FLUID FLOW IDEAL FLUID EQUATION OF CONTINUITY

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1 VISUAL PHYSICS School of Physics University of Sydney Australia FLUID FLOW IDEAL FLUID EQUATION OF CONTINUITY? How can the blood deliver oxygen to body so successfully? How do we model fluids flowing in streamlined motion? IDEAL FLUID Fluid motion is usually very complicated. However, by making a set of assumptions about the fluid, one can still develop useful models 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. EQUATION OF CONTINUITY (conservation of mass) Consider an ideal fluid flowing through a pipe of varying cross sectional area A. The volume V 1 of fluid and mass m 1 flowing past (1) in a very small time interval t is a03/p1/fluids/flow2.doc 1

2 V 1 = A 1 v 1 t m 1 = ρ 1 A 1 v 1 t Similarly the volume and mass of fluid flowing past (2) in time t is V 2 = A 2 v 2 t m 2 = ρ 2 A 2 v 2 t When the flow is steady all the material which goes past (1) must go past (2) in the same time (or else it will be continually piling up somewhere) and since the fluid is incompressible its density does not change ρ 1 = ρ 2 = ρ Therefore we must have m 1 = m 2 ρ A 1 v 1 t = ρ A 2 v 2 t A 1 v 1 = A 2 v 2 If the fluid is approximately incompressible, i.e. if its density never changes by very much, then the equation of continuity, as we quoted it, is approximately true. The quantity A v which measures the volume of the fluid that flows past any point of the tube divided by time is called the volume flow rate Q = dv/dt. The equation of continuity is often expressed as Q = A v = constant if A decreases then v increases if A increases then v decreases a03/p1/fluids/flow2.doc 2

3 A 1 A2 ρ ρ v 2 v 1 In complicated patterns of streamline flow, the streamlines effectively define flow tubes. So the equation of continuity says that where streamlines crowd together the flow speed must increase. Applications In flowing rivers, when going from deep to shallow, the flow speed increases (often becoming turbulent) "still water runs deep". A river flows slowly and languidly through a meadow where it is broad, but speeds up to torrential speed when passing a narrows. In the circulatory system of the blood there is a branching effect. When a fluid flows past a Y-junction made up of pipes of the same diameter, the total cross-sectional area after the branch is twice that before the branch, so the flow speed must fall to half. Conversely, if it is important to keep the flow speed up, the pipes after the branch must have half the cross-sectional area of those before. (Note: blood will clot if its speed falls too low.) Blood flow blood flows from the heart into the aorta then into the 32 major arteries. These branch into smaller arteries (arterioles) that branch into a myriad of tiny capillaries and then the blood returns to the heart via the veins. Air conditioning systems must also be built with a03/p1/fluids/flow2.doc 3

4 consideration for the branch effect. Also the tube structure of the respiratory system is remarkably similar to that of the circulatory system.? 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. Setup radius of aorta R A = 10 mm = m radius of capillaries R C = mm = m speed of blood thru. aorta v A = 300 mm.s -1 = m.s -1 speed of blood thru. capillaries R 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 where A A and A C are cross sectional areas of aorta & capillaries respectively. Action A C = A A (v A / v C ) = πr A 2 (v A / v C ) A C = π( ) 2 (0.300 / ) m 2 = 0.20 m 2 If N is the number of capillaries then A C = N π R C 2 N = A C / (π R C 2 ) = 0.2 / {π ( ) 2 } N = a03/p1/fluids/flow2.doc 4

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