GATE Civil Engineering (CE) 2019 Shift 2 Solved Paper

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Question : 25 of 65

Marks: +1, -0

The velocity field in a flow system is given by

{v}=2 {i}+( {x}+ {y}) {j}+( {xyz}) {k}

. The acceleration of the fluid at

(1,1,2)

is

${j}+ {k}$
$2 {i}+10 {k}$
$4 {j}+10 {k}$
$4 {i}+12 {k}$

Solution:

{a}_{ {x}}= {u} \;{∂ {u}}/{∂ {x}}+ {v} \;{∂ {u}}/{∂ {y}}+ {w} \;{∂ {u}}/{∂ {z}}+\;{∂ {u}}/{∂ {t}}

⇒ \;\; a_{x}=\;{2 × ∂(2)}/{∂ x}+(x+y) \;{∂(2)}/{∂ y}+(x y z)

\;{∂(2)}/{∂ z}+\;{∂(2)}/{∂ t}

{a}_{ {x}}=0

and

a_{y}=u \;{∂ v}/{∂ x}+v \;{∂ v}/{∂ y}+w \;{∂ v}/{∂ z}+\;{∂ v}/{∂ t}

=\;{2 ∂(x+y)}/{∂ x}+(x+y) \;{∂(x+y)}/{∂ y}+(x y z)

\;{∂(x+y)}/{∂ z}+\;{∂(x+y)}/{∂ t}

=2+( {x}+ {y})+( {xyz})(0)+0

a_{y}=2+x+y

⇒ \;\; a_{y}=2+1+

⇒ \;\; {a}↖{→}_{y}=4j↖{⋀}

and

{a}_{2}= {u} \;{∂ {W}}/{∂ {x}}+ {v} \;{∂ {W}}/{∂ {y}}+ {w} \;{∂ {W}}/{∂ {z}}+\;{∂ {W}}/{∂ {t}}

=2( {yz})+( {x}+ {y})( {xz})+( {xyz})( {xy})+0

{a}_{ {z}}=2(1 × 2)+(1+1)(1 × 2)+(1 × 1× 2)(1 × 1)

=10 {k}

Therefore

{ {a}↖{→}}=4 { {j↖{⋀}}}+10{ {k↖{⋀}}}

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