Week 8. Topics: Next deadline: Viscous fluid flow (Study guide 14. Sections 12.4 and 12.5.) Bolus flow (Study guide 15. Section 12.6.

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1 8/1 Topics: Week 8 Viscous fluid flow (Study guide 14. Sections 12.4 and 12.5.) Bolus flow (Study guide 15. Section 12.6.) Pulsatile flow (Study guide 15. Section 12.7.) Next deadline: Friday October 31 Quiz #8 (12-13); requires labs 12 and 13

2 8/2 Viscosity (Sec. 12.4) The fluids we have studied so far (those that obey Bernoulli s equation) didn t have any viscosity they didn t stick. Now we shall study sticky fluids those that have viscosity. Viscosity is an internal frictional force acting on the fluid. It tends to slow the fluid down. The degree of viscosity (how much it sticks) of a given fluid is measured by its coefficient of viscosity η (eta). Its unit is the N s / m 2.

3 8/3 The viscosity of a fluid depends on its temperature. For gases, η increases with increasing temperature (a cold gas is less viscous than a hot gas). For liquids, η decreases with increasing temperature (a cold liquid is more viscous than a hot liquid). Viscosity of various fluids (Table 12-1 in textbook) fluid η (N s/m 2 ) water at water at blood at castor oil at air at air at

4 8/4 The viscosity of a liquid can be measured with the following experiment: plate F v l ground A glass plate of area A is sliding with constant speed v on top of a layer of liquid with thickness l. The top layer of the liquid sticks to the plate and moves with the same speed v. The bottom layer of the liquid sticks to the ground and doesn t move. It is observed that a force F is required to make the plate move. (Without this force, the liquid s viscosity would bring the plate to a stop.)

5 8/5 It is also observed that the speed v is proportional to the force F : F A v l The constant of proportionality is defined to be the coefficient of viscosity η: F A = η v l Fluids for which this relation is valid are called Newtonian. (A more precise statement of this result is F/A = η dv/dy, where dv/dy is the velocity gradient within the liquid.) (For some fluids, the relation between F and v is more complicated. Such fluids are called non-newtonian.)

6 8/6 Flow of a viscous fluid (Sec. 12.5) Consider a viscous fluid flowing through a pipe of radius R and length L. L R To compensate for viscosity (which slows the fluid down), the flow must be maintained by a pressure difference across the length of the pipe. For example, the fluid will move from left to right if P (left) > P (right)

7 8/7 Inside the pipe, the fluid s outer layer sticks to the wall and doesn t move. The inner layers can move, however, and the central portion of the fluid moves with maximum speed v c. The fluid develops a parabolic velocity profile: r v c The velocity v at radial position r is given by v(r) = P (left) P (right) 4ηL ( R 2 r 2)

8 8/8 According to this, the velocity at the outer layer is v(r) = 0 and the maximum central velocity is The fluid s average velocity is The flow rate is then or v c = v(0) = R2 4η v = 1 2 v c = R2 8η P L P L Q = A v = (πr 2 ) R2 8η Q = πr4 8η P L P L

9 8/9 This result is known as Poiseuille s law. It means that the flow rate is proportional to the pressure gradient P L P (left) P (right) = L and increases very rapidly with the pipe s radius R. (Recall previous figure of velocity profile. The velocity gradient is largest near the walls and smallest at the centre. This is why a raft tends to lose directions near the edge of a fast-moving stream, but has relatively stable motion in the centre of the stream.)

10 8/10 Example #1: A vein of radius 2.0 mm and length 1.0 m is maintained at a pressure difference of 3.0 mm Hg. What is the average speed of the blood in this vein? Example #2: A certain diseased artery has a cross-sectional area of only 70.7% that of a health artery. Determine the ratio of the pressure gradients across the two arteries if the same volume of blood is to travel the arteries in the same time.

11 8/11 Bernoulli vs Poiseuille: Bernoulli s equation, P ρv2 + ρgy = constant applies to a nonviscous fluid. It must be used if the pipe changes in cross section or in elevation. Poiseuille s law, Q = πr4 8η [P (left) P (right)] L applies to a viscous fluid. It must be used if the fluid has viscosity and a pressure gradient, but the pipe must have constant cross section and elevation. The continuity equation, Q = A v = constant, can always be used.

12 8/12 Example #3 (similar to Text 12-2 and 12-10, SG-14 Self-Test I # 1) An artery of radius 3.0 mm divides into a number of capillaries. Each capillary has a radius of 6.0 µm. The flow speed in the capillaries is 2.5% of the flow speed in the artery. Into how many capillaries does the artery divide? Quiz 9 questions related to Lab 14: Only on measuring the viscosity of water based on Poiseuille s law, Eq. (1) in Lab page To calculate η, rewrite that equation as η = πr4 ρght 8LV How is this affected if h or t are changed?

13 8/13 Study guide 15 Bolus, pulsatile, and turbulent flows; aneurysms The fluid flows we have studied so far were rather simple. In real-life situations, fluid flows can be more complicated. We shall look into such possible complications in this study guide. The context for this is provided by an important application of fluid dynamics: blood circulation.

14 8/14 Bolus flow (Sec. 12.6) The flow of blood in capillaries is complicated by the fact that red blood cells are bigger than the blood vessels. A typical capillary has a diameter of 5-6 µm. Red blood cells (erythrocytes) have a diameter of about 8 µm. They must deform to go through the capillary, and the flow pattern is seriously affected. plasma red cell This type of fluid flow is known as bolus flow. It keeps the blood plasma well stirred for good distribution of nutrients; the plasma flow is like the motion of a bulldozer tread.

15 8/15 Pulsatile flow (Sec. 12.7) The flow of blood in large arteries or veins is complicated by the fact that the heart acts as a pump and produces a nonsteady flow. The beating action of the heart creates a pulsatile flow. Typically, during one heart beat, the blood advances 3 steps, then backs up one, and finally advances 1 step. Graph of flow rate vs time during one heart beat: Q max t 1 t 2 t 3 t p

16 8/16 The time t p represents a complete heart beat: it is the period associated with the heart cycle. The quantity Q max is the maximum value of the flow rate. (Positive Q means that the blood is moving forward. Negative Q means that the blood is moving backward.) The total volume V of blood displaced during one heart beat (the total flow) is given by the area under the curve: V = t p Q(t) dt. 0 Since the areas under t 2 and t 3, and under t 3 and t 4, approximately cancel out, V can be estimated by computing the area under the first peak. Approximating the shape of the peak by a triangle, we have V 1 2 (t 2 t 1 )Q max

17 8/17 Example: An analysis of blood flow through a blood vessel is carried out, and leads to the graph shown below. The vessel has a radius of 3.0 mm. The maximum flow speed of blood through the vessel is measured to be 50 cm/s. During the measurements the heart was beating at a rate of 60 beats per minute. What is the total flow of blood through the vessel after 1 minute? Q max o o 240 o 360 o

18 Note, not all questions have this shape of the graph; see, e.g., SG-15 Self-Test I. 8/18