# PHY121 Physics for the Life Sciences I

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1 PHY Physics for the Life Sciences I Lecture 0. Fluid flow: kinematics describing the motion. Fluid flow: dynamics causes and effects, Bernoulli s Equation 3. Viscosity and Poiseuille s Law for narrow tubes 4/7/04 Note: set your Clicker to Channel Lecture 0

2 Fluid Flow & Continuity Equation When fluid is in steady flow the amount of mass (or number of molecules!) that moes past any gien point in a flow system, must be constant otherwise we would hae a storage system which accumulates (molecules of) fluid somewhere this is an example of a continuity condition : what goes in comes out Consider two places, and, in a flow pipe, with cross sectional areas and, where the fluid has densities ρ and ρ, and speeds and. the continuity condition says: dm /dt = dm /dt, where dm is the mass of fluid passing either point or point in a time interal dt: cross section of the essel; for pipe R dm dt dv dt ds dt dm dt i.e.: the quantity ρ is constant eerywhere in the fluid: steady state flow If the fluid is INCOMPRESSIBLE (ρ constant eerywhere): =Q=constant 4/7/04 Confusion! Here: Q=Volume flow rate! Lecture 0

3 Water (considered an incompressible fluid) flows through a 3 diameter pipe at a flow speed of 0 ft/min. Down the line the 3 pipe reduces to a diameter pipe; the flow speed in the pipe must be (ft/min) Rank Responses 90=(3 / ) 0 ft/min Other 4% 7% 6% 4% 3% 8% Values: Value Matches: /7/04 Lecture 0 3

4 Summary: Fluids (so far) Fluids and gases are characterized by density and pressure (later we ll add temperature as well): Density: ρ M/V, units: kg/m 3 Pressure: p F / = Force rea / rea units: Pa (Pascal) = N/m other units: atm Pa; bar 0 5 Pa = 000 mbar Pressure is a SCLR (i.e. has NO DIRECTION) rather: the RE under pressure has a single direction; to the plane of the area: F = p Pressure in a fluid: p = p 0 + ρgd, with D depth, p 0 pressure aboe the fluid Buoyant force F B is due to the action of the surrounding fluid (differential pressure) on an immersed object: F B = ρ Fluid V object g j (upward) Continuity equation (for steady flow): ρ = constant 4/7/04 Lecture 0 4

5 Bernoulli s Equation In 738 Daniel Bernoulli, professor at Basel Uniersity, wrote his fluid dynamics book Hydrodynamica He used Newton s Law throughout, and simpliying assumptions: the fluid is incompressible: = constant absence of friction in the flow (typically a poor approximation, especially for thick or iscous fluids and thin pipes) p ρ p What does this equation remind us of? p gy constant 4/7/04 Lecture 0 5

6 Deriation of Bernoulli s Equation Now consider the full works: fluid in flow moing from point to point. t each point the pipe has a ertical height y, a cross section, and the fluid has speed. We impose two simplifying conditions: p the fluid is incompressible: = constant absence of friction in the flow 4/7/04 (typically a poor approximation, especially for thick or iscous fluids and thin pipes) WORK DONE ON THE FLUID ds by external forces (pressure and graity) equals the CHNGE IN KINETIC ENERGY of the fluid: dw Net dw p p ds K dv dw p dwg p ds dmg y dv gy dv dv dm y g p p y p y ρ Lecture 0 6 p gy y constant y=0

7 Example wide essel (cross section ) is filled with water at height h. faucet is opened at the bottom (cross section a<<). What is the speed of the water rushing out? h Use Bernoulli s equation and the continuity equation: a 4/7/04 at top of tank continuity: p gh a p p atmospheric pressure! a a gh combining: gh a p g0 at faucet Lecture 0 7 if a gh!!

8 Example tank (, H) is filled with water. Calculate the horizontal distance R between the down splash of water from a hole in the tank s wall ( <<, h) and the bottom of the tank 4/7/04 Note: the result is symmetric between h and H h; we would get the exact same result with the hole a distance h aboe the bottom! Lecture 0 8

9 ery wide water-filled tank deelops a small leak at the bottom. The water leel in the tank is 5 m. The flow speed of the water out of the leak is (m/s) Rank Responses = (gh) Other Values: Value Matches: 0 6% 7% % 5% 4% % /7/04 Lecture 0 9

10 Venturi Effect a narrowing throat in a pipe speeds fluid up (continuity equation), and REDUCES the local pressure (Bernoulli s equation): h Find and, gien h,, fluid accelerates here, so pressure on the left must be higher than on the right! p p y y p, p p gh p gh, gh similar for gases! 4/7/04 Lecture 0 0

11 To lift a piece of paper off a table without touching it or using instruments higher speed of air => lower pressure. blow gently OVER the paper, parallel to the paper B. blow gently, aiming at the crack between the paper and table 74% 6%. B. 4/7/04 Lecture 0

12 Real s. Ideal Fluid Flow Ideal fluids flow without friction (internal or with walls)! ccording to Newton, an ideal fluid flowing in a horizontal pipe will keep flowing at constant flow rate when there is no pressure gradient; if there is a pressure difference, the fluid will simply accelerate REL fluids only flow if there is a pressure gradient! p (=p ) iscosity characterizes the resistance to differential flow of fluids (and gases) e.g. honey is more iscous than water water is more iscous than air L ideal fluid real fluids p low iscosity (water) ll fluids hae non-zero iscosity Exception: superfluids, e.g. 4 He below ~ K high iscosity (honey) 4/7/04 Lecture 0

13 Fluid flow is most chaotic in. high iscosity liquids. low iscosity liquids see demo on preious slide 9% 9%.. 4/7/04 Lecture 0 3

14 Types of Fluid Flow REL fluids only flow if there is a pressure gradient! Flow is impeded by friction between the fluid and the wall(s) nearest the wall the fluid is moing ery slowly, further out, the fluid moes increasingly fast this is an example of LMINR flow, where the flow lines are perfectly parallel to the wall layers closer to the wall impede the next adjacent layer further out This can be easily erified by fluid trapped between two plates: the top plate moes with speed top, the bottom plate remains stationary: moing plate of area P h y stationary plate d dy the force that drags the successie layers along is due to friction: slippage! d top F P P P dy h h Here, is the fluid-dependent iscosity! semi-empirical relationship that defines iscosity! the horizontal shear stress is simply F/ P : F h Unit of iscosity : Pa s P 4/7/04 Lecture 0 4

15 Laminar Flow in a Round Pipe In laminar flow in a round pipe, in which a pressure difference Δp occurs between the ends, y friction acts between wall and fluid, and between successie fluid layers to gie a non-linear flow elocity profile: Empirically: for a round pipe of length L and cross section : the aerage flow speed depends on the pressure gradient Δp and the iscosity of the fluid: L p 8 this relationship holds for laminar flow in a round pipe, i.e. when the flow lines are perfectly parallel to the pipe walls but breaks down for turbulent flow, which starts at higher flow speeds! p 8L Fluid honey (5 C) 600 fluid elocity pipe wall water (0 C) blood (37 C) air (0 C) Viscosity [Pa s] 4/7/04 Lecture 0 5

16 Turbulent Flow t high elocities and/or low-iscosity, the flow transitions from laminar to turbulent. In the turbulent regime, inertia (characterized by the kinetic energy term in Bernoulli s Δp Equation) dominates the iscous shear stress F/ // dimensionless number, the Reynolds Number, characterizes their relatie importance for round pipes of diameter D, the Reynolds Number is: Inertia D Re Shear Stress D for flow in round pipes, flow is laminar for Re<300, and fully turbulent for Re>4000. Unstable transition region in between e.g. for a cm diameter pipe with water (η= ) flow of m/s: Re= 0 4 and the flow is turbulent for a mm diameter essel with blood (η= ) flow of cm/s: Re=4 and the flow is laminar for more, see: 4/7/04 Lecture 0 6

17 Poiseuille s Law Real fluids require a pressure difference Δp in order to p keep flowing (>p ) iscosity η strongly depends on Temperature! Flow rate through pipe of length L and radius R (laminar flow!): p 8 L 4 R Q 8 R p 8 L p L no flow at walls easy deposit of matter on the inside of essels Q R 4 any narrowing of the aperture requires a much larger Δp Rapid changes in blood essel diameter (e.g. by plaques) may lead to turbulent flow, for instance when a narrower essel widens into a larger one. theroma often causes turbulent flow; may be detected with a stethoscope... In turbulent flow the flow rate Q is proportional to Δ, as opposed to its direct proportionality in laminar flow; more pressure (and hence more power) is required to get the same flow rate as in the laminar flow regime! L iscous (for LMINR flow!) ttention: possible confusion with Heat flow!) (Poiseuille s Equation) p 4/7/04 Lecture 0 7

18 n artery is being narrowed by a plaque to half its original diameter; if the flow rate is to remain the same, the pressure difference across the region of the plaque must increase by a factor. /6 Q B. /8 4 R p C. /4 8 L D. / E. no difference F. G. 4 H. 8 I. 6 Jean-Louis Marie Poiseuille Response Counter ( ) % % % % % % % % %. B. C. D. E. F. G. H. I. 4/7/04 Lecture

19 Summary: Fluids Fluids (and gases) are characterized by density ρ, p, T Density: ρ M/V, units: kg/m 3 Pressure: p F / = Force rea / rea units: Pa (Pascal) = N/m other units: atm Pa; bar 0 5 Pa Pressure is a SCLR quantity (i.e. it has NO DIRECTION) rather the RE under pressure has direction; to the plane of the area: F = p Pressure in a fluid: p = ρgd+p 0, at depth D Buoyant force F B is due to the action of the surrounding fluid on an immersed object: F B = ρ Fluid V displaced fluid g j (up!) Continuity equation (for steady flow): ρ = constant Bernoulli s equation (for steady flow, NO FRICTION, and an incompressible fluid ( ρ constant) from W Net = ΔK: Poiseuille s Equation: with iscosity η Q 4 R 8 p L p gy constant 4/7/04 Lecture 0 9

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