Viscous Flow

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Viscous and turbulent flow Level: high school (16-17 years) 4 hours (2 hours class teaching, 2 hours practical excercises) Content: 1. Viscous flow 2. Poiseuille´s law 3. Passing from laminar to turbulent turbulent flow 4. Reynold´s number Aids needed for demonstrations: viscous fluid (water), tubes (pipes), injections, paper, stopwatches, vessels, , weights

Motivation Tornado in a bottle: How can I get the water from the upper bottle in the lower one? Air whirls: Put out the candle

Viscous Flow The idea of an ideal flow – every layer moves with the same velocity, there is no viscosity. The velocity in the middle of the pipe is the same as in layer next to the wall. In a real a real fluid , where viscosity is present, the velocities are not the same – in the middle of the pipe is the greatest velocity, the fluid layer next the wall has a velocity near 0.  How we will express, express, what viscosity is?

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Let´s have two parallel plates. The top plate is free to move, the bottom one is stationary. If the top plate is to move with a velocity v (relative to the bottom plate), a force F is required. It is needed another force in the case of water or in the case of honey or glycerin. A fluid can we model as a lot of plates which various velocities. The velocity of each layer is different. The greatest one is on the top, at bottom it is zero.This kind of flow is a laminar flow.

The tangential force F required to move a fluid layer at a constant speed v, when the layer has an area S and is located a perpendicular distance y from an immobile surface, is given by η Sv F  = , where η is the coefficient of viscosity.  y SI unit for viscosity: Pa.s Common unit: poise (P)

1 poise (P) = 0,1 Pa.s Jean Poiseulle (1797-1869) – french physician, he explored the movement of fluids in pipes to acquire laws or regularities of the blood flow in our body.

Values of viskosity Under ordinary conditions the viscosity of gases is smaller than those of liquids. The viscosity depends on temperture – the viscosities of liquids decrease with higher temperature, the viscosity of gases increase as the temperature is raised. Viscosity:

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Water (20 C) 1,00 .10 Pa.s -3 Benzen C6H6 0,65.10 Pa.s -3 Ethanol C2H6O 1,20.10 Pa.s -3 Glycerol C3H8O3 1480,00.10 Pa.s o -3 Blood (37 C) 5,00.10 Pa.s o -3 Aer (18 C) 0,019.10 Pa.s Viscous flow is common in various situations – for example oil moving through a pipeline. We have to identifie factors that determine the amount of liquid , that flows across a cross3 section of a pipe in a time interval. This is the volume flow rate Q (in m  /s). Q is proportional to P 2 – P1 – the pressure difference between any two locations along the pipe (greater pressure leads to a larger flow), a l ong pipe offers a larger resitance to the flow than a short pipe does (pumping station along the long pipelans). Hight viscosity fluids flow less readily than low viscosity fluids. The greatest importance has the dependence on the radius r 4 of the pipe – Q is proportional to the fourths power of the radius (r ). The mathematical relation is known as Poiseuille´s law:

A fluid whose viscosity is η, flowing through a pipe of radius r and lenght L , has a volume flow rate Q given by 4

Q=

π r 

( P2 − P1 )

, 8η  L P2 and P1 are poressures at the ends of the pipe. The central formula in the laminar flow model is the Poiseuille equation: 8η  L  R = 4 π r 

Origins of the Poiseuille equation The underlying assumption of laminar flow is the condition of uniform viscosity across the diameter of the conduit, ie, each fluid molecule within the pipe is exerting a similar force against its immediate neighbor towards the periphery. This yields the following: • Fluid adjacent to the conduit wall is motionless ( νo = 0). • Maximum fluid velocity ( νmax) is in the center of the conduit. • Fluid velocity is related to distance from the center ( νi) by a parabolic function. This is why laminar flow sometimes is termed parabolic flow. Another way to understand laminar flow conditions is to imagine concentric circles of fluid flow. Flow velocity is uniform within each annulus. Velocity increases as you approach the center of  the conduit.

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The idealized fluid flow model is used for its´ mathematical simplicity The science of fluid flow applies to many facets in the field of radiology, from arteriography and angioplasty to percutaneous drainage. Precise modeling of physiologic fluid flows under real-life flow conditions requires mathematics too complex for practical use in clinical medicine. A workable model may be devised from the first principles of engineering to create a first-order approximation that suits most purposes.

Further simplification: the Ohm law For clinical applications fluid properties may be taken as constant. The Poiseuille equation can be simplified to resemble the Ohm law of electrical resistance. This law relates voltage drop across an electric circuit (∆V ) to electric resistance ( R) and electric current ( I ), as follows: ∆V = IR. The greater the current (electron flow) or resistance, the higher the voltage drop required. For the sake of simplicity, ∆V  often is written as V , although it is the change in voltage and not the absolute voltage that matters. Fluid-flow calculations can be modelled as simple electric circuits. For this analogy (model) we 4 relate ∆V to ∆P, defining flow resistance as  R = 8 Lη / πr to obtain the following: ∆P= QR. Resistive losses (pressure drops) linearly are related to flow rate and flow resistance. While flow resistance linearly relates to conduit length, it is inversely related to t he fourth power of the radius (or diameter). For example, a 1-cm–diameter pipe has 16 times the flow resistance of a 2-cm– diameter pipe of the same length and carrying the same fluid.

Interdisciplinary relations - blood flow

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Giving an injection: Example: -3 A hypodermic syringe is filled with a solution whose viscosity is 1,5.10 Pa.s. As in fig. The -5 2 plunger area of the syringe is 8,0.10 m , the lenght of the needle is 0,025 m. The internal -4 radius of the needle is 4,0.10 m. The gauge pressure in a vein is 1900Pa. What force must be -6 -3 applied to the plunger, so that 1,0.10 m of solution can be injected in 3 s ?

Explanation: The necessary force is the pressure applied to the plunger times the area of the plunger. Since viscous flow is occurring, the pressure is different at different points along the syrine. The barrel of the syringe is so wide that little pressure difference is required to sustain the flow up to point 2, where the fluid encounters the narrow needle. Consequently, the pressure applied to the plunger is nearly equal to the pressure P2 at point 2. To find this pressure, we apply 4 Poiseuille´s law to the needle: P2 – P1 = 8ηLQ/(πr ). The pressure P1 is given as a gauge pressure, which, in this case, is the amount of pressure in excess of atmospheric pressure. This causes no difficulty, because we need to find the amount of force in excess of that applied to the plunger by the atmosphere. The volume flow rate Q can be obtained from the time needed to inject the known volume of solution. Solution: -6

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Q = 1,0.10 m  /3,0 s = 3,3.10 m  /s 8η  LQ 8(1,5.10 −3 Pa.s)(0,025m)(3,3.10−7 m 3 /s) P2 − P1 = = = 1200Pa −4 4 4 π r  π (4,0.10 m) Since P1 is 1900 Pa, the pressure P 2 must be P2 = 1200 Pa + 1900 Pa = 3100 Pa The force that must be applied to the plunger is the pressure times the plunger area: -5 2 F = 3100 Pa.8,0.10 m = 0,25 N

Turbulent flow Turbulent flow is when particles in a fluid move in many different directions and at many different velocities. Turbulent flow is the opposite of laminar flow.

Turbulent flow is a type of fluid (gas or liquid) flow in which the fluid undergoes irregular fluctuations, or mixing, in contrast tolaminar flow, in which the fluid moves in smooth paths

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or layers. In turbulent flow the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction. The flow of wind and rivers is generally turbulent in this sense, even if the currents are gentle. The air or water swirls and eddies while its overall bulk moves along a specific direction. Most kinds of fluid flow are turbulent, except for laminar flow at the leading edge of solids moving relative to fluids or extremely close to solid surfaces, such as the inside wall of a pipe, or in cases of fluids of high viscosity (relatively great sluggishness) flowing slowly through small channels. Common examples of turbulent flow are blood flow in arteries, oil transport in pipelines, lava flow, atmosphere and ocean currents, the flow through pumps and turbines, and the flow in boat wakes and around aircraft-wing tips.

Give me an example! Have you ever seen rapids in a river? Obstacles like rocks and fallen trees cause the water t o move in many different directions and at many different velocities, which creates turbulent flow. A profile of a car – turbulent flow

Turbulence flow can leeds to whirls.

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Next question – when comes to overlapping from laminar to turbulent flow – Reynolds  ρ rv  number. The Reynolds number is a dimensionless constant R, where  R = η 

where v is the velocity (critical velocity). For cylindrical pipes, the Reynolds number corresponding to the critical velocity is about 2000. Thus for water flowing through a pipe of diameter 2 cm (for example garden hose), the critical speed is vc = 2000

1.10 −3 N.s/m 2

10 3 kg/m 3 0.02m flow is turbulent, it has v = 1 m/s)

= 0,1m/s = 10cm/s . ( A low speed, ordinary the

Animations on www: http://www.physik.uni-wuerzburg.de/physikonline.html http://pen.physik.uni-kl.de/medien/MM_Videos/index.html

Farther applications : horizontal and vertical transport of pollutants in the atmosphere – time scales  Horizontal transport: Distance Time scales 1-10 m seconds 3-30 km hours 10-1000 km days hemisphere months Vertical: Layer Boundary Troposphere Stratosphere

Extension ground to 100 m 3 km 100 m 3 km to 10 km 15 km 10 km 15 km to 50 km

(plume shapes from a chimney) The troposphere determines the weather

Time scale minutes to hours days to weeks years

greenhouse gases, greenhouse effect

Problems 3

1. A pressure diference of 1,5.10 Pa is needed to drive water through a pipe whose -3 -4 3 radiu sis 6,8.10 m. The volume flow rate of the water is 3,2.10 m  /s. What is the -3 lenght of the pipe? The water viscosity η  = 1.10 Pa.s. -3 -3 2. A blood vesel is 0,1 m in lenght and has a radiu sof 1,5.10 m. Blood (η  = 4.10 Pa.s) -7 3 flows at a rate of 1.10 m  /s. Determine the difference in pressure that must be maintained between the two dnes of the vesel. 3. Calculate the highest average speed that blood could have and still Romain in laminar -3 -3 flow when it flows through the aorta ( R = 8.10 m,  ρ  = 1060 kg.m ).

Laboratory experiment

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The purpose is to study viscous fluid flow. The resistance to flow of single capillaries, 2 capillaries in a series and 2 capillaries in a parallel configurations is measured and compared to predictions. Equipment: 2 cups, 1 panel with 3 glass capillaries (various diametres), 2 identical capillaries in series and in parallel, tubes with valves, 1 beaker, 1 ruler Introduction: We study the flow of viscous fluids through glass capillaries. The flow rate through the capillary is proportional to the pressure difference across the capillary ∆P=R.Q, 8η  L where R is the resistance of the capillary to the fluid flow  R = . A beaker is positioned a 4 π r  heigh h above the cup. The capillary is kept horizontal, and vonected with the overflow in beaker. Then the pressure difference on the capillary is P = h ρg, where ρ is the density of  water. The water flows in a calibrated baker, or the mass of the water is determined on laboratory weights. By measuring the collection time, the flow rate Q can be computed. The water lavel in beaker have to be constant. The lenght of the capillaries is to determine. Various measurements can be obtained: measurement with capillaries of the same diameter but various lenght, equal lenghts and various diameters, one capillary in different hight (different pressure). The other conditions will be constant (temperature, density of water, humidity atc.). In the second part of the lab experiment is to measure 2 equal capillaries connected in series and parallel (analogy with Ohm law).