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# MCQs on Crocco’s Theorem

What does the curl of velocity field give us?

a) Moment of Inertial

b) Angular velocity

c) Angular acceleration

d) Moment

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b

Explanation: If the translation velocity of the fluid particle is given by V, then the angular velocity ω is given as the curl of velocity field. It tells us the measure of rotation for the fluid particle at a particular point.

ω = 12 ∇ × V

How is the vorticity related to the angular velocity?

a) Twice of angular velocity

b) Thrice of angular velocity

c) No difference

d) Half of angular velocity

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a

Explanation: Vorticity is a measure of tendency of the fluid particle to rotate. It is given by:

∇ × V

The value of vorticity is twice as much as the angular velocity whose formula is:

ω = 12 ∇ × V

What does Crocco’s theorem establish?

a) Relationship between inertial and fluid properties

b) Relationship between fluid and thermodynamic properties

c) Give the body forces

d) Determine the relation between vorticity and angular velocity

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b

Explanation: Crocco’s theorem was established by Luigi Crocco who found a relationship between vorticity, velocity and stagnation pressure of the fluid which is a kinematic property and the thermodynamic properties.

Which of these is Crocco’s theorem?

a) V × (∇ × V) = ∇h0 – T∇s

b) V(∇ × V) = ∇h0

c) (∇ × V) = T∇s

d) (∇ × V) = ∇h0 + T∇s

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a

Explanation: Crocco’s theorem is formulated for inviscid flow having no external body forces. According to the theorem, the vorticity is equal to the difference between total enthalpy gradient and the gradient of entropy. It is given by:

V × (∇ × V) = ∇h0 – T∇s

According to Crocco’s theorem, the vorticity behind the curved shockwave is zero.

a) True

b) False

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b

Explanation: Across a stationary shock wave, the change in enthalpy is zero therefore ∇s = 0. But for a curved shockwave, the enthalpy does not remain constant making ∇s ≠ 0. This results in V × (∇ × V) ≠ 0 according to the Crocco’s theorem. Therefore the vorticity behind a curved shock wave is not zero and the flow behind the curved shock wave is rotational.

Crocco’s theorem gives a relation between which of the following?

a) Entropy and flow velocity

b) Gradient of entropy and vorticity

c) Gradient of angular acceleration and enthalpy

d) Flow velocity and enthalpy

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b

Explanation: According to the Crocco’s theorem, there is a relationship established between vorticity, gradient of enthalpy and gradient of entropy.

V × (∇ × V) = ∇h0 – T∇s

Where, ∇ × V is the vorticity

∇h0 is gradient of total enthalpy

T∇s is gradient of entropy

Flow passing through curved shock creates vorticity in the downstream.

a) True

b) False

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a

Explanation: A fluid while passing through a curved shock wave, experiences a strong entropy change due to the presence of shock wave. This leads to a rotational fluid causing vorticity behind the shock wave and the gradient in entropy change is non zero.