Chapter Objectives Study fluid dynamics. Understanding Bernoulli s Equation.
Chapter Outline 1. Fluid Flow. Bernoulli s Equation 3. Viscosity and Turbulence
1. Fluid Flow An ideal fluid is a fluid that is incompressible, that is density do not change, and has no internal friction (viscosity). The path of an individual particle in a moving fluid is called a flow line. The overall flow pattern does not change with time, the flow is called steady flow. A streamline is a curve whose tangent at any point is in the direction of the fluid velocity at that point. In the figure, the flow lines passing through the edge of an imaginary element area and form a tube called a flow tube.
1. Fluid Flow Steady flow
1. Fluid Flow The figure shows pattern of fluid flow from left to right round a number of shapes. These patterns are typical of laminar flow. At sufficient high flow rates, the flow can become irregular and chaotic and is called turbulent flow.
Turbulent flow
1. Fluid Flow The continuity equation The mass of a moving fluid doesn t change as it flows. This leads to a quantitative relationship called continuity equation. The figure shows a flow tube with changing cross sectional area. If the fluid is incompressible, the product Av has the same value at all points along the tube.
1. Fluid Flow The continuity equation In steady flow the total mass in the tube is constant, so m = m 1 ρav t = A v t ρ 1 1 A = v A v 1 1 The product Av is the volume flow rate V/ t, the rate which volume crosses a section of the tube
14.4 Fluid Flow The continuity equation The mass flow rate is the mass flow per unit time through a cross section. This is equal to the density times the volume flow rate. We can generalize Eq. (14.10) for the case in which fluid is not incompressible. If and ρ are the densities at section 1 and, then ρ1 Av ρ A v 1 1 1 ρ =
Example 1. Incompressible fluid flow As part of a lubricating system for heavy machinery; oil of density 850kg/m 3 is pumped through a cylindrical pipe of diameter 8.0cm at a rate of 9.5 liters per second. (1L = 0.001 m 3 ) A) What is the speed of the oil? B) If the pipe diameter is reduced to 4.0cm, what are the new values of the speed and volume flow rate? Assume that the oil is incompressible.
Example 1. (SOLN) A) The volume flow rate is equal the product A 1 v 1 where A 1 is the cross-sectional are of the pipe of diameter 8.0cm and radius 4.0cm. Hence v volume flow rate A 9.5 10 π 4 10 3 = = = 1 1 ( ) = 1.9 m/s
Example 1. (SOLN) B) Since the oil is incompressible, the volume flow rate has the same value of (8.5L/s) in both sections of pipe. ( ) A π 4.0 10 1 1 A v = v = 1.9 = 7.6 m / s π ( ) ( ).0 10
. Bernoulli s Equation Bernoulli s equation states that the relationship of pressure, flow speed, and height for flow of an ideal, incompressible fluid.
. Bernoulli s Equation The subscript 1 and refer to any point along the flow tube, p 1 + ρgy + ρv = constant
Example. Bernoulli s equation A water tank has a spigot near its bottom. If the top of the tank is open to the atmosphere, determine the speed at which the water leaves the spigot when the water level is 0.500 m above the spigot.
Example. (SOLN)
Example. (SOLN)
Example. (SOLN) Torricelli s theorem. That is the speed of efflux from an opening at a distance h before the top surface of liquid is the same as the speed a body would acquire in falling freely through a height h.