Chapter 4, Problem 4.11P

Interpretation Introduction

Interpretation:

The continuity equation for the given shock wave is to be written and a relationship between the variables ${\rho }_2,{\rho }_1,v_2,\text{ and }{v}_w$ is to be obtained.

Concept Introduction:

The general continuity equation to be used for a controlled volume is:

  ${\displaystyle {\iint }_{C.S}\rho \left(v\cdot n\right)dA}+\frac{\partial }{\partial t}{\displaystyle {\iiint }_{C.V}\rho dV}=0$.... (1)

Here, C.S is the control surface over which the integral dA is taken, C.V is the control volume over which the integral dV is taken, $\rho$ is the density of the fluid, $v$ is the velocity vector, $n$ is the direction of the vector $v$ , The product $v\cdot n$ is scalar defined as:

  $v\cdot n=\left|v\right|\left|n\right|\cos \theta$