Answer to Question #296973 in Mechanical Engineering for bhargav

Question

If for a two-dimensional potential flow, the velocity potential is given by ϕ = 4x(3y - 4), determine the velocity at the point (2, 3) . Determine also the value of stream function ψ at the poins (2, 3).

Answer 1

Velocity potential Function

$u = - \frac{{\partial \phi }}{{\partial x}};\;v = - \frac{{\partial \phi }}{{\partial y}}$

Stream Function

$u = \frac{{\partial ψ }}{{\partial y}};\;v = -\frac{{\partial ψ }}{{\partial x}}$

$ϕ = 4x(3y - 4)$

$u=-\frac{\delta \phi}{\delta x}$ and $v=-\frac{\delta \phi}{\delta y}=-\frac{\delta ψ}{\delta x}$

$u = -4 (3y - 4) =\frac{\delta ψ}{\delta y}$

$ψy = 16y - 6y2 + C1$

and

$v = -12x =-\frac{\delta ψ}{\delta x}$

$ψx = 6x2 + C2$

Therefore, $ψ = ψx + ψy$

$ψ = 6(x2 – y2) + 16y$

${\left. ψ \right|_{\left( {2,3} \right)}} = 6\left( {4 - 9} \right) + 16 \times 3$

$= 18$ units