1. Viscous Fluid Flow (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 Umm Al-Qura University جا معة أم القرى Faculty of Applied Science كـلية العلوم التطبيقية Department of Physics قسم الفيزياء General Physics (For medical students): 403104-3 403104-3 : المدخل للفيزياء الطبية

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 m 2 in area. If the viscosity of the air is 1.8 x 10 -5 Pa.s, find the force required to move the cart at a constant speed of 0.2 m/s. Solution 14.1: 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. 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. 14.4 Flow in The Circulatory System

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 arteriovenous anastomosis (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 R f is defined in general, as the ratio of the pressure drop to the flow rate: - R f defines the flow resistance whether the flow is laminar or not. - The basic S.I. Unit of R f 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 10 5 kPa 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 m 3 s -1 ?

13. Solution 14.5: 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 R f1 and Q 1 :  P is the pressure across all of the arteries. -If there are N identical arteries, the total flow is Where R p 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 10 7 capillaries in the mesenteric vascular bed of a dog if they are assumed to be parallel?

16. With  =2.084x10 -3 Pa s, the resistance of one capillary is: Solution 14.6 There are N= 4.73x10 7 capillaries in parallel, so there effective resistance is :

17. Flow Rate m 3 /s Flow Resistance kPa.s/m 3 Brain 12.5 x 10 -6 9.3 x 10 5 Heart 4.2 x 10 -6 2.8 x 10 6 Legs 13.7 x 10 -6 8.5 x 10 5 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 R s of these sections which are said to be in series, is the sum of the resistances.