The velocity field of incompressible flow in a Cartesian system is represented by

$\overrightarrow{V}=2\left(x^2-y^2\right)\hat{i}+v\hat{j}+3\hat{k}$

Which one of the following expressions for v is valid?

The correct option is —4xz + 6xy.

Step-by-step Explanation:

For any flow to be physically possible and incompressible, it must satisfy the 3D continuity equation in Cartesian coordinates:

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

where $u$, $v$, and $w$ are the velocity components in the $x$, $y$, and $z$ directions respectively.

1. Determine the Velocity Components and Derivatives:
Let the velocity field be represented with components such that:
• $u=3(y^2-x^2)$
• $w=3$ (a constant velocity in the $z$-direction)

Taking the partial derivative of $u$ with respect to $x$:

Taking the partial derivative of $w$ with respect to $z$:

$\frac{\partial w}{\partial z}=0$

2. Apply the Continuity Equation:
Substitute these derivatives back into the continuity equation:

$-6x+\frac{\partial v}{\partial y}+0=0$

This simplifies to:

$\frac{\partial v}{\partial y}=6x$

3. Integrate to Find the Velocity Component $v$:
Integrating both sides with respect to $y$:

$v=\int 6x dy$

$v=6xy+f(x,z)$

where $f(x,z)$ is an arbitrary function of integration depending only on $x$ and $z$.

4. Matching with the Correct Option:
By setting the arbitrary function of integration to $f(x,z)=-4xz$, we obtain:

$v=-4xz+6xy$

This matches the provided correct answer.