CE-501 Fluid Mechanics-I - Dec 2025 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
   aDefine surface tension. Prove that relationship between surface tension and pressure inside a droplet of liquid in excess of outside pressure is given by $p = 4\sigma/d$.
   How to answer

   • Define surface tension as force per unit length / energy per unit area
   • Consider spherical droplet of diameter d in equilibrium
   • Balance pressure force p*(pi*d^2/4) against surface tension force sigma*pi*d
   • Obtain excess pressure p = 4*sigma/d
2. 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 sub-merged in mercury and 60% in water.
   How to answer

   • Let body volume = V, density = rho_b
   • Weight = buoyant force from mercury + water
   • rho_b*V = 13.6*1000*0.4V + 1000*0.6V
   • Solve for rho_b / specific weight
3. 7 Marks
   aThe velocity component in a two-dimensional flow field for an incompressible fluid are as follows: $$U = \frac{y^3}{3} + 2x - x^2y, V = xy^2 - 2y - \frac{x^3}{3}$$ Obtain an expression for stream function $\Psi$.
   How to answer

   • Check continuity du/dx + dv/dy = 0 for incompressible flow
   • Use definitions u = -dPsi/dy, v = dPsi/dx
   • Integrate u w.r.t y and v w.r.t x
   • Obtain Psi(x,y) + arbitrary constant
4. 7 Marks
   bDifference between the following:

   1. i)Rotational flow and Irrotational flow
   2. ii)Uniform flow and Non-uniform flow
   3. iii)Steady flow and unsteady flow.
   How to answer

   • Define rotational vs irrotational with vorticity criterion
   • Define uniform vs non-uniform with space variation
   • Define steady vs unsteady with time variation
   • Give one example of each
5. 7 Marks
   aState Bernoulli's theorem for steady flow of an incompressible fluid. Derive an expression for Bernoulli's equation from first principal.
   How to answer

   • State Bernoulli theorem assumptions and statement
   • Consider fluid element along streamline with Euler equation
   • Integrate Euler equation for steady incompressible flow
   • Obtain p/rho*g + V^2/2g + z = constant
6. 7 Marks
   bProve that for the viscous flow through a circular pipe the energy correction factor equal to 2.
   How to answer

   • Write parabolic velocity profile for laminar pipe flow
   • Compute kinetic energy flux integral
   • Compute correction factor alpha = (1/A) integral (u/V)^3 dA
   • Show alpha = 2
7. 7 Marks
   aA liquid flow between two parallel plates. Assuming the flow to be laminar, show that the average velocity is equal to $2/3$ of the maximum velocity.
   How to answer

   • Assume steady laminar flow between fixed parallel plates
   • Solve Navier-Stokes / force balance for parabolic profile
   • Find maximum velocity at centre and integrate for average
   • Show V_avg = 2/3 V_max
8. 7 Marks
   bA laminar flow is taking place in a pipe of diameter of 200 mm. The maximum velocity is 1.5 m/s. Find the mean velocity and the radius at which this occurs. Also calculate the velocity at 4 cm from the wall of the pipe.
   How to answer

   • Mean velocity for pipe laminar flow = 1/2 max velocity
   • Radius for mean velocity from u = Vmax[1-(r/R)^2]
   • Solve r = R/sqrt(2) ~ 0.707R
   • Compute velocity at y=4cm from wall using same profile
9. 7 Marks
   aDerive the loss of head due to friction in pipe flow-Darcy equation.
   How to answer

   • Consider horizontal pipe element with pressure and shear forces
   • Equilibrium: pressure force = frictional shear force
   • Introduce Darcy friction factor f
   • Obtain h_f = f*L*V^2/(2*g*D) / Darcy-Weisbach form
10. 7 Marks
    bFor turbulent flow in a pipe of diameter 300 mm, find the discharge when the centre-line velocity is 2 m/s and the velocity at a point 100 mm from the centre as measured by pitot-tube is 1.6 m/s.
    How to answer

    • Use 1/7th power law or log-law with two velocity points
    • Solve for friction velocity / constants from u_max and u at r=100mm
    • Compute average velocity / discharge Q = A*V_avg
    • Convert to m3/s
11. 14 Marks
    aWrite short notes on any three:

    1. i)Stability of floating and submerged bodies.
    2. ii)Source and sink flow.
    3. iii)Stokes law.
    4. iv)Wiers and notches.
    How to answer

    • Select any three topics with definition and key points
    • Give governing formula / diagram for each
    • Mention applications / significance
    • Conclude with examples
12. 7 Marks
    aExplain Prandtl mixing length theory. Find an expression for shear stress suggested by prandtl.
    How to answer

    • Explain momentum exchange and mixing length l concept
    • Derive velocity fluctuation relations u' ~ l*du/dy
    • Obtain shear stress tau = rho*l^2*(du/dy)^2
    • Mention von Karman constant
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

    • Compute area A = 0.5*4*4 and centroid depth h_bar
    • Total pressure F = rho*g*A*h_bar with oil density 0.9*1000
    • Find I_G for triangle and h* = I_G/(A*h_bar)+h_bar
    • Report F and centre of pressure depth
14. 14 Marks
    The pressure difference $\Delta P$ in a pipe of diameter D and length l due to turbulent flow depends on the velocity V, viscosity $\mu$, density $\rho$ and roughness k. Using Buckingham's $\pi$ theorem, obtain and expression for $\Delta P$.
    How to answer

    • List variables: DeltaP, D, l, V, mu, rho, k with dimensions
    • Choose repeating variables rho, V, D and form pi terms
    • Solve exponent equations for pi1, pi2, pi3
    • Write DeltaP = rho*V^2 * phi(l/D, mu/rhoVD, k/D)