Chapter 8: The Motion of Fluids PDF

Chapter 8 The Motion of Fluids Assignment (1,2,3,4,5, 6,7,8) 8.1 BERNOULLI’S EQUATION The equation of continuity The rate of flow or flow rate Q Q is defined as : the volume of fluid flowing past a point in a pipe per unit time....

Chapter 8 The Motion of Fluids Assignment (1,2,3,4,5, 6,7,8) 8.1 BERNOULLI’S EQUATION The equation of continuity The rate of flow or flow rate Q Q is defined as : the volume of fluid flowing past a point in a pipe per unit time. ∆𝑉 Q= … … … (1) ∆𝑡 3 The SI units of flow rate are 𝑚. 𝑠𝑒𝑐 Consider a section of a tube with a constant cross sectional area A (Fig. 8.1). In a time ∆ , the fluid moves a distance ∆𝑥 = 𝑣 ∆𝑡 , and the volume fluid leaving the tube is ∆𝑉 = 𝐴 ∆𝑥 = 𝐴𝑣 ∆𝑡. Comparing this equation with equation (1), we see that ∆𝑉 𝐴∆𝑥 𝐴𝑣∆𝑡 𝑄= = = ∆𝑡 ∆𝑡 ∆𝑡 Q = 𝐴𝑣 … … … (2) The flow rate equals the cross sectional area of the tube times the velocity of the fluid. If the fluid is incompressible, in a unit time as much fluid must flow out of the pipe as flows into it. Therefore, the rates of flow in segments 1 and 2 are equal (Fig. 8.2). This principle is called the equation of continuity. That is, 𝑄1 = 𝑄2 𝐴1𝑣1 = 𝐴2 𝑣2 … … … (3) The product of the cross sectional area and the velocity of the fluid is constant. If at 1  some point A decreases, v must increases. For example, if the area is halved, then the velocity doubles. Continuity equation follows from the law of mass conservation. Exercise 8-6.ompute the average velocity of the blood in the aorta of radius 1 cm if the flow rate is 5 liter/min. 𝑄 = 𝐴𝑣 = 𝜋𝑟2 𝑣 Exercise 8-7. When the rate of blood flow in the aorta is 5 liter/min, the velocity of the blood in the capillaries is about 0.33 mm/sec. If the average diameter of a capillary is 0.008 mm, calculate the number of capillaries in the circulatory system. BERNOULLI’S EQUATION The Bernoulli’s equation can be used under the following conditions: 1- The fluid is incompressible; (its density remain constant). 2- The fluid is non-viscous; (no mechanical energy is lost due to friction). 3- The flow is streamline (not turbulent ) 4- Steady state flow. ( The velocity of the fluid at any point is constant with time) 2 Bernoulli’s equation gives the relationship between velocity, pressure, and elevation in a line of flow. Bernoulli’s equation states that at any point in the channel of a flowing fluid the following relationship holds: Here P is the pressure in the fluid, h is the height, ρ is the density, and v is the velocity at any point in the flow channel.  The first term in the equation is the potential energy per unit volume of the fluid due to the pressure in the fluid. (Note that the unit for pressure, which is dyn/cm2, is identical to erg/cm3, which is energy per unit volume.)  The second term is the gravitational potential energy per unit volume.  The third term is the kinetic energy per unit volume.  Bernoulli’s equation follows from the law of energy conservation. W=PΔV 3   Bernoulli’s equation states that in the absence of friction, the total energy in the fluid at any point in the channel remain constant. In general, as a fluid moves through a region where its speed or elevation above the Earth’s surface changes, the pressure in the fluid varies with these changes (Fig. 8.3). The relationship between fluid speed (v), pressure (P), and elevation (h) was first derived in 1738 by Swiss physicist Daniel Bernoulli. Consider a fluid flowing through a pipe as shown in (Fig. 8.3). Bernoulli’s equation is often expressed as: For a horizontal pipe (Fig.8.2.) (ℎ1 = ℎ2) Bernoulli’s equation shows that the pressure of a fluid decreases as the speed of the fluid increases. Drop in pressure This relationship shows that while the flow velocity in segment 2 increases, the pressure in that segment decreases. 4 Exercise 8-8. Compute the decrease in the blood pressure of the blood flowing through an artery the radius of which is constricted by a factor of 3. Assume that the average flow velocity in the un-constricted region is 50 cm/sec. 5 6  8.2 VISCOSITY AND POISEUILLE’S LAW  In an ideal fluid, it flows without friction.  In a real fluid, the molecules attract each other; consequently, relative motion between the fluid molecules is opposed by a frictional force, which is called viscous friction.  Viscous friction is proportional to the velocity of flow and to the coefficient of viscosity for the given fluid.  As a result of viscous friction, the velocity of a fluid flowing through a pipe varies across the pipe.  The velocity is highest at the center and decreases toward the walls;  at the walls of the pipe, the fluid is stationary. Such fluid flow is called laminar. Fig. 8.2.  The length of the arrows indicates the magnitude of the velocity of the fluid  If viscosity is taken into account, the rate of laminar flow Q through a cylindrical tube of radius r and length L is given by Poiseuille’s law, which is 𝜋 𝑟4(𝑃1 − 𝑃2) 𝑐𝑚3 𝑄= 𝑖𝑛 8𝜂 𝐿 𝑠𝑒𝑐  Where 𝑃1 − 𝑃2 is the difference between the fluid pressures at the two ends of the cylinder and η : is the coefficient of viscosity measured in units of 𝑑𝑦𝑛.𝑠𝑒𝑐 = 𝑃𝑜𝑖𝑠𝑒 𝑐𝑚2  Note of : 7    Poiseuille’s law describes the flow rate of fluid through a channel.  Viscosity characterized the force between moving layers of fluid.  Viscosity increases as the fluid temperature decreases.  High fluid viscosity (large 𝜂 leads to low flow rates Q), i.e., as viscosity of a fluid increases, its flow decreases. ( which is reasonable)  Q is extremely depends on the radius of the tube (𝑄 ∝ 𝑟4), i.e., the most effective parameter affecting blood flow in an artery is its internal diameter. For example, increasing R to 1.19 R, flow rate Q is doubled. 𝜋 𝑟14(𝑃1 − 𝑃2) 𝑄1 = 8𝜂 𝐿 𝜋 𝑟4(𝑃1 − 𝑃2) 𝜋 (1.19)4𝑟4(𝑃1 − 𝑃2) 𝑄1 = 2 = 1 = 2 𝑄2 8𝜂 𝐿 8𝜂 𝐿  A frictionless fluid will flow steadily without an external force applied to it.   This fact is evident from Bernoulli’s equation, which shows that if the height (ℎ1 = ℎ2) and velocity of the fluid remain constant(𝑣1 = 𝑣2), there is no pressure drop along the flow path.  But Poiseuille’s equation for viscous flow states that a pressure drop always accompanies viscous fluid flow 𝑄 8𝜂 𝐿 𝑃1 − 𝑃 2 = 𝜋 𝑟4 Exercise: A large artery in a dog has an inner radius of 4 x 10-3 m. Blood flows through the artery at the rate of 1 cm3 sec-1.Find the pressure drop in a 10 cm long segment of the artery. (Given η = 2.084 x 10-3 Pa. sec). (Answer: 2.07 Pa)  The product of the pressure drop and the area of the pipe is the force required to overcome the frictional forces that tend to retard the flow in the pipe segment. 𝐹 = (𝑃1 − 𝑃2) 𝜋 𝑟2 8 9 Exercise 8-1 (a) Calculate the pressure drop per centimeter length of the aorta when the blood flow rate is 25 liter/min. The radius of the aorta is about 1 cm, and the coefficient of viscosity of blood is 4x10-2 pois 1 torr = 1.33 x 103 dyn/cm2 x = 42.5 dyn/ cm2 x = 3.19 x 10-2 torr/cm Exercise 8-2 Compute the drop-in blood pressure along a 30-cm length of artery of radius 0.5 cm. Assume that the artery carries blood at a rate of 8 liter/min. Exercise 8-5. (a) Show that if the pressure drop remains constant, reduction of the radius of the arteriole from 0.1 to 0.08 mm decreases the blood flow by more than a factor of 2. (b) Calculate the decrease in the radius required to reduce the blood flow by 90%. Solution: (a) Given: P1 –P2, 𝜂 and L are constants, r1 = 0.1 mm reduced to r2 = 0.08 mm, Find Q2 in terms of Q1. ( note: the radius decreases 20% ) 10  𝑟2 0.084 = = = 0.4096 𝑟4 0.104 𝑄2 = 0.4096 𝑄𝑄1 𝑄𝑄1 = 2.44 𝑄𝑄2 (a) Given: P1 –P2 , 𝜂 and L are constants, Q2 = 0.1Q1, Find r2 in terms of r1. (𝑃1 − 𝑃2)𝜋 𝑟4 8𝜂𝐿 (𝑃1 − 𝑃2)𝜋 𝑟4 8𝜂𝐿 𝑟 = 𝑟4 𝑟 = 𝑟4 𝑟2 1/ = (0.1) 4 = 0.56 𝑟1 8.3 TURBULENT FLOW Laminar flow Turbulent flow Quite and Silent Not a silent (noises) Characterized by the smooth flow Characterized by eddies and whirls that mix layers of the fluid in layers that do not of fluid together. mix. Streamline flow. Less resistance and less heating More resistance and greater heating.  Turbulence caused mainly from: 1. Obstruction or sharp corner 2. High speed of fluid. 11   If the velocity of a fluid is increased past a critical point, the laminar flow becomes turbulent.  In a cylindrical pipe the critical flow velocity vc above which the flow is turbulent, is given by:  Where, D is the diameter of the cylinder,  ρ is the density of the fluid, η is the viscosity and ℛis the Reynold’s number.  Reynold’s number ℛ for most fluids has a value between 2000 and 3000.   Through most of the circulatory system the blood flow is laminar. Only in the aorta does the flow occasionally become turbulent.  The frictional force (or mechanical energy loss) in turbulent flow is greater > than in laminar flow. 12 8.4 CIRCULATION OF THE BLOOD  The circulation of blood through the body is usually compared to a plumbing system with the heart as the pump and the veins, arteries, and capillaries as the pipes through which the blood flows.  This analogy is not entirely correct.  Blood is not a simple fluid (it contains cells that complicate the flow) especially when the passages become narrow. Furthermore, the veins and arteries are not rigid pipes but are elastic and alter their shape in response to the forces applied by the fluid.  Blood in arteries ( away from the heart), while blood in veins (moves towards the heart)  it is possible to analyze the circulatory system with reasonable accuracy using the concepts developed for simple fluids flowing in rigid pipes. 13  Fig. 8.4 is a drawing of the human circulatory system. The blood in the circulatory system brings oxygen, nutrients, and various other vital substances to the cells and removes the metabolic waste products from the cells.  The blood is pumped through the circulatory system by the heart, and it leaves the heart through vessels called arteries and returns to it through veins.  The mammalian heart consists of two independent pumps, each made of two chambers called the atrium and the ventricle.  The entrances to and exits from these chambers are controlled by valves that are arranged to maintain the flow of blood in the proper direction. Blood from all parts of the body except the lungs enters the right atrium, which contracts and forces the blood into the right ventricle.  The ventricle then contracts and drives the blood through the pulmonary artery into the lungs. In its passage through the lungs, the blood releases carbon dioxide and absorbs oxygen.  The blood then flows into the left atrium via the pulmonary vein.  The contraction of the left atrium forces the blood into the left ventricle, which on contraction drives the oxygen-rich blood through the aorta into the arteries that lead to all parts of the body except the lungs.  Thus, the right side of the heart pumps the blood through the lungs, and the left side pumps it through the rest of the body.  The large artery, called the aorta, which carries the oxygenated blood away from the left chamber of the heart, branches into smaller arteries, which lead to the various parts of the body.  These in turn branch into still smaller arteries, the smallest of which are called arterioles. 14   As we will explain later, the arterioles play an important role in regulating the blood flow to specific regions in the body.  The arterioles branch further into narrow capillaries that are often barely wide enough to allow the passage of single blood cells.  The capillaries are so profusely spread through the tissue that nearly all the cells in the body are close to a capillary.  The exchange of gases, nutrients, and waste products between the blood and the surrounding tissue occurs by diffusion through the thin capillary walls (see Chapter 9).  The capillaries join into tiny veins called venules, which in turn merge into larger and larger veins that lead the oxygen-depleted blood back to the right atrium of the heart. Aorta( O2) smaller arteries arterides capillaries Spread through the tissue close to almost every single cell ( nutrients, gases, waste,,, exchanged through diffusion Capillaries venuels veins(CO2) right atrium. 15 8.5 BLOOD PRESSURE  electrical pulses cause the contraction of the heart chambers that are applied simultaneously both to the left and to the right halves of the heart.  First the atria contract, the blood flows into the ventricles; then the ventricles contract, the blood leaves the heart to the body.  Because of the pumping action of the heart, blood enters the arteries in spurts or pulses.  systolic pressure: The maximum pressure driving the blood at the peak of the pulse.  the systolic pressure is about 120 torr (mm Hg)  diastolic pressure The lowest blood pressure between the pulses.  the diastolic pressure is about 80 torr usually expressed as 120/80. Therefore the average pressure of the pulsating blood at heart level is 100 torr.  As the blood flows through the circulatory system, its initial energy, provided by the pumping action of the heart, is dissipated by two loss mechanisms: 1) losses associated with the expansion and contraction of the arterial walls 2) viscous friction associated with the blood flow.  Due to these energy losses, the initial pressure fluctuations are smoothed out as the blood flows away from the heart, and the average pressure drops.  By the time the blood reaches the capillaries, the flow is smooth and the blood pressure is only about 30 torr.  The pressure drops still lower in the veins and is close to zero just before returning to the heart.  In this final stage of the flow, the movement of blood through the veins is aided by the contraction of muscles that squeeze the blood toward the heart. Also One-way flow is assured by unidirectional valves in the veins. 16   The main arteries in the body have a relatively large radius., is about 1 cm; therefore, the pressure drop along the arteries is small.  We can estimate this pressure drop using Poiseuille’s law , we must know the rate of blood flow.  The rate of blood flow 𝑄 through the body depends on the level of physical activity.  At rest, the total flow rate is about 5 liter/min.  During intense activity the flow rate may rise to about 25 liter/min. Exercise 8-1 shows that at peak flow the pressure drop per centimeter length of the aorta is only 42.5dyn/cm2 (3.19×10-2 torr), which is negligible compared to the total blood pressure. Resistance to flow R As the aorta branches, the size of the arteries decreases, resulting in an increased resistance to flow. Although the blood flow in the narrower arteries is also reduced, the pressure drop is no longer negligible, as the resistance to flow increase (see Exercise 8-2). The average pressure at the entrance to the arterioles is about 90 torr. Still, this is only a 10% drop from the average pressure at the heart. 17   The flow through the arterioles is accompanied by a much larger pressure drop, about 60 torr. As a result, the pressure at the capillaries is only about 30 torr.  Since the pressure drop in the main arteries is small, when the body is horizontal, the average arterial pressure is approximately constant throughout the body. (100 torr)  The arterial blood pressure, which is on the average 100 torr, can support a column of blood 129 cm high (see Eq. 7.1 and Exercise 8-3). Exercise 8-3. How high a column of blood can an arterial pressure at the heart of 100 torr support? (The density of blood is 1.05 g/cm3.) 100 𝑡𝑜𝑟𝑟 = 100 𝑚𝑚𝐻𝑔 18 This means that if a small tube were introduced into the artery, the blood in it would rise to a height of 129 cm (see Fig. 8.5). To calculate the average pressure at the head and the feet for a person standing erect as shown in the figure. Phead = P heart - 𝜌 𝑔 ℎ Pfeet = P heart + 𝜌 𝑔 ℎ The average pressure at the heart, the head and the feet for a person sleeping as shown in the fig. Phead = P heart = Pfeet 19 Exercise 8-4. (a) Calculate the arterial blood pressure in the head of an erect person. Assume that the head is 50 cm above the heart. (The density of blood is 1.05 g/cm3.) (b) Compute the average arterial pressure in the legs of an erect person, 130 cm below the heart Phead = P heart - 𝜌 𝑔 ℎ = 13.3 x 103 Pa - 1050 kg/m3 x 9.8 m/s2 x 0.5m = 8.2 x 103 Pa Pfeet = P heart + 𝜌 𝑔 ℎ = 13.3 x 103 Pa + 1050 kg/m3 x 9.8 m/s2 x 1.3 m = 26.8 x 103 Pa If a person is standing erect, the blood pressure in the arteries is not uniform in the various parts of the body. The weight of the blood must be taken into account in calculating the pressure at various locations. 20  The cardiovascular system has various flow-control mechanisms that can compensate for the large arterial pressure changes that accompany shifts in the position of the body.  Still, it may take a few seconds for the system to compensate. Thus, a person may feel momentarily dizzy as he/she jumps up from a prone position.  This is due to the sudden decrease in the blood pressure of the brain arteries, which results in a temporary decrease of blood flow to the brain.  The same hydrostatic factors operate also in the veins, and here their effect may be more severe than in the arteries.  The blood pressure in the veins is lower than in the arteries.  When a person stands motionless, the blood pressure is barely adequate to force the blood from the feet back to the heart. Thus, when a person sits or stands without muscular movement, blood gathers in the veins of the legs. This increases the pressure in the capillaries and may cause temporary swelling of the legs.  With some exceptions, the blood pressure of most animals is in the same range as that of humans. For example, the systolic blood pressure of a pig, cat, and dog is for all three about 120 torr. The giraffe with its head positioned high above the heart is an exception. Its blood pressure is significantly higher, typically 240/160. 21 8.6 CONTROL OF BLOOD FLOW The pumping action of the heart (that is, blood pressure, flow volume and rate of heart beat) is regulated by a variety of hormones. Hormones are molecules, often proteins, that are produced by organs and tissues in different parts of the body. They are secreted into the blood stream and carry messages from one part of the body to another. Hormones affecting the heart are produced in response to stimuli such as need for more oxygen, changes in body temperature, and various types of emotional stress. The flow of blood to specific parts of the body is controlled by the arterioles. Arterioles radius is small construction control blood flow These small vessels that receive blood from the arteries have an average diameter of about 0.1 mm. The walls of the arterioles contain smooth muscle fibers that contract when stimulated by nerve impulses and hormones. The contraction of the arterioles in one part of the body reduces the blood flow to that region and diverts it to another. Since the radius of the arterioles is small, constriction is an effective method for controlling blood flow. Poiseuille’s equation shows that if the pressure drop remains constant, a 20% decrease in the radius reduces the blood flow by more than a factor of 2 (see Exercise 8-5). 22  A stress-induced heart condition called stress cardiomyopathy (broken heart syndrome) has only recently been clearly identified by Western medicine. The syndrome occurs most frequently after a sudden intense emotional trauma such as death in the family, an experience of violence, or extreme anger. The symptoms are similar to an acute heart attack, but the coronary arteries are found to be normal and the heart tissue is not damaged. It has been suggested that the condition is triggered by an excessive release of stress-related hormones called chatecholamines. 8.8 TURBULENCE IN THE BLOOD 23  If the velocity of a fluid is increased past a critical point, the laminar flow becomes turbulent.  The critical velocity is given by: 𝑅𝜂 𝑣𝑐 = 𝜌𝐷  Through most of the circulatory system the blood flow is laminar.  Only in the aorta does the flow occasionally become turbulent. Example: Assuming a Reynold’s number of 2000, What is the critical velocity for the onset of turbulence in the 2-cm-diameter aorta? ( Given: and the coefficient of viscosity of blood is 4x10-2 poise and the density of the blood is 1.05 gm/cm3)  At rest, the flow velocity in the aorta is below this value.