[amp_mcq option1=”x2y” option2=”x2 – y2″ option3=”cosx” option4=”x2 + y2″ correct=”option4″]
The correct answer is D.
The continuity equation states that the divergence of the velocity field is zero. In other words, the amount of fluid entering a given volume must equal the amount of fluid leaving that volume.
The velocity potential is a scalar field that can be used to represent the velocity field. The velocity field can be obtained by taking the gradient of the velocity potential.
The gradient of a scalar field is a vector field that points in the direction of the greatest increase in the scalar field. The magnitude of the gradient is equal to the rate of change of the scalar field in the direction of the gradient.
The divergence of a vector field is a scalar field that represents the amount of fluid leaving a given volume. The divergence is equal to the sum of the components of the vector field that point in the outward direction.
The velocity potential satisfies the continuity equation if the divergence of the velocity field is zero. In other words, the amount of fluid entering a given volume must equal the amount of fluid leaving that volume.
The velocity potential $x^2 + y^2$ satisfies the continuity equation because the divergence of the velocity field is zero. The velocity field is given by $\mathbf{v} = \nabla \phi$, where $\phi = x^2 + y^2$. The divergence of the velocity field is $\nabla \cdot \mathbf{v} = 2x + 2y = 0$. Therefore, the velocity potential $x^2 + y^2$ satisfies the continuity equation.
The other options do not satisfy the continuity equation. The velocity potential $x^2y$ has a divergence of $2x + y$. The velocity potential $x^2 – y^2$ has a divergence of $2x – 2y$. The velocity potential $\cos x$ has a divergence of $-\sin x$.