Chapter 12 Fluid Dynamics PDF

Chapter 12: Fluid Dynamics Mrs. Pooja Brahmaiahchari Introduction We have dealt with many situations in which fluids are static. But by their very definition, fluids flow. Example- A column of smoke rises from a campfire, blood courses through your veins. Why does rising smoke curl and...

Chapter 12: Fluid Dynamics Mrs. Pooja Brahmaiahchari Introduction We have dealt with many situations in which fluids are static. But by their very definition, fluids flow. Example- A column of smoke rises from a campfire, blood courses through your veins. Why does rising smoke curl and twist? How does a nozzle increase the speed of water emerging from a hose? How does the body regulate blood flow? The physics of fluids in motion—fluid dynamics—answers all these questions. Flow Rate and Its Relation to Velocity Flow rate Q is defined to be the volume of fluid passing by some location through an area during a period of time, as seen in Figure. In symbols, this can be written as 𝑉 𝑄= 𝑡 where V is the volume t is the elapsed time.  The SI unit for flow rate is m3/s , but a number of other units for Q are in common use. Flow rate is the volume of fluid per unit time flowing past a point through the area A. Here the shaded cylinder of fluid flows past point P in a uniform pipe in time t. The volume of the cylinder is Ad and the average velocity is 𝑣̅ = 𝑑/𝑡 so that the flow rate is 𝑄 = = 𝐴𝑣̅ The greater the velocity of the water, the greater the flow rate of the river. But flow rate also depends on the size of the river.  Figure shows an incompressible fluid flowing along a pipe of decreasing radius. Because the fluid is incompressible, the same amount of fluid must flow past any point in the tube in a given time to ensure continuity of flow. In this case, because the cross-sectional area of the pipe decreases, the velocity must necessarily increase. In particular, for points 1 and 2, 𝑄 = 𝑄 𝐴 𝑣 = 𝐴 𝑣  Previous equation in slide is called as the equation of continuity and is valid for any incompressible fluid. The consequences of the equation of continuity can be observed when water flows from a hose into a narrow spray nozzle: it emerges with a large speed—that is the purpose of the nozzle. In other words, speed increases when cross-sectional area decreases, and speed decreases when cross-sectional area increases. 1. Blood is pumped from the heart at a rate of 5.0 L/min into the aorta (of radius 1.0 cm). Determine the speed of blood through the aorta. 2. Blood is flowing through an artery of radius 2 mm at a rate of 40 cm/s. Determine the flow rate and the volume that passes through the artery in a period of 30 s. Bernoulli’s Equation The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli’s equation, named after its discoverer, the Swiss scientist Daniel Bernoulli (1700–1782). Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant: 1 𝑃 + 𝜌𝑣 + 𝜌𝑔ℎ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2 where P is the absolute pressure, ρ is the fluid density, v is the velocity of the fluid, h is the height above some reference point, and g is the acceleration due to gravity.  Let the subscripts 1 and 2 refer to any two points along the path that the bit of fluid follows; Bernoulli’s equation becomes, 1 1 𝑃 + 𝜌𝑣 + 𝜌𝑔ℎ = 𝑃 + 𝜌𝑣 + 𝜌𝑔ℎ 2 2 Bernoulli’s equation is a form of the conservation of energy principle. Note that the second and third terms are the kinetic and potential energy with m replaced by ρ. In fact, each term in the equation has units of energy per unit volume.  We can prove this for the second term by substituting 𝜌 = , 1 1 𝑚𝑣 𝐾𝐸 𝜌𝑣 = 2 = 2 𝑉 𝑉 Making the same substitution into the third term in the equation, we find, 𝑚𝑔ℎ 𝑃𝐸 𝜌𝑔ℎ = = 𝑉 𝑉 Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction 3. Water flows through horizontal pipe with cross-sectional area of 4 m2 at a speed of 5 m/s with pressure 300,000 Pa at point A. At point B the area is 2 m2. a. What is the speed of water at B? b. Calculate Pressure at B? Which of the following assumption is required for Bernoulli's equation to be valid? A. The fluid must be compressible B. The fluid must experience frictional forces C. The fluid must be incompressible and have no viscosity D. None of the above Bernoulli’s Equation for Static Fluids Let us first consider the very simple situation where the fluid is static—that is,𝑣 = 𝑣 = 0. Bernoulli’s equation in that case is, 𝑃 + 𝜌𝑔ℎ = 𝑃 + 𝜌𝑔ℎ We can further simplify the equation by taking ℎ = 0, we get: 𝑃 = 𝑃 + 𝜌𝑔ℎ This equation tells us that, in static fluids, pressure increases with depth. As we go from point 1 to point 2 in the fluid, the depth increases by ℎ , and consequently, 𝑃 is greater than 𝑃 by an amount 𝜌𝑔ℎ. Bernoulli’s Principle—Bernoulli’s Equation at Constant Depth Another important situation is one in which the fluid moves but its depth is constant—that is, ℎ = ℎ = 0. Under that condition, Bernoulli’s equation becomes 1 1 𝑃 + 𝜌𝑣 = 𝑃 + 𝜌𝑣 2 2 Situations in which fluid flows at a constant depth are so important that this equation is often called Bernoulli’s principle. It is Bernoulli’s equation for fluids at constant depth. Applications of Bernoulli’s Principle People have long put the Bernoulli principle to work by using reduced pressure in high-velocity fluids to move things about. With a higher pressure on the outside, the high-velocity fluid forces other fluids into the stream. This process is called entrainment. Entrainment devices have been in use since ancient times, particularly as pumps to raise water small heights, as in draining swamps, fields, or other low-lying areas.  Examples of entrainment devices that use increased fluid speed to create low pressures, which then entrain one fluid into another. (a) A Bunsen burner uses an adjustable gas nozzle, entraining air for proper combustion. (b) An atomizer uses a squeeze bulb to create a jet of air that entrains drops of perfume. Paint sprayers and carburetors use very similar techniques to move their respective liquids. (c) A common aspirator uses a high-speed stream of water to create a region of lower pressure. Aspirators may be used as suction pumps in dental and surgical situations or for draining a flooded basement or producing a reduced pressure in a vessel. (d) The chimney of a water heater is designed to entrain air into the pipe leading through the ceiling. The Most General Applications of Bernoulli’s Equation Torricelli’s Theorem: 𝒗𝟐𝟏 = 𝒗𝟐𝟐 + 𝟐𝒈𝒉 where h is the height dropped by the water. This is simply a kinematic equation for any object falling a distance with negligible resistance. In fluids, this equation is called Torricelli’s theorem. Note that the result is independent of the velocity’s direction, just as we found when applying conservation of energy to falling objects. Power in Fluid Flow Power is the rate at which work is done or energy in any form is used or supplied. To see the relationship of power to fluid flow, consider Bernoulli’s equation: 1 𝑃 + 𝜌𝑣 + 𝜌𝑔ℎ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2 Now, considering units, if we multiply energy per unit volume by flow rate (volume per unit time), we get units of power. That is,(𝐸 ⁄𝑉) = 𝐸/𝑡. This means that if we multiply Bernoulli’s equation by flow rate Q, we get power. In equation form, this is 1 𝑃 + 𝜌𝑣 + 𝜌𝑔ℎ 𝑄 = 𝑃𝑜𝑤𝑒𝑟 2 Viscosity and Laminar Flow; Poiseuille’s Law Laminar Flow and Viscosity When you pour yourself a glass of juice, the liquid flows freely and quickly. But when you pour syrup on your pancakes, that liquid flows slowly and sticks to the pitcher. The difference is fluid friction, both within the fluid itself and between the fluid and its surroundings. We call this property of fluids viscosity. Juice has low viscosity, whereas syrup has high viscosity.  The precise definition of viscosity is based on laminar, or nonturbulent, flow. Before we can define viscosity, then, we need to define laminar flow and turbulent flow. Laminar flow is characterized by the smooth flow of the fluid in layers that do not mix. Turbulent flow, or turbulence, is characterized by eddies and swirls that mix layers of fluid together.  Streamlines are smooth and continuous when flow is laminar, but break up and mix when flow is turbulent. Turbulence has two main causes. First, any obstruction or sharp corner, such as in a faucet, creates turbulence by imparting velocities perpendicular to the flow. Second, high speeds cause turbulence. The drag both between adjacent layers of fluid and between the fluid and its surroundings forms swirls and eddies, if the speed is great enough.  The graphic shows laminar flow of fluid between two plates of area. The bottom plate is fixed. When the top plate is pushed to the right, it drags the fluid along with it. A force F is required to keep the top plate in Figure moving at a constant velocity v, and experiments have shown that this force depends on four factors. 1. F is directly proportional to v. 2. F is proportional to Area A of plate. 3. F is inversely proportional to the distance between the plates L. 4. F is directly proportional to the coefficient of Viscosity η.  The greater the viscosity, the greater the force required. These dependencies are combined into the equation 𝑣𝐴 𝐹= 𝜂 𝐿 which gives us a working definition of fluid viscosity η. Solving for η gives: 𝐹𝐿 𝜂= 𝑣𝐴 which defines viscosity in terms of how it is measured. The SI unit of viscosity is N.m/[(m/s) m2] = (N/m2)s or Pa. s 4. What force is needed to pull one microscope slide over another at a speed of 1.00 cm/s, if there is a 0.500-mm-thick layer of 20o C water (𝜂 = 1.002 x 10-3) between them and the contact area is 8.00cm2 ? 5. A layer of oil 1.50 mm thick is placed between two microscope slides. Researchers find that a force of 5.50x10-4 N is required to glide one over the other at a speed of 1.00 cm/s when their contact area is 6.00 cm2. What is the oil’s viscosity? Laminar Flow Confined to Tubes—Poiseuille’s Law Flow rate Q is in the direction from high to low pressure. The greater the pressure differential between two points, the greater the flow rate. This relationship can be stated as 𝑃 − 𝑃 𝑄= 𝑅 where 𝑃 and 𝑃 are the pressures at two points, such as at either end of a tube, and R is the resistance to flow. The resistance includes everything, except pressure, that affects flow rate.  The resistance R to laminar flow of an incompressible fluid having viscosity η through a horizontal tube of uniform radius r and length l, such as the one in Figure 12.14, is given by 8𝜂𝑙 𝑅= 𝜋𝑟 This equation is called Poiseuille’s law for resistance after the French scientist J. L. Poiseuille (1799–1869), who derived it in an attempt to understand the flow of blood, an often turbulent fluid. 6. Concrete is pumped from a cement mixer to the place it is being laid, instead of being carried in wheelbarrows. The flow rate is 200 L/min through a 50.0-m-long, 8.00-cm-diameter hose, and the pressure at the pump is 8.00 x 106 N/m2. (a) Calculate the resistance of the hose. (b) What is the viscosity of the concrete, assuming the flow is laminar? The Onset of Turbulence Sometimes we can predict if flow will be laminar or turbulent. We know that flow in a very smooth tube or around a smooth, streamlined object will be laminar at low velocity. We also know that at high velocity, even flow in a smooth tube or around a smooth object will experience turbulence. In between, it is more difficult to predict. In fact, at intermediate velocities, flow may oscillate back and forth indefinitely between laminar and turbulent.  An occlusion, or narrowing, of an artery, such as shown in Figure 12.17, is likely to cause turbulence because of the irregularity of the blockage, as well as the complexity of blood as a fluid.  Turbulence in the circulatory system is noisy and can sometimes be detected with a stethoscope, such as when measuring diastolic pressure in the upper arm’s partially collapsed brachial artery. These turbulent sounds, at the onset of blood flow when the cuff pressure becomes sufficiently small, are called Korotkoff sounds. Aneurysms, or ballooning of arteries, create significant turbulence and can sometimes be detected with a stethoscope. Heart murmurs, consistent with their name, are sounds produced by turbulent flow around damaged and insufficiently closed heart valves. Ultrasound can also be used to detect turbulence as a medical indicator in a process analogous to Doppler-shift radar used to detect storms.  An indicator called the Reynolds number 𝑵𝑹 can reveal whether flow is laminar or turbulent. For flow in a tube of uniform diameter, the Reynolds number is defined as 2𝜌𝑣𝑟 𝑁 = (𝑓𝑙𝑜𝑤 𝑖𝑛 𝑡𝑢𝑏𝑒) 𝜂 The Reynolds number is a unitless quantity. For 𝑁 below about 2000, flow is laminar. For 𝑁 above about 3000, flow is turbulent. For 𝑁 values of between about 2000 and 3000, flow is unstable. 7. Calculate the Reynolds number for flow in the needle of radius 0.150 mm and speed 1.70 m/s to verify the assumption that the flow is laminar. Assume that the density of the saline solution is 1025 kg/m3. Motion of an Object in a Viscous Fluid A moving object in a viscous fluid is equivalent to a stationary object in a flowing fluid stream. Flow of the stationary fluid around a moving object may be laminar, turbulent, or a combination of the two. Just as with flow in tubes, it is possible to predict when a moving object creates turbulence. We use another form of the Reynolds number , defined for an object moving in a fluid to be 𝜌𝑣𝐿 𝑁′ = ( 𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑛 𝑓𝑙𝑢𝑖𝑑) 𝜂  One of the consequences of viscosity is a resistance force called viscous drag Fv that is exerted on a moving object. For the special case of a small sphere of radius R moving slowly in a fluid of viscosity 𝜂, the drag force Fs is given by 𝐹 = 6𝜋𝑅𝜂𝑣 Instead, viscous drag increases, slowing acceleration, until a critical speed, called the terminal speed, is reached and the acceleration of the object becomes zero. Once this happens, the object continues to fall at constant speed. Terminal speed will be greatest for low-viscosity fluids and objects with high densities and small sizes. Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes Diffusion is the movement of substances due to random thermal molecular motion. Fluids, like fish fumes or odors entering ice cubes, can even diffuse through solids. Diffusion is a slow process over macroscopic distances. The densities of common materials are great enough that molecules cannot travel very far before having a collision that can scatter them in any direction, including straight backward. It can be shown that the average distance that a molecule travels is proportional to the square root of time: 𝑥 = 2𝐷𝑡  Membranes are generally selectively permeable, or semipermeable. One type of semipermeable membrane has small pores that allow only small molecules to pass through. In other types of membranes, the molecules may actually dissolve in the membrane or react with molecules in the membrane while moving across. Osmosis is the transport of water through a semipermeable membrane from a region of high concentration to a region of low concentration. Osmosis is driven by the imbalance in water concentration. For example, water is more concentrated in your body than in Epsom salt  Similarly, dialysis is the transport of any other molecule through a semipermeable membrane due to its concentration difference. Both osmosis and dialysis are used by the kidneys to cleanse the blood. Osmosis can create a substantial pressure. Water moves by osmosis from the left into the region on the right, where it is less concentrated, causing the solution on the right to rise. This movement will continue until the pressure ρgh created by the extra height of fluid on the right is large enough to stop further osmosis. This pressure is called a back pressure. The back pressure that stops osmosis is also called as relative osmotic pressure. Reverse osmosis and reverse dialysis (also called filtration) are processes that occur when back pressure is sufficient to reverse the normal direction of substances through membranes. THANK YOU