Optional Topic Engineering Physics Fluid Mechanics Subdivisions of

Optional Topic: Engineering Physics - Fluid Mechanics. Subdivisions of matter solids liquids gases rigid will flow fluids dense and incompressible condensed matter Q: what about thick liquids and soft solids? low density compressible

Fluid mechanics Ordinary mechanics Mass and force identified with objects Fluid mechanics Mass and force “distributed”

Density and Pressure Density r for element of fluid mass DM volume DV for uniform density mass M volume V units kg m-3

Density and Pressure p force per unit area for uniform force units N m-2 or pascals (Pa) Atmospheric pressure at sea level on average 101. 3 x 103 Pa or p 0 101. 3 k. Pa Gauge pressure pg excess pressure above atmospheric p = pg + p 0

Density and Pressure Gauge pressure pg pressure in excess of atmospheric gauge atmospheric total p = pg + p 0 typical pressures total gauge atmospheric 1. 0 x 105 Pa 0 car tire 3. 5 x 105 Pa 2. 5 x 105 Pa deepest ocean 1. 1 x 108 Pa best vacuum 10 -12 Pa - 100 k. Pa

Example to pump 30 cms 15 cms The can shown has atmospheric pressure outside. The pump reduces the pressure inside to 1/4 atmospheric • What is the gauge pressure inside? • What is the net force on one side?

Fluids at rest (hydrostatics) Hydrostatic equilibrium laws of mechanical equilibrium pressure just above surface is atmospheric, p 0 hence, pressure just below surface must be same, i. e. p 0 surface is in equilibrium

Fluids at rest (hydrostatics) Hydrostatic equilibrium element of fluid surface area A height Dy laws of mechanical equilibrium S Fy =0 p. A - (p+Dp)A - mg = 0 -Dp A - r. ADyg = 0 (p+Dp)A Dp =- rg. Dy Dy Pressure at depth h at distance h below surface, pressure is larger by rgh p = p 0+rgh p. A mg = r. ADyg

Question How far below surface of water must one dive for the pressure to increase by one atmosphere? What is total pressure and what is the gauge pressure, at this depth? ?

Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point

Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point Open tube manometer (i) If h=6 cm and the fluid is mercury (r=13600 kg m-3) find the gauge pressure in the tank (ii) Find the absolute pressure if p 0 =101. 3 k. Pa

Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point Barometer Find p 0 if h=758 mm

Pascal’s principle The pressure at a point in a fluid in static equilibrium depends only on the depth of that point A change in the pressure applied to an enclosed incompressible fluid is transmitted to every point in the fluid Hydraulic press alternative argument based on conservation of energy work out = work in volume moved is same on each side

Archimedes’s principle When a body is fully or partially submerged in a fluid, a buoyant force from the surrounding fluid acts on the body. The force is upward and has a magnitude equal to the weight of fluid displaced. Fg imagine a hole in the water-a buoyancy force exists fill it with fluid of mass mf and equilibrium will exist Fb=mfg stone more dense than water so sinks Fb Fg wood less dense than water so floats now the water displaced is less -just enough buoyancy force to balance the weight of the wood Fb=Fg

Flotation volume immersed Vi Fb For object of uniform density r Fb=Fg rfluid Vi g= r V g Vi/V = r/rfluid Fg Example 1 What fraction of an iceberg is submerged? (rice for sea ice =917 kg m-3 and rsea for sea water = 1024 kg m-3) Example 2 A “gold” statue weighs 147 N in vacuum and 139 N when immersed in salt water of density 1024 kg m-3. What is the density of the “gold”? total volume V

Fluid Dynamics The study of fluids in motion. turbulent laminar Ideal Fluid 1. Steady flow Velocity of the fluid at any point fixed in space doesn’t change with time. This is called “ laminar flow”, and for such flow the fluid follows “streamlines”. 2. Incompressible We will assume the density is fixed. Accurate for liquids but not so likely for gases. 3. Inviscid “Viscosity” is the frictional resistance to flow. Honey has high viscosity, water has small viscosity. We will assume no viscous losses. Our approach will only be true for low viscosity fluids.

Equation of continuity Streamlines tube of flow Conservation of mass in tube of flow means mass of fluid entering A 1 in time Dt = mass of fluid leaving A 2 in time Dt For incompressible fluid this means volume is also conserved. Volume entering and leaving in time Dt is DV DV = A 1 v 1 Dt =A 2 v 2 Dt Therefore A 1 v 1 = A 2 v 2 Equation of continuity (Streamline rule)

Bernoulli’s equation (Daniel Bernoulli, 1700 -1782) For special case of fluid at rest (Hydrostatics!) For special case of height constant (y 1=y 2) The pressure of a fluid decreases with increasing speed

Proof of Bernoulli’s equation Note: same volume DV with mass Dm enters A 1 as leaves A 2 in time Dt Work done at A 1 in time Dt (p 1 A 1)v 1 Dt =p 1 DV Use work energy theorem work done by external force (pressure) =change in KE + change in PE W=DK+ DU Work done Change in KE Change in PE

Problem Titanic had a displacement of 43 000 tonnes. It sank in 2. 5 hours after being holed 2 m below the waterline. Calculate the total area of the hole which sank Titanic.

Examples of Bernoulli’s relation at work Venturi meter Aircraft lift

Examples of Bernouilli’s relation at work “spin bowling”