GATE Marine Engineering Fluid Mechanics and Marine Hydrodynamics Syllabus

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 $m$.

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.

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.

How is flow over a circular cylinder (without lift) constructed using superposition?

Superpose a uniform flow with a doublet: $$w = U\left(z + \frac{a^{2}}{z}\right)$$ where $a$ is the cylinder radius and the doublet strength is $\mu = 2\pi U a^{2}$. The circle $r=a$ becomes a streamline.

State Stokes' integral theorem and its physical meaning in fluid mechanics.

$$\oint_C \vec{V}\cdot d\vec{l} = \iint_S (\nabla\times\vec{V})\cdot d\vec{A}$$ The circulation $\Gamma$ around a closed curve equals the flux of vorticity $\vec{\omega} = \nabla\times\vec{V}$ through any surface bounded by the curve.

Define circulation and relate it to vorticity.

Circulation: $$\Gamma = \oint_C \vec{V}\cdot d\vec{l}$$ By Stokes' theorem it equals the surface integral of vorticity $\vec{\omega}=\nabla\times\vec{V}$. For irrotational flow over a simply connected region, $\Gamma = 0$.

State the generalised (unsteady) Bernoulli equation for irrotational incompressible flow.

$$\frac{\partial\phi}{\partial t} + \frac{p}{\rho} + \frac{V^{2}}{2} + gz = f(t)$$ where $\phi$ is the velocity potential. For steady flow $\frac{\partial\phi}{\partial t}=0$ and the sum equals a constant along the whole field.

State the steady-flow Bernoulli equation and list the assumptions behind it.

$$\frac{p}{\rho g} + \frac{V^{2}}{2g} + z = \text{constant}$$ Assumptions: steady, incompressible, inviscid (frictionless) flow along a streamline (or whole field if irrotational), with no shaft work or heat addition.

How does adding a vortex produce flow with circulation around a cylinder, and what is its complex potential?

Add a vortex to the cylinder flow: $$w = U\left(z + \frac{a^{2}}{z}\right) - \frac{i\Gamma}{2\pi}\ln z$$ The circulation $\Gamma$ shifts the stagnation points and produces an asymmetric pressure distribution generating lift.

In flow with circulation around a cylinder, how do the stagnation points move as $\Gamma$ increases?

Stagnation points lie at $\sin\theta_s = -\frac{\Gamma}{4\pi U a}$. They move together on the lower surface as $\Gamma$ grows; when $\Gamma = 4\pi U a$ they merge at the bottom, and beyond this they leave the cylinder surface.

What characterizes axisymmetric (rotationally symmetric) potential flow and what is the Stokes stream function?

Flow is independent of the azimuthal angle. It is described by the Stokes stream function $\psi$ in spherical/cylindrical coordinates, e.g. for a point source: $\psi = -\frac{m}{4\pi}\cos\theta$. Used for flow past bodies of revolution like spheres and airship hulls.

Give the potential-flow solution for uniform flow past a sphere of radius $a$ (rotationally symmetric).

Superposition of uniform flow and a 3-D doublet gives Stokes stream function: $$\psi = \frac{1}{2}U r^{2}\sin^{2}\theta\left(1 - \frac{a^{3}}{r^{3}}\right)$$ Surface speed: $v_\theta = \frac{3}{2}U\sin\theta$ at $r=a$.

What is hydrodynamic lift and on what condition does it depend in potential flow?

Hydrodynamic lift is the force perpendicular to the free-stream velocity. In 2-D potential flow it arises only when there is net circulation $\Gamma$ around the body; without circulation a symmetric body produces zero lift.

State the Kutta–Joukowski lift theorem.

For 2-D potential flow, the lift per unit span on a body is $$L' = \rho U \Gamma$$ directed perpendicular to the free stream, where $\rho$ is density, $U$ free-stream speed and $\Gamma$ the circulation. Drag is zero (d'Alembert's paradox).

What is the Kutta condition and why is it needed for an airfoil?

The Kutta condition states that flow must leave the sharp trailing edge smoothly, fixing the rear stagnation point at the trailing edge. It selects the unique physically correct value of circulation $\Gamma$, making lift determinate.

State the fundamental concept of vortex motion and define a vortex line and vortex tube.

Vortex motion involves rotation of fluid elements (nonzero vorticity $\vec{\omega}$). A vortex line is a curve tangent to the vorticity vector everywhere; a vortex tube is the surface formed by vortex lines through a closed curve.

State Helmholtz's vortex theorems.

1) The strength (circulation) of a vortex tube is constant along its length. 2) A vortex tube cannot end in the fluid—it forms a closed loop, ends at a boundary, or extends to infinity. 3) Vortex lines move with the fluid and their strength is conserved in time (inviscid, barotropic flow).

State Kelvin's circulation theorem.

In an inviscid, barotropic fluid with conservative body forces, the circulation around a closed material curve is constant in time: $$\frac{D\Gamma}{Dt} = 0$$ Hence circulation is conserved following the fluid.

Explain the vortex analogy to the Biot–Savart law.

The velocity induced by a vortex filament mirrors the magnetic field of a current-carrying wire: vorticity $\leftrightarrow$ current, induced velocity $\leftrightarrow$ magnetic field. $$d\vec{V} = \frac{\Gamma}{4\pi}\frac{d\vec{l}\times\vec{r}}{|\vec{r}|^{3}}$$

Give the induced velocity from a straight semi-infinite and infinite vortex filament using the Biot–Savart law.

For an infinite straight filament: $V = \frac{\Gamma}{2\pi h}$. For a semi-infinite filament starting at the foot of the perpendicular: $V = \frac{\Gamma}{4\pi h}$, where $h$ is the perpendicular distance from the point to the filament.

Describe the motion of two straight parallel vortex filaments of equal and opposite strength.

A counter-rotating pair (strengths $+\Gamma$ and $-\Gamma$) separated by distance $d$ translates together (perpendicular to the line joining them) at constant speed $$V = \frac{\Gamma}{2\pi d}$$ without rotating about each other.

Describe the motion of two straight parallel vortex filaments of equal strength and same sign.

Two co-rotating vortices of equal strength $\Gamma$ separated by $d$ rotate about their common centroid (midpoint) with angular velocity $$\omega = \frac{\Gamma}{\pi d^{2}}$$ the mutual induced speed being $\frac{\Gamma}{2\pi d}$.

What is a vortex sheet and what flow property is discontinuous across it?

A vortex sheet is a surface across which the tangential velocity is discontinuous, modeled as a continuous distribution of vorticity. The local sheet strength $\gamma$ equals the jump in tangential velocity: $\gamma = u_1 - u_2$.

State the Navier–Stokes equations for an incompressible, constant-property Newtonian fluid.

$$\rho\frac{D\vec{V}}{Dt} = -\nabla p + \mu\nabla^{2}\vec{V} + \rho\vec{g}$$ together with continuity $\nabla\cdot\vec{V} = 0$. They express momentum conservation including viscous diffusion.

What does each term in the incompressible Navier–Stokes equation physically represent?

$\rho\frac{D\vec{V}}{Dt}$: inertia (local + convective acceleration); $-\nabla p$: pressure gradient force; $\mu\nabla^{2}\vec{V}$: viscous (diffusion) force; $\rho\vec{g}$: body force per unit volume.

Define plane Couette flow and give its velocity profile.

Plane Couette flow is steady viscous flow between two parallel plates, one moving at speed $U$, with no pressure gradient. The velocity profile is linear: $$u(y) = U\frac{y}{h}$$ where $h$ is the gap. Shear stress $\tau = \mu\frac{U}{h}$ is constant.

Give the velocity profile for generalized (pressure-driven) Couette flow between parallel plates.

With both plate motion $U$ and pressure gradient $\frac{dp}{dx}$: $$u(y) = U\frac{y}{h} - \frac{1}{2\mu}\frac{dp}{dx}\,y(h-y)$$ It superposes linear Couette flow and parabolic Poiseuille flow.

Define the stream function $\psi$ for 2-D incompressible flow and state how velocity components are obtained.

For 2-D incompressible flow: $$u = \frac{\partial\psi}{\partial y},\qquad v = -\frac{\partial\psi}{\partial x}$$ Lines of constant $\psi$ are streamlines; $\psi$ automatically satisfies continuity. The difference $\Delta\psi$ equals volume flow rate between streamlines.

Define the velocity potential $\phi$ and state the condition for its existence.

For irrotational flow ($\nabla\times\vec{V}=0$), velocity is the gradient of a scalar potential: $\vec{V} = \nabla\phi$, i.e. $u=\frac{\partial\phi}{\partial x}$, $v=\frac{\partial\phi}{\partial y}$. For incompressible irrotational flow $\phi$ satisfies Laplace's equation $\nabla^{2}\phi = 0$.

Why do the velocity potential and stream function form an orthogonal flow net, and what equations do they satisfy?

For 2-D incompressible irrotational flow, $\phi$ and $\psi$ satisfy the Cauchy–Riemann relations $\frac{\partial\phi}{\partial x}=\frac{\partial\psi}{\partial y}$, $\frac{\partial\phi}{\partial y}=-\frac{\partial\psi}{\partial x}$. Hence equipotential lines and streamlines intersect orthogonally, and both satisfy Laplace's equation.

What is d'Alembert's paradox in potential flow theory?

For steady, incompressible, irrotational (inviscid) flow past a closed body without circulation, the predicted net drag force is zero. This contradicts observation and is resolved by accounting for viscosity, boundary layers and flow separation.