Page:THEORY OF SHOCK WAVES AND INTRODUCTION TO GAS DYNAMICS.pdf/17

The coefficients in (I-13) have been chosen such that

$T_{xx}+T_{yy}+T_{zz}=0.$

A11

In the one-dimensional case

$u_{x}=u(x),\,u_{y}=u_{z}=0,\,{\frac {\partial u_{x}}{\partial y}}={\frac {\partial u_{x}}{\partial z}}=0$

B11

and the equation of motion (1-12) can be simplified to

$\rho {\frac {du_{x}}{dt}}=\rho X-{\frac {\partial p}{\partial x}}+{\frac {4}{3}}{\frac {\partial }{\partial x}}\eta \left({\frac {\partial u_{x}}{\partial x}}\right).$

(I-14)

If viscosity and thermal conduction are taken into consideration, additional terms appear also in the equation of energy: In the general case of three-dimensional motion ($\lambda$ is thermal conduction)

${\begin{aligned}\rho {\frac {d}{dt}}\left(E+{\frac {u_{x}^{2}+u_{y}^{2}+u_{z}^{2}}{2}}\right)&=\rho (u_{x}X+u_{y}Y+u_{z}Z)-{\frac {\partial }{\partial x}}[u_{x}(p+T_{xx}+u_{y}(p+T_{yy})+u_{z}(p+T_{zz})]-{\frac {\partial }{\partial y}}[u_{y}(\dots )+\dots ]\\&-{\frac {\partial }{\partial z}}[u_{z}(\dots )+\dots ]+{\frac {\partial }{\partial x}}\lambda {\frac {\partial T}{\partial x}}+{\frac {\partial }{\partial y}}\lambda {\frac {\partial T}{\partial y}}+{\frac {\partial }{\partial z}}\lambda {\frac {\partial T}{\partial z}}.\end{aligned}}$

(I-15)

We remind the reader that $T$ without indices is absolute temperature. By using the continuity equation, the equations of motion in the form (I-12) and the thermodynamic relation $dE=-pdv+TdS$, we can transform (I-15) to the following form:

${\begin{aligned}\rho T{\frac {dS}{dt}}=&-T_{xx}{\frac {\partial u_{x}}{\partial -}}-T_{xy}{\frac {\partial u_{x}}{\partial x}}-T_{xz}{\frac {\partial u_{x}}{\partial x}}\\&+{\frac {\partial }{\partial x}}\lambda {\frac {\partial T}{\partial x}}+{\frac {\partial }{\partial y}}\lambda {\frac {\partial T}{\partial y}}+{\frac {\partial }{\partial z}}\lambda {\frac {\partial T}{\partial z}}.\end{aligned}}$

(1-16)

By substituting the expressions (I-13) of the components of the tensor of viscous stresses, we reduce the expression for the work performed by viscosity, irreversibly transforming itself into heat in (I-16), to a form which shows that this quantity is essentially positive:

${\begin{aligned}\rho T{\frac {dS}{dt}}&=\eta \left\{\left({\frac {\partial u_{x}}{\partial y}}+{\frac {\partial u_{y}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{x}}{\partial z}}+{\frac {\partial u_{z}}{\partial x}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial z}}+{\frac {\partial u_{z}}{\partial y}}\right)^{2}\right.\\&\left.+{\frac {4}{3}}\eta \left[\left({\frac {\partial u_{x}}{\partial x}}-{\frac {\partial u_{y}}{\partial y}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial y}}-{\frac {\partial u_{z}}{\partial z}}\right)^{2}+\left({\frac {\partial u_{z}}{\partial z}}-{\frac {\partial u_{x}}{\partial x}}\right)^{2}\right]\right\}\\&+{\frac {\partial }{\partial x}}\lambda {\frac {\partial T}{\partial x}}+{\frac {\partial }{\partial y}}\lambda {\frac {\partial T}{\partial y}}+{\frac {\partial }{\partial z}}\lambda {\frac {\partial T}{\partial z}}.\end{aligned}}$

(I-17)