Fluid Kinematics Questions and Answers

Answer: C

The function is \(f(x,y) = -2xy+g(x)\).

For incompressible two-dimensional flow, continuity requires \(\partial u/\partial x + \partial v/\partial y = 0\):

\[ \begin{aligned} \frac{\partial u}{\partial x} &= 4x \\ \frac{\partial v}{\partial y} &= -2x+\frac{\partial f}{\partial y} \\ 4x-2x+\frac{\partial f}{\partial y} &= 0 \end{aligned} \]

So \(\partial f/\partial y = -2x\), and integrating in y gives \(f = -2xy + g(x)\), the function of integration depending on x only.

Check the others: \(-xy^{2}\) gives \(\partial f/\partial y = -2xy\), \(-2x\) gives 0, and \(-2x^{2}y\) gives \(-2x^{2}\); none cancels the remaining 2x.

Answer: A

The velocity components are \(u = U+\frac{mx}{x^{2}+y^{2}}\), \(v = \frac{my}{x^{2}+y^{2}}\).

Stream functions add, so \(\psi = m\tan^{-1}(y/x) + Uy\). With \(u = \partial\psi/\partial y\) and \(v = -\partial\psi/\partial x\):

\[ \begin{aligned} u &= m\frac{x}{x^{2}+y^{2}} + U \\ v &= -m\frac{-y}{x^{2}+y^{2}} = \frac{my}{x^{2}+y^{2}} \end{aligned} \]

A check: the source part points radially outward, \((x, y)/(x^{2}+y^{2})\) times m, and the uniform stream adds U in x only.

Putting U in the y-component treats the stream as vertical, and the minus sign in option C comes from forgetting the minus in \(v = -\partial\psi/\partial x\).

Answer: D

They are identical when the flow is steady.

A streamline shows the direction of velocity at an instant, a pathline traces one particle over time, and a streakline joins all particles that have passed through one point. In steady flow the velocity field does not change with time, so every particle from a point follows the same path, and all three lines coincide.

Uniform flow means no change with position, which is a different idea. Laminar flow can still be unsteady, and compressibility has no bearing on whether the three lines coincide.

Answer: B

Flow fields are visualised with streamlines.

A streamline is a curve everywhere tangent to the velocity vector, so a family of streamlines shows the direction of flow at every point, and their spacing shows where the flow speeds up or slows down. Flow patterns are drawn and analysed this way.

The venturi meter and orifice meter are devices that measure flow rate in a pipe, and the manometer measures pressure. None of them pictures the whole flow field.

Answer: D

The answer is external flow and internal flow.

Flow of a fluid that is unbounded and passes over a surface, such as air over a wing or a plate, is called external flow. Flow that is completely enclosed by solid walls, such as flow in a pipe or duct, is called internal flow. That pair fills the two blanks in order.

The other pairs classify flow on other grounds: uniform and non-uniform by change with position, viscous and inviscid by whether friction is counted, and laminar and turbulent by the regime of motion.

Answer: A

The continuity equation is based on the principle of conservation of mass.

For steady flow, the mass entering a control volume per second must equal the mass leaving it. For an incompressible fluid in a pipe this gives \(A_1V_1 = A_2V_2\): where the section narrows, the velocity rises.

Conservation of momentum gives the momentum equation, and energy conservation gives Bernoulli's equation. Weight is not a separately conserved quantity, and kinetic energy alone is not conserved in real flow.

Answer: C

The velocity of flow is 8.83 m/s.

Velocity is discharge divided by the cross-sectional area:

\[ \begin{aligned} A &= \frac{\pi}{4} (0.6)^2 = 0.2827\ \text{m}^2 \\ V &= \frac{2.5}{0.2827} = 8.84\ \text{m/s} \end{aligned} \]

The paper's 8.83 m/s is the same value rounded. Taking 0.6 m as the radius would give 2.21 m/s, and 2.5 m/s simply repeats the discharge figure. 5.67 and 12.32 m/s match no correct step.

Answer: B

Such a curve is a stream line.

A streamline is drawn so that at every point the velocity vector at that instant is tangent to it, so no fluid crosses a streamline.

A path line is the track of one particle over time, and a streak line joins all particles that have passed through one point. In steady flow all three coincide. A stream tube is a bundle of streamlines through a closed curve, not a single curve.

Answer: A

A streamline is a line tangent to which is the direction of velocity at every point.

At a given instant, a streamline is drawn so that the velocity vector at each point on it lies along the line. Because the velocity is always along the streamline, there is no flow across it, which is why two streamlines can never cross.

A line drawn normal to the velocity is the opposite of a streamline, it is how equipotential lines lie in potential flow. Dividing the passage into equal parts describes stream tubes of equal discharge, not the definition. The path of a particle is a pathline, which coincides with a streamline only in steady flow.

Answer: D

Mass conservation does not imply \(\nabla \times \bar{q} = 0\).

The equation of continuity is \(\partial\rho/\partial t + \nabla \cdot (\rho\bar{q}) = 0\). For an incompressible fluid with constant density this reduces to \(\nabla \cdot \bar{q} = 0\), and \(\partial\rho/\partial t = 0\) gives the form in option B as well. So A, B and C all follow from conservation of mass.

\(\nabla \times \bar{q} = 0\) says the flow is irrotational. That is a separate kinematic property, not a consequence of mass conservation: a rigid rotation \(\bar{q} = (-y, x, 0)\) is incompressible but has vorticity 2.