ME-404 Fluid Mechanics - Dec 2024 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
   aDetermine the Bulk modulus of elasticity of a liquid, if the pressure of the liquid is increased from $70 N/cm^2$ to $130 N/cm^2$. The volume of the liquid decreases by $0.15\%$.
   How to answer

   • Define bulk modulus $K = -V dp/dV = dp/(-dV/V)$
   • Convert pressure rise $dp = 130-70 = 60\, N/cm^2$
   • Use volumetric strain $dV/V = -0.15\% = -0.0015$
   • Compute $K$ and express in $N/cm^2$ / $MN/m^2$
2. 7 Marks
   bDefine the terms Dynamic viscosity and Kinematic viscosity. State their units of measurement.
   How to answer

   • Define dynamic viscosity $\mu$ from Newton's law of viscosity
   • Define kinematic viscosity $\nu = \mu/\rho$
   • State SI units: $Pa\cdot s$ and $m^2/s$
   • Mention measurement units poise, stokes
3. 7 Marks
   aDefine stream function and velocity potential function. Prove that stream lines and equipotential lines meet each other orthogonally.
   How to answer

   • Define stream function $\psi$ and velocity potential $\phi$ with $u,v$ relations
   • Write $d\psi=0$ for streamline and $d\phi=0$ for equipotential
   • Derive slopes $dy/dx$ for both lines
   • Show product of slopes $=-1$, hence orthogonal
4. 7 Marks
   bA stream functions in two dimensional flow $\psi = 2xy$. Determine the corresponding velocity potential $\Phi$.
   How to answer

   • Obtain $u,v$ from $\psi = 2xy$ using $u=\partial\psi/\partial y$, $v=-\partial\psi/\partial x$
   • Use Cauchy-Riemann $u=\partial\Phi/\partial x$, $v=\partial\Phi/\partial y$
   • Integrate to get $\Phi(x,y)$
   • Check irrotational / Laplace equation
5. 7 Marks
   aA man weighing 90 kgf descends to the ground from an aeroplane with the help of a parachute against the resistance of air the velocity with which the parachute, which the hemispherical in shape comes down is 20 m/s. Find the diameter of parachute. Assume CD= 0.5 and density of air $1.25 kg/m^3$.
   How to answer

   • Equate weight $W=90\,kgf$ to drag $F_D = C_D A \rho V^2/2$
   • Take projected area of hemispherical parachute $A=\pi D^2/4$
   • Substitute $V=20\,m/s$, $C_D=0.5$, $\rho=1.25\,kg/m^3$
   • Solve for $D$
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

   • State Bernoulli's theorem $p/\rho g + V^2/2g + z =$ constant
   • List assumptions: steady, incompressible, inviscid, along streamline
   • Derive from Euler's equation by integration
   • Add loss term $h_L$ and coefficients for real fluid application
7. 7 Marks
   aWhat are the minor losses and major losses in a pipe flow? Derive the chazy's formula for the loss of head due to friction.
   How to answer

   • Define major loss due to friction and list minor losses
   • Consider force balance on pipe element
   • Relate wall shear to pressure drop
   • Obtain Chezy's $V = C\sqrt{mi}$ and head loss form
8. 7 Marks
   bA laminar flow is taking place in a pipe of diameter 200 mm. The maximum velocity is 1.5m/s. Find the mean velocity and the radius at which this occurs. Also calculate the velocity at 4 cm from the wall of pipe. Find :

   1. i)Mean velocity, ū
   2. ii)A Radius at which ū occurs
   3. iii)Velocity at 4 cm from the wall
   How to answer

   • Use parabolic profile $u = U_{max}[1-(r/R)^2]$ so $\bar u = U_{max}/2$
   • Find radius where $u=\bar u$: $r = R/\sqrt{2}$
   • Compute velocity at $4\,cm$ from wall i.e. $r = R-0.04$
   • Substitute $D=200\,mm$, $U_{max}=1.5\,m/s$
9. 7 Marks
   aExplain separation of boundary layer. What is the effect of pressure gradient on boundary layer separation?
   How to answer

   • Define boundary layer separation and separation point
   • Explain adverse pressure gradient $dp/dx>0$ effect
   • Describe backflow and wake formation
   • Mention control methods
10. 7 Marks
    bDerive expression for the velocity distribution for viscous flow through a circular pipe.
    How to answer

    • Consider steady laminar flow in circular pipe, Hagen-Poiseuille assumptions
    • Apply Navier-Stokes / force balance on fluid element
    • Integrate with no-slip at wall and symmetry at centre
    • Obtain parabolic $u(r)$ and mean velocity
11. 14 Marks
    Find the Momentum 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

    • Recall $\theta = \int_0^\delta (u/U)(1-u/U)dy$, $\delta^*_e = \int_0^\delta (u/U)(1-(u/U)^2)dy$
    • Substitute $u/U = 2\eta-\eta^2$ with $\eta=y/\delta$
    • Integrate term-wise from 0 to 1
    • Express results as fractions of $\delta$
12. 7 Marks
    aExplain the following terms:

    1. i)Magnus effect
    2. ii)Stagnation point
    3. iii)Airfoil
    How to answer

    • Define Magnus effect with rotating cylinder lift
    • Define stagnation point where velocity is zero
    • Define airfoil and its lift generation
    • Give examples / diagrams
13. 7 Marks
    bFind the density of a metallic body which floats at the interface of mercury of specific gravity 13.6 and water such that 40% of its volume is submerged in mercury and 60% in water.
    How to answer

    • Equate weight of body to buoyancy from mercury + water
    • Write $\rho V = 0.4V\rho_{Hg}+0.6V\rho_{w}$
    • Use $\rho_{Hg}=13.6\times1000\,kg/m^3$
    • Solve for $\rho$
14. 7 Marks
    aWrite about Venturimeter. Derive the expression for rate of flow of fluid through Venturimeter.
    How to answer

    • Describe Venturimeter parts: convergent, throat, divergent + manometer
    • Apply Bernoulli and continuity between inlet and throat
    • Include discharge coefficient
    • Obtain $Q = C_d A_1A_2\sqrt{2gh}/\sqrt{A_1^2-A_2^2}$
15. 7 Marks
    bA 30 cm diameter pipe, conveying water, branches into two pipes of diameters 20 cm and 15 cm respectively. The average velocity in the 30 cm diameter pipe is 2.5 m/s, find the discharge in this pipe. Also determine the velocity in 15 cm pipe if the average velocity in 20 cm diameter pipe in 2 m/s.
    How to answer

    • Compute $Q = A_{30}V_{30}$ with $D=30\,cm$, $V=2.5\,m/s$
    • Apply continuity $Q = Q_{20}+Q_{15}$
    • Compute $Q_{20}=A_{20}\times2\,m/s$
    • Solve for $V_{15} = (Q-Q_{20})/A_{15}$