GATE ME Fluid Mechanics Formula Sheet 2027: Formulas, Examples, Traps

Which fluid mechanics formulas do you need for GATE ME 2027? A compact set, each safe only when you know the condition it carries. This sheet gives them topic by topic, with a worked example and the trap for each, checked against the last-minute sheet of the GATE ME 2027 book.

Key takeaways

• Every fluid formula carries a condition, such as steady or isentropic, and used outside it gives a confident wrong answer.
• The 2027 syllabus adds velocity potential and names converging and converging-diverging nozzles, so those rows are marked New in 2027.
• Bernoulli's equation joins two points on different streamlines only when the flow is irrotational.
• Many pipe questions reduce to a scaling law, such as $f = 64/\text{Re}$ in laminar flow or $h_f \propto Q^2/D^5$ at a fixed friction factor.
• Compressible-flow relations need absolute pressure and temperature in kelvin.
• A converging nozzle chokes when the back pressure falls to $0.528\,p_0$ for $\gamma = 1.4$, and its exit Mach number never exceeds 1.

The terms and notation this sheet uses

A fluid deforms continuously under shear. Density $\rho$ is mass per unit volume, dynamic viscosity $\mu$ links shear stress to the velocity gradient, and kinematic viscosity is $\nu = \mu/\rho$. Units are SI, with $g = 9.81\ \text{m/s}^2$.

A flow is steady when nothing changes with time at a point, incompressible when density is constant, inviscid when viscosity is neglected, and irrotational when fluid elements do not spin. A streamline is everywhere tangent to the velocity. Gauge pressure is measured above atmospheric; absolute pressure is gauge plus atmospheric.

The stream function $\psi$ is constant along each streamline; the velocity potential $\phi$ is a function whose gradient is the velocity. Subscript 1 marks an inlet, 2 an exit, 0 the stagnation state.

How pressure acts in a fluid at rest

Pressure in a still fluid rises linearly with depth. The force on a submerged plane is the centroid pressure times the area, acting lower down at the centre of pressure.

Worked example. A vertical gate 2 m deep and 1 m wide has its top edge at the free surface of water:

It acts at two-thirds of the depth. For capillary rise of water in a 1 mm tube with $\sigma = 0.073$ N/m and $\theta = 0$, $\displaystyle h = \frac{4 \times 0.073}{1000 \times 9.81 \times 0.001} = 0.0298$ m, about 29.8 mm.

Trap: a diameter left in millimetres inside the capillary formula is out by a factor of 1000.

When a floating body is stable

Buoyancy equals the weight of fluid displaced, acting through the centre of buoyancy B. Stability depends on the metacentric height:

$I$ is the second moment of the waterplane area about the tilt axis; $V$ is the displaced volume. A floating body is stable if $GM > 0$; a submerged one if B lies above G. The rolling period is $T = 2\pi k/\sqrt{g\,GM}$.

Worked example. A pontoon 10 m long and 4 m wide floats at a 1 m draft, with G 1.2 m above the keel:

It is stable. Trap: for rolling, the width is cubed, not the length.

How a fluid moves: acceleration, stream function and velocity potential

A particle accelerates in steady flow by moving to where the velocity differs: the convective part. The local part is change with time at a point.

Worked example. Take $\psi = 3xy$:

At $(1, 2)$: $a_x = u\,\partial u/\partial x + v\,\partial u/\partial y = 9$ and $a_y = u\,\partial v/\partial x + v\,\partial v/\partial y = 18$, so $\vert \vec{a} \vert = \sqrt{405} = 20.1\ \text{m/s}^2$. The flow is steady, yet it accelerates.

In one line: The stream function exists for any two-dimensional incompressible flow, the velocity potential only for an irrotational one; where both exist, lines of constant $\phi$ cross streamlines at right angles.

Momentum on a control volume and forces on vanes

For steady flow, the net force on the fluid in a control volume equals momentum out minus momentum in, per component:

Worked example. A water jet 50 mm in diameter at 20 m/s has $A = 1.963 \times 10^{-3}\ \text{m}^2$. On a fixed plate, $F = 785.4$ N. On a single plate moving at 8 m/s, $F = 282.7$ N; on a series of plates at that speed, $F = 471.2$ N.

Trap: the force on the vane is minus the force on the fluid, and pressure forces $pA$ at each cut section belong in the balance.

Bernoulli's equation and flow measurement

Along a streamline, the pressure, velocity and elevation heads add to a constant:

It needs steady, incompressible, inviscid flow with no shaft work. The energy equation adds pump head, turbine head and losses, with kinetic-energy factor $\alpha = 2$ for laminar pipe flow, about 1.05 for turbulent.

Worked example. A pitot tube in air ($\rho = 1.2\ \text{kg/m}^3$) reading 800 Pa gives $V = \sqrt{1333.3} = 36.5$ m/s. A 0.1 m mercury deflection under water is $0.1 \times 12.6 = 1.26$ m of water.

Remember: Bernoulli's equation holds along a streamline; it joins points on different streamlines only when the flow is irrotational, and never across a pump, a loss or a shock.

The book's last-minute sheet covers all ten technical sections this way, and the GATE ME 2027 book adds 942 questions with worked solutions and 10 full mock tests, built on all six GATE ME papers 2022-2026 (both 2022 sets).

Dimensional analysis and model laws

Buckingham's $\pi$ theorem: $n$ variables in $m$ fundamental dimensions form $n - m$ dimensionless groups.

Worked example. A 1:25 spillway model runs at 2 m/s and 0.01 $\text{m}^3/\text{s}$. The prototype velocity is $2\sqrt{25} = 10$ m/s, and the discharge is $0.01 \times 25^{2.5} = 31.25\ \text{m}^3/\text{s}$.

Laminar flow and losses in pipes

In laminar pipe flow, viscosity alone sets the pressure drop, so $f$ depends only on Re.

Worked example. Oil with $\rho = 900\ \text{kg/m}^3$ and $\mu = 0.1$ Pa s flows at 1 m/s in a 50 mm pipe:

Check: Darcy-Weisbach gives $h_f = 1.450$ m and $\rho g h_f = 12.8$ kPa. A sudden expansion from 4 m/s to 1 m/s loses $9/19.62 = 0.459$ m.

Trap: halving the diameter at fixed flow and $f$ raises the head loss $2^5 = 32$ times.

Boundary layers on a flat plate

A boundary layer is the thin region near a wall where viscosity slows the flow. On a flat plate it turns turbulent near $\text{Re}_x = 5 \times 10^5$.

Worked example. Air with $\nu = 1.5 \times 10^{-5}\ \text{m}^2/\text{s}$ at 3 m/s, 0.5 m from the leading edge: $\text{Re}_x = 10^5$, laminar, so $\delta = 5 \times 0.5/316.2 = 7.9$ mm. Trap: laminar $\delta$ grows as $\sqrt{x}$; wall shear falls as $1/\sqrt{x}$.

Compressible flow and nozzles

Every row here is New in 2027: the syllabus now reads "one dimensional high speed compressible fluid flow; flow through converging and converging-diverging nozzles".

Worked example 1. Air at 300 K moves at $M = 2$. Then $c = \sqrt{1.4 \times 287 \times 300} = 347.2$ m/s and $V = 694.4$ m/s. Also $T_0/T = 1 + 0.2 \times 4 = 1.8$, so $T_0 = 540$ K and $p_0/p = 1.8^{3.5} = 7.82$.

Worked example 2. A nozzle with a 10 $\text{cm}^2$ throat draws air from $p_0 = 500$ kPa absolute and $T_0 = 400$ K:

It chokes at back pressures up to $0.528 \times 500 = 264$ kPa. For a normal shock at $M_1 = 2$: $M_2 = 0.577$, $p_2/p_1 = 4.5$, $T_2/T_1 = 1.6875$ and $p_{02}/p_{01} = 0.721$. Practise powers such as $1.8^{3.5}$ on the GATE virtual calculator.

A choked converging-diverging nozzle delivers shock-free supersonic flow at its design back pressure. At higher back pressures that still choke it, a normal shock stands in the diverging part, moving towards the throat as back pressure rises.

Trap: A converging nozzle never gives an exit Mach number above 1, and every ratio here needs absolute pressure and kelvin; a gauge or Celsius value gives a plausible wrong answer.

More formula sheets: all of GATE ME · Heat Transfer · Strength of Materials · Thermodynamics

Quick revision

1. Force on a plane: $\rho g \bar{h} A$ at depth $\bar{h} + I_G \sin^2\theta/(A\bar{h})$.
2. Floating stability: $GM = I/V - BG > 0$, with $I$ about the tilt axis.
3. $u = \partial \psi/\partial y$, $v = -\partial \psi/\partial x$; $\phi$ exists only if $\omega_z = 0$, and then $\nabla^2 \phi = 0$ for incompressible flow.
4. Bernoulli: along a streamline; across streamlines only if irrotational; never through a pump, loss or shock.
5. Laminar pipe: $f = 64/\text{Re}$ and $V_{\max} = 2V$; at fixed $f$, $h_f \propto Q^2/D^5$.
6. Laminar plate: $\delta/x \approx 5/\sqrt{\text{Re}_x}$, transition near $5 \times 10^5$.
7. Nozzles: choke at $0.528\,p_0$; at $M_1 = 2$ a normal shock gives $M_2 = 0.577$ and $p_2/p_1 = 4.5$.
8. Recheck units, kelvin and absolute pressure before typing a NAT answer; NATs carry no negative marking under the marking scheme common to every GATE paper. See the exam-day rules and the GATE 2027 exam dates.