GATE Mechanical Engineering Fluid Mechanics and Thermal Sciences Syllabus

State Bernoulli's equation and list the assumptions behind it.

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

Define stream function $\psi$ and velocity potential $\phi$ for 2-D incompressible flow.

Stream function: $u = \frac{\partial \psi}{\partial y},\; v = -\frac{\partial \psi}{\partial x}$ (satisfies continuity automatically). Velocity potential: $u = \frac{\partial \phi}{\partial x},\; v = \frac{\partial \phi}{\partial y}$ (exists only for irrotational flow). Lines of constant $\psi$ and $\phi$ are orthogonal.

Define vorticity and the condition for irrotational flow.

Vorticity $\vec{\omega} = \nabla\times\vec{V}$. Flow is irrotational when $\vec{\omega} = 0$. In 2-D, the rotation is $\omega_z = \frac{1}{2}\left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right)$; irrotational requires $\frac{\partial v}{\partial x} = \frac{\partial u}{\partial y}$.

State Buckingham's $\pi$-theorem.

If a physical problem involves $n$ variables with $m$ independent fundamental dimensions, it can be reduced to $(n-m)$ independent dimensionless $\pi$ groups: $$f(\pi_1, \pi_2, \ldots, \pi_{n-m}) = 0$$

Define the Reynolds number and state its physical meaning.

$$Re = \frac{\rho V L}{\mu} = \frac{V L}{\nu}$$ It is the ratio of inertial forces to viscous forces, and governs the transition between laminar and turbulent flow.

Give the definitions of Froude, Mach, Euler, and Weber numbers.

Froude $Fr = \frac{V}{\sqrt{gL}}$ (inertia/gravity); Mach $Ma = \frac{V}{c}$ (inertia/elastic); Euler $Eu = \frac{V}{\sqrt{p/\rho}}$ (inertia/pressure); Weber $We = \frac{\rho V^{2} L}{\sigma}$ (inertia/surface tension).

For laminar fully developed flow in a circular pipe (Hagen–Poiseuille), give the velocity profile and ratio of max to mean velocity.

Parabolic profile: $$u(r) = u_{max}\left[1 - \left(\frac{r}{R}\right)^{2}\right]$$ with $u_{max} = 2 V_{avg}$, so $\frac{u_{max}}{V_{avg}} = 2$.

Give the Hagen–Poiseuille expression for pressure drop and discharge in laminar pipe flow.

$$\Delta p = \frac{32 \mu L V}{d^{2}}, \qquad Q = \frac{\pi \Delta p\, d^{4}}{128 \mu L}$$ Discharge varies with the fourth power of diameter.

For laminar flow between two stationary parallel plates, give max velocity ratio and the friction factor relation.

For flow between fixed parallel plates, $\frac{u_{max}}{V_{avg}} = \frac{3}{2}$. Velocity profile is parabolic. For pipe laminar flow the Darcy friction factor is $f = \frac{64}{Re}$.

Define the hydrodynamic boundary layer and its thickness $\delta$.

The boundary layer is the thin region near a solid surface where viscous effects are significant and velocity rises from zero (no-slip) to the free-stream value. $\delta$ is the distance from the wall where $u = 0.99\,U_\infty$.

Define displacement thickness and momentum thickness of a boundary layer.

Displacement thickness $$\delta^{*} = \int_0^\infty \left(1 - \frac{u}{U}\right)dy$$ Momentum thickness $$\theta = \int_0^\infty \frac{u}{U}\left(1 - \frac{u}{U}\right)dy$$ They quantify mass-flow and momentum deficits due to the boundary layer.

Give the Blasius laminar boundary-layer results for $\delta$ and local skin-friction coefficient over a flat plate.

$$\frac{\delta}{x} = \frac{5.0}{\sqrt{Re_x}}, \qquad C_{f,x} = \frac{0.664}{\sqrt{Re_x}}$$ valid for laminar flow with $Re_x = \frac{U x}{\nu}$.

What causes boundary-layer separation and what is the condition at the separation point?

Separation occurs under an adverse pressure gradient ($\frac{dp}{dx} > 0$), where flow decelerates and reverses. At the separation point the wall velocity gradient vanishes: $$\left(\frac{\partial u}{\partial y}\right)_{y=0} = 0$$

State the approximate Reynolds number ranges for laminar, transitional, and turbulent flow in a pipe.

Laminar: $Re < 2000$; Transitional: $2000 < Re < 4000$; Turbulent: $Re > 4000$. (Critical $Re \approx 2300$.) For a flat plate, transition occurs near $Re_x \approx 5\times10^{5}$.

Describe the turbulent velocity profile and the role of Reynolds (turbulent) shear stress.

Turbulent profile is fuller (flatter) than laminar, approximated by the $\frac{1}{7}$ power law $\frac{u}{u_{max}} = \left(\frac{y}{R}\right)^{1/7}$. Additional apparent stress (Reynolds stress) $\tau_t = -\rho\,\overline{u'v'}$ arises from velocity fluctuations and is modeled via eddy viscosity.

State the Darcy–Weisbach equation for major head loss in pipes.

$$h_f = f\frac{L}{d}\frac{V^{2}}{2g}$$ where $f$ is the Darcy friction factor, $L$ length, $d$ diameter, $V$ mean velocity. This accounts for frictional (major) losses.

Give the general form of minor losses in pipe fittings and the loss at a sudden expansion.

Minor loss: $$h_m = K\frac{V^{2}}{2g}$$ Sudden expansion (Borda–Carnot): $$h_L = \frac{(V_1 - V_2)^{2}}{2g}$$ where $K$ is the loss coefficient for the fitting/bend.

Define the Darcy friction factor for laminar flow and relate it to the Fanning friction factor.

Laminar: $f = \frac{64}{Re}$. The Darcy friction factor is four times the Fanning friction factor: $f_{Darcy} = 4 f_{Fanning}$.

Compare pipes in series and in parallel for flow and head loss.

Series: same discharge $Q$ through each pipe, total head loss adds: $h_L = h_{L1}+h_{L2}+\ldots$. Parallel: head loss across each pipe is equal, discharges add: $Q = Q_1 + Q_2 + \ldots$.

Define the hydraulic gradient line (HGL) and total energy line (TEL).

HGL plots $\frac{p}{\rho g} + z$ (pressure + elevation head). TEL plots $\frac{p}{\rho g} + z + \frac{V^{2}}{2g}$ (HGL plus velocity head). TEL always lies above HGL by the velocity head and slopes downward due to losses.

Define the speed of sound in a gas and the Mach number flow regimes.

$$c = \sqrt{\frac{\partial p}{\partial \rho}} = \sqrt{\gamma R T}$$ Regimes: subsonic $Ma<1$, sonic $Ma=1$, supersonic $Ma>1$, hypersonic $Ma\gtrsim5$.

Give the stagnation-to-static temperature ratio for isentropic compressible flow.

$$\frac{T_0}{T} = 1 + \frac{\gamma - 1}{2}Ma^{2}$$ Similarly $\frac{p_0}{p} = \left(1 + \frac{\gamma-1}{2}Ma^{2}\right)^{\frac{\gamma}{\gamma-1}}$.

State the three modes of heat transfer and the governing law for each.

Conduction (Fourier's law $q = -kA\frac{dT}{dx}$), Convection (Newton's law of cooling $q = hA\Delta T$), Radiation (Stefan–Boltzmann $q = \varepsilon\sigma A T^{4}$).

State Fourier's law of heat conduction in one dimension.

$$q = -kA\frac{dT}{dx}$$ where $q$ is heat rate, $k$ thermal conductivity, $A$ area, and $\frac{dT}{dx}$ the temperature gradient. The minus sign indicates heat flows down the temperature gradient.

State Newton's law of cooling for convection and define the convective resistance.

$$q = hA(T_s - T_\infty)$$ where $h$ is the convective heat transfer coefficient. Convective thermal resistance $$R_{conv} = \frac{1}{hA}$$

Write the general 3-D heat conduction (heat diffusion) equation with internal generation.

$$\nabla^{2}T + \frac{\dot{q}_{gen}}{k} = \frac{1}{\alpha}\frac{\partial T}{\partial t}$$ where $\alpha = \frac{k}{\rho c_p}$ is thermal diffusivity. For steady, no generation: $\nabla^{2}T = 0$ (Laplace's equation).

Using the electrical analogy, give the conduction resistance for a plane wall, cylinder, and sphere.

Plane wall: $R = \frac{L}{kA}$. Hollow cylinder: $R = \frac{\ln(r_2/r_1)}{2\pi k L}$. Hollow sphere: $R = \frac{r_2 - r_1}{4\pi k\, r_1 r_2}$. Heat flow $q = \frac{\Delta T}{R}$ (analogous to Ohm's law).

For composite walls/series resistances, how is overall heat transfer computed and what is overall conductance?

Resistances in series add: $R_{total} = R_1 + R_2 + \ldots$, giving $q = \frac{\Delta T_{overall}}{R_{total}}$. Overall heat transfer coefficient $U$ satisfies $\frac{1}{UA} = R_{total}$.

Define thermal diffusivity and explain its physical significance.

$$\alpha = \frac{k}{\rho c_p}\quad(\text{m}^2/\text{s})$$ It measures how quickly heat diffuses through a material relative to its heat storage capacity. High $\alpha$ means the material responds rapidly to temperature changes.

Explain the concept of critical radius of insulation for a cylinder and give its formula.

For a cylinder, adding insulation increases conduction resistance but decreases the outer convection resistance (larger area). The critical radius maximizes heat loss: $$r_c = \frac{k}{h}$$ Insulation reduces heat loss only when the outer radius exceeds $r_c$ (for a sphere, $r_c = \frac{2k}{h}$).