$\text {If the velocity potential in an inviscid flow is given by $\phi = A y$, where $\mathrm { A }$ is a constant, the}$ $\text {stream function is:}$
A) $- A y ^ { 2 } + B$
B) $A x ^ { 2 } + B$
C) $A y + B$
D) $- A x + B$

Question

Quizplus · Accepted Answer

$- A x + B$
$\text {Equation (8.5.11) relates the velocity potential function and the stream function:}$
$$\frac { \partial \psi } { \partial x } = - \frac { \partial \phi } { \partial y } = - A . \quad \therefore \psi ( x , y ) = \int - A d x = - A x + f ( y )$$
$\text {Then $\frac { \partial \psi } { \partial y } = \frac { d f } { d y }$ which must, by Eq. 8.5.11, equal $\frac { \partial \phi } { \partial x } = 0$.}$$\text { $\therefore \frac { d f } { d y } = 0$ so $f = \mathrm { const } = B$ and $\psi = \underline { - A x + B }$. Most often, we let $B = 0$.}$