The velocity field of an incompressible flow in a Cartesian system is represented by \overrightarrow{ {V}=2( {x}^{2}- {y}^{2}) {i}+ {v} {j}+3 {k} Which one of the following expressions for v is valid?

Correct Answer is : −4xy−4xz

GATE Mechanical Engineering (ME) 2020 Shift 1 Solved Paper

Question : 11 of 65

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The velocity field of an incompressible flow in a Cartesian system is represented by

{ {V}↖{→}=2( {x}^{2}- {y}^{2}) {i}+ {v} {j}+3 {k}

Which one of the following expressions for

v

is valid?

Solution:

Velocity for incompressible fluid flow,

{ {V}↖{→}=2( {x}^{2}- {y}^{2}) { {i}↖{∧}+ {V} { {j}↖{∧}+3 {k}

{V}↖{→}=2(x^{2}-y^{2}) {i}↖{∧}+V {j}↖{∧}+3 k

From above velocity relation

u=2(x^{2}-y^{2})

V=V

ω=3

If the flow is incompressible continuity equation has to be satisfied,

\;{∂ {u}}/{∂ {x}}+\;{∂ {V}}/{∂ {y}}+\;{∂ ω}/{∂ {z}}=0

\;{∂}/{∂ {x}}(2( {x}^{2}- {y}^{2}))+\;{∂ {V}}/{∂ {y}}+\;{∂}/{∂ {z}}(3)=0

⇒ 4 {x}+\;{∂ {V}}/{∂ {y}}+0=0

⇒ \;{∂ {V}}/{∂ {Y}}=-4 {x}

⇒ V=-4 x y+C

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