## Q. 5.41

For a two-dimensional irrotational flow, the velocity potential is given by $\phi=2 x y$, find the velocity at (1, 2) and (2, 2). Determine also the discharge passing between streamlines passing through these points.

## Verified Solution

Given data:
Velocity potential        $\phi=2 x y$

For irrotational flow, the velocity potential (Φ) is defined as

$u=\frac{\partial \phi}{\partial x}$

$v=\frac{\partial \phi}{\partial y}$

Thus, the velocity components become

$u=\frac{\partial \phi}{\partial x}=\frac{\partial}{\partial x}(2 x y)=2 y$

$v=\frac{\partial \phi}{\partial y}=\frac{\partial}{\partial y}(2 x y)=2 x$

The velocity is then        $\vec{V}=u \hat{i}+v \hat{j}=2 y \hat{i}+2 x \hat{j}$

The velocity at (1, 2) is          $\left.\vec{V}\right|_{(1.2)}=(2 \times 2) \hat{i}+(2 \times 1) \hat{j}=4 \hat{i}+2 \hat{j}$

The velocity at (2, 2) is      $\left.\vec{V}\right|_{(2,2)}=(2 \times 2) \hat{i}+(2 \times 2) \hat{j}=4 \hat{i}+4 \hat{j}$

Hence,                    $\frac{\partial u}{\partial x}=0$

$\frac{\partial v}{\partial y}=0$

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

The above velocity field satisfies the continuity equation for incompressible flow. Hence, stream function exists.

From the definition of stream function ψ, we get

$u=\frac{\partial \psi}{\partial y}$

or                          $\psi=\int u d y=\int 2 y d y$

or                      $\psi=2 \frac{y^2}{2}+f(x)=y^2+f(x)$          (5.75)

and                                          $v=\frac{\partial \psi}{\partial x}$

$\psi=-\int v d x=-\int 2 x d x$

or                      $\psi=-2 \frac{x^2}{2}+g(y)=-x^2+g(y)$          (5.76)

Comparing Eqs. (5.75) and (5.76), we have

$\psi=y^2-x^2$

Hence, the stream function for the flow is

$\psi=y^2-x^2$

The stream function at (1, 2)        $\left.\psi\right|_{(1,2)}=2^2-1^2=3$ The stream function at (2, 2) is  $\left.\psi\right|_{(2,2)}=2^2-2^2=0$

The discharge per unit width passing between streamlines passing through these points is given by Eq. (5.64) as

$q=\psi_2-\psi_1=3-0=3 \text{ units }$]