + \mathbf u \times (\nabla \times \mathbf u) - \frac{\partial \mathbf u}{\partial t} $$

In fact, in case of an external conservative field, by defining its potential φ:

$$\nabla \left( \frac 1 2 u^2 + \phi + \frac p \rho \right) = -\frac{p}{\rho^2} \nabla \rho  + \mathbf u \times (\nabla \times \mathbf u) - \frac{\partial \mathbf u}{\partial t}$$

In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes:

$$\nabla \left( \frac 1 2 u^2 + \phi + \frac p \rho \right) =  -\frac{p}{\rho^2} \nabla \rho  + \mathbf u \times (\nabla \times \mathbf u)$$

And by projecting the momentum equation on the flow direction, i.e. along a streamline, the cross product disappears because its result is always perpendicular to the velocity:

$$\mathbf u \cdot \nabla \left(\frac{1}{2}u^2 + \phi + \frac{p}{\rho}\right) = -\frac{p}{\rho^2} \mathbf u \cdot \nabla\rho$$

In the steady incompressible case the mass equation is simply:

$$\mathbf u \cdot \nabla \rho = 0,$$that is the mass conservation for a steady incompressible flow states that the density along a streamline is constant. Then the Euler momentum equation in the steady incompressible case becomes:

$$\mathbf u \cdot \nabla \left( \frac 1 2 u^2 + \phi + \frac p \rho \right) = 0$$

The convenience of defining the total head for an inviscid liquid flow is now apparent:

$$ b_l \equiv \frac 1 2 u^2 + \phi + \frac p \rho, $$

which may be simply written as:

$$\mathbf u \cdot \nabla b_l = 0$$

That is, the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant.

Compressible case
In the most general steady (compressible) case the mass equation in conservation form is:

$$ \nabla \cdot \mathbf j = \rho \nabla \cdot \mathbf u + \mathbf u \cdot \nabla \rho = 0.$$Therefore, the previous expression is rather

$$\mathbf{u} \cdot \nabla \left({\frac{1}{2}}u^2 + \phi + \frac{p}{\rho}\right) = \frac{p}{\rho}\nabla \cdot \mathbf{u}$$

The right-hand side appears on the energy equation in convective form, which on the steady state reads:

$$\mathbf u \cdot \nabla e =  - \frac{p}{\rho}  \nabla \cdot \mathbf u $$

The energy equation therefore becomes:

$$\mathbf u \cdot \nabla \left( e + \frac p \rho + \frac 1 2 u^2 + \phi \right) = 0, $$

so that the internal specific energy now features in the head.

Since the external field potential is usually small compared to the other terms, it is convenient to group the latter ones in the total enthalpy:

$$ h^t \equiv e + \frac p \rho + \frac 1 2 u^2$$

and the Bernoulli invariant for an inviscid gas flow is:

$$ b_g \equiv h^t + \phi = b_l + e, $$

which can be written as:

$$\mathbf u \cdot \nabla b_g = 0$$

That is, the energy balance for a steady inviscid flow in an external conservative field states that the sum of the total enthalpy and the external potential is constant along a streamline.

In the usual case of small potential field, simply:

$$\mathbf u \cdot \nabla h^t \sim 0$$

Friedmann form and Crocco form
By substituting the pressure gradient with the entropy and enthalpy gradient, according to the first law of thermodynamics in the enthalpy form:

$$v \nabla p = -T \nabla s + \nabla h$$

in the convective form of Euler momentum equation, one arrives to:

$$\frac{D\mathbf u}{Dt}=T \nabla\,s-\nabla \,h$$

Friedmann deduced this equation for the particular case of a perfect gas and published it in 1922. However, this equation is general for an inviscid nonconductive fluid and no equation of state is implicit in it.

On the other hand, by substituting the enthalpy form of the first law of thermodynamics in the rotational form of Euler momentum equation, one obtains:

$$\frac{\partial\mathbf{u}}{\partial t} + \frac{1}{2} \nabla\left(u^2\right) + (\nabla \times \mathbf{u}) \times \mathbf{u} + \frac{\nabla p}{\rho} = \mathbf{g}$$

and by defining the specific total enthalpy:

$$h^t = h + \frac{1}{2}u^2$$

one arrives to the Crocco–Vazsonyi form (Crocco, 1937) of the Euler momentum equation:

$$\frac{\partial \mathbf{ u}}{\partial t} + (\nabla \times \mathbf u) \times \mathbf u  - T \nabla s + \nabla h^t = \mathbf{g}$$

In the steady case the two variables entropy and total enthalpy are particularly useful since Euler equations can be recast into the Crocco's form:

$$\left\{\begin{align} \mathbf{u} \times \nabla \times \mathbf{u} + T\nabla s - \nabla h^t &= \mathbf{g} \\ \mathbf{u} \cdot \nabla s &= 0 \\ \mathbf{u} \cdot \nabla h^t &= 0 \end{align}\right.$$

Finally if the flow is also isothermal:

$$T \nabla s = \nabla (T s)$$

by defining the specific total Gibbs free energy:

$$ g^t \equiv h^t + Ts $$

the Crocco's form can be reduced to:

$$\left\{\begin{align} \mathbf{u} \times \nabla \times \mathbf{u} - \nabla g^t &= \mathbf{g} \\ \mathbf{u} \cdot \nabla g^t &= 0 \end{align}\right.$$

From these relationships one deduces that the specific total free energy is uniform in a steady, irrotational, isothermal, isoentropic, inviscid flow.

Discontinuities
The Euler equations are quasilinear hyperbolic equations and their general solutions are waves. Under certain assumptions they can be simplified leading to Burgers equation. Much like the familiar oceanic waves, waves described by the Euler Equations 'break' and so-called shock waves are formed; this is a nonlinear effect and represents the solution becoming multi-valued. Physically this represents a breakdown of the assumptions that led to the formulation of the differential equations, and to extract further information from the equations we must go back to the more fundamental integral form. Then, weak solutions are formulated by working in 'jumps' (discontinuities) into the flow quantities – density, velocity, pressure, entropy – using the Rankine–Hugoniot equations. Physical quantities are rarely discontinuous; in real flows, these discontinuities are smoothed out by viscosity and by heat transfer. (See Navier–Stokes equations)

Shock propagation is studied – among many other fields – in aerodynamics and rocket propulsion, where sufficiently fast flows occur.

To properly compute the continuum quantities in discontinuous zones (for example shock waves or boundary layers) from the local forms (all the above forms are local forms, since the variables being described are typical of one point in the space considered, i.e. they are local variables) of Euler equations through finite difference methods generally too many space points and time steps would be necessary for the memory of computers now and in the near future. In these cases it is mandatory to avoid the local forms of the conservation equations, passing some weak forms, like the finite volume one.

Rankine–Hugoniot equations
Starting from the simplest case, one consider a steady free conservation equation in conservation form in the space domain:

$$\nabla \cdot \mathbf F = \mathbf 0 $$

where in general F is the flux matrix. By integrating this local equation over a fixed volume Vm, it becomes:

$$ \int_{V_m} \nabla \cdot \mathbf F \,dV = \mathbf 0.$$

Then, basing on the divergence theorem, we can transform this integral in a boundary integral of the flux:

$$ \oint_{\partial V_m} \mathbf F \,ds = \mathbf 0.$$

This global form simply states that there is no net flux of a conserved quantity passing through a region in the case steady and without source. In 1D the volume reduces to an interval, its boundary being its extrema, then the divergence theorem reduces to the fundamental theorem of calculus:

$$ \int_{x_m}^{x_{m+1}} \mathbf F(x') \,dx' = \mathbf 0,$$

that is the simple finite difference equation, known as the jump relation:

$$ \Delta \mathbf F = \mathbf 0.$$

That can be made explicit as:

$$ \mathbf F_{m+1} - \mathbf F_m = \mathbf 0$$

where the notation employed is:

$$ \mathbf F_{m} = \mathbf F(x_m).$$

Or, if one performs an indefinite integral:

$$ \mathbf F - \mathbf F_0 = \mathbf 0.$$

On the other hand, a transient conservation equation:

$${\partial y \over \partial t} + \nabla \cdot \mathbf F = \mathbf 0 $$

brings to a jump relation:

$$ \frac{dx}{dt} \, \Delta u = \Delta \mathbf F.$$

For one-dimensional Euler equations the conservation variables and the flux are the vectors:

$$\mathbf y = \begin{pmatrix} \frac{1}{v} \\ j \\ E^t \end{pmatrix}, $$ $$\mathbf F = \begin{pmatrix} j \\ v j^2 + p \\ v j \left(E^t + p\right) \end{pmatrix}, $$

where:
 * $$v$$ is the specific volume,
 * $$j$$ is the mass flux.

In the one dimensional case the correspondent jump relations, called the Rankine–Hugoniot equations, are:<

$$\left\{\begin{align} \frac{dx}{dt}\Delta \left(\frac{1}{v}\right) &= \Delta j\\[1.2ex] \frac{dx}{dt} \Delta j &= \Delta(vj^2 + p)\\[1.2ex] \frac{dx}{dt}\Delta E^t &= \Delta(jv(E^t + p)). \end{align}\right.$$

In the steady one dimensional case the become simply:

$$\left\{\begin{align} \Delta j &= 0\\[1.2ex] \Delta\left(v j^2 + p\right) &= 0 \\[1.2ex] \Delta\left(j\left(\frac{E^t}{\rho} + \frac{p}{\rho}\right)\right) &= 0. \end{align}\right.$$

Thanks to the mass difference equation, the energy difference equation can be simplified without any restriction:

$$\left\{\begin{align} \Delta j &= 0 \\[1.2ex] \Delta\left(vj^2 + p\right) &= 0 \\[1.2ex] \Delta h^t &= 0 \end{align}\right.,$$

where $$h^t$$ is the specific total enthalpy.

These are the usually expressed in the convective variables:

$$\left\{\begin{align} \Delta j &= 0 \\[1.2ex] \Delta\left(\frac{u^2}{v} + p\right) &= 0 \\[1.2ex] \Delta\left(e + \frac{1}{2}u^2 + pv\right) &= 0, \end{align}\right.$$

where:
 * $$u$$ is the flow speed
 * $$e$$ is the specific internal energy.

The energy equation is an integral form of the Bernoulli equation in the compressible case. The former mass and momentum equations by substitution lead to the Rayleigh equation:

$$ \frac{\Delta p}{\Delta v} = - \frac {u_0^2}{v_0}. $$

Since the second term is a constant, the Rayleigh equation always describes a simple line in the pressure volume plane not dependent of any equation of state, i.e. the Rayleigh line. By substitution in the Rankine–Hugoniot equations, that can be also made explicit as:

$$\left\{\begin{align} \rho u &= \rho_0 u_0 \\[1.2ex] \rho u^2 + p &= \rho_0 u_0^2 + p_0 \\[1.2ex] e + \frac{1}{2}u^2 + \frac{p}{\rho} &= e_0 + \frac{1}{2}u_0^2 + \frac{p_0}{\rho_0}. \end{align}\right.$$

One can also obtain the kinetic equation and to the Hugoniot equation. The analytical passages are not shown here for brevity.

These are respectively:

$$\left\{\begin{align} u^2(v, p) &= u_0^2 + (p - p_0)(v_0 + v) \\[1.2ex] e(v, p) &= e_0 + \tfrac{1}{2} (p + p_0)(v_0 - v). \end{align}\right.$$

The Hugoniot equation, coupled with the fundamental equation of state of the material:

$$ e = e(v,p)$$

describes in general in the pressure volume plane a curve passing by the conditions (v0, p0), i.e. the Hugoniot curve, whose shape strongly depends on the type of material considered.

It is also customary to define a Hugoniot function:

$$ \mathfrak h (v,s) \equiv e(v,s) - e_0 + \tfrac{1}{2} (p(v,s) + p_0)(v - v_0)$$

allowing to quantify deviations from the Hugoniot equation, similarly to the previous definition of the hydraulic head, useful for the deviations from the Bernoulli equation.

Finite volume form
On the other hand, by integrating a generic conservation equation:

$$ \frac {\partial \mathbf y}{\partial t} + \nabla \cdot \mathbf F = \mathbf s $$

on a fixed volume Vm, and then basing on the divergence theorem, it becomes:

$$ \frac {d}{dt} \int_{V_m} \mathbf y dV + \oint_{\partial V_m} \mathbf F \cdot \hat n ds = \mathbf S .$$

By integrating this equation also over a time interval:

$$ \int_{V_m} \mathbf y(\mathbf r, t_{n+1}) \, dV - \int_{V_m} \mathbf y(\mathbf r, t_n) \, dV+ \int_{t_n}^{t_{n+1}} \oint_{\partial V_m} \mathbf F \cdot \hat n \, ds \, dt = \mathbf 0 .$$

Now by defining the node conserved quantity:

$$\mathbf y_{m,n} \equiv \frac 1 {V_m} \int_{V_m} \mathbf y(\mathbf r, t_n) \, dV ,$$

we deduce the finite volume form:

$$\mathbf{y}_{m,n+1}=\mathbf{y}_{m,n} - \frac{1}{V_m} \int_{t_n}^{t_{n+1}} \oint_{\partial V_m} \mathbf{F} \cdot \hat{n}\, ds \, dt .$$

In particular, for Euler equations, once the conserved quantities have been determined, the convective variables are deduced by back substitution:

$$\begin{cases} \displaystyle \mathbf u_{m,n} = \frac{\mathbf j_{m,n}}{\rho_{m,n}} \\[1.2ex] \displaystyle         e_{m,n} = \frac{E^t_{m,n}}{\rho_{m,n}} - \frac{1}{2} u^2_{m,n}. \end{cases}$$

Then the explicit finite volume expressions of the original convective variables are:<

{{Equation box 1 \rho_{m,n+1} &= \rho_{m,n} - \frac{1}{V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m}\rho\mathbf{u} \cdot \hat{n}\, ds\, dt \\[1.2ex] \mathbf u_{m,n+1} &= \mathbf u_{m,n} - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m} (\rho\mathbf{u} \otimes \mathbf{u} - p\mathbf{I}) \cdot \hat{n}\, ds\, dt \\[1.2ex] \mathbf e_{m,n+1} &= \mathbf e_{m,n} - \frac{1}{2}\left(u^2_{m,n+1} - u^2_{m,n}\right) - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m} \left(\rho e + \frac{1}{2}\rho u^2 + p\right)\mathbf{u} \cdot \hat{n}\, ds\, dt \\[1.2ex] \end{align}\right. .$$ }}
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 * title=Euler equations (Finite volume form)
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Constraints
It has been shown that Euler equations are not a complete set of equations, but they require some additional constraints to admit a unique solution: these are the equation of state of the material considered. To be consistent with thermodynamics these equations of state should satisfy the two laws of thermodynamics. On the other hand, by definition non-equilibrium system are described by laws lying outside these laws. In the following we list some very simple equations of state and the corresponding influence on Euler equations.

Ideal polytropic gas
For an ideal polytropic gas the fundamental equation of state is:

$$e(v, s) = e_0 e^{(\gamma-1)m\left(s-s_0\right)} \left({v_0 \over v}\right)^{\gamma-1}$$

where $$e$$ is the specific energy, $$v$$ is the specific volume, $$s$$ is the specific entropy, $$m$$ is the molecular mass, $$\gamma$$ here is considered a constant (polytropic process), and can be shown to correspond to the heat capacity ratio. This equation can be shown to be consistent with the usual equations of state employed by thermodynamics.

From this equation one can derive the equation for pressure by its thermodynamic definition:

$$p(v,e) \equiv - {\partial e \over \partial v} = (\gamma - 1) \frac e v$$

By inverting it one arrives to the mechanical equation of state:

$$e(v,p) = \frac {pv}{\gamma - 1}$$

Then for an ideal gas the compressible Euler equations can be simply expressed in the mechanical or primitive variables specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. At last, in convective form they result:

{{Equation box 1 {Dv \over Dt} &= v\nabla \cdot \mathbf{u} \\[1.2ex] \frac{D\mathbf{u}}{Dt} &= v\nabla p + \mathbf{g} \\[1.2ex] {Dp \over Dt} &= -\gamma p\nabla \cdot \mathbf{u} \end{align}\right.$$ }}
 * indent=:
 * title=Euler equations for an ideal polytropic gas (convective form){{sfn|Toro|1999|p= 91|loc=par 3.1.2 Nonconservative formulations}}
 * equation=$$\left\{\begin{align}
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and in one-dimensional quasilinear form they results:

$$ \frac{\partial \mathbf y}{\partial t} + \mathbf A \frac{\partial \mathbf y}{\partial x} = {\mathbf 0}. $$

where the conservative vector variable is:

$${\mathbf y}=\begin{pmatrix}v\\ u \\p \end{pmatrix}$$

and the corresponding jacobian matrix is:

$${\mathbf A}=\begin{pmatrix}u & -v & 0 \\ 0 & u & v \\ 0 & \gamma p & u \end{pmatrix}.$$

Steady flow in material coordinates
In the case of steady flow, it is convenient to choose the Frenet–Serret frame along a streamline as the coordinate system for describing the steady momentum Euler equation: $$ \boldsymbol{u}\cdot\nabla \boldsymbol{u} = - \frac{1}{\rho} \nabla p, $$

where $$\mathbf u$$, $$p$$ and $$\rho$$ denote the flow velocity, the pressure and the density, respectively.

Let $$\left\{ \mathbf e_s, \mathbf e_n, \mathbf e_b \right\}$$ be a Frenet–Serret orthonormal basis which consists of a tangential unit vector, a normal unit vector, and a binormal unit vector to the streamline, respectively. Since a streamline is a curve that is tangent to the velocity vector of the flow, the left-hand side of the above equation, the convective derivative of velocity, can be described as follows: $$\begin{align} \boldsymbol{u}\cdot\nabla \boldsymbol{u} \\[1ex] &= u\frac{\partial}{\partial s}(u\boldsymbol{e}_s) &(\boldsymbol{u} = u \boldsymbol{e}_s,~ {\partial / \partial s} \equiv \boldsymbol{e}_s \cdot \nabla) \\[1ex] &= u\frac{\partial u}{\partial s}\boldsymbol{e}_s + \frac{u^2}{R}\boldsymbol{e}_n &(\because~ \frac{\partial\boldsymbol{e}_s}{\partial s} = \frac{1}{R}\boldsymbol{e}_n), \end{align}$$

where $$R$$ is the radius of curvature of the streamline.

Therefore, the momentum part of the Euler equations for a steady flow is found to have a simple form: $$\begin{cases} \displaystyle u\frac{\partial u}{\partial s} = -\frac{1}{\rho}\frac{\partial p}{\partial s},\\ \displaystyle                 {u^2 \over R} = -\frac{1}{\rho}\frac{\partial p}{\partial n} &({\partial / \partial n}\equiv\boldsymbol{e}_n\cdot\nabla),\\ \displaystyle                             0 = -\frac{1}{\rho}\frac{\partial p}{\partial b} &({\partial / \partial b}\equiv\boldsymbol{e}_b\cdot\nabla). \end{cases}$$

For barotropic flow $$(\rho = \rho(p))$$, Bernoulli's equation is derived from the first equation: $$\frac{\partial}{\partial s}\left(\frac{u^2}{2} + \int\frac{\mathrm{d}p}{\rho}\right) = 0.$$

The second equation expresses that, in the case the streamline is curved, there should exist a pressure gradient normal to the streamline because the centripetal acceleration of the fluid parcel is only generated by the normal pressure gradient.

The third equation expresses that pressure is constant along the binormal axis.

Streamline curvature theorem

Let $$r$$ be the distance from the center of curvature of the streamline, then the second equation is written as follows: $$ \frac{\partial p}{\partial r} = \rho \frac{u^2}{r}~(>0), $$

where $${\partial / \partial r} = -{\partial /\partial n}.$$

This equation states: In a steady flow of an inviscid fluid without external forces, the center of curvature of the streamline lies in the direction of decreasing radial pressure.

Although this relationship between the pressure field and flow curvature is very useful, it doesn't have a name in the English-language scientific literature. Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem".

This "theorem" explains clearly why there are such low pressures in the centre of vortices, which consist of concentric circles of streamlines. This also is a way to intuitively explain why airfoils generate lift forces.

Exact solutions
All potential flow solutions are also solutions of the Euler equations, and in particular the incompressible Euler equations when the potential is harmonic.

Solutions to the Euler equations with vorticity are:
 * parallel shear flows – where the flow is unidirectional, and the flow velocity only varies in the cross-flow directions, e.g. in a Cartesian coordinate system $$(x,y,z)$$ the flow is for instance in the $$x$$-direction – with the only non-zero velocity component being $$u_x(y,z)$$ only dependent on $$y$$ and $$z$$ and not on $$x.$$
 * Arnold–Beltrami–Childress flow – an exact solution of the incompressible Euler equations.
 * Two solutions of the three-dimensional Euler equations with cylindrical symmetry have been presented by Gibbon, Moore and Stuart in 2003. These two solutions have infinite energy; they blow up everywhere in space in finite time.

title: Euler equations (fluid dynamics)


text:
Cowbell (instrument)

The cowbell is an idiophone hand percussion instrument used in various styles of music, such as Latin and rock. It is named after the similar bell used by herdsmen to keep track of the whereabouts of cows. The instrument initially and traditionally has been metallic; however, contemporarily, some variants are made of synthetic materials.

Origins
While the cowbell is commonly found in musical contexts, its origin can be traced to freely roaming animals. In order to help identify the herd to which these animals belonged, herdsmen placed these bells around the animal's neck. As the animals moved about the bell would ring, thus making it easier to know of the animal's whereabouts. Though the bells were used on various types of animals, they are typically referred to as "cowbells" due to their extensive use with cattle.

Tuned cowbells
Tuned cowbells or Almglocken (their German name, ‘Alm’ meaning a mountain meadow, and ‘Glocken’ bells), sometimes known by the English translation alpine bells (also Alpenglocken in German), typically refer to bulbous brass bells that are used to play music, sometimes as a novelty act or tourist attraction in the northern Alps, and sometimes in classical music, as in Richard Strauss's Alpine Symphony. Since they are tuned differently, in order to distinguish individual animals, they can be collected "from the pasture" in random tunings, but commercial sets in equal temperament are also available. The metal clapper is retained, and they sound much noisier than handbells, which are otherwise used similarly in ensembles.

Composers who included Almglocken among their musical palette include Tōru Takemitsu, Jo Kondo, Gustav Mahler, Richard Strauss, Roy Harter, John Adams, Thomas Ades, Joseph Schwantner, and Karlheinz Stockhausen. Olivier Messiaen used multiple chromatic sets of clapperless cowbells in several of his compositions, notably Et expecto resurrectionem mortuorum and Couleurs de la cité celeste.

Clapperless cowbells
Clapperless cowbells made of metal are an important element in Latin-American and go-go music. These cowbells are struck with a stick – the tone being modulated by striking different parts of the bell and by damping with the hand holding the bell.

In several parts of the world (notably in West Africa) pairs or trios of clapperless bells are joined in such a way that they can be struck separately or clashed together. The Brazilian name for these is "agogo" bells. Cylindrical wood blocks played in the same way are also called "Agogô". In Cuban music the cowbell is called cencerro and often played by the same player as the bongos. In Caribbean music two or three are often mounted together with a pair of timbales.

This type of cowbell can also be played with the foot using a modified bass drum pedal or bowed with a double bass bow.

Sports use
Cowbells are sometimes popular noisemakers at sporting events, despite attempts to suppress them. While different sports teams seem to have their own story explaining the use of cowbells at their games, it is unlikely that so many teams across varying levels of sports and geographical regions would not have a common cultural reason for the practice. Although the origin of cowbell noisemakers is unclear, the most plausible explanation has to do with early baseball. Baseball games in the 19th century were often played in cow pastures, as American society was agrarian-based at the time. Cowbells were easily found or taken from around the necks of cows in the pasture and were utilized as noisemakers. The tradition naturally grew to be common practice at various sporting events, not just baseball. In the United States, they are most closely identified with the Columbus Clippers minor league baseball team in Ohio and Mississippi State University, whose football fans once smuggled in cowbells by the thousands despite a ban on artificial noisemakers by its conference, the Southeastern Conference. SEC guidelines were changed in 2010 to permit limited cowbell use at Mississippi State football games, and again in 2014 to permit cowbells to be freely rung at any time except between the time that the offensive center is over the football until the play is whistled dead. Elsewhere in college football, the cowbell can be found at Penn State and Michigan football games, where it is played with a particular rhythm and accompanying chant.

Worldwide, in cross-country skiing, cowbells are often rung vigorously at the start and finishes of races, to cheer on the racers. They are used by cyclocross fans in much the same way.

Cornell ice hockey fans who are also known for their zealous support of their team have cheers that feature use of a cowbell while in Lynah Rink. The San Jose SaberCats of the Arena Football League are also infamous for their fans' use of cowbells. In New Zealand, supporters of the Waikato Rugby Union invariably use cowbells at home matches; this has been carried over to home matches of the Chiefs, the Super Rugby franchise centered on the Waikato region. During University of New Hampshire ice hockey games, a small group of fans at the base of the student section show their support for the UNH Wildcats with a cowbell. This group also leads the chants and shows their support with posters and other props. Finally the 2006–2007 American Basketball Association (ABA) champion Vermont Frost Heaves have a large cowbell following at their home games.

A small number of Toronto Blue Jays fans at Rogers Centre frequently bring cowbells to Blue Jays home games. They are common enough at Tampa Bay Rays home games that the stadium scoreboard graphics crew have a pre-built graphic that says "More Cowbell!!". The Everett Silvertips fans also use cowbells, after the team watched the Saturday Night Live skit while on their tour bus in their inaugural season, and said they wanted the fans to have cowbells. They have a "more cowbell" that sometimes shows on the jumbotron. The Belleville Bulls in the Ontario Hockey League used the skit on their video scoreboard and "More Cowbell" as a catchphrase during the 2007 playoffs. Their goalie Edward Pasquale has several Will Ferrell characters painted on his goalie mask, including the one from the Saturday Night Live skit with the cowbell and others.

The cowbell has become a tradition for the Melbourne Storm fans in the NRL and is rung by a group of 3 people in the active supporter bay behind the goals at AAMI Park, as well as a few other people around the ground.

The supporters of Huddersfield Giants also have a small, enthusiastic group of fans known as "the cowbell army" who try and create a fun and friendly atmosphere at home and away games in the Super League.

In popular culture
"More cowbell" is an American catchphrase spoken as a humorous recommendation for improving a music performance. The phrase originated from Christopher Walken saying it in a comedy sketch that aired on Saturday Night Live on April 8, 2000.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Cowbell (instrument)


text:
User:Gnu thomas~enwiki

fr:Utilisateur:Gnu_thomas oc:Utilisator:Gnu_thomas

title: User:Gnu thomas~enwiki


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Étienne Geoffroy Saint-Hilaire

Étienne Geoffroy Saint-Hilaire (15 April 1772 – 19 June 1844) was a French naturalist who established the principle of "unity of composition". He was a colleague of Jean-Baptiste Lamarck and expanded and defended Lamarck's evolutionary theories. Geoffroy's scientific views had a transcendental flavor (unlike Lamarck's materialistic views) and were similar to those of German morphologists like Lorenz Oken. He believed in the underlying unity of organismal design, and the possibility of the transmutation of species in time, amassing evidence for his claims through research in comparative anatomy, paleontology, and embryology. He is considered as a predecessor of the evo-devo evolutionary concept.

Life and early career
Geoffroy was born at Étampes (in present-day Essonne), and studied at the Collège de Navarre, in Paris, where he studied natural philosophy under M. J. Brisson. He then attended the lectures of Louis-Jean-Marie Daubenton at the College de France and Fourcroy at the Jardin des Plantes. In March 1793 Daubenton, through the interest of Bernardin de Saint-Pierre, procured him the office of sub-keeper and assistant demonstrator of the cabinet of natural history, made vacant by the resignation of Bernard Germain Étienne de la Ville, Comte de Lacépède. By a law passed in June 1793, Geoffroy was appointed one of the twelve professors of the newly constituted Muséum National d'Histoire Naturelle, being assigned the chair of zoology. In the same year he busied himself with the formation of a menagerie at that institution.

In 1794, Geoffroy entered into correspondence with Georges Cuvier. Shortly after the appointment of Cuvier as assistant at the Museum d'Histoire Naturelle, Geoffroy received him into his house. The two friends wrote together five memoirs on natural history, one of which, on the classification of mammals, puts forward the idea of the subordination of characters upon which Cuvier based his zoological system. It was in a paper entitled Histoire des Makis, ou singes de Madagascar, written in 1795, that Geoffroy first gave expression to his views on the unity of organic composition, the influence of which is perceptible in all his subsequent writings; nature, he observes, presents us with only one plan of construction, the same in principle, but varied in its accessory parts.

In 1798, Geoffroy was chosen a member of Napoleon's great scientific expedition to Egypt as part of the natural history and physics section of the Institut d'Égypte; 151 scientists and artists participated in the expedition, including Dominique-Vivant Denon, Claude Louis Berthollet, and Jean Baptiste Joseph Fourier. On the capitulation of Alexandria in August 1801, he took part in resisting the claim made by the British general to the collections of the expedition, declaring that, were that demand persisted in, history would have to record that he also had burnt a library in Alexandria. Early in January 1802 Geoffroy returned to Paris. He was elected a member of the French Academy of Sciences in September 1807. In March of the following year Napoleon, who had already recognized his national services by the award of the cross of the legion of honor, selected him to visit the museums of Portugal, for the purpose of procuring collections from them, and in the face of considerable opposition from the British he eventually was successful in retaining them as a permanent possession for his country.

Later career

In 1809, the year after his return to France, Geoffroy was made professor of zoology at the faculty of sciences at Paris, and from that period he devoted himself more exclusively than before to anatomical study. In 1818 he published the first part of his celebrated Philosophie anatomique, the second volume of which, published in 1822, and subsequent memoirs account for the formation of monstrosities on the principle of arrest of development, and of the attraction of similar parts.

Geoffroy's friend Robert Edmund Grant shared his views on unity of plan and corresponded with him while working on marine invertebrates in the late 1820s in Edinburgh (assisted in 1826 and 1827 by his student Charles Darwin) when Grant successfully identified the pancreas in molluscs. When, in 1830, Geoffroy proceeded to apply to the invertebrata his views as to the unity of animal composition, he found a vigorous opponent in Cuvier, his former friend.

Geoffroy, a synthesiser, contended, in accordance with his theory of unity of plan in organic composition, that all animals are formed of the same elements, in the same number; and with the same connections: homologous parts, however they differ in form and size, must remain associated in the same invariable order. With Johann Wolfgang von Goethe he held that there is in nature a law of compensation or balancing of growth, so that if one organ take on an excess of development, it is at the expense of some other part; and he maintained that, since nature takes no sudden leaps, even organs which are superfluous in any given species, if they have played an important part in other species of the same family, are retained as rudiments, which testify to the permanence of the general plan of creation. It was his conviction that, owing to the conditions of life, the same forms had not been perpetuated since the origin of all things, although it was not his belief that existing species are becoming modified.

Cuvier, who was an analytical observer of facts, admitted only the prevalence of laws of co-existence or harmony in animal organs, and maintained the absolute invariability of species, which he declared had been created with a regard to the circumstances in which they were placed, each organ contrived with a view to the function it had to fulfil, thus putting, in Geoffroy's considerations, the effect for the cause.

In 1836 he coined the term phocomelia. In 1838 he was named an Officer of the Légion d'honneur.

In July 1840, Geoffroy became blind, and some months later he had a paralytic attack. From that time his strength gradually failed him. He resigned his chair at the museum in 1841, and was succeeded by his son, Isidore Geoffroy Saint-Hilaire. He died in Paris on 19 June 1844 and is buried in Division 19 of the Cimetière du Père Lachaise.

Geoffroy's theory
Geoffroy was a deist, which is to say that he believed in a God, but also in a law-like universe, with no supernatural interference in the details of existence. This kind of opinion was common in the Enlightenment, and goes with a rejection of revelation and miracles, and does not interpret the Bible as the literal word of God. These views did not conflict with his naturalistic ideas about organic change.

Geoffroy's theory was not a theory of common descent, but a working-out of existing potential in a given type. For him, the environment causes a direct induction of organic change. This opinion Ernst Mayr labels as 'Geoffroyism'. It is definitely not what Lamarck believed (for Lamarck, a change in habits is what changes the animal). The direct effect of environment on heritable traits is not believed today to be a central evolutionary force; even Lawrence knew by 1816 that the climate does not directly cause the major differences between human races.

Geoffroy endorsed a theory of saltational evolution that "monstrosities could become the founding fathers (or mothers) of new species by instantaneous transition from one form to the next." In 1831 he speculated that birds could have arisen from reptiles by an epigenetic saltation. Geoffroy wrote that environmental pressures could produce sudden transformations to establish new species instantaneously. In 1864 Albert von Kölliker revived Geoffroy's theory that evolution proceeds by large steps, under the name of heterogenesis.

Geoffroy noted that the organization of dorsal and ventral structures in arthropods is opposite that of mammals. The inversion hypothesis was met with criticism and was rejected, however, some modern molecular embryologists have since resurrected this idea.

Taxa described

 * See Category:Taxa named by Étienne Geoffroy Saint-Hilaire

Legacy
The Geoffroy's cat (Leopardus geoffroyi) was named in his honour.

Étienne Geoffroy Saint-Hilaire is commemorated in the scientific name of a species of South American turtle, Phrynops geoffroanus.

His name is also honoured in that of a number of other species, including Geoffroy's spider monkey, Geoffroy's bat, and Geoffroy's tamarin.

The Catfish Corydoras geoffroy is named after him.

is a street in the 5ème arrondissement, Paris near the Jardin des Plantes and Muséum national d'histoire naturelle.

In popular culture
French author Honoré de Balzac dedicated his novel Le Père Goriot to Saint-Hilaire, "as a tribute of admiration for his labors and his genius."

title: Étienne Geoffroy Saint-Hilaire


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Oakworth Hall

Oakworth Hall is located in Oakworth, West Yorkshire, England. The manor house was rebuilt in the 17th century, but has a history dating back to 1066. The building overlooks the Worth Valley, facing south towards Haworth.

Nearby is Holden Park, which was the site of Oakworth House and its grounds, once owned by Sir Isaac Holden, an inventor who is said to have invented the lucifer match and revolutionised the process of carding wool. He also owned Oakworth Hall.

External features
The Hall is located on the corner of Providence Lane, which once ran through the Hall's farmyard. The cornerstone on the north rear wall, above the door of the farm cottage, is one of four found in the district commissioned by John Craven in July 1843.

Of important historical interest is a carved stone panel of a mounted huntsman of either Celtic origins or Romano-British of the first centuries CE. Accompanying the huntsman there are three small animals which might be taken to be dogs, but in the History of Keighley they are described as "three goats trippant". A goat is featured on the coat of arms on one branch of the Copley family who were early residents of the Hall. The panel may be even earlier, perhaps pre-Roman Iron Age. Two crude heads, which are thought to be of the same period, flank the carving.

The lintel above the door on the north side of the farm cottage is a curiously carved stone with triple rounded arches and double spiral decorative motifs. Both this stone and the panel came from the surrounding moors and it is believed they were incorporated into the building for good luck.

Internal features
All the original fireplaces have been preserved. There is a large inglenook fireplace, ten feet across, in the dining room. Next to the fireplace is a salt cellar. Salt was very important for the preservation of food and had to be stored in the driest place in the building. Stored in the salt cellar today is a pair of very small, leather, children's shoes which were discovered during restoration work on a chimney. Shoes were traditionally secreted away in the Middle Ages to ward off evil spirits and to bring good fortune to the residents of the Hall. Two oak beams in the dining room are 18 inches by 10 feet and actually extend throughout the Hall. The original adze marks are clearly in evidence.

In the kitchen there is an old oven. It has a cast iron interior and is surrounded by large stones. It is reputed to be the oldest oven in Yorkshire that is still capable of use. From the kitchen, steps lead down to a cellar. On the first floor, two of the six bedrooms have arched gothic style windows to provide light on the landing. The windows have stone mullions. All the doors are constructed of solid English oak and the handsome front door dates from when that part of the building was rebuilt in 1702.

In the study there is an original fireplace and cast iron stove. This is almost identical to the one in the later-built Brontë Parsonage Museum in nearby Haworth.

The Hall (now numbers 69 and 71 Colne road) is a Grade II listed building and is subject to a Preservation Order.

The History of the Hall

 * 1066 - Mentioned in the Domesday Book as the Manor of Oakworth. It was given by William the Conqueror to Gilbert Tyson.
 * 1366 - Property of the de Vaux family who were probably descendants of the Tyson family. The de Vaux families' main seat was Vauxhall (Vaux Hall, in south east London). They had large properties in northern England from South Yorkshire up to Hadrian's Wall. Thomas de Vaux was Chancellor of Richard I during the earlier Crusades to the Holy Land.
 * 1598 - Alvery Copley, the last member of the de Vaux family, died and the 32 tenants who farmed the lands were constituted Lords of Oakworth Manor. It is believed that the first building which stood on the site was built by the de Vaux family.
 * 1662 - The famous Baptist Minister John Moore was born at Oakworth Hall. He was a friend and close colleague of William Mitchell who was imprisoned for preaching in 1687.  Together they clandestinely crisscrossed the Pennines visiting various safe houses in both Lancashire and West Yorkshire.  It is thought that the previous building of Oakworth Hall was one of those "safe houses".  It is advanced that the destruction of the previous building of Oakworth Hall was as a result of the turbulent period of religious persecution.  It is also suggested that the old building was burned down to smoke out a "priest hole" that had existed beneath the Hall since the Reformation.
 * 1691 - Oakworth Hall is mentioned in the Keighley Rolls in the entry for 12 December 1691. Some travellers visited the Squire, Anthony Moore, at Oakworth Hall.
 * 1702 - The rebuilding of the Hall was probably completed by William Clough in its present form.  The date 1702 is over the door on the south side of the Farm Cottage.
 * 1708 - William Clough of Oakworth Hall is mentioned in the Haworth Court Rolls.
 * 1715 - Jamas Haggas of Oakworth Hall bound as an apprentice to a Halifax weaver. Later, he bought long wool in Lincolnshire and sorted wool with his son at Oakworth Hall.  These documents are in the deeds of the Hall.  The Hall  is mentioned in Heaton's Yorkshire Woollen and Worsted Industries.
 * 1883 - Sold by the descendants of the Haggas family, (Blamire and Craven) to Richard Longdon Hattersly.
 * 1913 - Trustees of the Hattersly family sold the Hall to Benjamin, William and John Speight who were plumbers.

 * (The families Clough, Haggas and Hattersly all became large textile manufacturers in the spinning and weaving industries in the West Riding of Yorkshire. The Cloughs became mohair spinners and weavers and the firm still exists today as British Mohair Spinners.  Haggas became the largest worsted spinners in the area and there is still a large manufacturing plant in Keighley. The Hattersleys owned a number of worsted and woollen Mills and factories producing textile and weaving booms.  There is still today Hattersley Narrow Fabrics Mill in Keighley.)

 * 1936 - The Speight family sold the Hall to the Yorkshire Penny Bank.
 * 1945 - When the Bank closed its Oakworth branch the Hall was bought by the Assistant Manager of the bank. It became a private residence once again. It changed hands a number of times and was a guest house from 1979–2002.
 * 2003 - The Hall is now a private residence.

title: Oakworth Hall


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Stanisław Ignacy Witkiewicz

Stanisław Ignacy Witkiewicz (24 February 1885 – 18 September 1939), commonly known as Witkacy, was a Polish writer, painter, philosopher, theorist, playwright, novelist, and photographer active before World War I and during the interwar period.

Life
Born in Warsaw, Stanisław Ignacy Witkiewicz was a son of the painter, architect and an art critic Stanisław Witkiewicz. His mother was Maria Pietrzkiewicz Witkiewiczowa. Both of his parents were born in the Samogitian region of Lithuania. His godmother was the internationally famous actress Helena Modrzejewska. Witkiewicz was reared at the family home in Zakopane. In accordance with his father's antipathy to the "servitude of the school," he was home-schooled and encouraged to develop his talents across a range of creative fields. Against his father's wishes he studied at the Kraków Academy of Fine Arts with Józef Mehoffer and Jan Stanisławski.

Witkiewicz was close friends with composer Karol Szymanowski and, from childhood, with Bronisław Malinowski and Zofia Romer. Romer was romantically linked to both Bronisław Malinowski and Witkiewicz. He had a tumultuous affair with prominent actress Irena Solska who according to Anna Micińska is represented as the heroine Akne Montecalfi in his first novel, The 622 Downfalls of Bungo or The Demonic Woman, 1911. According to Micińska he also represented himself as the character Bungo and Malinowski as the Duke of Nevermore. The unfinished novel, which was not published until 1972, also describes erotic encounters between Bungo and the Duke of Nevermore. Taught wet plate photography by his father, it was during this period that he also began producing the intimate portrait photography for which he is known; producing striking portraits of his circle in Zakopane and many self-portraits.

In 1914 following a crisis in Witkiewicz's personal life due to the suicide of his fiancée Jadwiga Janczewska, for which he blamed himself, he was invited by Malinowski to act as draftsman and photographer on his anthropological expedition to the then Territory of Papua, by way of Ceylon and Australia. The venture was interrupted by the onset of World War I. After quarrelling with Malinowski in Australia, Witkiewicz who was by birth a subject of the Russian Empire, travelled to St Petersburg (then Petrograd) from Sydney and was commissioned as an officer in the Pavlovsky Regiment of the Imperial Russian Army. His ailing father, a Polish patriot, was deeply grieved by his son's decision and died in 1915 without seeing him again.

In July 1916 he was seriously wounded in the battle on Stokhid River in what is now Ukraine and was evacuated to St Petersburg where he witnessed the Russian Revolution. He claimed that he worked out his philosophical principles during an artillery barrage, and that when the Revolution broke out he was elected political commissar of his regiment. His later works would show his fear of social revolution and foreign invasion, often couched in absurdist language.

He had begun to support himself through portrait painting and continued to do so on his return to Zakopane in Poland. He soon entered into a major creative phase, setting out his principles in New Forms in Painting and Introduction to the Theory of Pure Form in the Theatre. He associated with a group of "formist" artists in the early 1920s and wrote most of his plays during this period. Of about forty plays written by Witkiewicz between 1918 and 1925, twenty-one survive, and only Jan Maciej Karol Hellcat met with any public success during the author's lifetime. The original Polish manuscript of The Crazy Locomotive was also lost; the play, back-translated from two French versions, was not published until 1962.

After 1925, and taking the name 'Witkacy', the artist ironically re-branded his portrait painting which provided his economic sustenance as The S.I. Witkiewicz Portrait Painting Company, with the tongue in cheek motto: "The customer must always be satisfied". Several of the so-called grades of portraits were offered, from the merely representational to the more expressionistic and the narcotics-assisted. Many of his paintings were annotated with mnemonics listing the drugs taken while painting a particular painting, even if this happened to be only a cup of coffee. He also varied the spelling of his name, signing himself Witkac, Witkatze, Witkacjusz, Vitkacius and Vitecasse — the last being French for "breaks quickly".

In the late 1920s he turned to novel-writing, writing two works, Farewell to Autumn and Insatiability. The latter, his major work, encompasses geopolitics, psychoactive drugs, and philosophy. In 1935 he was awarded the Golden Laurel of the Polish Academy of Literature for his novels.

During the 1930s, Witkiewicz published a text on his experiences of narcotics, including peyote, and pursued his interests in philosophy writing, Concepts and Principles Implied by the Concept of Existence 1935. In 1934 he finishes perhaps his most famous literary work, the drama Szewcy, finally published in 1948. He also promoted emerging writers such as Bruno Schulz.

Death
Shortly after Poland was invaded by Germany in September 1939, Witkiewicz escaped with his young lover Czesława Oknińska to the rural frontier town of Jeziory, in what was then eastern Poland. After hearing the news of the Soviet invasion of Poland on 17 September 1939, Witkacy committed suicide on 18 September by taking a drug overdose and trying to slit his wrists. He convinced Czesława to attempt suicide with him by consuming Luminal, but she survived. The film Mystification (2010), written and directed by Jacek Koprowicz proposes, in surrealist fashion, that Witkiewicz faked his own death and lived secretly in Poland until 1968.

Legacy
Witkiewicz had died in some obscurity but his reputation began to rise soon after the war, which had destroyed his life and devastated Poland. Outside of Poland his work was discussed as a precursor to post-ww2 European drama in Martin Esslin's influential "Theatre of the Absurd" 1961, and later in Hans-Thies Lehmann's "Postdramatic Theatre" 2006. Konstanty Puzyna collected his surviving dramatic writings in two volumes in "Dramaty" (Dramas) 1962 which revived interest in his plays in Poland. Through his translations and scholarship, Daniel Gerould introduced English-language audiences to the writings of Witkiewicz.

Czesław Miłosz framed his argument in The Captive Mind around a discussion of Witkiewicz's novel, Insatiability. The artist and theater director Tadeusz Kantor was inspired by the Cricot group, through which Witkiewicz had presented his final plays in Kraków. Kantor brought many of the plays back into currency, first in Poland and then internationally, including The Cuttlefish (1956) and The Water Hen (1969). Visual artist Paulina Olowska produced Witkiewicz's The Mother: An Unsavoury Play in Two Acts and an Epilogue at the Tate Modern in 2015.

Films which have Witkiewicz as the subject include Tumor Witkacego 1985, Mystification 2010 and Witkacy and Malinowski: a cinematic séance in 23 scenes 2018. Films based on his works include Ludiot i kalugericata 1968, Farewell to Autumn 1990, Insatiability 2003, Madame Tutli-Putli 2007 and Nursery Rhyme of a Madman 2017.

Witkiewicz's paintings and pastel drawings are in the collections of the National Museum, Warsaw, the National Museum, Kraków, Museum of Literature, Warsaw and the Museum of Central Pomerania with 125 works in Słupsk Castle. The Metropolitan Museum of Art and Museum of Modern Art in New York, and the Art Gallery of New South Wales, Sydney hold important examples of his photography. The Villa Oksza Gallery of 20th century art of the Tatra Museum in Zakopane holds important examples of his photography and pastel drawings.

In the postwar period, People's Republic of Poland's Ministry of Culture decided to exhume Witkiewicz's body, move it to Zakopane, and give it a solemn funeral. This was carried out according to plan, though no one was allowed to open the coffin that had been delivered by the Soviet authorities.

On 26 November 1994, the Polish Ministry of Culture and Art ordered the exhumation of the presumed grave of Witkiewicz in Zakopane. Genetic tests on the remaining bones proved that the body had belonged to an unknown woman — a final absurdist joke, fifty years after the publication of Witkacy's last novel.

Art philosophy

 * Nowe formy w malarstwie (1919), translated into English as New Forms in Painting and the Misunderstandings Arising Therefrom (in The Witkiewicz Reader, Quartet, 1993)
 * Szkice estetyczne (Aesthetic Sketches, 1922)

Novels

 * 622 Upadki Bunga czyli demoniczna kobieta (1911) partial translation into English as The 622 Downfalls of Bungo or The Demonic Woman (in The Witkiewicz Reader)
 * Pożegnanie jesieni (1927) partial translation into English as Farewell to Autumn (in The Witkiewicz Reader)
 * Nienasycenie (1930) translated into English as Insatiability (Quartet Encounter, 1985)

Plays

 * Maciej Korbowa i Bellatrix (Maciej Korbowa and Bellatrix) (1918)
 * Pragmatyści (1919) (translated into English as The Pragmatists)
 * Mister Price, czyli Bzik tropikalny (1920) (translated into English as Mr Price, or Tropical Madness)
 * Tumor Mózgowicz (1920) (translated into English as Tumor Brainiowicz)
 * Nowe wyzwolenie (1920) (translated into English as The New Deliverance)
 * Oni (1920) (translated into English as They)
 * Panna Tutli-Putli (1920) (Miss Tootli-Pootli)
 * W małym dworku (1921) (translated into English as Country House)
 * Niepodległość trójkątów (1921) (translated into English as The Independence of Triangles)
 * Metafizyka dwugłowego cielęcia (1921) (translated into English as Metaphysics of a Two-Headed Calf)
 * Gyubal Wahazar, czyli Na przełęczach bezsensu (translated into English as Gyubal Wahazar, or Along the Cliffs of the Absurd: A Non-Euclidean Drama in Four Acts) (1921)
 * Kurka Wodna (1921) (Translated into English as The Water Hen)
 * Bezimienne dzieło (1921) (translated into English as The Anonymous Work: Four Acts of a Rather Nasty Nightmare)
 * Mątwa (1922) (translated into English as The Cuttlefish, or The Hyrcanian World View)
 * Nadobnisie i koczkodany, czyli Zielona pigułka (1922) (Translated into English as Dainty Shapes and Hairy Apes, or The Green Pill: A Comedy with Corpses)
 * Jan Maciej Karol Wścieklica (1922) (translated into English as Jan Maciej Karol Hellcat)
 * Wariat i zakonnica (1923) (translated into English as The Madman and the Nun)
 * Szalona lokomotywa (1923) (translated into English as The Crazy Locomotive)
 * Janulka, córka Fizdejki (1923) (translated into English as Janulka, Daughter of Fizdejko)
 * Matka (1924) translated into English as The Mother (in The Mother & Other Unsavoury Plays, Applause, 1993)
 * Sonata Belzebuba (1925) (translated into English as The Beelzebub Sonata)
 * Szewcy (1931–34) translated into English as The Shoemakers (in The Mother & Other Unsavoury Plays, Applause, 1993)

Filmography

 * Witkacy z Niną w Warszawie (1927), comedy film by Stanisław Ignacy Witkiewicz. Starring him and his wife Jadwiga in the city of Warsaw. Film housed in the Adam Mickiewicz Museum of Literature in Warsaw. Details: black and white film reel Pathé 9.5 mm, 5 minutes, silent film; remastered in 4K quality in 2015.

Other works

 * Narkotyki — niemyte dusze (1932), partial translation into English as Narcotics (in The Witkiewicz Reader)
 * Pojęcia i twierdzenia implikowane przez pojęcie istnienia (Concepts and Statements Implied by the Idea of Existence) (1935)
 * Jedyne wyjście
 * Kompozycia fantastyczna
 * Pocałunek mongolskiego księcia

Performances of work

 * The Crazy Locomotive (Szalona lokomotywa) received its New York premier at the Chelsea Theatre in 1977, under the direction of Des McAnuff. The Obie Award-winning production starred Dwight Schultz, Bob DeFrank and Glenn Close in leading roles.

 * Two New York premiers of Witkacy plays: The Madman and the Nun (Wariat i zakonnica) in 1979 under the direction of Paul Berman and The Water Hen (Kurka Wodna) directed by Brad Mays were staged by the Theatre Off-Park, in 1983. Broadway producer / Theatre Off-Park managing director Patricia Flynn Peate produced both plays, which were well received by critics and audiences alike. Future New York Times theatre critic Mark Matousek, then writing for the theatrical journal Other Stages, praised The Water Hen for "masterful comic direction," and the piece was videotaped for permanent inclusion in the Lincoln Center's Billy Rose Theatre Collection.
 * The British premiere of "They" "(Oni)" was presented at the Polish Theatre Hammersmith, London by POSK, directed by Paul Brightwell in 1984
 * The New York premiere of The Shoemakers (Szewcy) was presented by the Jean Cocteau Repertory under the direction of Włodzimierz Herman in 1987.
 * The Madman and the Nun was presented in 1989 by The Cosmic Bicycle Theatre at the Summer Music from Greensborough, a Classical Music Festival in Greensborough, Vermont, and in Boston, at The Charlestown Working Theatre. Directed by Jonathan Edward Cross [a.k.a. Jonny ClockWorks]. The production used Actors alongside Life-sized Puppets. Two of the original Puppet Figures are in the collection of the Witkacy Teatre in Zarkopane' Poland.
 * The New York premiere of Witkacy's Tumor Brainiowicz presented by La MaMa ETC was performed by The Theatre of a Two-Headed Calf (named after the Witkacy play Metaphysics of a Two-Headed Calf), under the direction of Brooke O'Harra. This production was followed by Witkacy's The Mother in 2003, also under O'Harra's direction and also a New York premier. The production featured puppets and video.
 * In 2019 Witkacy/Two-Headed Calf, a collaboration between CalArts Center for New Performance and STUDIO teatrgaleria, Warsaw, was directed by Natalia Korczakowska

title: Stanisław Ignacy Witkiewicz