Question

For a two-dimensional, incompressible flow having velocity components \(u\) and \(v\) in the x and y directions, respectively, the expression \[ \frac{\partial(u^2)}{\partial x} + \frac{\partial(uv)}{\partial y} \] can be simplified to:

• A. \( u \frac{\partial u}{\partial x} + u \frac{\partial v}{\partial y} \)
• B. \( 2u \frac{\partial u}{\partial x} + u \frac{\partial v}{\partial y} \)
• C. \( 2u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \)
• D. \( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \)

Hint:

In fluid mechanics, the convective acceleration terms in the Navier-Stokes equation are derived using the product rule and the chain rule for differentiation. This is important for analyzing fluid flow and understanding the forces at play.

Answer 1

The Correct Option is D

In fluid mechanics, the expression \( \frac{\partial(u^2)}{\partial x} + \frac{\partial(uv)}{\partial y} \) represents the terms in the Navier-Stokes equations related to the convective acceleration. Let's simplify the terms:
- The term \( \frac{\partial(u^2)}{\partial x} \) expands to \( 2u \frac{\partial u}{\partial x} \), using the chain rule of differentiation.
- The term \( \frac{\partial(uv)}{\partial y} \) expands to \( u \frac{\partial v}{\partial y} + v \frac{\partial u}{\partial y} \).
Thus, the full expression becomes: \[ \frac{\partial(u^2)}{\partial x} + \frac{\partial(uv)}{\partial y} = 2u \frac{\partial u}{\partial x} + u \frac{\partial v}{\partial y} \] So the simplified form is Option (D), which gives the correct expression as \( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \).
Thus, the correct answer is (D).