Lecture 9. Fluid flow Pressure Bernoulli Principle Surface Tension
A v L A is the area Fluid flow Speed of a fluid in a pipe is not the same as the flow rate Relating: Fluid flow rate to Average speed v is the average speed = L/t Volume V =AL Flow rate Q is the volume flowing per unit time (V/t) Q = (V/t) Q = AL/t = A v Q = A v Flow rate Q is the area times the average speed Depends on the radius of the pipe. example: Low speed Large flow rate Same low speed Small flow rate
Fluid flow -- Pressure Pressure in a moving fluid with low viscosity and laminar flow Bernoulli Principle Relates the speed of the fluid to pressure Daniel Bernoulli (Swiss Scientist 1700-1782) Speed of a fluid is high pressure is low Speed of a fluid is low pressure is high Bernoulli Equation 1 2 P v gh constant 2 P =pressure at some chosen point h= height of the point above some reference level
Fluid flow -- Pressure Bernoulli Principle Bernoulli s principle allows the combination of pressure, speed, and height of a fluid at one point to be compared to the same three properties at a different point in the fluid Bernoulli Equation 1 1 P v gh P v gh 2 2 2 2 1 1 1 2 2 2
Fluid flow -- Pressure Bernoulli Principle Fluid P 1 P 2 1 1 P v gh P v gh 2 2 2 2 1 1 1 2 2 2 if h h 1 2 v 1 v 2 1 1 P v P v 2 2 2 2 1 1 2 2 1 v 1 v P P 2 2 2 2 2 1 1 2 if v2 is higher then P2 is lower
Fluid flow Venturi Effect constricted tube enhances the Bernoulli effect P 1 P 2 P 3 v A 1 1 v 2 A 3 v 3 A 2 If fluid is incompressible, flow rate Q is the same everywhere along tube Q = A v therefore A 1 A 1 v 1 = A 2 v 2 v 2 = v 1 Continuity of flow A 2 Since A 2 < A 1 v 2 > v 1 Thus from Bernoulli s principle P 1 > P 2
Fluid flow Bernoulli s principle: Explanation Fluid P 1 P 2 Speed increases in smaller tube Therefore kinetic energy increases. (Tube horizontal so no change in gravitational potential energy) Potential energy associated with pressure is employed to increase kinetic energy. Therefore pressure decreases. Speed increases pressure decreases High speed low pressure
Fluid flow Example If the average speed of blood in a capillary of diameter 4 x10-4 cm is 3.5 x10-2 cms -1, calculate the flow rate in litres per second. Q =flow rate A = area v =average speed Q = A v A = pr 2 = (2 x10-4 cm) 2 p Q = [(2 x10-4 cm) 2 p](3.5 x10-2 cms -1 ) Q = 44 x 10-10 cm 3 s -1 Q = 44 x10-13 litres.s -1
Fluid flow Plaque build-up on an artery wall reduces its effective diameter from 1.1 cm to 0.75 cm. If the speed of the blood Is 15 cms -1 before reaching the region of plaque build-up. Find the speed of the blood within the plaque region? Q =flow rate A = area v =average speed Q = A v Assume blood is incompressible, flow rate Q is the same everywhere along artery A 1 v 1 = A 2 v 2 A 1 v 2 = v 1 A 2 A 2 d1 1 p 0.95 A 2 2 d2 2 p 0.44 0.95cm2 v2 15 cms 32 cms 0.44cms 2 1 1 cm 2 cm 2
Fluid flow Bernoulli effect not limited to fluid flow in tubes. Airplane wing profile Air moves faster over the upper side of the wing Pressure is lower, resulting in lift Shower curtain Curtain is sucked inwards when water is switched on Increased water/air speed inside curtain results in reduced pressure
Forces between Molecules Molecules close together: forces repulsive >>>Liquids and solids almost incompressible Otherwise forces are attractive Intermolecular forces mainly attractive Attractive forces >>>> phenomena such as surface tension Water droplet spherical shape why? Surface is subject to tension: makes surface area as small as possible Liquid surface behaves like a rubber membrane under tension
Forces between Molecules Surface Tension Molecule at B surrounded on all sides by A other similar molecules. Net attractive force is zero since it is attracted equally B in all directions Molecule at A no liquid molecules above, therefore net force exists which pulls it towards the interior of the liquid Net effect of the pull on all molecules at surface Surface of liquid contracts >>surface area becomes a minimum Minimum surface area for a given volume is when shape is a sphere Reason why drops of water have a spherical shape
Forces between Molecules Measuring surface tension Measure force (F) required to stretch liquid film F L 2 Surface Tension g = F/L SI unit of surface tension Newton per metre Nm -1 Surface Tension Liquid g (Nm -1 ) Blood 0.058 Ethyl Alcohol 0.023 Mercury 0.44 Water (0 o C) Water (20 o C) Water (100 o C) Soapy Water 0.076 0.072 0.059 0.037
Surface Tension Phenomena Needle on surface of water Force due to surface tension Density of steel s» w mg But steel needle does not sink Surface tension results in Upwards force on needle Liquid surface behaves like a rubber membrane under tension
Surface Tension Phenomena Insects can walk on water Depression in water surface (increases surface area) Surface tension opposes this, which results in an upwards force that tends to bring back surface to original flat shape. Liquid surface behaves like a rubber membrane under tension
Surface Tension Phenomena Temperature of liquid increases: Surface tension of liquids decreases Molecules moving faster bound together less tightly Surfactant (surface active) substances (soaps) When added to liquid will lower its surface tension Uses Soapy water can penetrate the fine structure of clothes or skin more easily than water and hence clean better
Lungs Surface Tension Phenomena Similar effect occurs Surface tissue of air sacs (alveoli) has a liquid with large surface tension that would result in difficulty in lungs expanding during inhalation the body secretes a fluid (surfactant) into the tissue of the air sacs that lowers the surface tension of the liquid and allows easy inflation of the air sacs Premature birth This surfactant produced late on in the development of the child Resultpremature infant suffer respiratory distress
Surface Tension Capillary action Forces between like molecules are called cohesive forces e.g. between water molecules Forces between unlike molecules are called adhesive forces e.g. between water and glass Capillary tube in water h F q q F meniscus H 2 O Adhesive forces (between water and glass) greater than the cohesive forces between waters molecules Result: water rises in the capillary tube until the weight of the water column supported = the upward force
Surface Tension Capillary action F q q F h F v q F H 2 O Surface Tension g = F/L F = gl = g2pr Vertical component of force F v = FCosq F v = g2prcosq This force must equal the weight w of the liquid which rises to height h, w = mg = Vg = ( pr 2 h) g Therefore pr 2 h g = g2prcosq h = 2g gr Cosq Therefore small radius (r) large h
Surface Tension Capillary action When adhesive forces between liquid and glass are less than the cohesive forces between liquid molecules h Mercury q F F Hg Cohesive forces are dominant. Liquid in capillary tube is depressed to a distance h below the surface of the surrounding liquid. h = 2g gr Cosq
Surface Tension Capillary action Applications Used to draw samples of blood Plants: feed using capillary action Kitchen towels: absorb using capillary action
Surface Tension Dental application: filling Adhesion: interaction force between two materials at their contact interface Chemical bond Adhesion of material to tooth surface Advantage conserves tooth structure Alternative Mechanical (amalgam) no bonding undercutting required: chamber that is smaller at the surface and wider inside. Mechanical interlock
Surface Tension Effectiveness of adhesion Important characteristic is the way in which the adhesive wets the surface Contact angles Water drop f f Water drop +wetting agent f wetting agent reduces surface tension Wetting characterised by the way in which the substance spreads out: f >90 o large surface tension f <90 o small surface tension
Surface Tension Dental application: Adhesion good intimate contact Large area Enamel normally covered with thin layer of pellicle (organic substance deposited from saliva) Clean surface to achieve good adhesion Low surface tension adhesive desirable in promoting adhesion
Surface Tension Chemical Adhesion Dental restoration Bond strength depends on contact area Rough surfaces when viewed on atomic scale Rough surfaces>>>small contact area Small force >>large stress at local points >>result failure Smooth surface large contact area lower stress Use fluid that flows into irregularities to provide intimate contact over larger surface area Example- glass slides with water
Surface Tension Chemical Adhesion Dental restoration Fluid must flow easily (wetting) to achieve bonding Bonding to tooth surfaces impaired by contamination -Etching debris and saliva Wetting of enamel and dentine surfaces reduced by application of aqueous fluoride solution less plaque adheres to enamel surface treated with fluoride Viscosity Adhesive should spread out (wet) therefore a low viscosity adhesive is important
Forces between Molecules Like water droplets, Bubbles are also spherical Inward force due to surface tension increases pressure of the gas inside Excess pressure DP inside bubble given by DP = 4g/r
Surface Tension Example Calculate the excess pressure in (a) SI units and (b) in mm Hg inside a water bubble of radius 0.25mm DP = 4g/r (a) DP = 4 (0.072N/m)/(0.25 x10-3 m) DP = 1.152 x10 3 Nm -2 DP = 1.152 x10 3 Pa (b) P = gh h = P/ g h = 1.152 x10 3 Pa (13.6 x10 3 kgm -3 )(9.8ms -2 ) h = 8.6 x10-3 m h = 8.6 mm Hg
Surface Tension Calculate the pressures inside bubbles of water and soapy water each of diameter 1.5cm. Surface tension of water(g w ) is 0.072Nm -1 Surface tension of soapy water (g sw ) is 0.037Nm -1 Pressure inside a bubble is given by Water P =(4g)/r Pressure (P) = (4 x 0.072Nm -1 )/(0.75 x10-2 m) P = 38.4Nm -2 Soapy water Pressure (P) = (4 x 0.037Nm -1 )/(0.75 x10-2 m) P = 19.6Nm -2