A two-dimensional incompressible frictionless flow field is given by \overrightarrow{u}=x \hat{i}-\hat{y} . If \rho is the density of the fluid, the expression for pressure gradient vector at any point in the flow field is given as

Correct Answer is : −ρ(xi∧+yj∧)

-\rho(x^{2} \hat{i}+y^{2} \hat{j})

GATE Mechanical Engineering (ME) 2019 Shift 2 Solved Paper

Question : 35 of 65

Marks: +1, -0

A two-dimensional incompressible frictionless flow field is given by

{u}↖{→}=x {i}↖{∧}-{y}↖{∧}

. If

ρ

is the density of the fluid, the expression for pressure gradient vector at any point in the flow field is given as

Solution:

Euler’s Equation of motion in 2D.

\;{∂ u}/{∂ t}+u \;{∂ u}/{∂ x}+v \;{∂ u}/{∂ y}+w \;{∂ u}/{∂ z}=f_{x}-\;{1}/{ρ} \;{∂ p}/{∂ x}

\;{∂ v}/{∂ t}+u \;{∂ v}/{∂ x}+v \;{∂ v}/{∂ y}+w \;{∂ v}/{∂ z}=f_{y}-\;{1}/{ρ} \;{∂ p}/{∂ y}

As there is no extra force acting on the fluid.

f_{x}

and

f_{y}=0

Now form the given expiration,

u=x \& v=y

∴ \;{∂ x}/{∂ t}+x \;{∂ x}/{∂ x}+y \;{∂ x}/{∂ y}+w \;{∂ x}/{∂ z}=0-\;{1}/{ρ} \;{∂ p}/{∂ x}

\;{∂ y}/{∂ t}+x \;{∂ y}/{∂ x}+y \;{∂ y}/{∂ y}+w \;{∂ v}/{∂ z}=0-\;{1}/{ρ} \;{∂ p}/{∂ y}

∴ 0+x × 1+0+0=0-\;{1}/{ρ} \;{∂ p}/{∂ x}

0+0+y × 1+0=0-\;{1}/{ρ} \;{∂ p}/{∂ y}

∴

Press Grad Vec

=\;{∂ p}/{∂ x} {ı}↖{∧}+\;{∂ p}/{∂ y} {i}↖{∧}=-ρ(x {ı}↖{∧}+yj↖{∧})

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