GATE Marine Engineering Fluid Mechanics and Marine Hydrodynamics Syllabus

Fluid Mechanics and Marine Hydrodynamics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Fluid Mechanics and Marine Hydrodynamics in GATE Marine Engineering, not a summary of it.

1. 1

   Fluid Mechanics

   52 topics

   • Fluid properties
   • Fluid statics
   • Stability of floating bodies
   • Conservation laws: Mass, momentum and energy (Integral and differential form)
   • Dimensional analysis and dynamic similarity
   • Sources, sinks, doublets, line vortex and their superposition
   • Stoke’s integral theorem
   • Generalised Bernoulli’s equation
   • Flow with circulation
   • Potential flow with rotational symmetry
   • Hydrodynamical lift
   • Kutta-Joukowski theorem
   • Vortex motion- Fundamental concepts
   • Vortex analogy to Biot-Savart’s law
   • Straight parallel vortex filaments
   • Vortex sheets
   • Viscous flow- Navier-Stokes equations
   • Couette flow
   • Plane poiseuille flow
   • Equation of continuity
   • Euler‘s equation
   • Bernoulli‘s equation
   • Viscous flow of incompressible fluids
   • Elementary turbulent flow
   • Boundary layer
   • Flow through pipes
   • Boundary layer theory- Prandtl’s boundary layer equations
   • Criterion for separation
   • Blasius solution
   • Skin friction
   • Displacement thickness
   • Momentum thickness
   • Turbulent boundary layer
   • Boundary layer control
   • Airfoils- Lift, drag, circulation, pressure distribution
   • Theory of thin aerofoils
   • Wings of infinite and finite span
   • Circulation distribution
   • Cavitation
   • Vorticity and Kelvin’s theorem
   • Potential flow theory
   • Sources, Sinks and Doublets
   • Hydrodynamic forces in potential flow
   • D’Alembert’s paradox
   • Added-mass
   • Slender-body theory
   • Hydrodynamic model testing
   • Scaling laws
   • Application of potential theory to surface waves
   • Energy transport
   • Wave/body forces
   • Linearised theory of lifting surfaces

Fluid Mechanics and Marine Hydrodynamics flashcards for GATE Marine Engineering

23 of 51 cards from the Fluid Mechanics and Marine Hydrodynamics deck — real questions with worked answers.

1. Define dynamic viscosity and state Newton's law of viscosity for a simple shear flow.

   Dynamic viscosity $\mu$ is the proportionality constant between shear stress and velocity gradient. Newton's law of viscosity: $$\tau = \mu \frac{du}{dy}$$ where $\tau$ is shear stress and $\frac{du}{dy}$ is the rate of shear strain. SI unit of $\mu$: $\text{Pa·s}$.

2. What is kinematic viscosity and how does it relate to dynamic viscosity?

   Kinematic viscosity $\nu$ is the ratio of dynamic viscosity to density: $$\nu = \frac{\mu}{\rho}$$ Its SI unit is $\text{m}^{2}/\text{s}$. It represents momentum diffusivity.

3. Define the bulk modulus of elasticity of a fluid and relate it to compressibility.

   Bulk modulus $K$ measures resistance to volumetric compression: $$K = -V\frac{dp}{dV} = \rho\frac{dp}{d\rho}$$ Compressibility is its reciprocal, $\beta = \frac{1}{K}$.

4. State the relation for surface tension across a spherical droplet and a soap bubble.

   For a liquid droplet (one surface): $$\Delta p = \frac{2\sigma}{R}$$ For a soap bubble (two surfaces): $$\Delta p = \frac{4\sigma}{R}$$ where $\sigma$ is surface tension and $R$ the radius.

5. Give the capillary rise/depression formula for a liquid in a tube.

   $$h = \frac{2\sigma\cos\theta}{\rho g r}$$ where $\theta$ is the contact angle and $r$ the tube radius. Rise occurs for $\theta < 90^\circ$ (wetting), depression for $\theta > 90^\circ$.

6. State the basic equation of fluid statics (hydrostatic equation) in differential form.

   In a fluid at rest, pressure varies only with depth: $$\frac{dp}{dz} = -\rho g$$ For incompressible fluid: $p = p_0 + \rho g h$, where $h$ is depth below the free surface.

7. How is the total hydrostatic force on a plane submerged surface and its centre of pressure located?

   Total force: $F = \rho g \bar{h}_c A$, acting at the pressure centre located below the centroid by $$h_{cp} = \bar{h}_c + \frac{I_{xc}\sin^{2}\theta}{\bar{h}_c A}$$ where $I_{xc}$ is the second moment of area about the centroidal axis.

8. State Archimedes' principle and define the centre of buoyancy.

   A body immersed in a fluid experiences an upward buoyant force equal to the weight of the displaced fluid: $F_B = \rho g V_{disp}$. The centre of buoyancy is the centroid of the displaced fluid volume.

9. Define metacentre and metacentric height for a floating body.

   The metacentre $M$ is the point where the line of action of buoyancy intersects the body's vertical axis after a small tilt. Metacentric height $\overline{GM}$ is the distance from the centre of gravity $G$ to $M$; it governs stability.

10. Give the formula for metacentric height of a floating body and the stability criteria.

    $$\overline{GM} = \overline{BM} - \overline{BG} = \frac{I}{V} - \overline{BG}$$ where $I$ is the second moment of the waterplane area and $V$ the displaced volume. Stable: $\overline{GM}>0$ ($M$ above $G$); unstable: $\overline{GM}<0$; neutral: $\overline{GM}=0$.

11. What distinguishes the stability of a fully submerged body from a floating body?

    For a fully submerged body, stability requires $B$ (centre of buoyancy) above $G$, since the metacentre coincides with $B$. For a floating body, $G$ may be above $B$ and still be stable provided $M$ lies above $G$ ($\overline{GM}>0$).

12. State the integral (control-volume) form of the conservation of mass (continuity).

    $$\frac{\partial}{\partial t}\int_{CV}\rho\, dV + \oint_{CS}\rho\,\vec{V}\cdot d\vec{A} = 0$$ Rate of accumulation of mass plus net mass efflux equals zero.

13. State the differential form of the continuity equation.

    $$\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\vec{V}) = 0$$ For incompressible flow it reduces to $\nabla\cdot\vec{V} = 0$.

14. Write the integral form of the linear momentum equation for a control volume.

    $$\sum\vec{F} = \frac{\partial}{\partial t}\int_{CV}\rho\vec{V}\, dV + \oint_{CS}\rho\vec{V}(\vec{V}\cdot d\vec{A})$$ The sum of external forces equals the time rate of change plus net momentum flux.

15. State the differential momentum equation for a frictionless flow (Euler's equation).

    $$\rho\frac{D\vec{V}}{Dt} = -\nabla p + \rho\vec{g}$$ where $\frac{D}{Dt} = \frac{\partial}{\partial t} + (\vec{V}\cdot\nabla)$ is the material derivative.

16. Write the integral form of the energy equation (first law) for a control volume.

    $$\dot{Q} - \dot{W} = \frac{\partial}{\partial t}\int_{CV}e\rho\, dV + \oint_{CS}\left(e + \frac{p}{\rho}\right)\rho\,\vec{V}\cdot d\vec{A}$$ where $e = u + \frac{V^{2}}{2} + gz$.

17. What is the purpose of dimensional analysis and what does Buckingham's $\pi$-theorem state?

    Dimensional analysis reduces variables to dimensionless groups. Buckingham's $\pi$-theorem: if a problem has $n$ variables and $m$ fundamental dimensions, it can be expressed in $(n-m)$ independent dimensionless $\pi$ groups.

18. State the three conditions required for complete dynamic similarity between model and prototype.

    Geometric similarity (same shape/scale ratios), kinematic similarity (similar velocity fields/streamline patterns), and dynamic similarity (same ratios of forces, i.e. equal relevant dimensionless numbers).

19. Define the Reynolds number and Froude number and state which force ratios they represent.

    Reynolds number: $Re = \frac{\rho V L}{\mu} = \frac{\text{inertia}}{\text{viscous}}$. Froude number: $Fr = \frac{V}{\sqrt{gL}} = \sqrt{\frac{\text{inertia}}{\text{gravity}}}$. Froude similarity governs ship/free-surface model testing.

20. Give the complex potential, velocity potential and stream function of a 2-D uniform flow of speed $U$ along the x-axis.

    Complex potential: $w(z) = Uz$. Velocity potential: $\phi = Ux$. Stream function: $\psi = Uy$. Velocity: $u = U$, $v = 0$.

21. State the velocity potential and stream function of a 2-D source of strength $m$ at the origin.

    Complex potential: $w = \frac{m}{2\pi}\ln z$. Potential: $\phi = \frac{m}{2\pi}\ln r$; stream function: $\psi = \frac{m}{2\pi}\theta$. Radial velocity: $v_r = \frac{m}{2\pi r}$. A sink has negative $m$.

22. Give the complex potential and stream function of a 2-D doublet of strength $\mu$ at the origin.

    Complex potential: $w = \frac{\mu}{2\pi z}$. Stream function: $\psi = -\frac{\mu}{2\pi}\frac{\sin\theta}{r}$; potential: $\phi = \frac{\mu}{2\pi}\frac{\cos\theta}{r}$. A doublet is the limit of a source-sink pair as separation $\to 0$ with constant strength.

23. Write the complex potential and velocity field of a 2-D free (line) vortex of circulation $\Gamma$.

    Complex potential: $w = -\frac{i\Gamma}{2\pi}\ln z$. Tangential velocity: $v_\theta = \frac{\Gamma}{2\pi r}$, $v_r = 0$. Stream function: $\psi = -\frac{\Gamma}{2\pi}\ln r$. The flow is irrotational everywhere except at the core.

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