Fluid dynamics - Equation of. continuity and Bernoulli s principle.

Transcription

1 Fluid statics Fluid dynamics - Equation of What is a fluid? Density Pressure Fluid pressure and depth Pascal s principle Buoyancy Archimedes principle continuity and Bernoulli s principle. Lecture 4 Dr Julia Bryant Fluid dynamics Reynolds number Equation of continuity Bernoulli s principle Viscosity and turbulent flow Poiseuille s equation web notes: Fluidslect4.pdf flow1.pdf flow.pdf

2 Fluid dynamics

3 REYNOLDS NUMBER A British scientist Osborne Reynolds ( ) established that the nature of the flow depends upon a dimensionless quantity, which is now called the Reynolds number R e. R e = ρ v L / η ρ density of fluid v average flow velocity over the cross section of the pipe L characteristic dimension η viscosity

4 R e = ρ v L / η [R e ] [kg.m -3 ] [m.s -1 ][m] [Pa.s] kg x m x m x s.m = [1] m 3 s kg.m.s R e is a dimensionless number As a rule of thumb, for a flowing fluid pascal 1 Pa = 1 N.m - newton 1 N = 1 kg.m.s - 1 Pa.s = kg.m.s -. m -.s R e < ~ 000 laminar flow ~ 000 < R e < ~ 3000 unstable laminar to turbulent flow R e > ~ 000 turbulent flow

5 Consider an IDEAL FLUID Fluid motion is very complicated. However, by making some assumptions, we can develop a useful model of fluid behaviour. An ideal fluid is Incompressible the density is constant Irrotational the flow is smooth, no turbulence Nonviscous fluid has no internal friction (η=0) Steady flow the velocity of the fluid at each point is constant in time.

6 Consider the average motion of the fluid at a particular point in space and time." An individual fluid element will follow a path called a flow line." Streamlines" - in steady flow, a bundle of streamlines makes a flow tube. Steady flow is when the pattern of flow lines does not change with time. v Velocity of particle is tangent to streamline" Streamlines cannot cross

7 Steady flow is also called laminar flow." If steady flow hits a boundary or if the rate of flow increases, steady flow can become chaotic and irregular. This is called turbulent flow." For now, we will consider only steady flow situations."

8

9 A model EQUATION OF CONTINUITY (conservation of mass) A 1 A v 1 v The mass of fluid in a flow tube is constant. Where streamlines crowd together the flow speed increases.

10 EQUATION OF CONTINUITY (conservation of mass) mass flowing in = mass flowing out m 1 = m ρ V 1 = ρ V ρ A 1 v 1 Δt = ρ A v Δt A 1 v 1 = A v =Q=V/t =constant Applications Q=volume flow rate m 3 s -1 Rivers Circulatory systems Respiratory systems Air conditioning systems V=A Δx v=δx/δt So V=A vδt

11 Y-junction with pipes of the same diameter A i v i = A f v f A f = A i So flow speed must drop to half v f = 1/ v i ms -1

12 A i v i = A f v f Y-junction with pipes of half the diameter The pipes after the branch must have half the cross-sectional area of those before if the flow speed is to stay the same.

13 How big does each of the tubes need to be if the different sized coke bottles are to fill in the same time? A 0 A 1 A If the velocity is constant, (A 1 + A ) v = A 0 v Volume flow rate is V/t = Av m 3 s -1 Bottles fill in time t t = V = = 1.5 s Av A 1 v A v A 1 = A 0 A A = 1.5 A A 375 ml 1.5 L A = 1.5 A 0 /1.65 m = 0.77 A 0 m A 1 = 0.3 A 0 m

14 What velocity must the flow of scotch and coke be to make a drink with the volume of scotch being 15% that of the coke, if the scotch tube is 3 times the area of the coke tube? From the continuity equation A 1 v 1 + A v = A f v f v 1 v The relative volume flow rate is V 1 /t 1 = 0.15 V /t m 3 s -1 A 1 A But volume flow rate V/t = Av so A 1 v 1 = 0.15 A v m 3 s -1 A f v f The areas of the pipes are different A 1 = 3 A m 3 A v 1 = 0.15 A v 3v 1 = 0.15 v v 1 = 0.05 v m.s -1

15 Blood flowing through our body The radius of the aorta is ~ 10 mm and the blood flowing through it has a speed ~ 300 mm.s -1. A capillary has a radius ~ mm but there are literally billions of them. The average speed of blood through the capillaries is ~ m.s -1. Calculate the effective cross sectional area of the capillaries and the approximate number of capillaries.

16 Setup aorta R A = m =10mm and A A =cross sectional area of aorta capillaries R C = m =0.004mm and A C =cross sectional area of capillaries aorta v A = m.s -1 capillaries v C = m.s -1 Assume steady flow of an ideal fluid and apply the equation of continuity Q = A v = constant A A v A = A C v C aorta capillaries Area of capilliaries A C A C = A A (v A / v C ) = π R A (v A / v C ) A C = 0.0 m Number of capillaries N A C = N π R C N = A C / (π R C ) = 0. / {π ( ) } N =

17 BERNOULLI'S PRINCIPLE An increase in the speed of fluid flow results in a decrease in the pressure. (In an ideal fluid.) How can a plane fly? What is the venturi effect? Why is Bernoulli s principle handy in a bar? Why does a cricket ball swing or a baseball curve? Daniel Bernoulli ( ) web notes: flow3.pdf

18 Bernoulli s floating ball - the vacuum cleaner shop experiment! Higher pressure High v Lower pressure Higher pressure DEMO

19 A 1" A" A 1" Fluid flow v 1" v " v 1" low speed" low KE" high pressure" high speed" high KE" low pressure" Streamlines closer" low speed" low KE" high pressure"

20 p large" p large" p small" v small" v large" v small"

21 In a storm how does a house lose its roof? Air flow is disturbed by the house. The "streamlines" crowd around the top of the roof. faster flow above house reduced pressure above roof to that inside the house roof lifted off because of pressure difference.

22 Why do rabbits not suffocate in the burrows? Air must circulate. The burrows must have two entrances. Air flows across the two holes is usually slightly different slight pressure difference forces flow of air through burrow. One hole is usually higher than the other and the a small mound is built around the holes to increase the pressure difference. p 1 p p < p 1

23 VENTURI EFFECT

24 VENTURI EFFECT Flow tubes velocity increased" pressure decreased"

25 VENTURI EFFECT Flow tubes Eq Continuity high! pressure! (p atm )! low pressure" velocity increased" pressure decreased" DEMO

26 VENTURI EFFECT Spray gun

27

28 force" high speed" low pressure" force"

29 Flow tubes Eq continuity force" high speed" low pressure" force" What happens when two ships or trucks pass alongside each other? Have you noticed this effect in driving across the Sydney Harbour Bridge?

30 artery" Flow speeds up at constriction" Pressure is lower" Internal force acting on artery wall is reduced" External forces are " unchanged" Artery can collapse" Arteriosclerosis and vascular flutter

31 Bernoulli s Equation Consider an element of fluid with uniform density. The change in energy of that element as it moves along a pipe must be zero - conservation of energy. This is the basis for Bernoulli s equation.

32 Continuity Volume passing from point c to d in time dt is dv = A ds Volume passing from point a to b in time dt is dv = A 1 ds 1 dv = A 1 ds 1 = A ds (Continuity)

33 Derivation of Bernoulli's equation Consider conservation of energy Work done = kinetic energy + potential energy W = K + U W = F 1 Δx 1 = p 1 A 1 Δx 1 = p 1 V K = 1 m v 1 = 1 ρ V v 1 U = m g y 1 = ρ V g y 1 p 1 Δ x 1 A 1 m v 1 p 1 V = - 1 ρ V v 1 - ρ V g y 1 Rearranging y 1 t i m e 1 p ρ v 1 + ρ g y 1 = constant Applies only to an ideal fluid (zero viscosity)

34 Derivation of Bernoulli's equation Mass element m moves from (1) to () m = ρ A 1 Δx 1 = ρ A Δx = ρ ΔV where ΔV = A 1 Δx 1 - A Δx Equation of continuity A V = constant A 1 v 1 = A v A 1 > A v 1 < v Since v 1 < v the mass element has been accelerated by the net force F 1 F = p 1 A 1 p A Conservation of energy A pressurized fluid must contain energy by the virtue that work must be done to establish the pressure. A fluid that undergoes a pressure change undergoes an energy change.

35 Change in kinetic energy ΔK = 1 m v - 1 m v 1 = 1 ρ ΔV v - 1 ρ ΔV v 1 Change in potential energy ΔU = m g y m g y 1 = ρ ΔV g y - ρ ΔV g y 1 Work done W net = F 1 Δx 1 F Δx = p 1 A 1 Δx 1 p A Δx Work done = change in kinetic + potential energy W net = p 1 ΔV p ΔV = ΔK + ΔU p 1 ΔV p ΔV = 1 ρ ΔV v - 1 ρ ΔV v 1 + ρ ΔV g y - ρ ΔV g y 1 Rearranging p ρ v 1 + ρ g y 1 = p + 1 ρ v + ρ g y Applies only to an ideal fluid (zero viscosity)

36 Bernoulli s Equation for any point along a flow tube or streamline p + 1 ρ v + ρ g y = constant Between any two points along a flow tube or streamline p ρ v 1 + ρ g y 1 = p + 1 ρ v + ρ g y Dimensions p [Pa] = [N.m - ] = [N.m.m -3 ] = [J.m -3 ] 1 ρ v [kg.m -3.m.s - ] = [kg.m -1.s - ] = [N.m.m -3 ] = [J.m -3 ] ρ g h [kg.m -3 m.s -. m] = [kg.m.s -.m.m -3 ] = [N.m.m -3 ] = [J.m -3 ] Each term has the dimensions of energy / volume or energy density. 1 ρ v KE of bulk motion of fluid ρ g h GPE for location of fluid p pressure energy density arising from internal forces within moving fluid (similar to energy stored in a spring)

37 Ideal fluid" Real fluid" With viscosity" In a real fluid the pressure decreases along the pipe. Viscous fluids have frictional forces which dissipate energy through heating.