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.

Explanation:

From the given image, the velocity vector field of the flow is:
$\overrightarrow{u}=A(x\hat{i}-y\hat{j})$
Here, the velocity components in the $x$ and $y$ directions are:
$u=Ax$
$v=-Ay$

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

1. Continuity Equation:
For a two-dimensional incompressible flow, the continuity equation is:
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$
Differentiating the velocity components:
$\frac{\partial u}{\partial x}=\frac{\partial }{\partial x}\left(Ax\right)=A$
$\frac{\partial v}{\partial y}=\frac{\partial }{\partial y}(-Ay)=-A$
Substituting these values:
$A+(-A)=0$
Since the continuity equation is satisfied, the statement "It satisfies continuity equation" is TRUE.

2. Unidirectional Flow:
As $x\to 0$ and $y\to \infty$, the velocity components become:
$u=Ax\to 0$
$v=-Ay\to -\infty$
Since the velocity component $u$ becomes negligible compared to $v$, the flow becomes entirely directed along the y-axis (unidirectional). Therefore, the statement "It is unidirectional when x → 0 and y → ∞" is TRUE.

3. Equation of Streamlines:
The differential equation representing a streamline in a two-dimensional flow is:
$\frac{dx}{u}=\frac{dy}{v}$
Substituting the expressions for $u$ and $v$:
$\frac{dx}{Ax}=\frac{dy}{-Ay}\Rightarrow \frac{dx}{x}=-\frac{dy}{y}$
Integrating both sides:
$\int \frac{dx}{x}=-\int \frac{dy}{y}$
$lnx=-lny+lnC$
$ln\left(xy\right)=lnC\Rightarrow xy=C$
Thus, the streamlines are rectangular hyperbolas given by $xy=C$, where $C$ is a constant. The statement "Its streamlines are given by x = y" is FALSE.

4. Irrotational Flow:
The component of rotation about the z-axis for a 2D flow is:
${\omega }_z=\frac{1}{2}(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y})$
Finding the partial derivatives:
$\frac{\partial v}{\partial x}=\frac{\partial }{\partial x}(-Ay)=0$
$\frac{\partial u}{\partial y}=\frac{\partial }{\partial y}\left(Ax\right)=0$
Thus, the rotation is:
${\omega }_z=0$
Since the rotation is zero, the flow field is irrotational, so the statement "It is irrotational" is TRUE.