ME-404 Fluid Mechanics - Jun 2023 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

1. 7 Marks
   a1. a)

   1. i)Distinguish between ideal and real fluids.
   2. ii)How do you determine the hydrostatic force acting on an inclined surface?
   3. iii)What is metacentric height and what is its significance with respect to stability and periodic time of oscillation of a floating body?
   How to answer

   • Distinguish ideal (inviscid, incompressible, no shear) vs real (viscous, compressible) fluids
   • Hydrostatic force magnitude F = rho*g*h_c*A and line of action via centre of pressure
   • Metacentric height GM definition, stability criteria and formula for period of oscillation
2. 7 Marks
   bWhat is the physical significance of mathematical terms, $\Delta \cdot \mathbf{V}$ and $\Delta \times \mathbf{V}$ in fluid mechanics? Where, V is the 3D velocity vector of the fluid flow.
   How to answer

   • Define divergence of velocity vector
   • Physical meaning: volumetric dilatation rate, continuity
   • Define curl of velocity vector
   • Physical meaning: vorticity, rotation, irrotationality condition
3. 14 Marks
   For the velocity field, $\mathbf{V} = (y^3+6x-3x^2y)i + (3xy^2-6y-x^3)j$, check whether the flow is

   1. i)continuous
   2. ii)rotational or irrotational
   3. iii)if irrotational, find potential function.
   How to answer

   • Compute divergence to check continuity $\partial u/\partial x + \partial v/\partial y = 0$
   • Compute vorticity $\omega_z$ to check rotationality
   • If irrotational find velocity potential by integrating u, v
4. 7 Marks
   aExplain the terms:

   1. i)Path line
   2. ii)Streak line
   3. iii)Stream line
   4. iv)Stream tube
   How to answer

   • Define path line: trajectory of single fluid particle
   • Define streak line: locus of particles through a point
   • Define stream line and stream tube with properties
   • Distinguish steady vs unsteady case
5. 7 Marks
   bFor a discharge of $2.1\, \text{m}^3/\text{s}$ of air through a pipe of area $0.093\, \text{m}^2$, determine the area of throat of a venturimeter as shown in Figure 1. Take specific weight of air as $11.96\, \text{N}/\text{m}^3$ and neglect losses.

   

   How to answer

   • Apply continuity Q = A1*V1 = A2*V2
   • Apply Bernoulli with manometer head difference
   • Relate mercury deflection 25 mm to air-water head
   • Solve for throat area A2
6. 7 Marks
   aDistinguish between:

   1. i)Steady flow and un-steady flow
   2. ii)Uniform and non-uniform flow
   3. iii)Compressible and incompressible flow
   4. iv)Rotational and irrotional flow
   5. v)Laminar and turbulent flow
   How to answer

   • Define steady vs unsteady with time derivative
   • Define uniform vs non-uniform with space derivative
   • Compressible vs incompressible, rotational vs irrotational, laminar vs turbulent with examples
7. 7 Marks
   bA stream function is given by $(x^2-y^2)$. Find the potential function of the flow.
   How to answer

   • Check continuity via stream function
   • Derive velocity components u, v from psi
   • Integrate to get velocity potential phi
   • Verify Laplace equation
8. 7 Marks
   aA vertical isosceles triangular gate is immersed in water with its vertex up. The base width of gate is $2\, \text{m}$ and height $1.5\, \text{m}$. The vertex of gate is $1\text{m}$ below the free water surface. Determine the total pressure force and its location.
   How to answer

   • Compute centroid depth h_c of triangular gate
   • Total force F = rho*g*h_c*A
   • Centre of pressure h_p = h_c + I_xx_c/(h_c*A)
   • Locate distance from free surface / vertex
9. 7 Marks
   bDerive Bernoulli's equation for flow along a stream line.
   How to answer

   • Consider Euler equation along streamline
   • Assume steady, inviscid, incompressible flow
   • Integrate to get p/rho*g + V^2/2g + z = constant
   • State assumptions and limitations
10. 7 Marks
    aA conical thrust bearing idealized as a cone of vertex angle $60^\circ$, maximum cone diameter $200\, \text{mm}$, rests and revolves about vertex over a uniform fluid layer of thickness $1\, \text{mm}$ at $600\, \text{rpm}$. If viscosity of fluid is $1\, \text{Poise}$, calculate the power lost in overcoming the viscous resistance.
    How to answer

    • Shear stress in conical gap tau = mu*V/h
    • Integrate torque dT over slant element
    • Power P = 2*pi*N*T/60
    • Substitute vertex angle 60 deg, D=200 mm, N=600 rpm
11. 7 Marks
    bWater at $25^\circ\text{C}$ is flowing through a $1.0\, \text{km}$ long G.I. pipe of $200\, \text{mm}$ diameter at the rate of $0.07\, \text{m}^3/\text{s}$. If value of Darcy friction factor for this pipe is $0.02$ and density of water is $1000\, \text{kg}/\text{m}^3$, find the pumping power (in kW) required to maintain the flow?
    How to answer

    • Compute velocity V = Q/A and Reynolds regime
    • Head loss h_f = f*L*V^2/(2*g*D) using Darcy factor
    • Pumping power P = rho*g*Q*h_f
    • Convert to kW
12. 7 Marks
    aTwo pipes each $300\, \text{m}$ long are available for connecting to a reservoir from which a flow of $0.085\, \text{m}^3/\text{s}$ is required. If diameters of the two pipes are $300\, \text{mm}$ and $150\, \text{mm}$ respectively, determine the ratio of head lost when the pipes are connected in series to the head lost when connected in parallel. Neglect minor losses.
    How to answer

    • Head loss in series = sum of h_f1 + h_f2 for same Q
    • Head loss in parallel with flow division Q1+Q2=Q
    • Use Darcy h_f proportional to Q^2/D^5
    • Compute ratio h_series / h_parallel
13. 7 Marks
    bWrite the short note on Fanno and Reyleigh lines. Draw the Fanno and reyleigh lines on the h-s plot.
    How to answer

    • Define Fanno flow (adiabatic with friction) and Rayleigh flow (heat transfer)
    • Derive governing equations and choking condition
    • Sketch h-s diagram with Fanno and Rayleigh lines intersecting at sonic point
    • Explain subsonic and supersonic branches
14. 14 Marks
    Write the short note on the following:

    1. a)Stagnation properties
    2. b)Normal and oblique shocks
    3. c)Compressible flow
    4. d)Mach number
    How to answer

    • Stagnation properties: p0, T0, h0 definitions
    • Normal vs oblique shock relations and deflection
    • Compressible flow regimes by Mach number
    • Mach number definition and significance