Project ForgeFluid Mechanics
Bernoulli's Equation & Flow
GATE 2021
Hard
For a two-dimensional, incompressible flow having velocity components u and v in the x and y directions, respectively, the expression ∂(u²)/∂x + ∂(uv)/∂y can be simplified to
- Au∂u/∂x + u∂v/∂y
- B2u∂u/∂x + u∂v/∂y
- C2u∂u/∂x + v∂u/∂y
- u∂u/∂x + v∂u/∂y
Explanation
Expanding: ∂(u²)/∂x + ∂(uv)/∂y = 2u∂u/∂x + u∂v/∂y + v∂u/∂y. By 2D incompressible continuity, ∂u/∂x+∂v/∂y=0 ⟹ ∂v/∂y=−∂u/∂x, so u∂v/∂y=−u∂u/∂x. Substituting: 2u∂u/∂x − u∂u/∂x + v∂u/∂y = u∂u/∂x + v∂u/∂y.
Source: GATE ME 2021 Session 2 (ME-2) (2021)
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