For a two-dimensional incompressible flow field given by , where A>0, which one of the following statements is FALSE?

The correct option is: Its streamlines are given by x = y .

Step-by-step Explanation:

From the provided images, the velocity vector field $\overrightarrow{u}$ of the flow is given by:
$\overrightarrow{u}=A(x\hat{i}-y\hat{j})$

Here, the velocity components in the x-direction ($u$) and y-direction ($v$) are:
$u=Ax$
$v=-Ay$

Let us analyze each statement to determine which one is false:

1. Checking the Continuity Equation:
For a two-dimensional incompressible flow, the continuity equation is:
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

Substitute the velocity components into the continuity equation:
$\frac{\partial }{\partial x}\left(Ax\right)+\frac{\partial }{\partial y}(-Ay)=A-A=0$

Since the sum is zero, the continuity equation is satisfied. Thus, the statement "It satisfies continuity equation." is TRUE.

2. Checking Irrotationality:
A flow field is irrotational if the vorticity component in the z-direction (${\omega }_z$) is zero:
${\omega }_z=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$

Substituting the values of $u$ and $v$:
${\omega }_z=\frac{\partial }{\partial x}(-Ay)-\frac{\partial }{\partial y}\left(Ax\right)=0-0=0$

Since the curl of the velocity field is zero, the flow is irrotational. Thus, the statement "It is irrotational." is TRUE.

3. Checking Unidirectionality:
As $x\to 0$ and $y\to \infty$:
The x-component of velocity $u=Ax\to 0$.
The y-component of velocity $v=-Ay$ dominates, meaning the flow occurs almost exclusively along the y-direction (unidirectional). Thus, the statement "It is unidirectional when x → 0 and y → ∞." is TRUE.

4. Analyzing the Streamlines:
The differential equation for streamlines in a 2D flow is:
$\frac{dx}{u}=\frac{dy}{v}$

Substitute $u=Ax$ and $v=-Ay$ into the equation:
$\frac{dx}{Ax}=\frac{dy}{-Ay}$

Cancel the constant $A$ on both sides:
$\frac{dx}{x}=-\frac{dy}{y}$

Integrate both sides:
$\int \frac{dx}{x}=-\int \frac{dy}{y}$
$\ln \left(x\right)=-\ln \left(y\right)+\ln \left(C\right)$

Rearranging the terms yields:
$\ln \left(x\right)+\ln \left(y\right)=\ln \left(C\right)$
$\ln \left(xy\right)=\ln \left(C\right)$
$xy=C$

Therefore, the streamlines are rectangular hyperbolas given by $xy=C$, and not the line $x=y$. This makes the statement "Its streamlines are given by x = y." FALSE.