ME-404 Fluid Mechanics - Jun 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
   aState Newton's law of viscosity and discuss behaviour of Newtonian fluid.
   How to answer

   • State Newton's law: shear stress proportional to velocity gradient, $\tau = \mu du/dy$
   • Define dynamic and kinematic viscosity
   • Define Newtonian fluid with linear stress-strain rate relation and examples
   • Discuss behaviour: constant viscosity, straight line through origin on rheological diagram
2. 7 Marks
   bA cylindrical vertical container 50 cm ID is rotated about it's axis. The container has a height of 90 cm and was initially filled to 45cm. Calculate the speed of rotation at which water will spill and also pressure at a point 20 cm radial position and 5 cm above base.
   How to answer

   • Apply forced vortex free surface equation $z = \omega^2 r^2/(2g)$ and volume conservation
   • Calculate angular speed for water to reach top rim for spill
   • Convert to RPM / speed of rotation
   • Calculate pressure from $p = -\rho g z + \rho \omega^2 r^2/2 + const$ at given point
3. 7 Marks
   aExplain stability of a floating body.
   How to answer

   • Define floating stability, centre of gravity G, buoyancy B, metacentre M
   • Explain stable, unstable and neutral equilibrium via relative positions of G and M
   • Define metacentric height GM and righting moment
   • Include neat sketch of tilted floating body
4. 7 Marks
   bA solid cylinder of diameter 1m and height 1m floats in fresh water with it's axis vertical. The cylinder is made of a material of specific gravity 0.7. Determine the metacentric height and state the condition of it's equilibrium.
   How to answer

   • Find draft from Archimedes principle using specific gravity 0.7
   • Calculate $BM = I/V_{submerged}$ for circular section
   • Calculate $BG$ and $GM = BM - BG$
   • State stable if $GM>0$, unstable if $GM<0$
5. 14 Marks
   Distinguish between :

   1. i)Steady and unsteady flow
   2. ii)Uniform and non uniform flow
   3. iii)Incompressible and compressible flow
   How to answer

   • Define steady vs unsteady with $\partial/\partial t$ criterion and examples
   • Define uniform vs non-uniform with $\partial/\partial s$ criterion and examples
   • Define incompressible vs compressible with density variation and Mach number
   • Tabulate differences
6. 7 Marks
   aDerive Euler's equation along a streamline.
   How to answer

   • Consider fluid element along streamline and apply Newton's second law
   • Include pressure, gravity and acceleration components
   • Obtain $dp/\rho + VdV + gdz = 0$ form
   • State assumptions of inviscid steady flow
7. 7 Marks
   bWater is flowing in a pipe of 200 mm diameter with an average velocity of 3m/sec. If the pipe is 7m above the datum and pressure is measured to be 250 kN/m², Determine the total head of water.
   How to answer

   • Write Bernoulli total head $H = p/\rho g + V^2/(2g) + z$
   • Convert pressure $250\,kN/m^2$ to pressure head
   • Add velocity head and datum head $7\,m$
   • Compute total head in m of water
8. 14 Marks
   The $x$ and $y$ components of velocity in a two dimensional incompressible flow are given by $u=3x+y$ and $v=2x-3y$. Derive an expression for the stream function and hence show that flow is not irrotational. Calculate the velocity at point $(-1, 2)$.
   How to answer

   • Check continuity $\partial u/\partial x + \partial v/\partial y = 0$ for incompressible flow
   • Integrate $u = \partial \psi/\partial y$, $v = -\partial \psi/\partial x$ to get $\psi(x,y)$
   • Compute vorticity $\omega_z$ to show rotational flow
   • Evaluate velocity magnitude at $(-1,2)$
9. 14 Marks
   Explain the flow phenomenon between two parallel plates kept at distance 2h apart.
   How to answer

   • Describe plane Poiseuille / Couette setup with plates 2h apart
   • State assumptions: steady, laminar, fully developed, pressure gradient
   • Give velocity profile and shear distribution
   • Mention discharge and pressure drop relation
10. 14 Marks
    Evaluate the boundary layer thickness $\delta$ and skin friction coefficient $C_f$ for flow over a flat plate by using following profile.

    1. i)$\frac{u}{U_m} = \frac{Y}{\delta}$
    2. ii)$\frac{u}{U_m} = 2\left(\frac{y}{\delta}\right) - \left(\frac{Y}{\delta}\right)^2$
    How to answer

    • Apply von Karman momentum integral equation for flat plate
    • Integrate given $u/U_m$ profiles to get momentum thickness
    • Derive $\delta/x$ vs $Re_x$ relation for each profile
    • Derive skin friction coefficient $C_f$ vs $Re$
11. 14 Marks
    Write short notes on any three:

    1. a)Venturimeter
    2. b)Difference between velocity potential and stream function
    3. c)Flow through circular cylinder
    4. d)Head losses in turbulent flow
    5. e)Boundary Layer phenomenon
    How to answer

    • Select any three topics from list
    • Give definition, working principle and formula for each
    • Add neat sketch where applicable e.g. venturimeter, cylinder flow, boundary layer
    • Mention applications and losses