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Physics 221, March 15 Key Concepts: •Viscosity •Turbulence •Surface tension Viscosity Ideal fluids flow without friction and do not dissipate ordered energy. Real fluids convert ordered energy into disordered energy when flowing. Viscosity is a measure of the internal friction of a flowing fluid. It is a result of intermolecular forces. Ideal flow is always laminar. Viscous flow can be laminar or turbulent. Fluid density, flow speed, viscosity, and geometry of the flow determine when viscous flow transitions fro laminar to turbulent flow. A single number, called the Reynolds number, can be computed for any viscous flow. R = ρfluidDv/η Here ρfluid is the density of the fluid, v is its flow speed, η is its viscosity, and D is a characteristic length that depends on the geometry. When the Reynolds number R becomes greater than ~2300, the flow becomes turbulent. Laminar viscous flow Definition: η = viscosity (Pa-s) How can you measure it? Stokes’ law: F = 6πηrspherev Force required to move a sphere through a viscous fluid with velocity v. Falling sphere: vterminal = 2(ρsphere - ρfluid)(rsphere )2g)/(9 η) Consequences? Poiseuille’s law gives rate at which a fluid flows through a hose or a pipe. Q = π∆Pr4/(8ηL) Q = volume flow rate ∆P = pressure difference across L of the pipe r = radius of the pipe A thin boundary layer of fluid next to a walls does not move at all. Why can't you get all the dust off your car by just squirting water from a hose onto it? Why can't you simply remove dust just be blowing across a surface? Why does dust cling to a fast rotating fan? How can a leaf stay on a car moving at high speed? What is one common answer to all these questions? 1. 2. 3. 4. This is a consequence of Bernoulli’s equation. This is a consequence of the formation of stationary boundary layer. This is a consequence of Poiseuille’s law. This is a consequence of Stokes’ law. 0% 1 0% 2 0% 3 30 0% 4 Suppose you work in the E.R. and a choking patient comes in with their trachea half obstructed. (The diameter is half of what it was before the obstruction.) You manage to correct the problem and send them on their way safely, explaining that such obstructions are particularly dangerous because 1. 2. 3. 4. 5. a half – blocked airway passes only 1/4 the regular amount of air. a half – blocked airway passes only 1/16 the regular amount of air. a half – blocked airway passes only 1/2 the regular amount of air. a half – blocked airway passes only 1/32 the regular amount of air. a half – blocked airway passes only 1/8 the regular amount of air. 0% 1 0% 0% 0% 2 3 4 0% 30 5 Extra credit: Two marbles of radii r and √r fall through a high-viscosity fluid. What is the ratio of their terminal velocities? 1. 2. 3. 4. 5. 1:1 r:1 2:1 1:2 √r :1 0% 1 0% 2 0% 0% 3 4 0% 5 30 Newtonian and non-Newtonian Fluids If the viscosity of a fluid depends only on temperature, the fluid is said to be Newtonian. If the viscosity depends on flow speed, shear stress, etc, the fluid is said to be non-Newtonian. Examples: Shear thinning materials: paint, blood, ketchup (Viscosity decreases with increased stress) Shear thickening material: (Viscosity increases with increased stress) cornstarch dissolved in water Turbulence As the flow rate increases, a moving fluid can no longer sustain laminar flow. Turbulence appears when the Reynolds number is about 2300. For flow through a smooth cylindrical pipe D is the inner diameter of the pipe Consequences of turbulence: Ordered energy is “wasted” (converted into disordered energy.) Bernoulli’s equation not valid. Poiseuille’s law not valid, (Q is not proportional to ΔP.) The equation of continuity still valid for incompressible fluids. So for flow through a smooth cylindrical pipe R = ρfluidDv/η = ρfluidDvA/(ηA) = ρfluidDQ/(η π D2/4) = 4 ρfluidQ/(η π D) ∝ Q/D. A water pump generates a constant flow Q, which is causing a turbulent spray (Reynolds number R = 3200) out of this cylindrical nozzle: Which of the following three nozzles could be substituted to achieve a streamline flow (R < 2000)? 1. 2. 3. 4. Tube I Tube II Tube III Tube I and III 0% 1 0% 2 0% 3 0% 4 30 Compare the flight of two golf balls given the same impulse by the club. Which ball will travel a larger distance? 1. 2. 3. The smooth ball The dimpled ball They will travel the same distance since they have the same initial velocity. 30 0% 1 0% 2 0% 3 Surface tension Cohesion, or attraction between molecules of the same kind leads to surface tension. The surface tension γ is defined as the force along a line of unit length. γ = F/L (N/m) Laplace's law for a spherical membrane in equilibrium: Pin - Pout = 2γ/r The smaller a soap bubble, the higher is the pressure inside. A soap bubble with radius of 3.0 cm is floating in air. What is the pressure difference across the two surfaces of the bubble? The surface tension of soapy water is 69 * 10-3 N/m. 1. 2. 3. 4. 9.2 N/m 2.6 N/m 8.3*10-3 N/m 9.2*10-2 N/m 0% 1 0% 0% 2 3 0% 4 30 Extra credit: Two bubbles are connected by a hollow tube plus a valve. What will happen once the valve between the two is opened? 1. 2. 3. The small bubble will drain into the big bubble. The big bubble will empty into the small bubble until the two are equal in size. Nothing will happen – the bubbles will stay the same size. 0% 1 0% 2 0% 3 30 Capillary action • Adhesion is the attraction between unlike molecules. • Adhesive and cohesive forces determine the contact angle between a liquid and a solid surface. • Adhesion and surface tension together can produce capillary action. • The maximum height to which a liquid will rise through capillary action is given by h = 2γcosθ/(ρgr). – [mg = (F/L) cosθ 2πr (net force = 0) ρπr2h g = γ cosθ 2πr (solve for h)] In an experiment water rose by capillary attraction through two columns of soils, one with coarse grains (column 1) and the other with fine gains (column 2). In which column did the water rise faster and in which column did it rise to greater height? 1. faster: greater height: 2. faster: greater height: 3. faster: greater height: 4. faster: greater height: column 1 column 1 column 2 column 2 column 1 column 2 column 2 column 1 25% 25% 25% 25% 30 1 2 3 4 Extra credit: Blood and water have the same contact angle with glass. The density of blood is 5% higher than that of water. (ρblood = 1.05 ρwater) While you make your capillary action measurements you find that water rises in the same tube a factor 1.69 higher than a blood sample. The surface tension of water at the same temperature is 0.073 N/m. What is the surface tension of the blood sample? 1. 0.045 N/m 2. 0.117 N/m 3. 0.073 N/m 0% 1 0% 2 0% 3 30