MechanicsHigh Yield (2 Qs in JEE)

Fluid Mechanics & Surface Tension Formulas

Pascal's law, Archimedes principle, continuity equation, Bernoulli's theorem, viscosity (Stokes law, Poiseuille), and surface tension.

7 Formulas•368+ Questions in Bank

Core Formulas & Mathematical Identities

A_1 v_1 = A_2 v_2, \quad P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}

Mass and energy conservation in ideal streamline fluid flow.

Examiner Tip: Torricelli's speed of efflux: v = √(2gh).

F_v = 6\pi\eta r v, \quad v_t = \frac{2}{9}\frac{r^2(\rho - \sigma)g}{\eta}

Viscous drag force on sphere and steady terminal fall speed.

Examiner Tip: Terminal velocity is proportional to the square of sphere radius (v_t ∝ r²).

\Delta P_{\text{drop}} = \frac{2T}{R}, \quad \Delta P_{\text{soap bubble}} = \frac{4T}{R}

Pressure difference across curved liquid interface due to surface tension T.

Examiner Tip: Soap bubble has 2 free surfaces, so excess pressure is doubled.

h = \frac{2T\cos\theta}{\rho g r}

Height of liquid column climbing a capillary tube of radius r with contact angle θ.

Examiner Tip: For pure water in clean glass, contact angle θ ≈ 0° (cos θ = 1).

v = \sqrt{2gh}, \quad R = 2\sqrt{h(H - h)}, \quad R_{\text{max}} = H \; (\text{at } h = H/2), \quad t_{\text{empty}} = \frac{A}{a}\sqrt{\frac{2H}{g}}

Speed of efflux from orifice at depth h below surface, horizontal range on ground, and total emptying time of tank.

Examiner Tip: Two holes at depths h and (H - h) yield identical horizontal ranges.

Q = \frac{\pi \Delta P r^4}{8\eta L}, \quad R_e = \frac{\rho v D}{\eta}

Volume flow rate Q through a circular capillary pipe and dimensionless Reynolds number determining flow stability.

Examiner Tip: Flow is laminar for R_e < 2000, unstable for 2000 < R_e < 3000, and turbulent for R_e > 3000.

W = T \Delta A, \quad R = n^{1/3}r, \quad \Delta U_{\text{released}} = 4\pi T r^2 (n - n^{2/3}), \quad \Delta T = \frac{3T}{s \rho}\left(\frac{1}{r} - \frac{1}{R}\right)

Work done in changing surface area, and heat released when n identical droplets coalesce into a single large drop.

Examiner Tip: When droplets coalesce, surface area decreases, releasing energy and warming the liquid.

Frequently tested misconceptions in Fluid Mechanics & Surface Tension

•Bernoulli's theorem applies ONLY to non-viscous, incompressible, irrotational streamline flow.
•Excess pressure in soap bubble has TWO surfaces: ΔP = 4T/R (liquid drop has only ONE surface: ΔP = 2T/R).