Fluids Fluids in Motion or Fluid Dynamics Resources: Serway - Chapter 9: 9.7-9.8 Physics B Lesson 3: Fluid Flow Continuity Physics B Lesson 4: Bernoulli's Equation MIT - 8: Hydrostatics, Archimedes' Principle, and Fluid Dynamics Fluid Flow When real fluids flow through pipes, two distinct forces act on them. Frictional forces exerted on the fluid by the walls of the pipe. Viscous forces within the fluid. Viscosity describes the degree of internal friction. Energy is lost due to these forces. Two types of flow: Laminar (streamline) Smooth path, path called streamline Turbulent (irregular) Abrupt changes in velocity, irregular motions called eddy currents
Turbulent Types of Fluid Flow Turbulent Flow Laminar Streamline Laminar Flow Fluid Flow Wind Tunnels
Conditions for an Ideal Fluid Nonviscous flow Flow where there is no internal friction force between adjacent layers. Incompressible flow The density of the fluid remains constant. Steady flow Laminar flow where the velocity, density, and pressure at each point in the fluid are constant. Irrotational flow (no turbulence) Each element of the fluid has zero angular velocity about its center. No eddy currents or whirlpools. Fluid Flow Rate Where, V = volume (m 3 ) t = time (s) A = area (m ) v = speed of fluid (m/s) Note: Q is sometimes used to represent the flow rate. 3
Fluid Flow Rate Where, m = mass (kg) t = time (s) ρ = density (kg/m 3 ) A = area (m ) v = speed of fluid (m/s) Fluid Flow Continuity V = V ρ V = ρ V m = m V = V A v t = A v t A L = A L A v = A v The volume per unit time of a liquid flowing in a pipe is constant throughout the pipe. A v = A v AP Continuity Equation A, A : cross sectional areas at points and v, v : speed of fluid flow at points and 4
Practice Problem Spray Water travels through a 9.6 cm diameter fire hose with a speed of.3 m/s. At the end of the hose the water flows out through a nozzle whose diameter is.5 cm. What is the speed of the water exiting the nozzle? Solution: A v = A v v = v (A /A ) = v (πd /4)/(πd /4) v = v (d )/(d ) = (.3 m/s)[(0.096 m) /(0.05 m) v = 9. m/s Bernoulli s Theorem The sum of the pressure (P ), the potential energy per unit volume (ρgh ) and the kinetic energy per unit volume (½ρv ) has the same value at all points along a streamline. All other considerations being equal, when fluid moves faster, the pressure drops. This equation is essentially a statement of conservation of energy in a fluid. 5
Bernoulli s Theorem P + ρgy + ½ρv = const. P : pressure (Pa) AP ρ : density of fluid (kg/m 3 ) g: gravity (9.8 m/s ) y: height above lowest point (m), might appear as h v: speed of fluid flow at a point in the pipe (m/s) const. constant Bernoulli s Equation: P + ρgy + ρv = P + ρgy + ρv Applications of Bernoulli s Principle The Bernoulli effect is simple to demonstrate all you need is a sheet of paper. Try this! Hold the paper by its end, so that it would be horizontal if it were stiff, and blow across the top. BLOW HARD! What happens? The paper will rise, due to the higher speed, and therefore lower pressure, above the sheet. 6
Airplane Wing Applications of Bernoulli s Principle The Bernoulli effect is used in many common applications. Airplane Wing Lift Designed so that the air speed above the wing is great than the air speed below which causes a pressure difference resulting in an upward lift force. Carburators (Venturi Tube) As a fluid flows through a tube or pipe, when the pipe narrows the speed of the flow increases, but its pressure drops. A venturi is the name for the restricted, or narrowed, area of a container through which a fluid flows. 7
Applications of Bernoulli s Principle The Bernoulli effect is used in many common applications. Curve of a Ball Bernoulli s principle creates an imbalance on the forces, cause the ball to deflect, aka the "Magnus Effect". Atomizers The stream of air passing over the tube reduces the pressure above the tube, this causes the liquid to rise into the airstream. Example Venturi Meter A Venturi meter is used to measure fluid speed in a pipe. Suppose that the pipe in question carries water, A =.0A, and the fluid heights in the vertical tubes are h =.0 m and h = 0.80 m. Find the following: (a) The ratio of the flow speeds, v /v (b) The gauge pressures P and P (c) The flow speed v in the pipe. 8
Practice Problem Spray Water travels through a 9.6 cm diameter fire hose with a speed of.3 m/s. At the end of the hose the water flows out through a nozzle whose diameter is.5 cm. Suppose the pressure in the fire hose is 350 kpa. What is the pressure in the nozzle? Solution: P + ρgh + ½ρv = P + ρgh + ½ρv P + ½ρv = P + ½ρv P = P + ½ρv - ½ρv P = P + ½ρ(v - v ) P = 350 kpa + ½ (000 kg/m 3 )[(.3 m/s) (9. m/s) ] P = 67 kpa 9