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Surface Tension Bernoulli Principle Fluid Flow Pressure

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Surface Tension Bernoulli Principle Fluid Flow Pressure Lecture 9. Fluid flow Pressure Bernoulli Principle Surface Tension Fluid flow Speed of a fluid in a pipe is not the same as the flow rate Depends on the radius of the pipe. example: Low speed Same low speed Large flow rate Small flow rate Relating: Fluid flow rate to Average speed L v is the average speed = L/t A v Volume V =AL A is the area 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 Fluid flow -- Pressure Pressure in a moving fluid with low viscosity and laminar flow Bernoulli Principle Relates the speed of the fluid to the pressure Speed of a fluid is high—pressure is low Speed of a fluid is low—pressure is high Daniel Bernoulli (Swiss Scientist 1700-1782) Bernoulli Equation 1 Prr v2 gh constant 2 Fluid flow -- Pressure Bernoulli Equation 11 Pr v22 r gh P r v r gh 122 1 1 2 2 2 Fluid flow -- Pressure Bernoulli Principle P1 P2 Fluid v2 v1 if h12 h 11 Prr v22 P v 122 1 2 2 1122 P11r v22 r gh P r v r gh 1rrv 1 v 1 P 2 P 2 2 22222 1 1 2 if v22 is higher then P is lower Fluid flow Venturi Effect –constricted tube enhances the Bernoulli effect P1 P3 P2 A A v1 1 v2 3 v3 A2 If fluid is incompressible flow rate Q is the same everywhere along tube Q = A v A v 1 therefore A1v 1 = A2 2 v2 = v1 A2 Continuity of flow Since A2 < A1 v2 > v1 Thus from Bernoulli’s principle P1 > P2 Fluid flow Bernoulli’s principle: Explanation P1 P2 Fluid 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 Applications of Bernoulli principle (creating a low pressure) Atomiser air Perfume entrained in air flow Rubber bulb Liquid (perfume) Water aspirator uses the Venturi effect to produce low pressure Principle Increased speed Lower pressure Fluid flow Example If the average speed of blood in a capillary of diameter 4 x10-4cm is 3.5 x10-2cms-1, calculate the flow rate in litres per second. Q =flow rate A = area v =average speed Q = A v A = pr2 = (2 x10-4cm)2 p Q = [(2 x10-4cm)2 p](3.5 x10-2cms-1) Q = 44 x 10-10 cm3s-1 Q = 44 x10-13 litres.s-1 Fluid flow Plaque build-up on an artery wall reduces its diameter from 1.1cm to 0.75cm. If the speed of the blood is 15cms-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 A1v 1 = A2 v2 2 d1 2 A1 p 0.95 cm A1 2 v 2 = v1 2 A2 d2 2 A2 p 0.44 cm 2 (a) 0.95cm 2 v15 cms11 32 cms 2 0.44cms 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. Why? Forces between Molecules Molecules close together: forces repulsive Otherwise forces are attractive Molecules close together: forces repulsive >>>Liquids and solids almost incompressible Attractive forces >>>> phenomena such as surface tension Water droplet spherical shape why? Surface is subject to tension makes surface area as small as possible Forces between Molecules Surface Tension A Molecule at B surrounded on all sides by other similar molecules. Net attractive force B is zero since it is attracted equally 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 Forces between Molecules Measuring surface tension Measure force (F) required to stretch liquid film L F Surface Tension g = F/L Surface Tension Liquid g(Nm-1) SI unit of surface tension Blood 0.058 Newton per metre Nm-1 Ethyl 0.023 Alcohol Mercury 0.44 Water 0.076 (0oC) Water 0.072 (20oC) Water 0.059 (100oC) Soapy 0.037 Water Surface Tension Phenomena Needle on surface of water Force due to surface tension Density of steel rs » rw mg 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 Surface Tension Phenomena Lungs 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 As lungs expand (push against the liquid’s surface tension) during inhalation, the body secretes a fluid (surfactant) into the tissue of the air sacs that gradually 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 fetus Result- premature 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 F q q h F meniscus Adhesive forces H O 2 (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 F h v q F Surface Tension g = F/L H2O F = gL = g2pr Vertical component of force Fv = FCosq Fv = g2prCosq This force must equal the weight w of the liquid which rises to height h, w = mg = rVg = rpr2h g Thereforerpr2h g = g2prCosq 2g h = rgr 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 Mercury q h F F Hg 2g h = rgr 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 f Water drop +wetting agent f f Water drop wetting agent reduces surface tension Wetting characterised by the way in which the substance spreads out: f>90o large surface tension f<90o 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 (good wetting ability) adhesive is 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 Wetting of enamel and dentine 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 to a value greater than atmospheric pressure 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 x10-3m) DP = 1.152 x103 Nm-2 DP = 1.152 x103 Pa (b) P = rgh h = P/rg 1.152 x103 Pa h = (13.6 x103 kgm-3)(9.8ms-2) h = 8.6 x10-3m h = 8.6 mm Hg Surface Tension Calculate the pressures inside bubbles of water and soapy water each of diameter 1.5cm.
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