Mechanical Properties of Fluids

2.1–2.2 Fluid and Ideal Fluid

• A fluid is any substance that can flow — it deforms continually under an external force. Liquids, gases and plasmas are fluids.
• The shear modulus of a fluid is zero: a fluid cannot resist a shear (tangential) stress in static equilibrium. Ideal fluids can be subjected only to normal, compressive stress, which is pressure.
• Properties of fluids: they do not oppose deformation (they get permanently deformed), they can flow, and they take the shape of the container.
• An ideal fluid is (i) incompressible — constant density, (ii) irrotational — smooth flow, no turbulence, (iii) non-viscous — no internal friction, (iv) steady — velocity at each point constant in time.
• Real fluids display viscosity and can therefore sustain low levels of shear stress. Liquid helium at about 2 K is the only almost non-viscous fluid.
• Hydrostatics = study of fluids at rest. Hydrodynamics = study of fluids in motion.

2.3 Pressure

• Pressure = normal force per unit area of contact, $p = F/A$. Scalar. SI unit pascal; $[L^{-1}M^{1}T^{-2}]$.
• Pressure due to a liquid column: $p = h\rho g$. It does not depend on the area of the imaginary column, nor on the shape or size of the container, and holds for gases as well.
• Absolute pressure at depth $h$: $p = p_{0} + h\rho g$. Gauge pressure $= p - p_{0} = h\rho g$.
• Atmospheric pressure = weight of a column of air of unit cross section extending from that point to the top of the atmosphere. Highest at sea level; the effective air column is about 8 km. Normal atmospheric pressure = $1.013\times10^{5}$ Pa = 76 cm of Hg.
• Vacuum = region where the gas pressure is much less than atmospheric. Perfect vacuum = no matter at all.
• Hydrostatic paradox: in interconnected vessels of different shapes the liquid stands at the same level, because the pressure at a point depends only on the height of the liquid column above it. The walls of a slanted vessel push on the liquid with forces normal to the wall whose vertical components support the extra weight.
• Pascal's law: pressure applied at any point of an enclosed fluid at rest is transmitted equally and undiminished to every point of the fluid and to the walls of the container, provided gravity is neglected.
• Applications: hydraulic lift and hydraulic brakes, both using $F_{2} = (A_{2}/A_{1})F_{1}$.
• Mercury barometer (Torricelli): $p_{0} = h\rho g$, with Torricelli's vacuum above the column. Open tube manometer: $p = p_{0} + h\rho g$, so it reads the gauge pressure $h\rho g$.

2.4 Surface Tension — definitions

• Cohesive force: force of attraction between molecules of the same substance. Adhesive force: between molecules of different substances.
• Range of molecular force: the maximum distance up to which the molecular force is effective — of the order of a few nanometres, hence a short range force.
• Sphere of influence: an imaginary sphere with a molecule at its centre and radius equal to the molecular range.
• Surface film: the surface layer of a liquid of thickness equal to the range of intermolecular force.
• Free surface: the surface of a fluid which experiences no shear stress.
• Surface tension $T$: the tangential force acting per unit length on both sides of an imaginary line drawn on the free surface of a liquid. $T = F/L$; N m$^{-1}$; $[L^{0}M^{1}T^{-2}]$.
• Surface energy: the extra potential energy of the molecules in the surface layer of a liquid. Work done isothermally to increase the area by $dA$ is $dW = T\,dA$, so surface tension = surface energy per unit area (N m$^{-1}$ = J m$^{-2}$).
• Angle of contact $\theta$: the angle between the tangent to the free liquid surface and the tangent to the solid surface at the point of contact, measured within the liquid.
• Capillarity: the rise or fall of a liquid inside a capillary tube dipped in it.

2.4 Surface Tension — standard results

2.4 Shape of meniscus and angle of contact

• Typical values: pure water on clean glass $0^\circ$; ether–glass $16^\circ$; kerosene–glass $26^\circ$; water–paraffin $107^\circ$; mercury–glass $140^\circ$.
• Angle of contact is constant for a given solid–liquid pair at a given temperature; it changes with impurity and decreases with rise in temperature; a small contamination changes it greatly.
• Drop on a solid surface: $\cos\theta = (T_{2} - T_{1})/T_{3}$. If $(T_{2} - T_{1}) > T_{3}$ then $\cos\theta > 1$, which is impossible, and the liquid spreads instead of forming a drop.

2.4 Effect of impurity and temperature on surface tension

• Soluble impurity (common salt in water): surface tension increases.
• Sparingly soluble impurity (phenol, detergent): surface tension decreases — that is why detergent is added while washing clothes.
• Insoluble impurity (dust on mercury): surface tension decreases.
• For most liquids surface tension decreases as temperature rises; for molten copper and molten cadmium it increases. Surface tension becomes zero at the critical temperature.

2.4 Excess pressure across a curved surface

• Plane surface: the resultant force of surface tension on a molecule just below the surface is zero, so the pressure just above and just below is the same.
• Upper convex surface: the resultant force of surface tension is vertically downwards, adds to the atmospheric force, so the pressure just below (concave side) is greater.
• Upper concave surface: the resultant force of surface tension is vertically upwards, opposes the atmospheric force, so the pressure just below (convex side) is smaller.
• In every case: pressure on the concave side is greater than the pressure on the convex side.

2.5 Fluids in Motion

• Flow line: the path of an individual particle in a moving fluid.
• Streamline: a curve whose tangent at any point is along the velocity of flow there. For a steady flow, streamlines and flow lines are identical.
• Flow tube: an imaginary bundle of flow lines bounded by an imaginary wall; in steady flow the fluid cannot cross its walls and adjacent tubes cannot mix.
• Steady flow: a measurable property such as pressure or velocity at a given point is constant in time.

2.6 Viscosity, critical velocity and Reynolds number

• Viscosity is that property of a fluid by virtue of which the relative motion between different layers of the fluid experiences a dragging force, called the viscous drag.
• In liquids the viscous drag is due to short range molecular cohesive forces; in gases it is due to collisions between fast moving molecules.
• Velocity gradient: the rate of change of velocity $dv$ with distance $dx$ measured from a stationary layer, $dv/dx$.
• Newton's law of viscosity: $f = \eta A (dv/dx)$, with $A$ the area of the layer parallel to the flow.
• Coefficient of viscosity = viscous force per unit area per unit velocity gradient. SI unit N s m$^{-2}$ (decapoise); $1$ poise $= 0.1$ N s m$^{-2}$.
• With rise in temperature the coefficient of viscosity of a liquid decreases but that of a gas increases.
• Critical velocity: the velocity beyond which a streamline flow becomes turbulent, $v_{c} = R_{n}\eta/\rho d$.
• Reynolds number $R_{n} = \rho v_{c} d/\eta$ is a pure number: $R_{n} < 1000$ streamline, $R_{n} > 2000$ turbulent, in between unsteady.

2.7 Stokes' law and terminal velocity

• Stokes' law: the viscous force on a small sphere falling through a viscous medium is directly proportional to the radius of the sphere, its velocity and the coefficient of viscosity — $F_{v} = 6\pi\eta r v$, the constant $6\pi$ being obtained empirically.
• Three forces act on a sphere falling through a viscous fluid: weight downwards, viscous force upwards (increasing with speed) and buoyant force upwards (constant).
• Terminal velocity is the constant downward velocity reached when $F_{g} = F_{v} + F_{u}$; the whole discussion applies to streamline flow only.
• $v = \dfrac{2}{9}\dfrac{r^{2}g(\rho - \sigma)}{\eta}$, so $v \propto r^{2}$.
• A body can never move through a viscous fluid faster than the terminal velocity for that body in that fluid.

2.8 Equation of continuity

• Statement: for a steady flow of an incompressible fluid, $A_{1}v_{1} = A_{2}v_{2}$, i.e. $Av$ = constant. It expresses conservation of mass.
• The volume rate of flow $dV/dt = Av$ is the volume flux; $\rho\,dV/dt = dm/dt$ is the mass flux.
• A particle moves faster through a narrow section and slower through a wider section, because there is less space to accommodate the particles behind it; otherwise the incompressible fluid would have to be compressed.

2.9 Bernoulli's equation

• Statement: the work done per unit volume of a fluid by the surrounding fluid is equal to the sum of the changes in kinetic energy and potential energy per unit volume during the flow. It is an application of the work–energy theorem, and is consistent with conservation of energy.
• $p + \dfrac{1}{2}\rho v^{2} + \rho g h =$ constant, for an ideal fluid in steady flow.
• Bernoulli's principle: where the speed of a fluid is greater, its internal pressure is smaller.
• Applications: speed of efflux (Torricelli's law $v = \sqrt{2gh}$), Venturi tube, dynamic lift on an aeroplane wing, atomizer / paint gun / carburettor, blowing off of roofs by stormy wind.
• Aeroplane: streamlines crowd above the aerodynamically shaped wing, so the air moves faster and the pressure above falls while that below is unchanged; the resulting upward dynamic lift raises the plane when it exceeds its weight.
• Roof in a storm: fast wind over the roof lowers the pressure above it while the air inside stays at atmospheric pressure, so the roof is lifted and blown off.

Answer-in-brief points the board repeats

• Surface tension of paints and lubricating oils is kept low so that they spread easily over a large area.
• Work done in forming a soap bubble of radius $r$ is $W = 8\pi r^{2}T$ (two surfaces).
• The basis of Bernoulli's principle is the conservation of energy (the work–energy theorem applied to a fluid element).
• A low density manometric liquid is used because it gives a larger difference of levels for the same pressure difference, so the reading is more sensitive.
• An incompressible fluid is one whose density is constant, i.e. does not change with pressure.
• Two or more mercury drops merge into one because a single drop has a smaller total surface area, and hence a smaller surface energy; the system moves to minimum potential energy and energy is released.
• Water speeds up on entering a narrow pipe because of the equation of continuity, $Av$ = constant.
• At a constriction in a horizontal pipe the speed increases by continuity, and Bernoulli's equation then requires the pressure to fall.

Traps that cost marks

• A soap bubble has two surfaces, a liquid drop or an air bubble inside a liquid has one. Use $4T/r$ only for a soap bubble, and $2T/r$ for a drop or for a bubble in liquid.
• In work-done problems, remember the factor 2 for a soap film or bubble; and always convert dyne cm$^{-1}$ to N m$^{-1}$ ($1$ dyne cm$^{-1} = 10^{-3}$ N m$^{-1}$) and erg to joule ($1$ erg $= 10^{-7}$ J).
• The excess pressure formula gives a pressure difference. If the question asks for the pressure inside a raindrop, add the atmospheric pressure.
• $p = h\rho g$ is the gauge pressure. If the question says "find the pressure" and gives the pressure on the free surface, the answer is $p_{0} + h\rho g$.
• In the terminal velocity formula $\rho$ is the density of the body and $\sigma$ that of the medium. For a rising bubble the answer is negative and the sign is part of the answer.
• In $\eta = f/[A(dv/dx)]$, $A$ is the area of the layer parallel to the flow, not the cross section, and $dx$ is the thickness of the fluid layer.
• Radius, not diameter, goes into $2T/r$, $6\pi\eta r v$ and $h = 2T\cos\theta/(r\rho g)$. Halve the given diameter first.
• Reynolds number has no unit and no dimension — do not attach one.
• Capillary rise depends on the radius, not the area. If the area is halved, $r$ falls by $\sqrt{2}$ and $h$ rises by $\sqrt{2}$.