Fluids, Surface Tension, Viscosity & Elasticity :

Hydraulic Press:

$P=\frac{f}{a}=\frac{F}{A} \text { or } F=\frac{A}{a} \times f$

Hydrostatic Paradox: $P_{A}=P_{B}=P_{C}$

(i) Liquid placed in elevator : When elevator accelerates upward with acceleration $a_{0}$ then pressure in the fluid, at depth ’ $h$ ’ may be given by,

$ P=\rho h\left[g+a_{0}\right] $

and force of buoyancy, $B=m\left(g+a_{0}\right)$

(ii) Free surface of liquid in horizontal acceleration :

$\tan \theta=\frac{a_{0}}{g}$

$P_{1}-P_{2}=\rho v a_{0}$

where $P_{1}$ and $P_{2}$ are pressures at points 1 and 2.

Then $h_1-h_2=\frac{v a_0}{g}$

(iii) Free surface of liquid in case of rotating cylinder:

$h=\frac{v^{2}}{2 g}=\frac{\omega^{2} r^{2}}{2 g}$

Equation of Continuity:

$a_{1} v_{1}=a_{2} v_{2}$

In general, av = constant.

Bernoulli’s Theorem:

$\frac{P}{\rho}+\frac{1}{2} v^{2}+g h= \text{constant}$

Torricelli’s theorem:

Speed of efflux: $v=\sqrt{\frac{2 g h}{1-\frac{A_{2}^{2}}{A_{1}{ }^{2}}}}$

$A_{2}=$ area of hole, ${A}_{1}=$ area of vessel.

Elasticity and Viscosity:

$\text{Stress}=\frac{\text { restoring force }}{\text { area of the body }}=\frac{\mathrm{F}}{\mathrm{A}}$

Strain, $\in=\frac{\text { change in configuration }}{\text { original configuration }}$

(i) Longitudinal strain $=\frac{\Delta \mathrm{L}}{\mathrm{L}}$

(ii) $\epsilon_{\mathrm{v}}=$ volume strain $=\frac{\Delta \mathrm{V}}{\mathrm{V}}$

(iii) Shear Strain : $\tan \phi$ or $\phi=\frac{\mathrm{X}}{\ell}$

Young’s Modulus Of Elasticity:

$ \mathrm{Y}=\frac{\mathrm{F} / \mathrm{A}}{\Delta \mathrm{L} / \mathrm{L}}=\frac{\mathrm{FL}}{\mathrm{A} \Delta \mathrm{L}}$

Potential Energy Per Unit Volume:

$u =\frac{1}{2}(\text{stress} \times \operatorname{strain})=\frac{1}{2}\left(\mathrm{Y} \times \operatorname{strain}^{2}\right)$

Inter-Atomic Force-Constant:

$\mathrm{k}=\mathrm{Yr}_{0}$

Newton’s Law of viscosity:

$F \propto A \frac{d v}{d x} \text { or } F=-\eta A \frac{d v}{d x}$

Stoke’s Law:

$F=6 \pi \eta r v $

Terminal velocity:

$ v_T=\frac{2}{9} \frac{r^{2}(\rho-\sigma) g}{\eta}$

Surface Tension

$T=\frac{\text { Total force on either of the imaginary line }(F)}{\text { Length of the line }(\ell)}$

$\mathrm{T}=\mathrm{S}=\frac{\Delta \mathrm{W}}{\mathrm{A}}$

Thus, surface tension is numerically equal to surface energy or work done per unit increase surface area.

• Inside a bubble : $\left(p-p_{a}\right)=\frac{4 T}{r}=p_{\text {excess }} ;$

• Inside the drop : $\left(p-p_{a}\right)=\frac{2 T}{r}=p_{\text {excess }}$

• Inside air bubble in a liquid : $\left(p-p_{a}\right)=\frac{2 T}{r}=p_{\text {excess }}$

• Capillary Rise $ h=\frac{2 T \cos \theta}{r \rho g}$

Relations Between Elastic Constants

E = Young’s modulus of elasticity,

G = Modulus of rigidity,

K = Bulk modulus,

$\nu$ = Poisson’s ratio

• Relationships between Young’s Modulus and Modulus of Rigidity: $E = 2G(1 + \nu)$

• Relationships between Young’s Modulus and Bulk Modulus: $E = 3K(1 - 2\nu)$

• Relationships between Modulus of Rigidity and Bulk Modulus: $G = \frac{3K(1 - 2\nu)}{2(1 + \nu)}$

• Relationships between Poisson’s Ratio $\nu = \frac{3K - 2G}{2(3K + G)}$

• Relation involving all three moduli: $E = 9KG / (3K + G)$

Bulk Modulus

$B=-P/(\Delta V / V)$

Compressibility

$k=(1 / B)=-(1 / \Delta P) \times(\Delta V / V)$