Question

In a steady two-dimensional {compressible} flow, $u$ and $v$ are the $x$- and $y$-components of velocity, and $\rho$ is the fluid density. Among the following pairs of relations, which one(s) perfectly satisfies/satisfy the definition of stream function, $\psi$, for this flow?

• A. $u=\dfrac{\partial \psi}{\partial y}$ and $v=-\dfrac{\partial \psi}{\partial x}$
• B. $u=-\dfrac{\partial \psi}{\partial x}$ and $v=-\dfrac{\partial \psi}{\partial y}$
• C. $\rho u=\dfrac{\partial \psi}{\partial y}$ and $\rho v=-\dfrac{\partial \psi}{\partial x}$
• D. $\rho u=-\dfrac{\partial \psi}{\partial y}$ and $\rho v=\dfrac{\partial \psi}{\partial x}$

Hint:

For compressible 2D flow, define the mass-flux stream function: $\rho u=\psi_y$, $\rho v=-\psi_x$. For incompressible flow ($\rho=\text{const}$), this reduces to $u=\psi_y$, $v=-\psi_x$.

Answer 1

The Correct Option is C, D

Step 1: Continuity equation for steady 2D compressible flow.
\[ \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y}=0. \]
Step 2: Define a {compressible stream function $\psi$.}
Choose $\psi$ such that \[ \rho u=\frac{\partial \psi}{\partial y}, \qquad \rho v=-\frac{\partial \psi}{\partial x}. \] Then \[ \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} =\frac{\partial^2 \psi}{\partial x \partial y} -\frac{\partial^2 \psi}{\partial y \partial x}=0 \] identically, so continuity is automatically satisfied.
Step 3: Match with options.
The above is exactly option (C). Options (A), (B) correspond to incompressible form ($\rho$ constant) and (D) has opposite signs, which would violate the standard convention. Final Answer: \[ \boxed{\text{(C)}} \]