Fluids DynamicsDue: 8:00pm on Thursday, September 22, 2011 Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy [Switch to Standard Assignment View] Streamlines and Fluid Flow Learning Goal: To understand the continuity equation. Streamlines represent the path of the flow of a fluid. You can imagine that they represent a time-exposure photograph that shows the paths of small particles carried by the flowing fluid. The figure shows streamlines for the flow of an incompressible fluid in a tapered pipe of circular cross section. The speed of the fluid as it enters the pipe on the left is entrance on the left and Part A . Assume that the cross-sectional areas of the pipe are at its at its exit on the right. Find , the volume of fluid flowing into the pipe per unit of time. This quantity is also known as the volumetric flow rate. Hint A.1 Find the volume of fluid entering the pipe Hint not displayed Express the volumetric flow rate in terms of any of the quantities given in the problem introduction. ANSWER: = Correct Part B the amount of fluid that flows into the pipe must equal the amount of fluid that flows out. find the velocity of the fluid flowing out of the right end of the pipe. ANSWER: velocity Correct Thus the velocity of the flow increases with increasing density (number per unit area) of streamlines. This fact is embodied in the continuity equation. and subscripts 1 and 2 refer to any two points along the streamline. however. that is. For a fluid element of density that flows along a streamline.1 Find the volumetric flow rate out of the pipe Hint not displayed Hint B. Bernoulli's equation states that . is the flow speed. whose flow patterns do not change with time. Hint B.2 Apply the continuity equation Hint not displayed Express your answer in terms of any of the quantities given in the problem introduction. Bernoulli's equation is an essential tool in understanding the behavior of fluids in many practical applications. where is the pressure. Enter a one-word answer. and elevation. Using the continuity equation. from plumbing systems to the flight of airplanes. It applies exclusively to ideal fluids with steady flow. is the acceleration due to gravity.Because the fluid is assumed to be incompressible and mass is conserved. flow speed. Despite its limitations. you can conclude that the __________ of the fluid is greater where the streamlines are closer together. at a particular moment in time. The physical interpretation of Bernoulli's equation becomes clearer if we rearrange the terms of the equation as follows: . ANSWER: = Correct Part C If you are shown a picture of streamlines in a flowing fluid. Understanding Bernoulli's Equation Bernoulli's equation is a simple relation that can give useful insight into the balance among fluid pressure. is the height. fluids with a constant density and no internal friction forces. The term on the left-hand side represents the total work done on a unit volume of fluid by the pressure forces of the surrounding fluid to move that volume of fluid from point 1 to point 2. what can you say about the pressure at point 2? Hint A. respectively. and the change in kinetic energy.1 How to approach the problem Hint not displayed Hint A. In other words. This is nothing more than the statement of conservation of mechanical energy for an ideal fluid flowing along a streamline. . . Bernoulli's equation states that the work done on a unit volume of fluid by the surrounding fluid is equal to the sum of the change in potential and kinetic energy per unit volume that occurs during the flow. The two terms on the right-hand side represent. .. Part A Consider the portion of a flow tube shown in the figure. If the cross section of the flow tube at point 1 is greater than that at point 2.2 Apply Bernoulli's equation Hint not displayed Hint A. Point 1 and point 2 are at the same height. of the unit volume during its flow from point 1 to point 2.3 Determine with respect to Hint not displayed ANSWER: The pressure at point 2 is lower than the pressure at point 1. An ideal fluid enters the flow tube at point 1 and moves steadily toward point 2. the change in potential energy. equal to the pressure at point 1.When the cross section of the flow tube decreases. and therefore the pressure decreases. by combining the continuity equation and Bernoulli's equation. higher than the pressure at point 1. In other words. Since the cross section of the flow tube is decreasing. Part B As you found out in the previous part. . the flow speed increases. Physically. then and The ends of the flow tube have the same areas as the ends of the horizontal flow tube shown in Part A. as shown in the figure. Correct Thus. Bernoulli's equation tells us that a fluid element flowing toward point 2 from point 1 moves toward a region of lower pressure. what is the . Hint B. Bernoulli's equation tells us that a fluid element that flows through a flow tube with decreasing cross section moves toward a region of lower pressure. In this case.1 Effects from conservation of mass Hint not displayed ANSWER: decrease in speed increase in speed remain in equilibrium Correct Part C Now assume that point 2 is at height with respect to point 1. the pressure drop experienced by the fluid element between points 1 and 2 acts on the fluid element as a net force that causes the fluid to __________. if . one can characterize the flow of an ideal fluid. so she decides to order a glass of orange juice and a glass of cranberry juice and do the mixing herself. larger than the pressure drop occurring in a purely horizontal flow. and while covering the top of the straw with her thumb. how do you explain the fact that the pressure drop at the ends of the elevated flow tube from Part C is larger than the pressure drop occurring in the similar but purely horizontal flow from Part A? Hint D.1 Physical meaning of the pressure drop in a tube Hint not displayed ANSWER: increase in potential energy from the elevation change. She drinks about 1/8 of the orange juice. carefully bends the straw and places the end over the orange juice glass. 7/8 orange juice and 1/8 cranberry juice. therefore a higher pressure is needed for the flow to occur. Correct Part D From a physical point of view. larger decrease in kinetic energy. The pressure drop is A greater amount of work is needed to balance the A Siphon at the Bar Jane goes to a juice bar with her friend Neil. the difference in pressure must also balance the increase in potential energy of the fluid. but the drink is not on the menu. In an elevated flow tube. decrease in potential energy from the elevation change. equal to the pressure drop occurring in a purely horizontal flow. the . Correct In the case of purely horizontal flow. then takes the straw from the glass containing cranberry juice. After she releases her thumb.pressure drop experienced by the fluid element? Hint C. The drinks come in two identical tall glasses. Jane shows Neil something she learned that day in class.1 How to approach the problem Hint not displayed ANSWER: smaller than the pressure drop occurring in a purely horizontal flow. to avoid spilling while mixing the two juices. the difference in pressure between the two ends of the flow tube had to balance only the increase in kinetic energy resulting from the acceleration of the fluid. larger increase in kinetic energy. sucks up just enough cranberry juice to fill the straw. She is thinking of ordering her favorite drink. . The speed of fluid flowing from the outlet of a siphon tube is the same as the speed that a body would acquire in falling from rest through a distance fluid flowing from an opening in a container at distance Part B Given the information found in Part A. This result is valid also for below the surface of the fluid. What is the initial velocity of the cranberry juice as it flows out of the straw? Let denote the magnitude of the acceleration due to gravity.2 Apply Bernoulli's principle Hint not displayed Express your answer in terms of and ANSWER: = Correct . Jane has successfully designed a siphon. Let the atmospheric pressure be and assume that the cranberry juice has negligible viscosity. Hint A. find the time it takes to Jane to transfer enough .1 How to approach the problem Hint not displayed Hint A. Assume that the glass containing cranberry juice has a very large diameter with respect to the diameter of the straw and that the cross-sectional area of the straw is the same at all points. Part A Consider the end of the straw from which the cranberry juice is flowing into the glass containing orange juice. and let be the distance below the surface of cranberry juice at which that end of the straw is located: .cranberry juice flows through the straw into the orange juice glass. 4 Rewrite the expression for the outflow speed as a function of the crosssectional areas of the tank and the opening Hint not displayed .0 centimeters and are filled to height 14. Assume that the flow rate of the liquid is constant. The cross-sectional area of the tank is meters.) Hint A. and that the glasses are cylindrical with a diameter of 7. square does it take to empty half the tank? (Note: A useful antiderivative is .3 Find the outflow speed as a function of the fluid speed at the surface Hint not displayed Hint A. By simply dividing the volume you need to remove by the volume flow rate.2 Find the volume flow rate Hint not displayed Express your answer numerically in seconds to two significant figures. Hint B.4 centimeters. so you decide to empty it by letting the water flow steadily from an opening at the side of the tank.1 How to approach the problem To make her favorite drink. Hint B. while that of the opening is Part A How much time square meters.2 Find the discharge rate Hint not displayed Hint A. you will find the time needed to complete the fluid transfer. The tank is filled with water to a height meter.1 How to approach the problem Hint not displayed Hint A. Take the diameter of the straw to be 0.cranberry juice into the orange juice glass to make her favorite drink if centimeters. ANSWER: 3. Jane has to transfer 1/8 of the cranberry juice into the orange juice glass.0 centimeters. located near the bottom. To calculate the time needed to remove 1/8 of the cranberry juice.8 = Correct A Water Tank That Needs Cleaning A cylindrical open tank needs cleaning. you need to know at what volume flow rate the cranberry juice flows into the glass containing orange juice. 7 The limits of integration Hint not displayed Express your answer numerically in seconds. and Part A Find . The fluid rises to heights is and in the two open-ended tubes (see figure). ANSWER: 51.2 Simplified Bernoulli's equation .1 How to approach the problem Hint not displayed Hint A. at the position of tube 1. the gauge pressure at the bottom of tube 1.6 How to solve a separable first-order ODE Hint not displayed Hint A.) Hint A. Take the free-fall acceleration due to gravity to be meters per second per second.5 Find the rate of change of the level of water in the tank Hint not displayed Hint A. a configuration called a Venturi meter.9 = Correct Venturi Meter with Two Tubes A pair of vertical.Hint A. open-ended glass tubes inserted into a horizontal pipe are often used together to measure flow velocity in the pipe. Consider such an arrangement with a horizontal pipe carrying fluid of density . (Gauge pressure is the pressure in excess of outside atmospheric pressure. The cross-sectional area of the pipe at the position of tube 2. However. a quantity that is more easily measured than the heights themselves. and either and or . Part B Find .3 Find in terms of Hint not displayed in terms of given quantities Hint not displayed Express your answer in terms of ANSWER: = Correct Note that this result depends on the difference between the heights of the fluid in the tubes. which is equal to . the magnitude of the acceleration due to gravity. the speed of the fluid in the left end of the main pipe. ANSWER: = Correct The fluid is pushed up tube 1 by the pressure of the fluid at the base of the tube.Hint not displayed Express your answer in terms of quantities given in the problem introduction and . it must be applied separately to the fluid in the tube and the fluid flowing in the main pipe. Water Flowing from a Tank Water flows steadily from an open tank as shown in the figure. . . Thus energy is not conserved (there is turbulence at the edge of the tube) at the entrance of the tube.2 Find Hint B.1 How to approach the problem Hint not displayed Hint B. . Since Bernoulli's law is essentially a statement of energy conservation. since there is no streamline around the sharp edge of the tube. and not by its kinetic energy. the pressure in the fluid is the same just inside and just outside the tube. . Hint B. 200 = Correct Part B What is the gauge pressure at point 2? Hint B. . ANSWER: 0.0160 square meters.0 meters.2 The volume flow rate Hint not displayed Hint A.1 Definition of gauge pressure Hint not displayed Hint B. compute the discharge rate Hint A. The cross-sectional area at point 2 is 0. where the water is discharged. Part A Assuming that Bernoulli's equation applies.The elevation of point 1 is 10.2 How to approach the problem Hint not displayed Hint B. at point 3.0480 square meters.1 How to approach the problem Hint not displayed Hint A.00 meters.3 Apply Bernoulli's principle Hint not displayed . it is 0. and the elevation of points 2 and 3 is 2. The crosssectional area of the tank is very large compared with the cross-sectional area of the pipe.3 Find the fluid speed at the end of the pipe Hint not displayed Express your answer in cubic meters per second. Hint B. ANSWER: 6.0 meters above the ground and is trying to spray Ferdinand. per second. Isabella is holding the hose in her hand 1.2 Projectile motion Hint not displayed ANSWER: Yes No Correct Part B To increase the range of the water.81 meters per second. Ferdinand and Isabella.4 Find the fluid speed at point 2 Hint not displayed Hint B.98×104 Correct ± Playing with a Water Hose Two children.1 General approach: considerations on fluid mechanics Hint not displayed Hint B. Isabella places her thumb on the hose hole and partially covers it. Part A Will Isabella be able to spray Ferdinand if the water is flowing out of the hose at a constant speed of 3. Hint A.5 meters per second? Assume that the hose is pointed parallel to the ground and take the magnitude of the acceleration due to gravity to be 9. what fraction of the cross-sectional area of the hose hole does she have to cover to be able to spray her friend? Assume that the cross section of the hose opening is circular with a radius of 1. Assuming that the flow remains steady.2 Find the outflow speed needed .1 General approach: considerations on particle motion Hint not displayed Hint A.5 centimeters.0 meters away.Hint B.5 Density of Water Hint not displayed Express your answer in pascals. who is standing 10. are playing with a water hose on a sunny summer day. 4: Fluid Flow An ideal incompressible fluid flows through a horizontal tube of radius 2.3 Find the cross-sectional area needed Hint not displayed Express your answer as a percentage to the nearest integer.00 . At one point along the tube's length there is a constriction where the radius is only 1. leads out of the bottom of tank A. to what height will liquid rise in pipe E? Express your answer using one significant figure. Assume streamline flow and no viscosity. ANSWER: 3 = Correct below the Test Your Understanding 14.Hint not displayed Hint B. Part A Compared to the volume flow rate in the 2. A horizontal pipe BCD. having a constriction at C and open to the air at D. the volume .00-radius portion of the tube. ANSWER: 84 = Correct % Problem 14. Part A If the cross-sectional area at C is one-half the area at D and if D is a distance level of the liquid in A. and a vertical pipe E opens into the constriction at C and dips into the liquid in tank F.00 .91 Two very large open tanks A and F (the figure ) contain the same liquid. flow rate in the 1.0 Both points are at the same height. so and the third term on the left-hand side cancels the third term on the right-hand side: . The fluid speed is greater where the radius is smaller and vice versa. Part A . Compared to the pressure at point #1. Test Your Understanding 14. whereas at point #2 the fluid is moving at 20. no matter what the tube's radius. but the volume flow rate is unaffected by changes in the radius. At point #1 the fluid is moving at 10.0 .00. the pressure at point #2 is ANSWER: 4 times as great as great 2 times as great as great the same not enough information given to decide Correct Bernoulli's equation states that The height is the same at both points.-radius portion of the tube is ANSWER: 2 times as great the same as great 4 times as great as great not enough information is given to decide Correct The volume flow rate of an incompressible fluid flowing in a tube is the same at points in the tube.5: Bernoulli's Equation Consider two points along a certain streamline in a pattern of incompressible fluid flow. so the term is four times greater than the term .The fluid speed at point #2 ( ) is twice as great as the fluid speed at point #1 ( ). we cannot determine the ratio of the pressure at point #2 to the pressure at point #1 . however.9%. You received 97. Score Summary: Your score on this assignment is 102. [ Print ] . Hence Bernoulli's equation becomes or This result shows that the pressure at point #2 ( ) is less than the pressure at point #1 ( ).73 out of a possible total of 95 points.which is what the question asks us to do. Since we are not given this information. we would need to know the value of and the value of the fluid density ( ). To determine the value of .