Exercises

1. Prove the following useful theorems regarding partial derivatives:

   $$\left(\frac{\partial X}{\partial Y}\right)_Z = 1\left/\left(\frac{\partial Y}{\partial X}\right)_Z\right.,$$

   $$\left(\frac{\partial X}{\partial Y}\right)_Z = \left(\frac{\partial X}{\partial W}\right)_Z\left/\left(\frac{\partial Y}{\partial W}\right)_Z\right.,$$

   $$\left(\frac{\partial X}{\partial Y}\right)_Z = -\left(\frac{\partial Z}{\partial Y}\right)_X\left/\left(\frac{\partial Z}{\partial X}\right)_Y\right.\$$

   
   

2. 1. The specific internal energy of a (not necessarily ideal) gas is defined by
      $$d{\cal E} = T\,d{\cal S} -p\,dv.$$
      Demonstrate that

      $$T = \left(\frac{\partial {\cal E}}{\partial {\cal S}}\right)_v, p =- \left(\frac{\partial {\cal E}}{\partial v}\right)_{\cal S},$$

      
      
      and
      $$\left(\frac{\partial T}{\partial v}\right)_{\cal S} = -\left(\frac{\partial p}{\partial {\cal S}}\right)_v.$$

   2. The specific enthalpy of a gas is defined by
      $${\cal H} = {\cal E} + p\,v.$$
      Demonstrate that

      $$T = \left(\frac{\partial {\cal H}}{\partial {\cal S}}\right)_p, v = \left(\frac{\partial {\cal H}}{\partial p}\right)_{\cal S},$$

      
      
      and
      $$\left(\frac{\partial T}{\partial p}\right)_{\cal S} = \left(\frac{\partial v}{\partial {\cal S}}\right)_p.$$

   3. The specific Helmholtz free energy of a gas is defined by
      $${\cal F} = {\cal E} - T\,{\cal S}.$$
      Demonstrate that

      $${\cal S} = -\left(\frac{\partial {\cal F}}{\partial T}\right)_v, p =- \left(\frac{\partial {\cal F}}{\partial v}\right)_{\cal T},$$

      
      
      and
      $$\left(\frac{\partial {\cal S}}{\partial v}\right)_{T} = \left(\frac{\partial p}{\partial T}\right)_v.$$

   4. The specific Gibbs free energy of a gas is defined by
      $${\cal G} = {\cal H} - T\,{\cal S}.$$
      Demonstrate that

      $${\cal S} = -\left(\frac{\partial {\cal G}}{\partial T}\right)_p, v =\left(\frac{\partial {\cal G}}{\partial p}\right)_{T},$$

      
      
      and
      $$\left(\frac{\partial {\cal S}}{\partial p}\right)_{T} = -\left(\frac{\partial v}{\partial T}\right)_p.$$

3. Demonstrate that the specific heat at constant volume of a (not necessarily ideal) gas can be written
   $$c_v =\left(\frac{\partial {\cal E}}{\partial T}\right)_v = T\left(\frac{\partial{\cal S}}{\partial T}\right)_v.$$
   Likewise, show that the specific heat at constant pressure takes the form
   $$c_p =\left(\frac{\partial {\cal H}}{\partial T}\right)_v = T\left(\frac{\partial{\cal S}}{\partial T}\right)_p.$$

4. The quantities

   $$\alpha = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_p,$$

   $$\kappa_T = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_T,$$

   $$\kappa_{\cal S} = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_{\cal S},$$

   
   
   are known as the coefficient of thermal expansion, the isothermal compressibility, and the adiabatic compressibility, respectively. Demonstrate that for a (not necessarily ideal) gas,

   $$c_p - c_v = \frac{T\,v\,\alpha^{\,2}}{\kappa_T},$$

   $$\kappa_T - \kappa_{{\cal S}} = \frac{T\,v\,\alpha^{\,2}}{c_p}.$$ (14.151)

   
   
   Hence, deduce that
   $$\frac{c_p}{c_v} = \frac{\kappa_T}{\kappa_{\cal S}}.$$
   Show that for the special case of an ideal gas, $\alpha=1/T$ , $\kappa_T = 1/p$ , and $\kappa_{\cal S}=1/(\gamma\,p)$ . Hence, obtain the following standard results for an ideal gas:

   $$\frac{c_p}{c_v} = \gamma,$$

   $$c_p - c_v = {\cal R},$$

   $$c_p =\left(\frac{\gamma}{\gamma-1}\right){\cal R},$$

   
   

5. Show that for an ideal gas

   $${\rm Ma} = \frac{v}{c_0}\left[1-\frac{1}{2}\,(\gamma-1)\left(\frac{v}{c_0}\right)^2\right]^{\,-1/2},$$

   $$\frac{T}{T_0} = 1-\frac{1}{2}\,(\gamma-1)\left(\frac{v}{c_0}\right)^2,$$

   $$\frac{p}{p_0} = \left[1-\frac{1}{2}\,(\gamma-1)\left(\frac{v}{c_0}\right)^2\right]^{\,\gamma/(\gamma-1)},$$

   $$\frac{\rho}{\rho_0} = \left[1-\frac{1}{2}(\gamma-1)\left(\frac{v}{c_0}\right)^2\right]^{\,1/(\gamma-1)},$$

   
   
   where ${\rm Ma}$ is the Mach number, $v$ the flow speed, and $T_0$ , $p_0$ , $\rho_0$ , and $c_0$ , are the temperature, pressure, density, and sound speed, respectively, at the stagnation point.

6. Consider the flow of an isentropic ideal gas down a straight nozzle with a slowly-varying cross-sectional area, $A(x)$ , where $x$ measures distance along the nozzle. Let $u(x)$ , $\rho(x)$ , $c(x)$ , and ${\rm Ma}(x)$ be the flow speed, density, sonic speed, and Mach number, respectively. Demonstrate that
   $$c^{\,2}\,\frac{d\rho}{\rho}+u\,du=0,$$
   and
   $$\frac{d\rho}{\rho} + \frac{dA}{A}+\frac{du}{u} =0.$$
   Hence, show that
   $$\frac{dA}{A} =\frac{du}{u}\,({\rm Ma}^{\,2}-1).$$
   Deduce that the throat of the nozzle (where $A$ attains its minimum value) either corresponds to a sonic point (i.e, ${\rm Ma}=1$ ), or a point of maximum or minimum flow speed. Finally, demonstrate that
   $$\frac{d{\rm Ma}}{{\rm Ma}} = \left[\frac{1+(1/2)\,(\gamma-1)\,{\rm Ma}^{\,2}}{1-{\rm Ma}^{\,2}}\right]\frac{dA}{A}.$$

   Figure 14.4: A shock tube.

   

7. As indicated in Figure 14.4, a shock tube is a tube of uniform cross section that is divided by a diaphragm into two chambers that contain different gases at different pressures. Let $x$ measure distance along the tube, and let the diaphragm be located at $x=0$ . Suppose that the chamber to the left of the diaphragm (which lies at $x<0$ ) is filled with stationary gas of pressure, density, temperature, and ratio of specific heats, $p_1$ , $\rho_1$ , $T_1$ , and $\gamma_1$ , respectively. Likewise, suppose that the chamber to the right of the diaphragm (which lies at $x>0$ ) is filled with stationary gas of pressure, density, temperature, and ratio of specific heats, $p_4$ , $\rho_4$ , $T_4$ , and $\gamma_4$ , respectively. It is assumed that $p_4>p_1$ . At $t=0$ , the diaphragm is ruptured. As indicated in the figure, a shock wave subsequently travels to the left with speed $\vert V_s\vert$ , and an expansion wave to the right with speed $c_4$ . The so-called contact surface marks the boundary between the two different gases that were originally on either side of the diaphragm. Neglecting diffusion, the gases do not mix, but are permanently separated by the contact surface. On either side of the contact surface, which moves to the left with speed $\vert u_2\vert$ , the temperatures and densities can be different, but the pressures and flow velocities must be the same. We can divide the gas in the tube into four regions. Regions 1 lies to the left of the shock wave. Region 2 lies between the shock wave and the contact surface. Region 3 lies between the contact surface and the expansion wave. Region 4 lies to the right of the expansion wave. Thus, we expect the flow velocity, pressure, density, temperature, and ratio of heats to be $u_1=0$ , $p_1$ , $\rho_1$ , $T_1$ , and $\gamma_1$ , respectively, in Region 1; $u_2$ , $p_2$ , $\rho_2$ , $T_2$ , and $\gamma_1$ , respectively in Region 2; $u_2$ , $p_2$ , $\rho_3$ , $T_3$ , and $\gamma_4$ , respectively in Region 3; and $u_4=0$ , $p_4$ , $\rho_4$ , $T_4$ , and $\gamma_4$ , respectively, in Region 1. Note that $u_2$ and $V_s$ are negative. Demonstrate that
   $$\vert u_2\vert= c_1\left(\frac{p_2}{p_1}-1\right)\left[\frac{2/\gamma_1}{2\,\gamma_1+ (\gamma_1+1)\,(p_2/p_1-1)}\right]^{\,1/2},$$
   and
   $$\vert u_2\vert = \frac{2\,c_4}{\gamma_4-1}\left[1-\left(\frac{p_2}{p_4}\right)^{(\gamma_4-1)/\gamma_4}\right],$$
   where $c_1$ and $c_4$ are the sound speeds in Regions 1 and 4, respectively. Hence, obtain
   $$\frac{p_4}{p_1}=\frac{p_2}{p_1}\left[1-\frac{(\gamma_4-1)\,(c_1/c... ...t{2\,\gamma_1 +(\gamma_1+1)\,(p_2/p_1-1)}}\right]^{-2\,\gamma_4/(\gamma_4-1)}.$$
   This expression give the shock strength, $p_2/p_1$ , implicitly as a function of the diaphragm pressure ratio, $p_4/p_1$ . All other quantities of interest can be expressed in terms of the shock strength. Show that

   $$\vert V_s\vert = c_1\left[1+\left(\frac{\gamma_1+1}{2\,\gamma_1}\right)\left(\frac{p_2}{p_1}-1\right)\right]^{\,1/2},$$

   $$\frac{\rho_2}{\rho_1} = \frac{2\,\gamma_1+(\gamma_1+1)\,(p_2/p_1-1)}{2\,\gamma_1+(\gamma_1-1)\,(p_2/p_1-1)},$$

   $$\frac{T_2}{T_1} = \frac{p_2}{p_1}\left[\frac{2\,\gamma_1+(\gamma_1-1)\,(p_2/p_1-1)}{2\,\gamma_1+(\gamma_1+1)\,(p_2/p_1-1)}\right],$$

   $$\frac{\rho_3}{\rho_4} = \left(\frac{p_2/p_1}{p_4/p_1}\right)^{\,1/\gamma_4},$$

   
   
   and
   $$\frac{T_3}{T_4} = \left(\frac{p_2/p_1}{p_4/p_1}\right)^{\,(\gamma_4-1)/\gamma_4}.$$

8. Show that the maximum shock strength and shock speed attainable in a (uniform) shock tube, in the limit $p_4/p_1\rightarrow\infty$ , are
   $$\sqrt{\frac{p_2}{p_1}} = \frac{c_4}{c_1}\frac{\sqrt{2\,\gamma_1\,(\gamma_1+1)}}{(\gamma_4-1)},$$
   and
   $$\vert V_s\vert = \left(\frac{\gamma_1+1}{\gamma_4-1}\right)c_4,$$
   respectively. Show, further, that the contact surface moves at the speed
   $$\vert u_2\vert = \left(\frac{2}{\gamma_4-2}\right)c_4.$$

9. Show that Equations (14.38) and (14.39) can be written in the form

   $$\frac{2}{\gamma-1}\left(\frac{\partial c}{\partial t} + u\,\frac{\partial c}{\partial x}\right)+c\,\frac{\partial u}{\partial x} = 0,$$

   $$\frac{\partial u}{\partial t} + u\,\frac{\partial u}{\partial x} +\frac{2\,a}{\gamma-1}\,\frac{\partial c}{\partial x} = 0,$$

   
   
   where $c$ is the sound speed. By adding and subtracting the previous equations, obtain

   $$\left[\frac{\partial}{\partial t} + (u+c)\,\frac{\partial}{\partial x}\right]\left(u+\frac{2\,c}{\gamma-1}\right) = 0,$$

   $$\left[\frac{\partial}{\partial t} + (u-c)\,\frac{\partial}{\partial x}\right]\left(u-\frac{2\,c}{\gamma-1}\right) =0.$$

   
   
   These equations indicate that the quantities $P=u+2\,c/(\gamma-1)$ and $Q=u-2\,c/(\gamma-1)$ are constant on curves that have the slope $dx/dt=u+c$ and $dx/dt=u-c$ , respectively. These curves are called characteristics, and $P$ and $Q$ are known as Riemann invariants. In situations in which all the $Q$ characteristics originate from regions where the gas is at rest, we expect $Q$ to be constant throughout the gas. Deduce that [cf., Equation (14.50)]
   $$c= c_0 + \left(\frac{\gamma-1}{2}\right)u,$$
   where $c_0$ is the stagnation sound speed. Show that
   $$\left[\frac{\partial}{\partial t} + (u+c)\,\frac{\partial}{\partial x}\right](u+c) = 0.$$
   Hence, conclude that the $P$ characteristics are straight-lines.

10. Consider a one-dimensional sound wave propagating through an ideal gas whose unperturbed sound speed is $c_0$ . At time $t=0$ , the velocity perturbation, $u(x,t)$ , due to the wave, has the form $u_0\,\sin(k\,x)$ , where $u_0$ and $k$ are both positive. Demonstrate that shocks form after a time
    $$t_\ast \simeq \frac{2}{(\gamma+1)\,k\,u_0}$$
    has elapsed. [Hint: Shocks are associated with the crossing of different $P$ characteristics.] Show that in the time interval $t=0$ to $t=t_\ast$ a local maximum of $u(x,t)$ travels a distance
    $$\left[1+\frac{2\,c_0}{(\gamma+1)\,u_0}\right]\frac{1}{k},$$

11. An ideal gas is initially at rest in a uniform tube, and occupies the region to the right of a tight-fitting piston whose position is $X(t)=(1/2)\,a\,t^{\,2}$ for $t>0$ . Here, $a>0$ . Show that a shock first forms at $t=t_\ast$ and $x=x_\ast$ , where

    $$t_\ast = \left(\frac{2}{\gamma+1}\right)\,\frac{c_0}{a},$$

    $$x_\ast = \left(\frac{2}{\gamma+1}\right)\,\frac{c_0^{\,2}}{a}.$$

    
    
    Here, $c_0$ is the stagnation sound speed, and $\gamma$ the ratio of specific heats.