(14 p.341) Viscous Fluid Flow

1. Umm Al-Qura University جا معة أم القرى Faculty of Applied Scienceكـلية العلوم التطبيقية Department of Physics قسم الفيزياء General Physics (For medical students): 403104-3 403104-3 : المدخل للفيزياء الطبية (14 p.341)Viscous Fluid Flow A. Prof. Hamid NEBDI hbnebdi@uqu.edu.sa Faculty of Applied Science. Department of Physics. Room: 315 second floor. Phone: 3192

2. Course’s Topics 14.1- Viscosity 14.4- Flow in the Circulatory System

3. 14.1 Viscosity - The viscosity of a fluid is a measure of its resistance to flow under an applied force. - The greater the viscosity, the larger the force required to maintain the flow, and the more energy that is dissipated. - Molasses has a high viscosity, water a smaller viscosity, and air a still smaller viscosity.

4. Viscosity is readily defined by considering a simple experiment. Figure 1 shows two flat plates separated by a thin fluid layer. Figure 1 • The lower plate is held fixed, a force is required to move the upper plate at a constant speed. • This force is needed to overcome the viscous forces due to the liquid and is greater for a highly viscous fluid Moving plate Fixed plate

5. - The force F is observed to be proportional to the area of the plates A and to the velocity of the upper plate ∆v (the moving one) and inversely proportional to the plate separation ∆y. (1) • The proportionality constant  (Greek letter “eta”) is called the viscosity. - The larger the viscosity, the larger force needed to move the plate at a constant speed.

6. Dimensions of Viscosity M, L, and T stand for mass, length, and time respectively. The S.I. Unit of viscosity is: 1 kg m-1 s-1 = 1 Pa s.

7. The Relation between Viscosity and Temperature • As the temperature increases viscosity decreases, for liquids. • As the temperature increases viscosity increases, for gases. • Because viscous forces are usually small, fluids are often used as lubricants to reduce friction.

8. Example 14.1 An air track used in physics lecture demonstrations, supports a cart that rides on a thin cushion of air 1mm thick and 0.04 m2 in area. If the viscosity of the air is 1.8 x 10-5Pa.s, find the force required to move the cart at a constant speed of 0.2 m/s. Solution 14.1:The force required is: This is a very small force and is consistent with the observation that an air track is nearly frictionless

9. 14.4 Flow in The Circulatory System The blood - The circulatory system transports the substances required by the body and take off the waste products of metabolism. - In order to perform a large number of functions, the blood contains many different constituents, including red blood cells, white blood cells, platelets, and proteins. - However, for our purposes, it is sufficient to treat the blood as a uniform fluid with viscosity and density.

10. The Cardiovascular System - This system includes the heart, and an extensive system of arteries, vascular beds containing capillaries, and veins. • A particularly interesting compound of the cardiovascular system is the arteriovenousanastomosis (AVA), or shunt. - These shunts are particularly important, since the surrounding muscle tissue can adjust the diameter of the blood flow to various organs as conditions change. - Smaller shunts in the skin are open if the body needs to release heat or to increase skin temperature.

11. Flow Resistance - The flow resistance Rf is defined in general, as the ratio of the pressure drop to the flow rate: - Rfdefines the flow resistance whether the flow is laminar or not. - The basic S.I. Unit of Rf is the : Pa.s.m-3 But we use :kPa s m-3 In text and literature on physiology pressures are usually measured in torr and lengths in cm: 1 torr s cm-3 = 1.333 x 105kPa s m-3 - Usually the flow resistance in a large artery is small. Consequently, the pressure drop in such arteries is small. • The following relation is applicable only if the flow is laminar: Where l is the length of the tube, and R is the Radius of the tube.

12. Example 14.5: • The aorta of an average adult human has a radius 1.3 x 10-2 m. What are the resistance and the pressure drop over a 0.2 m distance, assuming a flow rate of 10-4 m3s-1?

13. Solution 14.5:From Table 14.3, =2.084 x 10-3 Pa s, so the flow resistance of the aorta is: The pressure drop over the 0.2 m distance is then: • - This is very small value of the pressure drop, compared to the total pressure drop in the system, which is about 13.3 kPa. • - Most of the flow resistance and pressure drops occur in the smaller arteries and vascular beds of the body (Table 14.4).

14. - The flow resistance of a collection of arteries can be calculated. The calculation can be done by considering each category of artery separately. • We assume that all of the arteries of a given size are in parallel; each artery carries its equal share of the Rf1 and Q1: Pis the pressure across all of the arteries. • If there are N identical arteries, the total flow is Where Rp is the equivalent flow resistance of this arrangement.

15. Example 14.6 From Table 14.2, the radius of a single capillary is 4 x 10-6 m and its length is 10-3 m. What is the resistance of 4.73 x 107 capillaries in the mesenteric vascular bed of a dog if they are assumed to be parallel?

16. Solution14.6 With =2.084x10-3 Pa s, the resistance of one capillary is: There are N= 4.73x107 capillaries in parallel, so there effective resistance is :

17. Some approximate flow rates and resistances for the resting, reclining adult.

18. Suppose we know the resistances of N sections, each of which leads into the next. • The total pressure drop is as follows - Each pressure drop is the total flow rate Q times the resistance of that section. - The effective flow resistance Rs of these sections which are said to be in series, is the sum of the resistances.