Governing Equations

Conservation laws ( the conservation of mass, the conservation of momentum, the conservation of momentum of momentum and the conservation of energy), which accompanied by the constitutive and thermodynamical state equations lead to the system of the Navier - Stokes equations describing the motion of viscous compressible flows. The precise mathematical determinatin can be found in many textbooks, e.g. [#!Feistauer!#] so that we present only the mathematical formulation of physical conservation laws:

The conservation law of density has the form  

$$\frac{\partial\rho}{\partial t}+ \sum_{i=1}^{d} \frac{\par... ... x_i} = 0 \qquad \mbox{in}\ \Omega \times \left ( 0,T\right),$$ (1)

the conservation law of momentum  

$$\frac{\partial\rho u_i}{\partial t}+ \sum_{j=1}^{d} \frac{... ...box{in}\ \Omega \times \left ( 0,T\right),\qquad i = 1,\dots,d$$ (2)

and the conservation of energy  

$$\frac{\partial e}{\partial t} + \sum_{i=1}^{d}\frac{\part... ...rtial x_i} \qquad \mbox{in}\ \Omega \times \left ( 0,T\right).$$ (3)

Further form the conservation of momentum of momentum we derive that stress tensor $\tau_{ij}$ is symmetric, i.e. $\tau_{ij}=\tau_{ji},\ i,j=1,\dots,d$.

We see that the number of unknown quantities is larger than the number of equations. To complete the whole system of conservation laws we have to add some constitutive or closing equations that will specify our fluids. At first we must find relation between the stress tensor $\left ( \tau_{ij}\right )_{i,j=1}^{d}$ and other quantities describing the fluid motion. We will suppose Newtonian fluid for which stress tensor can be evaluated in the form  

$$\tau_{ij} = -p\delta_{ij} + \tau_{ij}^{V},\quad i,j=1,\dots,d$$ (4)

 

$$\tau_{ij}^{V} = \lambda\ {\bf div\ u}\ \delta_{ij} + 2\mu e_{ij},\quad i,j=1,\dots,d\ , \nonumber$$

where $\delta_{ij}$ is the Kronecker delta and ${\bf e} = (e_{ij})_{i,j=1}^{d}$ is the so called deformation velocity tensor

$$e_{ij} = \frac 12 \left (\frac {\partial u_{i}}{\partial x_j}+ \frac{\partial u_j}{\partial x_i} \right ).$$ (5)

Here $\lambda,\mu$ are viscosity coefficients. For them we will use the relation derived from the kinetic theory for the one-atomic gas:

$$3\lambda+2\mu = 0,\qquad \mu \geq 0$$ (6)

The next constitutive relation is obtained from Fourier's law  

$${\bf q} = -k\ {\bf grad }\ \theta,$$ (7)

where $k\geq 0$ is called the heat conductivity and is supposed to be constant.

We will always consider the perfect gas, which means  

$$p=R\rho\theta.$$ (8)

This equation is called the state equation of the perfect gas. R>0 is the specific gas constant and it can be expressed in the form

R=c_p-c_v,

where c_p and c_v are the specific heat at constant pressure and volume, respectively. We assume that c_p and c_v are constants. It follows from theory that c_p>c_v. The quantity

$$\kappa = \frac{c_p}{c_v}\gt 1$$ (10)

is called the Poisson constant.

The total energy e consists of the internal energy u and the kinetic energy. We suppose the perfect polytropic gas, i.e. $u = c_v\theta$. We obtain  

$$e= \rho c_v\theta + \frac 12\rho\left \vert {\bf u}\right\vert^2.$$ (11)

Now putting (4), (5), (8), (12) in (1), (2) and (3) we obtain  

$$\frac{\partial\rho}{\partial t}+ \sum_{i=1}^{d} \frac{\partial\left (\rho u_i\right )} {\partial x_i} = 0,$$ (12)

 

$$\frac{\partial\rho u_i}{\partial t}+ \sum_{j=1}^{d} \frac{... ...1}^d\frac{\partial}{\partial x_j}\left ( \mu e_{ij} \right) ,$$ (13)

Now together with (9) we have the closed system for the unknowns $\rho,{\bf u},\theta,p$.