CE-604-A Fluid Mechanics-II - May 2022 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
   aExplain in brief:

   1. i)Specific weight
   2. ii)Cohesion
   3. iii)Capillarity
   How to answer

   • Define specific weight as weight per unit volume, formula $\gamma = \rho g$, units $N/m^3$.
   • Define cohesion as intermolecular attraction in same fluid causing surface tension and resistance to separation.
   • Define capillarity as rise/fall in small tube due to cohesion-adhesion balance, formula $h=4\sigma\cos\theta/\gamma d$.
   • Give examples and significance in fluid mechanics.
2. 7 Marks
   bDescribe in brief Reynolds experiment for determining critical velocity in pipe flow.
   How to answer

   • Describe Reynolds apparatus: glass pipe, dye filament, water tank and valve.
   • Explain procedure to vary velocity and observe laminar, transition and turbulent dye patterns.
   • Define critical velocity and critical Reynolds number for pipe flow.
3. 7 Marks
   aWhat is meant by Turbulence? How does it affect the flow properties?
   How to answer

   • Define turbulence as irregular, random, eddying fluid motion with velocity fluctuations.
   • Explain increased mixing, shear stress and energy loss.
   • Discuss effects on velocity profile, pressure drop, heat and mass transfer.
4. 7 Marks
   bDerive continuity equation for a three dimensional steady or unsteady flows in a Cartesian coordinate system.
   How to answer

   • Consider infinitesimal control volume $dx\,dy\,dz$ in Cartesian system.
   • Write mass inflow minus outflow in x, y, z directions plus accumulation.
   • Obtain $\partial\rho/\partial t + \nabla\cdot(\rho \mathbf{V})=0$ and steady/incompressible forms.
5. 7 Marks
   aWith the help of a diagram explain Streamlines, equipotential lines and flow net. Prove that equipotential lines and steam lines intersect each other orthogonally.
   How to answer

   • Define streamline, equipotential line and flow net with neat diagram.
   • Write velocity potential and stream function relations $u=-\partial\phi/\partial x$, etc.
   • Show slopes product $=-1$ to prove orthogonal intersection.
6. 7 Marks
   bTwo velocity components are given in the following equations, find the third component such that they satisfy the continuity equation. $u = x^3 + y^2 + 2z^2$ ; $v = -x^2y + yz - xy$
   How to answer

   • Write 3D continuity equation $\partial u/\partial x+\partial v/\partial y+\partial w/\partial z=0$.
   • Compute $\partial u/\partial x = 3x^2$ and $\partial v/\partial y = -x^2+z-x$.
   • Integrate for $w$ with respect to $z$ to satisfy continuity.
7. 7 Marks
   aWhat should be the diameter of a Droplet of Water? If the pressure inside is to be $0.0018$ kg(f)/cm$^2$ greater than the outside? Given the value of surface tension of water in contact with air at $20^\circ$C as $0.0075$ kg(f)/m.
   How to answer

   • Write droplet pressure relation $\Delta p = 4\sigma/d$.
   • Convert $\Delta p = 0.0018\,kg(f)/cm^2$ to $kg(f)/m^2$ and use $\sigma=0.0075\,kg(f)/m$.
   • Solve for diameter $d$ and check units.
8. 7 Marks
   bFor laminar flow of an oil having dynamic viscosity $u = 1.766$ Pa.s in a $0.3$m diameter pipe, the velocity distribution is parabolic with a maximum point velocity of $3$ m/s at the center of the pipe. Calculate the shear stresses at the pipe wall and within the fluid $50$ mm from the pipe wall.
   How to answer

   • Assume parabolic profile $u=U_{max}[1-(r/R)^2]$ with $U_{max}=3\,m/s$, $R=0.15\,m$.
   • Compute velocity gradient $du/dr = -2U_{max}r/R^2$ and shear $\tau=\mu|du/dr|$.
   • Evaluate at wall $r=R$ and at $50\,mm$ from wall ($r=0.10\,m$).
9. 7 Marks
   aDerive Euler's equation of motion along a stream line and hence derive the Bernoulli's theorem.
   How to answer

   • Consider fluid element along streamline and apply Newton's second law.
   • Neglect viscous forces to get Euler's equation $dp/\rho+gdz+VdV=0$.
   • Integrate along streamline for steady incompressible flow to get Bernoulli $p/\rho g+V^2/2g+z=$ constant.
10. 7 Marks
    bWrite the differentiate between steady and unsteady flow.
    How to answer

    • Define steady flow: properties at a point invariant with time; unsteady: vary with time.
    • Compare velocity, pressure, streamlines and examples.
    • Mention mathematical condition $\partial/\partial t =0$ vs $\ne 0$.
11. 7 Marks
    aWhat do you understand by the Boundary Layer? Explain the development of boundary layer along a thin flat and smooth plate held parallel to uniform flow. Point out the salient features.
    How to answer

    • Define boundary layer as thin region near solid where velocity varies from zero to free stream.
    • Explain laminar, transition and turbulent zones along thin flat plate.
    • Point out thickness growth, shear stress, separation salient features with diagram.
12. 7 Marks
    bDefine a fluid and distinguish between compressible and incompressible fluids.
    How to answer

    • Define fluid as substance deforming continuously under shear stress.
    • Define compressible fluid with density variation and incompressible with constant density.
    • Distinguish with examples, Mach number, applications.
13. 7 Marks
    aA barometer shows $75$ mm of mercury (Hg) on the surface of a lake. Calculate the absolute pressure at a depth of $4$ m from the surface of lake.
    How to answer

    • Convert barometer $75\,mm$ Hg to water column / pressure head.
    • Add $4\,m$ water column for lake depth to get absolute head.
    • Compute absolute pressure $p=\rho g h$ in $kPa$.
14. 7 Marks
    bDiscuss the application of fluid mechanics to the flow of blood in our veins and to the pumping action of heart.
    How to answer

    • Relate blood flow to pipe flow: viscosity, laminar/turbulent, pressure gradient.
    • Explain heart as pump generating pulsatile pressure.
    • Mention Bernoulli, Poiseuille and clinical applications.
15. 7 Marks
    aExplain in briefly reaction turbine with neat sketch diagram.
    How to answer

    • Define reaction turbine: pressure and velocity both change, runner full of water.
    • Sketch Francis/Kaplan turbine with guide vanes, runner, draft tube.
    • Brief working and types.
16. 7 Marks
    bExplain the total energy line (energy gradient line) and the hydraulic gradient line for fluid flow through a piping system.
    How to answer

    • Define TEL as total head line $p/\rho g+V^2/2g+z$ and HGL as $p/\rho g+z$.
    • Explain relative positions and drop due to losses in piping system.
    • Illustrate with diagram for pipe with pump/turbine.