Project ForgeFluid Mechanics
Bernoulli's Equation & Flow
GATE 2022
Hard
The steady velocity field in an inviscid fluid of density 1.5 is given to be V⃗ = (y²−x²)î + (2xy)ĵ. Neglecting body forces, the pressure gradient at (x=1, y=1) is __________.
- A10ĵ
- B20î
- −6î−6ĵ
- D−4î−4ĵ
Explanation
For inviscid flow with no body forces, Euler's equation gives −∇p=ρa⃗, where a⃗ is the convective acceleration. With u=y²−x², v=2xy: ax=u∂u/∂x+v∂u/∂y=(y²−x²)(−2x)+(2xy)(2y)=2x(x²+y²); ay=u∂v/∂x+v∂v/∂y=(y²−x²)(2y)+(2xy)(2x)=2y(x²+y²). At (1,1): ax=2(1)(2)=4, ay=2(1)(2)=4. ∇p=−ρa⃗=−1.5×(4,4)=(−6,−6), i.e. −6î−6ĵ.
Source: GATE ME 2022 Session 2 (ME-2) (2022)
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