Question

1) What is the equation for the volumetric strain rate? (Hint: Uses dot product). incompressible flow be determined from the volumetric strain rate? Given the velocity field V= (u, v) (0.75+1.2x) +(2.25-1.2y ), determine if the flow is incompressible or compressible. (20 pts) How can 2) What is the equation for vorticity? (Hint: Uses cross product). How can irrotational flow be determined from vorticity? Given the velocity field V = (u, v) (0.75+ 1.2x)t+ (2.25 1.2y), determine if the flow is rotational or irrotational. (20 pts)

Answer 1

1.) Equation for the volumetric strain rate.

а а д Д д +j дх де ду (л+ + ).

а а д - (иi + vj + wk) де Д д +j дх ду

Ου δυ Οu 4 Δ θε 0ε θυ Β13

☆ The above equation is also termed as Divergence of Velocity vector.

☆ If the divergence of Velocity vector is zero, it is incompressible and if the divergence of Velocity vector is not zero, it is compressible in nature.

☆ Now analyse with given question,

(u, v) =(0.75+ 1.2x)i (2.25- 1.2y)j

Ου δυ Οu 4 Δ θε 0ε θυ Β13

0.751.2 a(2.25 1.2y) a(0)

y. V= 1.2 1.2 0

☆ The given velocity equation is incompressible in nature, The density of the fluid doesn't affect the nature of the flow.

2.) Equation for vorticity

i

X A ay x (uivjwk) +j k

X

du k V X 7 y

☆ Given equation

(u, v) =(0.75+ 1.2x)i (2.25- 1.2y)j

du k V X 7 y

0(2.25 1.2y) 0(0.751.2) (0)e a(0.751.2x) V x - j a(2.25 1.2y)

$\Rightarrow \triangledown\times \overrightarrow{V} = i \left ( 0 - 0 \right) - j \left ( 0 - 0 \right ) + k \left ( 0 - 0 \right )$

$\Rightarrow \triangledown\times \overrightarrow{V} = 0$

☆ Given velocity vector is irrotational in nature.

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