existence of stream function

Question

Show from $\frac{\partial}{\partial t}\int_{\Omega}\rho dV = -\int_{\partial \Omega} \rho v\cdot n dS$ that a stream function $\psi(x, y)$ exists, in two-dimensional steady flow such that the difference between the respective values of $\psi$ at any two points of a connected fluid domain $\Omega_t$ equals the mass flow rate across an arbitrary (rectifiable) curve segment $\subset \Omega_t$ joining the two points. Here, $\Omega$ is the space that contains the fluid, $\partial \Omega$ the boundary, $v$ is the velocity of the fluid, and $n$ is normal to $\partial \Omega$. I'm not sure how to approach this problem, some guidance would be greatly appreciated!

Answer 1

How to approach the problem

You are asked to prove there exists a function $\psi(x,y)$ such that for any path $C$ joining points $(x_1,y_1)$ and $ (x_2,y_2)$ we have $Q_C = \psi(x_2,y_2) - \psi (x_1,y_1)$, where the mass flow rate $Q_C$ across $C$ is given by the line integral

$$Q_{C} =\int_C (\rho \mathbf{v}) \cdot \mathbf{n} \, dl = \int_C (\rho v_x \,dy - \rho v_y \,dx)$$

Hint: Consider the union of any two paths joining $(x_1,y_1)$ and $(x_2,y_2)$