ME-404 Fluid Mechanics - Jun 2026 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 is the continuum hypothesis in fluid mechanics? Under what conditions does it break down? Define metacentre and metacentric height, and explain the conditions for the stable, unstable, and neutral equilibrium of a floating body.
   How to answer

   • Define continuum hypothesis: fluid treated as continuous medium ignoring molecular structure
   • Condition for validity: Knudsen number <<1; breakdown in rarefied gas, shock waves, micro/nano flows
   • Define metacentre, metacentric height GM
   • Conditions: GM>0 stable, GM<0 unstable, GM=0 neutral with explanation
2. 7 Marks
   bA rectangular plate $3\,\text{m} \times 1.5\,\text{m}$ is immersed vertically in water such that its longer side is parallel to the free water surface. Determine the total pressure force and the position of the center of pressure on the plate when:

   1. i)The free surface coincides with the top edge of the plate.
   2. ii)The free surface is 2 m above the top edge of the plate.
   How to answer

   • Compute area $A=3\times 1.5$ and centroid depth $\bar{h}$ for each case
   • Apply $F = \rho g A \bar{h}$ for total pressure force
   • Compute $I_{G}$ about centroidal axis parallel to free surface
   • Apply $h_p = \bar{h}+I_G\sin^2\theta/(A\bar{h})$ for centre of pressure
3. 7 Marks
   aDistinguish between the Lagrangian and Eulerian methods of describing fluid flow. Derive the three-dimensional continuity equation in Cartesian coordinates for steady, incompressible flow.
   How to answer

   • Distinguish Lagrangian (follow fluid particle) vs Eulerian (fixed control volume/field)
   • Consider differential control volume mass balance
   • Apply steady incompressible assumptions
   • Obtain $\partial u/\partial x + \partial v/\partial y + \partial w/\partial z = 0$
4. 7 Marks
   bIn a two-dimensional incompressible fluid flow, the velocity components are given by $u = x^2-y^2$ and $v = -2xy$.

   1. i)Determine if this flow is physically possible.
   2. ii)If the flow is irrotational, find the velocity potential function ($\Phi$) and the stream function ($\psi$).
   How to answer

   • Check continuity $\partial u/\partial x + \partial v/\partial y = 0$ for physical possibility
   • Check vorticity/irrotationality $\partial v/\partial x - \partial u/\partial y = 0$
   • Integrate $u=\partial\phi/\partial x$, $v=\partial\phi/\partial y$ to get velocity potential $\Phi$
   • Integrate $u=\partial\psi/\partial y$, $v=-\partial\psi/\partial x$ to get stream function $\Psi$
5. 7 Marks
   aState Bernoulli's theorem. Derive Bernoulli's equation from Euler's equation of motion along a streamline, listing all the assumptions made. How is it modified for real (viscous) fluid flows?
   How to answer

   • State Bernoulli's theorem for steady, incompressible, inviscid flow along streamline
   • Start from Euler's equation of motion and integrate along streamline
   • List assumptions: steady, incompressible, inviscid, along streamline, no shaft work
   • Add head loss term for real viscous flow
6. 7 Marks
   bA horizontal venturimeter with inlet diameter $200\,\text{mm}$ and throat diameter $100\,\text{mm}$ is used to measure the flow rate of oil of specific gravity $0.8$ A differential U-tube mercury manometer connected between the inlet and throat shows a deflection of $250\,\text{mm}$. Find the rate of flow of oil through the pipeline if the coefficient of discharge of the venturimeter is $0.98$ (Take specific gravity of mercury $=13.6$).
   How to answer

   • Compute inlet and throat areas from $D_1=200\,mm$, $D_2=100\,mm$
   • Convert manometer deflection to differential head using $SG_{oil}=0.8$, $SG_{Hg}=13.6$
   • Apply venturimeter formula $Q=C_d A_1A_2\sqrt{2gh}/\sqrt{A_1^2-A_2^2}$
   • Substitute $C_d=0.98$ to get oil discharge
7. 7 Marks
   aDerive the Hagen-Poiseuille equation for the pressure drop in a steady, laminar flow through a horizontal circular pipe of constant diameter. Show that the friction factor for laminar flow is given by $f = 64/Re$.
   How to answer

   • Consider force balance on fluid cylinder in fully developed laminar pipe flow
   • Derive parabolic velocity profile and integrate for discharge
   • Obtain Hagen-Poiseuille pressure drop $\Delta p = 32\mu LV/D^2$
   • Show Darcy friction factor $f = 64/Re$
8. 7 Marks
   bA pipeline of $300\,\text{mm}$ diameter and $1200\,\text{m}$ length connects two reservoirs whose water surface levels differ by $15\,\text{m}$. If the friction factor for the pipe is $f=0.02$, calculate the discharge through the pipe taking into account:

   1. i)Only major frictional losses.
   2. ii)All losses (including entry loss, exit loss, and friction).
   How to answer

   • Write energy equation between reservoirs with head difference 15 m
   • For (i) equate head to friction loss $h_f = f(L/D)(V^2/2g)$ and solve for V, Q
   • For (ii) add entrance $0.5V^2/2g$ and exit $V^2/2g$ losses
   • Compare discharge with and without minor losses
9. 7 Marks
   aWrite the Navier- Stokes equation for viscous fluid flow in Cartesian coordinates, explaining the physical significance of each term. Explain the concept of the Hydraulic Gradient Line (HGL) and the Total Energy Line (TEL).
   How to answer

   • Write Navier-Stokes equations in x, y, z Cartesian form
   • Explain local, convective, pressure, viscous and body force terms
   • Define HGL as $p/\rho g + z$ line
   • Define TEL as $p/\rho g + V^2/2g + z$ line and relation
10. 7 Marks
    bThree pipes of lengths $800\,\text{m}$, $500\,\text{m}$, and $400\,\text{m}$, and diameters $400\,\text{mm}$, $300\,\text{mm}$, and $200\,\text{mm}$ respectively, are connected in series. The difference in water surface levels of the two reservoirs connected by this pipe system is $16\,\text{m}$. Determine the rate of flow of water if the friction factor $f = 0.02$ is the same for all three pipes (neglect minor losses). Also, find the equivalent diameter of a single pipe of length $1700\,\text{m}$ to replace this compound pipe system.
    How to answer

    • Write total head loss as sum $\sum f(L/D)(V^2/2g)$ with continuity $Q=A_1V_1=...$
    • Solve for $Q$ from $H=16\,m$ neglecting minor losses
    • Equate to single pipe loss with $L=1700\,m$ for equivalent diameter
    • Compute $D_{eq}$ from $H = fLQ^2/(12.1 D_{eq}^5)$ form
11. 7 Marks
    aWhat is boundary layer separation? Explain the physical mechanism behind boundary layer separation on a curved surface under an adverse pressure gradient. List the common methods used to prevent or delay boundary leyer separation in engineering applications.
    How to answer

    • Define separation as detachment of boundary layer from surface
    • Explain adverse pressure gradient decelerating near-wall fluid to zero wall shear
    • Describe reverse flow and vortex formation on curved surface
    • List control methods: suction, blowing, streamlining, vortex generators, etc.
12. 7 Marks
    bAir flows over a flat plate of length $2\,\text{m}$ and width $1.5\,\text{m}$ at a free-stream velocity of $5\,\text{m/s}$. The velocity distribution within the boundary layer is given by the cubic profile: $$\frac{u}{U} = \frac{3}{2}\left(\frac{y}{\delta}\right)-\frac{1}{2}\left(\frac{y}{\delta}\right)^3$$ Calculate:

    1. i)The displacement thickness ($\delta^*$) and momentum thickness ($\theta$) in terms of the boundary layer thickness ($\delta$).
    2. ii)The drag force on one side of the plate if the boundary layer thickness at the trailing edge is $30\,\text{mm}$. (Take density of air $=1.2\,\text{kg/m}^3$ and dynamic viscosity $=1.8\times 10^{-5}\,\text{Pa.s}$)
    How to answer

    • Integrate cubic profile to get $\delta^* = \int_0^\delta (1-u/U)dy$ and $\theta = \int_0^\delta (u/U)(1-u/U)dy$
    • Express $\delta^*=3\delta/8$, $\theta=39\delta/280$ approx
    • Compute wall shear via $\tau_0 = \mu (du/dy)_{y=0}$ and momentum integral
    • Compute drag $F_D = b\int \tau_0 dx$ or $\rho b U^2\theta$ with $\delta=30\,mm$
13. 7 Marks
    aDefine stagnation temperature, stagnation pressure and stagnation density. Derive the expression for stagnation temperature ($T_o$) in terms of static temperature (T), Mach number (M), and specific heat ratio ($\gamma$) for a compressible isentropic flow.
    How to answer

    • Define stagnation properties as isentropic deceleration to zero velocity
    • Apply steady flow energy equation $h+T$ relation
    • Introduce $M$ and $\gamma$ to get $T_0/T = 1+(\gamma-1)M^2/2$
    • Mention $p_0/p$ and $\rho_0/\rho$ isentropic relations
14. 7 Marks
    bExplain normal shock waves in compressible flows. Why does a shock wave only occur in supersonic flows? Briefly explain the significance of Fanno and Rayleigh lines with the help of an h-s (enthalpy-entropy) diagram.
    How to answer

    • Define normal shock as thin discontinuity with supersonic to subsonic transition
    • Explain why compression shock requires supersonic upstream Mach
    • Introduce Fanno (adiabatic friction) and Rayleigh (heat transfer) lines
    • Use h-s diagram intersection to show shock process and choking
15. 7 Marks
    aA hemispherical parachute is being designed to decelerate a pay load of mass $120\,\text{kg}$. If the parachute falls vertically through air (density $=1.2\,\text{kg/m}^3$) at a terminal velocity of $8\,\text{m/s}$, determine the required diameter of the parachute. Take the drag coefficient ($C_D$) for the hemispherical parachute as $1.33$. If a flat circular parachute of the same diameter is used instead (with $C_D = 1.15$), what would be its terminal velocity?
    How to answer

    • Equate weight $mg$ to drag $(1/2)\rho V^2 A C_D$ at terminal velocity
    • Solve for area and diameter $D$ with $C_D=1.33$, $V=8\,m/s$
    • Reuse same diameter with $C_D=1.15$ to solve for new $V$
    • Comment on higher terminal velocity for lower $C_D$
16. 7 Marks
    bWhat is a flow net? Explain the graphical and analytical methods of constructing flow nets. State their primary applications and limitations in analyzing fluid flows.
    How to answer

    • Define flow net as grid of streamlines and equipotential lines
    • Explain graphical sketching and analytical/complex potential methods
    • State applications: seepage, uplift, exit gradient, discharge estimation
    • State limitations: 2D, steady, incompressible, irrotational, homogeneous