For 1D constant area isentropic flow(assume inviscid and adiabatic). Given the following relations: $$\rho = \frac{\partial \phi}{\partial x} $$ $$\rho u = -\frac{\partial \phi}{\partial t} $$ where $a$ is the speed of sound and $u$ is the fluid speed. Prove the following:
- This equation (which is the governing equation) is satisfied: $$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x} = 0 $$
- This governing equation can be rewritten in the following way: $$ \frac{\partial^2 \phi}{\partial t^2}+2u\frac{\partial^2 \phi}{\partial x \partial t}+ (u^2-a^2)\frac{\partial^2 \phi}{\partial x^2} =0 $$
- Finally, prove that 2) equation is "always" hyperbolic
My attempt:
- First, I would appreciate it if someone elaborated what is it meant by the this equations being satisfied, what I did is I subbed in my relations into the governing equation and used the info that cross derivatives are equal or: $\frac{\partial^2}{\partial x\partial t} = \frac{\partial^2}{\partial t\partial x}$ and got the following: $ \frac{\partial^2 \phi}{\partial x \partial t} = \frac{\partial^2 \phi}{\partial t \partial x} $
- For the second part, I subbed in for the relations and used product rule but I am still getting an incomplete expression so I am not sure what I am missing. and finally for 3) if I use the following to classify my PDE: $$ A_{2}^2 -4A_{1}A_{3} > 0 $$ then PDE is Hyperbolic and what I found is this is equal to $4a^2$. Thanks for the help