Answered: "The velocity components in an incompressible, two-dimensional flow..."

Answered question

2022-03-07

The velocity components in an incompressible, two-dimensional flow field are given by the equations u = x^2 and v=-2xy + x. Determine, if possible, the corresponding stream function.

Answer & Explanation

Ian Adams

Skilled2022-03-14Added 163 answers

Any psysically possible incompressible flow filed must satisfy the conservation of mass is followed by

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 ∵ \frac{\partial w}{\partial z} = 0 = \frac{\partial}{\partial x} (x^{2}) + \frac{\partial}{\partial y} (- 2 x y + x) = 0 = 2 x + (- 2 x) = 0 = 0$

The flow field is possible

$\to$ from the definition o tream function

$\frac{\partial ψ}{\partial y} = u \frac{\partial ψ}{\partial x} = v \Rightarrow \frac{\partial ψ}{\partial y} = x^{2} \Rightarrow \int \partial ψ = \int x^{2} d y = ψ = x^{2} y + f_{1} (x) (1)$

Similarily

$u = \frac{\partial ψ}{\partial x} \Rightarrow \partial ψ = - v \partial x = \int d ψ = \int - v d x = \int - (- 2 x y + x) d x ψ = \int (2 x y + x) d x = \frac{2 x^{2}}{2} y - \frac{x^{2}}{2} + f_{2} (y) ψ = x^{2} y - \frac{x^{2}}{2} + f_{2} (y) (2)$

To satisfy the both equations (1) and (2) we can equate that

$ψ = x^{2} y$

$∴$ Stream fucntion of the given flow field

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Analysis

Didn't find what you were looking for?