Page:THEORY OF SHOCK WAVES AND INTRODUCTION TO GAS DYNAMICS.pdf/16

far-out field. On the other hand, in most typical problems of gas dynamics we are faced with short pipes and nozzles, short bodies to be flowed around; in a short pipe turbulence has no time to develop, even if the $Re$ number is high. Finally, in the hydrodynamics of small velocities, with $M<1$, the formation of eddies and turbulence is the only resistance mechanism for $Re\gg 1$; their consideration is absolutely necessary for studying the forces affecting a body moving in a fluid. In the case of supersonic speeds there occurs what is known as wave resistance and the possibility of irreversible dissipation of energy in steady-state shock waves; a resistance different from $0$ may be found also without studying turbulence.

Appendix

In order to determine the applicability of Eq. (I-1) - (I-6), let us take the general form of gas dynamic equations (see, for instance, [23, 27]).

The equation of motion, has the form:

$\rho {\frac {du_{x}}{dt}}=\rho X-{\frac {\partial p}{\partial x}}-{\frac {\partial T_{xx}}{\partial x}}-{\frac {\partial T_{xy}}{\partial y}}-{\frac {\partial T_{xz}}{\partial z}},$

(I-12)

where the quantities $X$, $Y$, $Z$ are components of volumetric force applied to a unit of mass, and the quantities $T_{xx}$, $T_{xy}$, and so forth, are components of the tensor of stresses due to the effect of viscosity. The effect of viscosity depends on the relative motion of neighboring fluid particles. From the conditions of tensor symmetry, confining ourselves to terms proportional to the first derivatives of velocity with respect to the coordinate, taking the invariant sum of normal stresses on three mutually perpendicular platforms to be equalled to the three-fold pressure, and isolating pressure from the stress tensor, as this already has been done in formula (I-12), we arrive at the following expression for the stress tensor:

$T_{xx}={\frac {2}{3}}\eta \left({\frac {\partial u_{x}}{\partial x}}+{\frac {\partial u_{y}}{\partial y}}+{\frac {\partial u_{z}}{\partial z}}\right)-2\eta {\frac {\partial u_{x}}{\partial x}};\quad T_{xz}=-\eta \left({\frac {\partial u_{z}}{\partial x}}+{\frac {\partial u_{x}}{\partial z}}\right);\quad T_{xy}=-\eta \left({\frac {\partial u_{y}}{\partial x}}+{\frac {\partial u_{x}}{\partial x}}\right).$

(I-13)

The equations of motion with respect to the two other coordinates are found from (I-12) and (I-13) by a cyclic shifting of indices.

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