JEE Physics Mechanical Properties Of Fluids Notes Set 08

Fluid Pressure and Hydrostatics

A fluid at rest exerts a force perpendicular to any surface in contact with it. Pressure is force per unit area, \( P=\dfrac{F}{A} \), a scalar with SI unit pascal. At depth \( h \) in a liquid of density \( \rho \) the absolute pressure is \( P=P_0+\rho gh \), and the gauge pressure is \( \rho gh \). Pressure is the same at all points on one horizontal level in a connected liquid at rest. Useful units: \( 1\ \text{atm}=1.013\times10^{5}\ \text{Pa}=76\ \text{cm Hg} \).

Pascal's law: a pressure change applied to an enclosed fluid is transmitted undiminished to every part of it. A hydraulic lift uses this: \( \dfrac{F_1}{A_1}=\dfrac{F_2}{A_2} \), so a small force on the small piston balances a large load on the large one, while the displacements obey \( A_1x_1=A_2x_2 \) (work is conserved).

In a U-tube with two immiscible liquids, the heights of the columns above the common interface are inversely proportional to the densities: \( h_1\rho_1=h_2\rho_2 \). The force on a vertical wall of width \( b \) and depth \( H \) is \( \tfrac12\rho gbH^{2} \) and acts at \( \tfrac23H \) below the surface.

Liquids in Accelerated Containers

In a frame with acceleration \( a \) the liquid behaves as if gravity were replaced by an effective gravity \( \vec g_{eff}=\vec g-\vec a \). In a lift accelerating upward, \( P=\rho(g+a)h \); downward, \( P=\rho(g-a)h \), and in free fall the pressure difference vanishes. For horizontal acceleration the free surface tilts through \( \tan\theta=\dfrac{a}{g} \) and the pressure at depth \( h \) is \( \rho h\sqrt{g^{2}+a^{2}} \) measured along the effective vertical. In a vessel rotating with angular speed \( \omega \) the surface is a paraboloid with \( y=\dfrac{\omega^{2}r^{2}}{2g} \). For a tank of length \( l \) accelerated along its length, the level rises at the rear by \( \dfrac{al}{2g} \) above the mean level.

Buoyancy and Floating

Archimedes' principle: a body wholly or partly immersed experiences an upward force equal to the weight of the displaced fluid, acting at the centre of buoyancy (the centroid of the displaced volume). For a body of density \( \rho_b \) in a liquid of density \( \sigma \): it sinks if \( \rho_b>\sigma \), floats with a fraction \( \dfrac{\rho_b}{\sigma} \) of its volume submerged if \( \rho_b<\sigma \), and stays at any depth if the densities are equal. Apparent weight in the liquid is \( W\left(1-\dfrac{\sigma}{\rho_b}\right) \).

1. A floating body in a lift or an accelerated vessel keeps the same submerged fraction, since weight and buoyancy both scale with \( g_{eff} \).
2. When ice floating in water melts, the water level does not change; if the ice holds a heavier object the level falls, and if it holds a lighter object the level rises.
3. A body hanging by a string of tension \( T \) in a liquid satisfies \( T+B=W \); a body held down by a string satisfies \( B=W+T \).
4. A body floating between two immiscible liquids settles at their interface when its density lies between the two densities.
5. A floating body is stable when the metacentre lies above the centre of gravity.

Streamline Flow, Continuity and Bernoulli's Equation

For steady, incompressible, non-viscous flow the volume flow rate is the same through every cross-section: \( A_1v_1=A_2v_2 \). Bernoulli's equation, a statement of energy conservation per unit volume, is

\( P+\tfrac12\rho v^{2}+\rho gh=\text{constant} \)

The terms are static pressure, kinetic energy density and potential energy density. Where the speed is high the pressure is low, which explains the lift on a wing, the curving of a spinning ball and the spray from an atomiser.

1. Torricelli's theorem: speed of efflux from a hole at depth \( h \) below the open surface is \( v=\sqrt{2gh} \). If the surface falls with speed \( V \) (for a large hole), use \( v^{2}-V^{2}=2gh \) with \( AV=av \).
2. The horizontal range of the jet from a hole at height \( y \) above the ground when the liquid depth above the hole is \( h \) is \( R=2\sqrt{h\,y} \); it is the same for holes at depths \( h \) and \( H-h \) in a tank of depth \( H \), and is maximum when the hole is at mid-height, \( R_{max}=H \).
3. A freely falling stream narrows because its speed increases: \( A_2=A_1\dfrac{v_1}{v_2} \).
4. The Venturi meter gives the speed from the pressure drop between a wide and a narrow section: \( v_1=A_2\sqrt{\dfrac{2\Delta P}{\rho(A_1^{2}-A_2^{2})}} \).
5. Reaction force of a jet: \( F=\rho Av^{2} \) for a jet of cross-section \( A \) and speed \( v \).

Surface Tension and Surface Energy

The surface layer of a liquid behaves like a stretched elastic membrane. Surface tension \( T \) is the force per unit length acting along a line drawn in the surface, perpendicular to it and tangential to the surface, so \( T=\dfrac{F}{l} \) (N/m). It is also the surface energy per unit area, \( T=\dfrac{W}{\Delta A} \) (J/m²). It falls with rising temperature and is reduced by detergents and impurities that dissolve.

A soap film has two surfaces, so a film pulling on a wire of length \( l \) exerts \( 2Tl \) and the work for an area increase \( \Delta A \) of the frame is \( 2T\,\Delta A \). When \( n \) small drops of radius \( r \) merge into one drop of radius \( R \) (so \( R=n^{1/3}r \)) the energy released appears as heat: \( W=4\pi T(nr^{2}-R^{2})=4\pi TR^{3}\left(\dfrac1r-\dfrac1R\right) \), and the temperature rise is \( \Delta\theta=\dfrac{3T}{\rho s}\left(\dfrac1r-\dfrac1R\right) \). Splitting a drop into smaller drops costs the same amount of energy.

Excess Pressure and Angle of Contact

The pressure is greater on the concave side. Between two soap bubbles of radii \( r_1<r_2 \) air flows from the smaller to the larger bubble, and the common film has radius \( \dfrac{r_1r_2}{r_2-r_1} \). Inside a bubble at depth, the total gas pressure is \( P_0+h\rho g+\dfrac{2T}{r} \).

The angle of contact \( \theta \) is the angle between the liquid surface and the solid, measured inside the liquid. It is acute for wetting liquids (water on glass, meniscus concave, liquid rises) and obtuse for non-wetting liquids (mercury on glass, meniscus convex, liquid is depressed).

Capillarity

In a tube of radius \( r \) the liquid rises (or falls) by

\( h=\dfrac{2T\cos\theta}{r\rho g} \)

so the rise is inversely proportional to the radius, and is proportional to \( T\cos\theta/\rho \). The height measured is that of the lowest point of the meniscus. Two parallel plates at separation \( d \) give \( h=\dfrac{2T\cos\theta}{d\rho g} \). If the tube is shorter than \( h \) the liquid does not overflow; the meniscus flattens until its radius satisfies \( R=\dfrac{2T}{H\rho g} \) (for zero contact angle). In a freely falling vessel the rise fills the whole tube because \( g_{eff}=0 \). In a tube of non-uniform bore use the radius at the meniscus position, with the wall tilt added to the contact angle.

Viscosity and Stokes' Law

Viscosity is the internal friction between layers of a flowing fluid. For layers with velocity gradient \( \dfrac{dv}{dx} \) the viscous force is \( F=\eta A\dfrac{dv}{dx} \), with coefficient of viscosity \( \eta \) (SI unit Pa s or decapoise; 1 poise = 0.1 Pa s). It falls with temperature for liquids and rises for gases. Fluids obeying this linear relation are Newtonian.

Stokes' law: a sphere of radius \( r \) moving at speed \( v \) through a fluid of viscosity \( \eta \) feels the drag \( F=6\pi\eta rv \). A sphere of density \( \rho \) falling through a fluid of density \( \sigma \) reaches terminal velocity when weight equals buoyancy plus drag:

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

so \( v_T\propto r^{2} \) and the terminal velocity is negative (upward) when \( \sigma>\rho \), which is why air bubbles rise. Falling drops merging into a bigger drop gain terminal velocity as \( R^{2} \). If the drag is instead proportional to \( v^{2} \) (large bodies in air), then \( v_T\propto\sqrt{r} \) for a sphere. At the start of the fall the acceleration is \( g\left(1-\dfrac\sigma\rho\right) \).

Flow Regimes and Reynolds Number

Flow is streamline (laminar) at low speed and turbulent at high speed. The Reynolds number \( R_e=\dfrac{\rho vd}{\eta} \) decides the regime: for a pipe flow is laminar below about 1000 and turbulent above about 2000, with an unstable transition between. Poiseuille's formula for laminar flow through a pipe of radius \( r \) and length \( l \) under pressure difference \( P \) is \( Q=\dfrac{\pi Pr^{4}}{8\eta l} \), so the flow rate is proportional to \( r^{4} \) and the fluid resistance is \( \dfrac{8\eta l}{\pi r^{4}} \). Resistances in series or parallel combine as in electrical circuits.