Question 1

Which of the following conditions must be satisfied for a flow field to be considered incompressible at a point?

Accepted Answer

The divergence of the velocity vector is zero: $$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$$

Question 2

Given the velocity components $$u = x^2$ - y^2$, v = -2xy$, and w = 0$, what is the vorticity $$\zeta_z$$ at a point (x$$, y, z)$$?

Accepted Answer

$$\zeta_z = 0$$

Question 3

Which statement best describes the physical meaning of the stream function $$\psi(x, y) in a $$two-dimensional incompressible flow?

Accepted Answer

It describes the flow rate per unit width between streamlines.

Question 4

For a two-dimensional incompressible flow, the velocity components are $$u = \frac{\partial \psi}{\partial y}$$ and $$v = -\frac{\partial \psi}{\partial x}. $$Which equation must $$\psi(x, y)$$ satisfy to ensure incompressibility?

Accepted Answer

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$

Question 5

A flow field is given by $$u = 2xy$$, $$v = x^2$, w = 0$. Is this flow field physically possible for an incompressible fluid?

Accepted Answer

No, because $$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} 
eq 0$$

Question 6

Which of the following is the correct expression for the shear strain rate $$\gamma_{xy} at a $$point in a flow field?

Accepted Answer

$$\gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}$$

Question 7

If the stream function for a flow is $$\psi(x, y) = 2xy - y^2$, what are the velocity components u$ and v$?

Accepted Answer

Question 8

Consider a velocity field $$\vec{V} = x\hat{i} + y\hat{j} + z\hat{k}. $$What is the volumetric dilatation rate at a point?

Accepted Answer

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 3$$

Question 9

Which of the following statements about streamlines in a flow field is correct?

Accepted Answer

The velocity vector at every point on a streamline is tangent to the streamline.

Question 10

Given the velocity field $$u = y$$, $$v = -x$$, $$w = 0$$, what is the vorticity vector $$\vec{\zeta} at a $$point?

Accepted Answer

$$\vec{\zeta} = 2\hat{k}$$

Question 11

For a horizontal pipe flow (z-axis), which equation best describes the velocity profile according to the Stokes approximation?

Accepted Answer

$$\frac{d}{dr}\left( r \frac{d v_z}{dr} \right) = 0$$

Question 12

Which of the following is true regarding the stream function in cylindrical coordinates for axisymmetric flow?

Accepted Answer

The stream function $$\psi$$ satisfies $$v_r = \frac{1}{r} \frac{\partial \psi}{\partial 	heta}$$ and $$v_	heta = -\frac{\partial \psi}{\partial r}$$ for axisymmetric flow.

Question 13

A velocity field is given by $$u = ax + by$$, $$v = cx + dy$$, $$w = 0. $$For which condition on the constants $$a, b, c, d is $$the flow incompressible?

Accepted Answer

$$a + d = 0$$

Question 14

Which of the following is a necessary condition for the existence of a stream function in a flow field?

Accepted Answer

The flow must be two-dimensional and incompressible.

Question 15

If the stream function $$\psi(x, y) is $$known, how can the volumetric flow rate per unit width between two streamlines $$\$$psi_1$$ and $$\$$psi_2 be $$calculated?

Accepted Answer

$$q = \$$psi_2$$ - \$$psi_1$$

Question 16

Which equation represents the continuity equation for a two-dimensional incompressible flow in Cartesian coordinates?

Accepted Answer

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$

Question 17

For a flow field with velocity components $$u = 0$$, $$v = 0$$, $$w = 0$$, what is the vorticity at any point?

Accepted Answer

Vorticity is zero everywhere.

Question 18

Which of the following best describes the relationship between streamlines and the stream function in a two-dimensional flow?

Accepted Answer

Streamlines are lines of constant stream function.

Fluid Mechanics II: Kinematic Analysis and Deformation in Fluid Flow

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