ME-404 Fluid Mechanics - Nov 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
   aWhat do you mean by center of Pressure? Explain.
   How to answer

   • Definition of centre of pressure as point of application of resultant hydrostatic force.
   • Distinction from centroid / centre of gravity.
   • Formula $h_p = \bar{h} + I_{xx}\sin^2\theta/(A\bar{h})$ with terms explained.
   • Explanation why it lies below centroid for vertical/inclined surfaces.
2. 7 Marks
   bDefine Dynamic viscosity. Determine the viscosity of a liquid having kinematic viscosity 6 stokes and specific gravity 1.9.
   How to answer

   • Definition of dynamic viscosity from Newton's law $\tau = \mu du/dy$.
   • Relation $\nu = \mu/\rho$ and unit conversions: 1 stoke $=10^{-4} m^2/s$.
   • Density from specific gravity: $\rho = 1.9 \times 1000 kg/m^3$.
   • Calculation $\mu = \rho \nu$ with units $N\cdot s/m^2$ / poise.
3. 7 Marks
   aFor steady incompressible flow derive the continuity equation using 3-D rectangular co-ordinate system.
   How to answer

   • Consider 3-D fluid element $dx\,dy\,dz$ with steady flow.
   • Mass inflow vs outflow in x, y, z directions using $\rho u, \rho v, \rho w$.
   • Conservation of mass: net outflow $=0$ for steady incompressible flow.
   • Simplification to $\partial u/\partial x + \partial v/\partial y + \partial w/\partial z = 0$.
4. 7 Marks
   bA 25 cm diameter pipe carries oil of sp. gr. 0.9 at a velocity of 3 m/s. At another section the diameter is 20 cm. Find the velocity at this section and also mass rate of flow of oil.
   How to answer

   • Continuity $A_1V_1 = A_2V_2$ for incompressible oil.
   • Areas $A=\pi D^2/4$ with $D_1=25 cm$, $D_2=20 cm$.
   • Calculate $V_2 = V_1(D_1/D_2)^2$.
   • Mass flow rate $\dot{m}=\rho A_1V_1$ with $\rho =0.9\times1000 kg/m^3$.
5. 7 Marks
   aA kite weighing 0.8 kgf (7.848 N) has an effective area of $0.8\, m^2$. It is maintained in air at an angle of $10^{\circ}$ to the horizontal. The string attached to the kite makes an angle of $45^{\circ}$ to the horizontal and at this position the value of co-efficient of drag and lift are 0.6 and 0.8 respectively. Find the speed of the wind and the tension in the string. Take the density of air as $1.25\, kg/m^3$.
   How to answer

   • Resolve weight, lift, drag and string tension in equilibrium.
   • Lift $F_L = C_L A \rho V^2/2$, Drag $F_D = C_D A \rho V^2/2$.
   • Force equilibrium equations in x and y with $10^{\circ}$ and $45^{\circ}$ angles.
   • Solve for wind speed $V$ and tension $T$.
6. 7 Marks
   bState and prove Bernoulli's theorem. Mention the assumptions made. How is it modified while applying in real problem?
   How to answer

   • Statement of Bernoulli's $p/\rho g + V^2/2g + z = $ constant.
   • Assumptions: steady, incompressible, inviscid, along streamline, etc.
   • Proof via Euler's equation integration along streamline.
   • Modification with head loss $h_f$ and kinetic energy correction for real fluids.
7. 7 Marks
   aA laminar flow is taking place in a pipe of diameter 400 mm. The maximum velocity is 3 m/s. Find the mean velocity and the radius at which this occurs. Also calculate the velocity at 6 cm from the wall of pipe. Find:

   1. i)Mean velocity, $\bar{u}$
   2. ii)A Radius at which $\bar{u}$ occurs.
   3. iii)Velocity at 6 cm from the wall.
   How to answer

   • Laminar pipe profile $u = U_{max}(1-r^2/R^2)$ and mean $\bar{u}=U_{max}/2$.
   • Radius where $u=\bar{u}$: $r = R/\sqrt{2}$.
   • Velocity at distance $y$ from wall: use $r = R-y$.
   • Substitute $R=200 mm$, $U_{max}=3 m/s$, $y=6 cm$.
8. 7 Marks
   bDefine Mach number. What is the significance of Mach number in compressible fluid flows?
   How to answer

   • Definition $M = V/c$ where $c=\sqrt{kRT}$.
   • Classification: subsonic, sonic, supersonic, hypersonic.
   • Significance for compressibility effects and pressure wave propagation.
   • Use in compressible flow similarity and choking/nozzle flows.
9. 7 Marks
   aExplain separation of boundary layer. What are the different methods of preventing the separation of boundary layers?
   How to answer

   • Concept of adverse pressure gradient and zero wall shear.
   • Causes and effects of separation: wake, drag increase.
   • Methods: suction/blowing, streamlining, guide vanes, boundary layer control.
   • Sketch of separated profile if needed.
10. 7 Marks
    bFind the expression for mass rate of flow of compressible fluid through a nozzle fitted to a large tank.
    How to answer

    • Large tank stagnation conditions $p_0, T_0$ and isentropic relations.
    • Velocity via energy equation and pressure ratio.
    • Mass flow $\dot{m}=\rho A V$ as function of pressure ratio.
    • Maximum/choked flow condition at sonic throat.
11. 14 Marks
    Find the displacement thickness and Energy thickness for the velocity distribution in the boundary layer is given by: $$\frac{u}{U} = 2\left\{\frac{y}{\delta}\right\} - \left\{\frac{y}{\delta}\right\}^2$$
    How to answer

    • Definitions $\delta^* = \int_0^\delta (1-u/U)dy$, $\delta_e = \int_0^\delta (u/U)(1-(u/U)^2)dy$.
    • Substitute $u/U = 2(y/\delta)-(y/\delta)^2$ and integrate.
    • Evaluate $\delta^* = \delta/3$.
    • Evaluate energy thickness $\delta_e = 2\delta/15$ (check integration).
12. 7 Marks
    aExplain the following terms:

    1. i)Stream-line body and bluff body
    2. ii)Friction drag and pressure drag
    How to answer

    • Streamlined body: flow attached, wake small, friction drag dominates.
    • Bluff body: separation, large wake, pressure drag dominates.
    • Friction drag due to wall shear vs pressure drag due to pressure difference.
    • Examples: aerofoil vs flat plate normal to flow.
13. 7 Marks
    bDetermine the total pressure and centre of pressure on an isosceles triangular plate of base 4 m and altitude 4 m when it is immersed vertically in an oil of sp. gr 0.9. The base of the plate coincides with the free surface of oil.
    How to answer

    • Oil density $\rho =0.9\times1000$, plate area $A=0.5\times4\times4$.
    • Depth of centroid $\bar{h}= (2/3)\times4$ from free surface with base at top.
    • Total pressure $F = \rho g A \bar{h}$.
    • Centre of pressure $h_p = \bar{h}+I_{xx}\sin^2\theta/(A\bar{h})$ with $I_{xx}$ for triangle.
14. 7 Marks
    aThe stream function for a two-dimensional flow is given by, $\psi = 2xy$, calculate the velocity at the point P (2, 3). Find also the velocity potential function at point P.
    How to answer

    • Velocity components $u = -\partial\psi/\partial y$, $v = \partial\psi/\partial x$.
    • For $\psi=2xy$: $u=-2x$, $v=2y$; at $P(2,3)$ magnitude $V=\sqrt{u^2+v^2}$.
    • Check irrotationality and integrate $u=-\partial\phi/\partial x$, $v=-\partial\phi/\partial y$.
    • Potential $\phi = x^2 - y^2 + C$ evaluated at P.
15. 7 Marks
    bA sudden enlargement of a water main from 240 mm to 480 mm diameter, the hydraulic gradient rises by 10 mm. Estimate the rate of flow.
    How to answer

    • Continuity $A_1V_1=A_2V_2=Q$ with $D_1=240 mm$, $D_2=480 mm$.
    • Sudden enlargement loss $h_e=(V_1-V_2)^2/2g$.
    • Hydraulic gradient rise $10 mm$ equals head loss difference.
    • Solve for $Q$ from $h_e = 0.01 m$.