UPSC ESE Mechanical Engineering Fluid Mechanics Flashcards

24 sample cards from the Fluid Mechanics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. State Archimedes' principle and the buoyant force formula.

   Archimedes' principle: a body wholly or partly immersed in a fluid experiences an upward buoyant force equal to the weight of fluid displaced. $$F_{B} = \rho_{fluid}\, g\, V_{displaced}$$

2. What conditions determine whether a body floats, sinks, or is neutrally buoyant?

   Compare body weight $W = \rho_{body} g V$ with buoyancy $F_{B} = \rho_{fluid} g V_{sub}$. If $\rho_{body} < \rho_{fluid}$ it floats; if $\rho_{body} > \rho_{fluid}$ it sinks; if equal, it is neutrally buoyant (suspended).

3. Define the centre of buoyancy.

   The centre of buoyancy is the point of application of the buoyant force; it coincides with the centroid (centre of gravity) of the displaced fluid volume.

4. Define metacentre and metacentric height.

   The metacentre $M$ is the point where the vertical line through the new centre of buoyancy (after a small tilt) intersects the original line through the centre of gravity. Metacentric height $GM$ is the distance between the centre of gravity $G$ and the metacentre $M$.

5. Give the formula for metacentric height $GM$ of a floating body.

   $$GM = \frac{I}{V} - BG$$ where $I$ is the second moment of area of the waterline plane about the tilt axis, $V$ is the submerged volume, and $BG$ is the distance between centre of buoyancy $B$ and centre of gravity $G$.

6. State the conditions for stable, unstable, and neutral equilibrium of a floating body.

   For a floating body: stable if $M$ is above $G$ ($GM > 0$); unstable if $M$ is below $G$ ($GM < 0$); neutral if $M$ coincides with $G$ ($GM = 0$). A larger $GM$ gives greater stability but more rapid (stiff) rolling.

7. State the condition for stability of a fully submerged body.

   For a fully submerged body, stability requires the centre of buoyancy $B$ to be above the centre of gravity $G$. (Metacentre coincides with $B$ since the displaced volume does not change on tilting.)

8. Give the formula for total hydrostatic force on a submerged plane surface.

   $$F = \rho g \bar{h} A$$ where $\bar{h}$ is the vertical depth of the centroid of the surface below the free surface and $A$ is the area. The force acts normal to the surface.

9. Locate the centre of pressure on a vertically submerged plane surface.

   $$h_{cp} = \bar{h} + \frac{I_{G}\sin^{2}\theta}{A\bar{h}}$$ (for a vertical surface $\sin\theta = 1$). Here $I_{G}$ is the second moment of area about the centroidal axis. The centre of pressure always lies below the centroid.

10. Why does the centre of pressure lie below the centroid of a submerged surface?

    Because pressure increases with depth, the lower portion of the surface experiences greater force. This shifts the resultant (centre of pressure) below the centroid by $\frac{I_{G}\sin^{2}\theta}{A\bar{h}}$. As depth increases, the centre of pressure approaches the centroid.

11. How are the horizontal and vertical components of hydrostatic force on a curved surface determined?

    Horizontal component $F_{H} = \rho g \bar{h} A_{proj}$ equals the force on the vertical projection of the curved surface. Vertical component $F_{V} = \rho g V$ equals the weight of fluid contained (or displaced) above the surface. Resultant $F = \sqrt{F_{H}^{2} + F_{V}^{2}}$.

12. Give the second moment of area $I_{G}$ about the centroid for a rectangle and a circle.

    Rectangle (width $b$, depth $d$): $I_{G} = \frac{b d^{3}}{12}$. Circle (diameter $D$): $I_{G} = \frac{\pi D^{4}}{64}$. These are used in centre-of-pressure calculations.

13. State the continuity equation for one-dimensional steady incompressible flow.

    $$A_{1} V_{1} = A_{2} V_{2} = Q = \text{constant}$$ The volume flow rate is constant; velocity is inversely proportional to cross-sectional area.

14. State the continuity equation for steady compressible one-dimensional flow.

    $$\rho_{1} A_{1} V_{1} = \rho_{2} A_{2} V_{2} = \dot{m} = \text{constant}$$ The mass flow rate $\dot{m}$ is conserved; density variation is included for compressible flow.

15. Write the differential (general) continuity equation for compressible flow.

    $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$$ For steady flow $\frac{\partial \rho}{\partial t}=0$; for incompressible flow it reduces to $\nabla \cdot \vec{V} = \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z} = 0$.

16. State Bernoulli's equation and name each term's physical meaning.

    $$\frac{p}{\rho g} + \frac{V^{2}}{2g} + z = \text{constant}$$ The terms are pressure head ($\frac{p}{\rho g}$), velocity (kinetic) head ($\frac{V^{2}}{2g}$), and elevation (potential) head ($z$). Their sum is the total head along a streamline.

17. List the assumptions underlying Bernoulli's equation.

    The flow is steady, incompressible, inviscid (frictionless / non-viscous), irrotational, along a single streamline, and only gravity (no external work/heat) acts. It expresses conservation of mechanical energy.

18. How is Bernoulli's equation modified for real (viscous) flow between two sections?

    $$\frac{p_{1}}{\rho g} + \frac{V_{1}^{2}}{2g} + z_{1} = \frac{p_{2}}{\rho g} + \frac{V_{2}^{2}}{2g} + z_{2} + h_{L}$$ where $h_{L}$ is the head loss due to friction between sections 1 and 2.

19. Derive the Torricelli velocity of efflux from a small orifice using Bernoulli's equation.

    For a tank with free surface a height $H$ above the orifice, applying Bernoulli gives $$V = \sqrt{2gH}$$ the theoretical velocity of efflux (same as a freely falling body through height $H$).

20. Give the discharge formula for a venturimeter.

    $$Q = C_{d}\,\frac{A_{1} A_{2}}{\sqrt{A_{1}^{2} - A_{2}^{2}}}\,\sqrt{2gh}$$ where $C_{d}$ is the coefficient of discharge, $A_{1}, A_{2}$ are inlet and throat areas, and $h$ is the manometer/pressure head difference.

21. How does a pitot tube measure flow velocity?

    A pitot tube measures stagnation pressure; combined with static pressure it gives the velocity head. From Bernoulli: $$V = C_{v}\sqrt{2gh}$$ where $h$ is the difference between stagnation and static heads and $C_{v}$ is the coefficient of velocity.

22. Write the Navier-Stokes equation for incompressible, constant-viscosity flow in vector form.

    $$\rho\left(\frac{\partial \vec{V}}{\partial t} + (\vec{V}\cdot\nabla)\vec{V}\right) = -\nabla p + \mu \nabla^{2}\vec{V} + \rho \vec{g}$$ It expresses Newton's second law (momentum balance) per unit volume for a viscous fluid.

23. Identify the physical meaning of each term in the Navier-Stokes equation.

    $\rho\frac{\partial \vec{V}}{\partial t}$ = local (unsteady) inertia; $\rho(\vec{V}\cdot\nabla)\vec{V}$ = convective inertia; $-\nabla p$ = pressure force; $\mu\nabla^{2}\vec{V}$ = viscous (diffusion) force; $\rho\vec{g}$ = body (gravity) force per unit volume.

24. What do the Euler equations of motion represent and how do they relate to Navier-Stokes?

    The Euler equations are the Navier-Stokes equations with the viscous term dropped ($\mu = 0$): $$\rho\frac{D\vec{V}}{Dt} = -\nabla p + \rho\vec{g}$$ They govern inviscid flow; integrating Euler's equation along a streamline yields Bernoulli's equation.

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