UPSC IES/ESE (Engineering Services) Thermodynamics, Fluid Mechanics and Heat Transfer Syllabus

What is the dual (mixed/limited-pressure) cycle?

An air-standard cycle in which heat is added partly at constant volume and partly at constant pressure, combining features of the Otto and Diesel cycles; it models modern compression-ignition engines.

Define density, specific weight, specific gravity and dynamic viscosity of a fluid.

Density $\rho = m/V$ (kg/m³); specific weight $\gamma = \rho g$ (N/m³); specific gravity $S = \rho/\rho_{water}$ (dimensionless); dynamic viscosity $\mu$ relates shear stress and velocity gradient via $\tau = \mu \dfrac{du}{dy}$ (Pa·s).

Distinguish dynamic viscosity from kinematic viscosity.

Dynamic viscosity $\mu$ (units Pa·s) measures resistance to shear: $\tau = \mu\,\dfrac{du}{dy}$. Kinematic viscosity $\nu = \dfrac{\mu}{\rho}$ (units m²/s) is the ratio of dynamic viscosity to density, representing momentum diffusivity.

State Pascal's law and the hydrostatic pressure variation with depth.

Pascal's law: pressure at a point in a fluid at rest is equal in all directions. Hydrostatic variation: $p = p_0 + \rho g h$, so pressure increases linearly with depth $h$ below a free surface.

How does a simple piezometer / U-tube manometer measure pressure?

A piezometer reads gauge pressure as $p = \rho g h$ from the liquid column height. A U-tube manometer balances columns: for a manometer fluid of density $\rho_m$, $p_A = \rho_m g h_m - \rho g a + \ldots$, equating pressures at a common datum level along the connected fluid.

What is the principle of a differential manometer and the expression for pressure difference?

It measures the difference between two points using a heavier manometric fluid. $p_A - p_B = g h \,(\rho_{m} - \rho)$ for a simple U-tube differential manometer with reading $h$, manometer fluid density $\rho_m$ and pipe fluid density $\rho$ (same fluid both limbs, same level).

Give the total hydrostatic force on a plane submerged surface and the depth of its centre of pressure.

Total force $F = \rho g \bar{h} A$, acting at centre of pressure $h^* = \bar{h} + \dfrac{I_G \sin^2\theta}{A\bar{h}}$, where $\bar{h}$ is the centroid depth, $A$ the area, $I_G$ the second moment about the centroidal axis, and $\theta$ the plane's inclination to the horizontal.

State Archimedes' principle and the condition for floating.

Archimedes' principle: a body wholly or partly immersed experiences an upward buoyant force equal to the weight of fluid displaced, $F_B = \rho g V_{disp}$. A body floats when its weight equals the buoyant force of the displaced fluid.

Define metacentre and metacentric height, and give the stability criterion for a floating body.

The metacentre $M$ is the point where the line of buoyancy intersects the body's axis after a small tilt. Metacentric height $GM = \dfrac{I}{V} - BG$, where $I$ is the waterplane second moment, $V$ displaced volume, $BG$ centre-of-buoyancy to centre-of-gravity distance. Stable if $M$ is above $G$ ($GM > 0$).

Write the continuity equation for steady incompressible flow (1-D and differential form).

1-D: $A_1 V_1 = A_2 V_2 = \text{constant}$. Differential (incompressible): $\dfrac{\partial u}{\partial x} + \dfrac{\partial v}{\partial y} + \dfrac{\partial w}{\partial z} = 0$, i.e. $\nabla \cdot \vec{V} = 0$.

Define the stream function $\psi$ and state two of its properties.

For 2-D incompressible flow, $u = \dfrac{\partial \psi}{\partial y}$, $v = -\dfrac{\partial \psi}{\partial x}$. Properties: (1) it exists for any incompressible flow (satisfies continuity automatically); (2) the difference $\psi_2 - \psi_1$ between two streamlines equals the volume flow rate between them; lines of constant $\psi$ are streamlines.

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

$u = -\dfrac{\partial \phi}{\partial x}$, $v = -\dfrac{\partial \phi}{\partial y}$, $w = -\dfrac{\partial \phi}{\partial z}$. It exists only for irrotational flow (zero vorticity). For incompressible irrotational flow $\phi$ satisfies the Laplace equation $\nabla^2 \phi = 0$.

What is the relationship between streamlines (constant $\psi$) and equipotential lines (constant $\phi$)?

In a flow net, streamlines and equipotential lines are mutually orthogonal (intersect at right angles). The product of their slopes is $-1$, and the Cauchy-Riemann relations $\dfrac{\partial \phi}{\partial x} = \dfrac{\partial \psi}{\partial y}$, $\dfrac{\partial \phi}{\partial y} = -\dfrac{\partial \psi}{\partial x}$ hold for irrotational incompressible flow.

Distinguish a streamline, pathline and streakline.

Streamline: a curve everywhere tangent to the instantaneous velocity vector. Pathline: the actual trajectory traced by a single fluid particle over time. Streakline: the locus of all particles that have passed through a fixed point. In steady flow all three coincide.

Classify types of fluid flow with examples.

Steady vs unsteady (time dependence), uniform vs non-uniform (spatial variation), laminar vs turbulent (orderly vs chaotic), compressible vs incompressible (density change), rotational vs irrotational (vorticity), and one-, two- or three-dimensional flow.

State Bernoulli's equation for steady, incompressible, inviscid flow and list its assumptions.

$\dfrac{p}{\rho g} + \dfrac{V^2}{2g} + z = \text{constant}$ along a streamline. Assumptions: steady flow, incompressible fluid, inviscid (no friction losses), flow along a streamline, and no energy added or removed.

Name three practical applications of Bernoulli's equation.

Flow measurement devices (venturimeter, orificemeter, Pitot tube), the theory of lift on aerofoils, and analysis of flow through nozzles, siphons, and pipe systems (with head-loss terms added).

Write the linear momentum equation for a control volume (steady flow).

$\sum \vec{F} = \dot{m}\,(\vec{V}_{out} - \vec{V}_{in})$, i.e. the net external force equals the rate of change of momentum flux: $\sum \vec{F} = \int_{CS} \vec{V}(\rho \vec{V}\cdot d\vec{A})$.

Give the force exerted by fluid flowing through a pipe bend (reducing bend, x and y components).

$F_x = (p_1 A_1 + \rho Q V_1) - (p_2 A_2 + \rho Q V_2)\cos\theta$ and $F_y = (p_2 A_2 + \rho Q V_2)\sin\theta$, where $\theta$ is the bend angle; the resultant on the bend is $R = \sqrt{F_x^2 + F_y^2}$ (reaction is opposite).

Define Reynolds number and give the laminar/turbulent transition values for pipe flow.

$Re = \dfrac{\rho V D}{\mu} = \dfrac{V D}{\nu}$. For pipe flow: laminar for $Re < 2000$, transitional for $2000 < Re < 4000$, and turbulent for $Re > 4000$.

For fully developed laminar flow in a circular pipe (Hagen-Poiseuille), give the velocity profile and friction factor.

Parabolic profile with $V_{max} = 2\,V_{avg}$; pressure drop $\Delta p = \dfrac{128 \mu L Q}{\pi D^4}$. Darcy friction factor $f = \dfrac{64}{Re}$ (independent of roughness).

Write the Darcy-Weisbach equation for major head loss in pipe flow.

$h_f = f \dfrac{L}{D}\dfrac{V^2}{2g}$, where $f$ is the Darcy friction factor, $L$ the pipe length, $D$ the diameter and $V$ the mean velocity. (With the Fanning factor $f' = f/4$.)

Distinguish major and minor losses in pipe flow and give the form of minor loss.

Major losses are due to wall friction over the pipe length (Darcy-Weisbach). Minor losses are due to fittings, bends, valves, sudden expansions/contractions and entrances: $h_L = K\dfrac{V^2}{2g}$, where $K$ is a loss coefficient. Sudden expansion: $h_L = \dfrac{(V_1 - V_2)^2}{2g}$.

List the common flow-measurement devices and what each fundamentally measures.

Venturimeter and orificemeter (measure discharge via a pressure-drop / differential head); Pitot tube (measures point/velocity via stagnation pressure); notches and weirs (measure open-channel discharge via head over the crest); rotameter (variable-area flow rate).

Write the discharge equation for a venturimeter.

$Q = C_d \dfrac{A_1 A_2}{\sqrt{A_1^2 - A_2^2}}\sqrt{2 g h}$, where $A_1$, $A_2$ are inlet and throat areas, $h$ the differential head, and $C_d \approx 0.95\text{-}0.98$ is the coefficient of discharge.

Compare a venturimeter and an orificemeter.

Both use a pressure drop across a constriction with $Q = C_d \dfrac{A_1 A_2}{\sqrt{A_1^2 - A_2^2}}\sqrt{2gh}$. The venturimeter has a gradual converging-diverging section, high $C_d$ ($\approx 0.98$) and low head loss but is bulky and costly. The orificemeter is a thin plate, cheap and compact, with low $C_d$ ($\approx 0.6$) and high permanent head loss due to flow separation.

Explain the principle of a Pitot tube and give the velocity expression.

It measures stagnation (total) pressure where the flow is brought to rest, and relates it to the dynamic head. Velocity $V = C_v\sqrt{2 g h}$, where $h$ is the difference between stagnation and static heads and $C_v$ the coefficient of velocity (a Pitot-static tube reads both static and stagnation pressures).

Give the discharge formula for a rectangular and a triangular (V) notch.

Rectangular notch: $Q = \dfrac{2}{3} C_d \, L \sqrt{2g}\; H^{3/2}$. Triangular (V-) notch of apex angle $\theta$: $Q = \dfrac{8}{15} C_d \sqrt{2g}\,\tan\!\dfrac{\theta}{2}\; H^{5/2}$, where $H$ is the head over the crest.

Define the boundary layer and state Prandtl's boundary-layer concept.

The boundary layer is the thin region near a solid surface where viscous effects are significant and velocity rises from zero at the wall to ~99% of the free-stream value. Prandtl's concept: the flow field can be split into a thin viscous boundary layer near the wall and an outer essentially inviscid (potential) flow.

What is boundary-layer separation and what causes it?

Separation is the detachment of the boundary layer from the surface, leaving a region of reversed/recirculating flow. It is caused by an adverse pressure gradient ($\dfrac{dp}{dx} > 0$) combined with wall friction, which decelerates near-wall fluid until $\left(\dfrac{\partial u}{\partial y}\right)_{y=0} = 0$; it increases pressure (form) drag.

Define drag and lift, and write the drag and lift force expressions for flow over a body.

Drag is the force component parallel to the relative flow; lift is perpendicular to it. $F_D = C_D \cdot \dfrac{1}{2}\rho V^2 A$ and $F_L = C_L \cdot \dfrac{1}{2}\rho V^2 A$, where $C_D$, $C_L$ are the drag and lift coefficients, $\rho$ the fluid density, $V$ the relative velocity and $A$ a reference area.