The velocity field for solid-body rotation in the r? plane

Problem 11P A weather balloon is launched into the atmosphere by meteorologists. When the balloon reaches an altitude where it is neutrally buoyant, it transmits information about weather conditions to monitoring stations on the ground (Fig. P11-10C). Is this a Lagrangian or an Eulerian measurement? Explain.

Problem 12P A Pitot-static probe can often be seen protruding from the underside of an airplane (Fig. P11–11C). As the airplane flies, the probe measures relative wind speed. Is this a Lagrangian or an Eulerian measurement? Explain.

Problem 13P Is the Eulerian method of fluid flow analysis more similar to study of a system or a control volume? Explain.

Problem 16P Consider steady, incompressible, two-dimensional flow through a converging duct (Fig. P11–15). A simple approximate velocity field for this flow is where U0is the horizontal speed at x= 0. Note that this equation ignores viscous effects along the walls but is a reasonable approximation throughout the majority of the flow field. Calculate the material acceleration for fluid particles passing through this duct. Give your answer in two ways: (1) as acceleration components axand ayand (2) as acceleration vector .

Problem 15P List at least three other names for the material derivative, and write a brief explanation about why each name is appropriate.

Problem 14P Define a steady flow field in the Eulerian reference frame. In such a steady flow, is it possible for a fluid particle to experience a nonzero acceleration?

Problem 18P A steady, incompressible, two-dimensional velocity field is given by the following components in the xy-plane: Calculate the acceleration field (find expressions for acceleration components a x and a y), and calculate the acceleration at the point(x, y) = (-1, 2).

Problem 17P Converging duct flow is modeled by the steady, two-dimensional velocity field of Prob. 11–15. The pressure field is given by where P0 is the pressure at x =0. Generate an expression for the rate of change of pressure following a fluid particle. PROBLEM: Consider steady, incompressible, two-dimensional flow through a converging duct (Fig. P11–15). A simple approximate velocity field for this flow is where U0 is the horizontal speed at x= 0. Note that this equation ignores viscous effects along the walls but is a reasonable approximation throughout the majority of the flow field. Calculate the material acceleration for fluid particles passing through this duct. Give your answer in two ways: (1) as acceleration components ax and ay and (2) as acceleration vector .

Problem 20P A steady, incompressible, two-dimensional (in the xy-plane) velocity field is given by Calculate the acceleration at the point (x, y) = (-1.55, 2.07).

Problem 19P A steady, incompressible, two-dimensional velocity field is given by the following components in the xy-plane: Calculate the acceleration field (find expressions for acceleration components a x and a y) and calculate the acceleration at the point (x, y) = (1, 3).

Problem 21P For the velocity field of Prob. 11–2, calculate the fluid acceleration along the nozzle centerline as a function of x and the given parameters. PROBLEM: Consider the steady flow of water through an axisymmetric garden hose nozzle (see figure). Along the centerline of the nozzle, the water speed increases from uentrance to uexit as sketched. Measurements reveal that the centerline water speed increases parabolically through the nozzle. Write an equation for centerline speed u(x), based on the parameter given here, from x=0 to x=L.

Problem 22P Consider steady flow of air through the diffuser portion of a wind tunnel (Fig. P11–20). Along the centerline of the diffuser, the air speed decreases from uentrance to uexit as sketched. Measurements reveal that the centerline air speed decreases parabolically through the diffuser. Write an equation for centerline speed u(x), based on the parameters given here, from x = 0 to x = L.

Problem 26P What is the definition of a pathline?What do pathlines indicate?

Problem 27P What is the definition of a streakline? How do streaklines differ from streamlines?

Problem 28P Consider the visualization of flow over a 15° delta wing in Fig. P11-28C. Are we seeing streamlines, streaklines, pathlines, or timelines? Explain. FIGURE P11-28C Visualization of flow over a 15° delta wing at a 20° angle of attack at a Reynolds number of 20.000. The visualization is produced by colored fluid injected into water from ports on the underside of the wing. Curtesy ONERA. Photograph by Werlé.

Problem 34P Consider the following steady, incompressible, two- dimensional velocity field: Generate an analytical expression for the flow streamlines and draw several streamlines in the upper-right quadrant from x = 0 to 5 and y= 0 to 6.

Problem 53P Converging duct flow is modeled by the steady, two- dimensional velocity field of Prob. 4?17. Since the flow is symmetric about the.x-axis. line segment AB along the x-axis remains on the axis, but stretches from length ? to length ?+?? as it flows along the channel centerline (Fig. P4?55). Generate an analytical expression for the change in length of the line segment. ?? .(Hint: Use the result of Prob. 4?54.) FIGURE P4?55

Problem 52P Converging duct flow is modeled by the steady, two dimensional velocity field of Prob. 4?17. A fluid particle (A) is located on the x-axis at x = xA at time t = 0 (Fig. P4-54). At some later time t, the fluid particle has moved downstream FIGURE P4?54 with the flow to some new location. x = xA, as shown in the figure. Since the flow is symmetric about the x-axis, the fluid particle remains on the.x-axis at all times. Generate an analytical expression for the.x-location of the fluid particle at some arbitrary time t in terms of its initial location xA and constants U0 and b. In other words, develop an expression for xA (Hint: We know that u = dxparticle/dt following a fluid particle. Plug in u. separate variables, and integrate.)

Problem 33P Converging duct flow (Fig. P4?17) is modeled by the steady, two-dimensional velocity field of Prob. 4?17. Generate an analytical expression for the flow streamlines. PROBLEM: Consider steady, incompressible, two-dimensional flow through a converging duct (Fig. P11–15). A simple approximate velocity field for this flow is where U0 is the horizontal speed at x= 0. Note that this equation ignores viscous effects along the walls but is a reasonable approximation throughout the majority of the flow field. Calculate the material acceleration for fluid particles passing through this duct. Give your answer in two ways: (1) as acceleration components ax and ay and (2) as acceleration vector .

Problem 55P Converging duct flow is modeled by the steady, two- dimensional velocity field of Prob. 4?17. A fluid particle (A) is located at x = xA and y = y A at time t = 0 (Fig. P4?57). At some later time t. the fluid particle has moved downstream with the flow to some new location x = x A., y =y A., as show n in the figure. Generate an analytical expression for the y-location of the fluid particle at arbitrary time t in terms of its initial y-location y A and constant b. In other words, develop an expression for y A.(Hint: We know that v = dyparticle/dt following a fluid particle. Substitute the equation for v. separate variables, and integrate.) FIGURE P4?57

Problem 56P Converging duct flow is modeled by the steady, two- dimensional velocity field of Prob. 4?17. As vertical line segment AB moves downstream it shrinks from length ? to length ?+ ?? as sketched in Fig. P4?58. Generate an analytical expression for the change in length of the line segment, ??. Note that the change in length, ??, is negative. (Hint: Use the result of Prob. 4?57.) FIGURE P4?58

Problem 54P Using the results from Prob. 4?55 and the fundamental definition of linear strain rate (the rate of increase in length per unit length), develop an expression for the linear strain rate in the x-direction (?xx) of fluid particles located on the centerline of the channel. Compare your result to the general expression for ?xx in terms of the velocity field, i.e., ?xx = ?u/?x. (Hint: Take the limit as time t? 0. You may need to apply a truncated series expansion for e bt.) PROBLEM 4-55: Converging duct flow is modeled by the steady, two-dimensional velocity field of problem 4 - 17. Since the flow is symmetric about the x-axis, line segment AB along the x-axis remains on the axis, but stretches from length to length as it flows along the channel centerline (see Figure A). Generate an analytical expression for the change in length of the line segment, . (Hint: Use the result of Prob. 4-51.) PROBLEM 4-17: Consider steady, incompressible, two-dimensional flow through a converging duct (Fig. B). A simple approximate velocity field for this flow is Where U0 is the horizontal speed at x = 0. Note that this equation ignores viscous effects along the walls but is a reasonable approximation throughout the majority of the flow field. Calculate the material acceleration for fluid particles passing through this duct. Give your answer in two ways: (1) as acceleration components ax and ay and (2) as acceleration vector FIGURE A: FIGURE B:

Problem 59P A general equation for a steady, two-dimensional velocity field that is linear in both spatial directions (x and y) is Where U and V and the coefficients are constants. Their dimensions are assumed to be appropriately defined. Calculate the x- and y-components of the acceleration field.

Problem 58P Converging duct flow is modeled by the steady, two- dimensional velocity field of Prob. 4?17. Use the equation for volumetric strain rate to verify that this flow field is incompressible.

Problem 57P Using the results of Prob. 4?58 and the fundamental definition of linear strain rate (the rate of increase in length per unit length), develop an expression for the linear strain rate in the y-direction (?yy) of fluid particles moving down the channel. Compare your result to the general expression for ?yy,in terms of the velocity field, i.e., ?yy = ?vl?y. (Hint: Take the limit as time t ? 0. You may need to apply a truncated series expansion for e ?bt.)

Problem 60P For the velocity field of Prob. 4?63, what relationship must exist between the coefficients to ensure that the flow field is incompressible? PROBLEM: A general equation for a steady, two-dimensional velocity field that is linear in both spatial directions (x and y) is Where U and V and the coefficients are constants. Their dimensions are assumed to be appropriately defined. Calculate the x- and y-components of the acceleration field.

Problem 62P For the velocity field of Prob. 4?63, calculate the shear strain rate in the xy-plane. PROBLEM: A general equation for a steady, two-dimensional velocity field that is linear in both spatial directions (x and y) is Where U and V and the coefficients are constants. Their dimensions are assumed to be appropriately defined. Calculate the x- and y-components of the acceleration field.

Problem 61P For the velocity field of Prob. 4?63, calculate the linear strain rates in the x- and y-directions. PROBLEM: A general equation for a steady, two-dimensional velocity field that is linear in both spatial directions (x and y) is Where U and V and the coefficients are constants. Their dimensions are assumed to be appropriately defined. Calculate the x- and y-components of the acceleration field.

Problem 63P Combine your results from Probs. 4?65 and 4?66 to form the two-dimensional strain rate tensor ?ij in the xy-plane, Under what conditions would the x- and y-axes be principal axes? PROBLEM 4 -65: For the velocity field of Prob. 4?63, calculate the linear strain rates in the x- and y-directions. PROBLEM 4-66: For the velocity field of Prob. 4?63, calculate the shear strain rate in the xy-plane. PROBLEM 4-63: A general equation for a steady, two-dimensional velocity field that is linear in both spatial directions (x and y) is Where U and V and the coefficients are constants. Their dimensions are assumed to be appropriately defined. Calculate the x- and y-components of the acceleration field.

Problem 73P Consider a steady, two-dimensional, incompressible flow field in the xy-plane. The linear strain rate in the x-direction is 2.5 s -1. Calculate the linear strain rate in the y-direction.

Problem 68P Consider the steady, incompressible, two-dimensional flow field of Prob. 4?69. Using the results of Prob. 4?69(a), do the following: (a) From the fundamental definition of the rate of rotation (average rotation rate of two initially perpendicular lines that intersect at a point), calculate the rate of rotation of the fluid particle in the xy-plane.?z (Hint: Use the lower edge and the left edge of the fluid particle, which intersect at 90° at the lower-left corner of the particle at the initial time.) (b) Compare your results with those obtained from the equation for ?z. in Cartesian coordinates, i.e.. PROBLEM: Consider steady, incompressible, two-dimensional shear flow for which the velocity field is where a and b are constants. Sketched in Fig. P4?69 is a small rectangular fluid particle of dimensions dx and dy at time t. The fluid particle moves and deforms with the flow such that at a later time (t + dt). the particle is no longer rectangular, as also shown in the figure. The initial location of each corner of the fluid particle is labeled in Fig. P4?69. The lower-left corner is at (x. y) at time t. where the x-component of velocity is u = a + by. At the later time, this corner moves to (x + u dt, y), or (a) In similar fashion, calculate the location of each of the other three corners of the fluid particle at time t ? dt. (b) From the fundamental definition of linear strain rate (the rate of increase in length per unit length). calculate linear strain rates ?xx and ?yy. (c) Compare your results with those obtained from the equations for ?xx and ?yy in Cartesian coordinates, i.e.. FIGURE P4?69

Problem 67P Consider the steady, incompressible, two-dimensional flow field of Prob. 4?69. Using the results of Prob. 4?69(a). do the following: (a) From the fundamental definition of shear strain rate (half of the rate of decrease of the angle between two initially perpendicular lines that intersect at a point), calculate shear strain rate ?xy in the xy-plane.(Hint: Use the lower edge and the left edge of the fluid particle, which intersect at 90° at the lower-left corner of the particle at the initial time.) (b) Compare your results with those obtained from the equation for ?xy in Cartesian coordinates, i.e., PROBLEM: Consider steady, incompressible, two-dimensional shear flow for which the velocity field is where a and b are constants. Sketched in Fig. P4?69 is a small rectangular fluid particle of dimensions dx and dy at time t. The fluid particle moves and deforms with the flow such that at a later time (t + dt). the particle is no longer rectangular, as also shown in the figure. The initial location of each corner of the fluid particle is labeled in Fig. P4?69. The lower-left corner is at (x. y) at time t. where the x-component of velocity is u = a + by. At the later time, this corner moves to (x + u dt, y), or (a) In similar fashion, calculate the location of each of the other three corners of the fluid particle at time t ? dt. (b) From the fundamental definition of linear strain rate (the rate of increase in length per unit length). calculate linear strain rates ?xx and ?yy. (c) Compare your results with those obtained from the equations for ?xx and ?yy in Cartesian coordinates, i.e.. FIGURE P4?69

Problem 88P Briefly explain the similarities and differences between the material derivative and the Reynolds transport theorem.

True or false: For each statement, choose whether the statement is true or false and discuss your answer briefly. (a) The Reynolds transport theorem is useful for transforming conservation equations from their naturally occurring control volume forms to their system forms. (b) The Reynolds transport theorem is applicable only to non deforming control volumes. (c) The Reynolds transport theorem can be applied to both steady and unsteady flow fields. (d) The Reynolds transport theorem can be applied to both scalar and vector quantities.

Problem 74P A cylindrical tank of water rotates in solid-body rotation, counterclockwise about its vertical axis (Fig. P4?78) at angular speed n = 260 rpm. Calculate the vorticity of fluid particles in the tank. FIGURE P4?78

Problem 90P Consider the general form of the Reynolds transport theorem (RTT) given by where is the velocity of the fluid relative to the control surface. Let Bsys be the mass m of a system of fluid particles. We know that for a system, dmldt =0 since no mass can enter or leave the system by definition. Use the given equation to derive the equation of conservation of mass for a control volume.

Problem 75P A cylindrical tank of water rotates about its vertical axis (Fig. P4?78). A PIV system is used to measure the vorticity field of the flow. The measured value of vorticity in the z-direction is ?45.4 rad/s and is constant to within ±0.5 percent everywhere that it is measured. Calculate the angular speed of rotation of the tank in rpm. Is the tank rotating clockwise or counterclockwise about the vertical axis?

Problem 91P Consider the general form of the Reynolds transport theorem (RTT) given by Prob. 11–45. Let Bsys be the linear momentum of a system of fluid particles. We know that for a system, Newton’s second law is Use the equation of Prob. 11–45 and this equation to derive the equation of conservation of linear momentum for a control volume.

Problem 92P Consider the general form of the Reynolds transport theorem (RTT) given in Prob. 11–45. Let Bsys be the angular momentum of a system of fluid particles, where is the moment arm. We know that for a system, conservation of angular momentum can be expressed as where is the net moment applied to the system. Use the equation given in Prob. 11—45 and this equation to derive the equation of conservation of angular momentum for a control volume.

Problem 93P Reduce the following expression as far as possible: (Hint: Use the one-dimensional Leibniz theorem.)

Problem 69P From the results of Prob. 4?72, (a) Is this flow rotational or irrotational? (b) Calculate the z-component of vorticity for this flow field. PROBLEM: Consider the steady, incompressible, two-dimensional flow field of Prob. 4?69. Using the results of Prob. 4?69(a), do the following: (a) From the fundamental definition of the rate of rotation (average rotation rate of two initially perpendicular lines that intersect at a point), calculate the rate of rotation of the fluid particle in the xy-plane.?z (Hint: Use the lower edge and the left edge of the fluid particle, which intersect at 90° at the lower-left corner of the particle at the initial time.) (b) Compare your results with those obtained from the equation for ?z. in Cartesian coordinates, i.e.. PROBLEM: Consider steady, incompressible, two-dimensional shear flow for which the velocity field is where a and b are constants. Sketched in Fig. P4?69 is a small rectangular fluid particle of dimensions dx and dy at time t. The fluid particle moves and deforms with the flow such that at a later time (t + dt). the particle is no longer rectangular, as also shown in the figure. The initial location of each corner of the fluid particle is labeled in Fig. P4?69. The lower-left corner is at (x. y) at time t. where the x-component of velocity is u = a + by. At the later time, this corner moves to (x + u dt, y), or (a) In similar fashion, calculate the location of each of the other three corners of the fluid particle at time t ? dt. (b) From the fundamental definition of linear strain rate (the rate of increase in length per unit length). calculate linear strain rates ?xx and ?yy. (c) Compare your results with those obtained from the equations for ?xx and ?yy in Cartesian coordinates, i.e.. FIGURE P4?69

Problem 94P Consider the integral Solve it two ways: (a) Take the integral first and then the time derivative. (b) Use Leibniz theorem. Compare your results.

Problem 95P Solve the integral as far as you are able.

Problem 96P Consider fully developed two-dimensional Poiseuille flow—flow between two infinite parallel plates separated by distance h,with both the top plate and bottom plate stationary, and a forced pressure gradient dP/dxdriving the flow as illustrated in Fig. P11–48. (dP/dxis constant and negative.) FIGURE P11-48 The flow is steady, incompressible, and two-dimensional in the xy-plane. The velocity components are given by where ? is the fluid’s viscosity. Is this flow rotational or irrotational? If it is rotational, calculate the vorticity component in the z-direction. Do fluid particles in this flow rotate clockwise or counterclockwise?

Problem 99P Consider the two-dimensional Poiseuille flow of Prob. 11–48. The fluid between the plates is water at 40°C. Let the gap height h = 1.6 mm and the pressure gradient dP/dx = ? 230 N/m3. Calculate and plot seven pathlinesfrom t= 0 to t= 10 s. The fluid particles are released at x = 0 and at y =0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4 mm.

Problem 103P Consider the two-dimensional Poiseuille flow of Prob. 11–48. The fluid between the plates is water at 40°C. Let the gap height h =1.6 mm and the pressure gradient dP/dx = ?230 N/m3 . Imagine a hydrogen bubble wire stretched vertically through the channel at x =0 (Fig. P11–52). The wire is pulsed on and off such that bubbles are produced periodically to create timelines. Five distinct timelines are generated at t = 0, 2.5, 5.0,7.5, and 10.0 s. Calculate and plot what these five timelines look like at time t = 12.5 s. FIGURE P11-52

Problem 104P Consider fully developed axisymmetric Poiseuille flow—flow in a round pipe of radius R(diameter D = 2 R), with a forced pressure gradient dP/dxdriving the flow as illustrated in Fig. P11–53. (dP/dxis constant and negative.) The flow is steady, incompressible, and axisymmetric about the x-axis. The velocity components are given by where ? is the fluid’s viscosity. Is this flow rotational or irrotational? If it is rotational, calculate the vorticity component in the circumferential (?) direction and discuss the sign of the rotation. FIGURE P11-53

Problem 98P Combine your results from Problem given below to form the two-dimensional strain rate tensor in the ?ij in the xy-plane, Are the x- and y-axes principal axes? PROBLEM: For the two-dimensional Poiseuille flow of Prob. 4-100, calculate the linear strain rates in the x- and y-directions, and calculate the shear strain rate ?xy. PROBLEM 4 - 100: Consider fully developed two-dimensional Poiseuille flow—flow between two infinite parallel plates separated by distance h,with both the top plate and bottom plate stationary, and a forced pressure gradient dP/dxdriving the flow as illustrated in Fig. P11–48. (dP/dxis constant and negative.) FIGURE P11-48 The flow is steady, incompressible, and two-dimensional in the xy-plane. The velocity components are given by where ? is the fluid’s viscosity. Is this flow rotational or irrotational? If it is rotational, calculate the vorticity component in the z-direction. Do fluid particles in this flow rotate clockwise or counterclockwise?

Problem 97P For the two-dimensional Poiseuille flow of Prob. 4-100, calculate the linear strain rates in the x- and y-directions, and calculate the shear strain rate ?xy. PROBLEM: Consider fully developed two-dimensional Poiseuille flow—flow between two infinite parallel plates separated by distance h,with both the top plate and bottom plate stationary, and a forced pressure gradient dP/dxdriving the flow as illustrated in Fig. P11–48. (dP/dxis constant and negative.) FIGURE P11-48 The flow is steady, incompressible, and two-dimensional in the xy-plane. The velocity components are given by where ? is the fluid’s viscosity. Is this flow rotational or irrotational? If it is rotational, calculate the vorticity component in the z-direction. Do fluid particles in this flow rotate clockwise or counterclockwise?

Problem 108P Consider the vacuum cleaner of Prob. 11–54. For the case where b = 2.0 cm, L = 35 cm, and = 0.1098 m3/s, create a velocity vector plot in the upper half of the xy-plane from x = ? 3 cm to 3 cm and from y = 0 cm to 2.5 cm. Draw as many vectors as you need to get a good feel of the flow field. Note:The velocity is infinite at the point (x, y) =(0,2.0 cm), so do not attempt to draw a velocity vector at that point.

Problem 107P We approximate the flow of air into a vacuum cleaner attachment by the following velocity components in the centerplane (the xy-plane): and where bis the distance of the attachment above the floor, L is the length of the attachment, and is the volume flow rate of air being sucked up into the hose (Fig. P11–54). Determine the location of any stagnation point(s) in this flow field. FIGURE P11-54

Problem 110P Consider a steady, two-dimensional flow field xy-plane whose x-component of velocity is given by where a, b,and care constants with appropriate dimensions. Of what form does the y-component of velocity need to be in order for the flow field to be incompressible? In other words, generate an expression for v as a function of x,y, and the constants of the given equation such that the flow is incompressible.

Problem 105P For the axisymmetric Poiseuille flow of Prob. 4-108. calculate the linear strain rates in the x- and r-directions, and calculate the shear strain rate ?xr. The strain rate tensor in cylindrical coordinates (r, ?. x) and (ur,u?. ux). is PROBLEM: Consider fully developed axisymmetric Poiseuille flow—flow in a round pipe of radius R(diameter D = 2 R), with a forced pressure gradient dP/dx driving the flow as illustrated in Fig. P11–53. (dP/dx is constant and negative.) The flow is steady, incompressible, and axisymmetric about the x-axis. The velocity components are given by where ? is the fluid’s viscosity. Is this flow rotational or irrotational? If it is rotational, calculate the vorticity component in the circumferential (?) direction and discuss the sign of the rotation. FIGURE P11-53

Problem 109P Consider the approximate velocity field given for the vacuum cleaner of Prob. 11–54. Calculate the flow speed along the floor. Dust particles on the floor are most likely to be sucked up by the vacuum cleaner at the location of maximum speed. Where is that location? Do you think the vacuum cleaner will do a good job at sucking up dust directly below the inlet (at the origin)? Why or why not? PROBLEM: We approximate the flow of air into a vacuum cleaner attachment by the following velocity components in the centerplane (the xy-plane): and where bis the distance of the attachment above the floor, L is the length of the attachment, and is the volume flow rate of air being sucked up into the hose (Fig. P11–54). Determine the location of any stagnation point(s) in this flow field. FIGURE P11-54

Problem 111P In a steady, two-dimensional flow field in the xy - plane, the x-component of velocity is where a, b, and c are constants with appropriate dimensions. Generate a general expression for velocity component v such that the flow field is incompressible.

Problem 23P For the velocity field of Prob. 4-23, calculate the fluid acceleration along the diffuser centerline as a function of x and the given parameters. For L = 1.56 m, uentrance = 24.3 m/s, and uexit=16.8 m/s, calculate the acceleration at x =0 and x = 1.0 m. PROBLEM: Consider steady flow of air through the diffuser portion of a wind tunnel (Fig. P11–20). Along the centerline of the diffuser, the air speed decreases from uentrance to uexit as sketched. Measurements reveal that the centerline air speed decreases parabolically through the diffuser. Write an equation for centerline speed u(x), based on the parameters given here, from x = 0 to x = L. FIGURE:

Problem 24P What is the definition of a streamline? What do streamlines indicate?

Problem 25P Consider the visualization of flow over a 12° cone in Fig. P11-25C. Are we seeing streamlines, streaklines pathlines, or timelines? Explain. Visualization of flow over a 12° cone at a 16° angle of attack at a Reynolds number of 15.000. The visualization is produced by colored fluid injected into water from ports is the body. Courtesy ONERA. Photograph by Werlé.

Problem 64P For the velocity field of Prob. 4?63. calculate the vorticity vector. In which direction does the vorticity vector point?

Problem 65P Consider steady, incompressible, two-dimensional shear flow for which the velocity field is where a and b are constants. Sketched in Fig. P4?69 is a small rectangular fluid particle of dimensions dx and dy at time t. The fluid particle moves and deforms with the flow such that at a later time (t + dt). the particle is no longer rectangular, as also shown in the figure. The initial location of each corner of the fluid particle is labeled in Fig. P4?69. The lower-left corner is at (x. y) at time t. where the x-component of velocity is u = a + by. At the later time, this corner moves to (x +u dt. y), or (a) In similar fashion, calculate the location of each of the other three corners of the fluid particle at time t ? dt. (b) From the fundamental definition of linear strain rate (the rate of increase in length per unit length). calculate linear strain rates ?xx and ?yy. (c) Compare your results with those obtained from the equations for ?xx and ?yy in Cartesian coordinates, i.e.. FIGURE P4?69

Problem 66P Use two methods to verify that the flow of Prob. 4?69 is incompressible: (a) by calculating the volume of the fluid particle at both times, and(b) by calculating the volumetric strain rate. Note that Prob. 4?69 should be completed before this problem.

Problem 100P Consider the two-dimensional Poiseuille flow of Prob. 11-48 . The fluid between the plates is water at 40°C. Let the gap height h = 1.6 mm and the pressure gradient dP/dx= ? 230 N/m3. Calculate and plot seven streaklinesgenerated from a dye rake that introduces dye streaks at x = 0 and at y =0.2, 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4 mm (Fig. P11–50). The dye is introduced from t =0 to t =10 s, and the streaklines are to be plotted at t =10 s. FIGURE P11-50

Problem 102P Compare the results of Probs 4-104 and 4-105 and comment about the linear strain rate in the x-direction.

Problem 101P Repeat Prob. 11–50 except that the dye is introduced from t= 0 to t= 10 s, and the streaklines are to be plotted at t =12 s instead of 10 s.

Problem 29P Consider the visualization of ground vortex flow in Fig. P11-29C. Are we seeing streamlines, streaklines, pathlines, or timelines? Explain. FIGURE P11-29C Visualization of ground vortex flow. A high-speed round air jet impinges on the ground in the presence of a free-stream flow of air from left to right. (The ground is at the bottom of the picture.) The portion of the jet that travels upstream forms a recirculating flow known as a ground vortex. The visualization is produced by a smoke wire mounted vertically to the left of the field of view. Photo by John M. Cimbala.

Problem 30P Consider the visualization of flow over a sphere in fig. P11-30C. Are we seeing streamlines, streaklines, pathlines. or timelines? Explain. FIGURE P11-30C Visualization of flow over a sphere at a Reynolds number of 15,000. The visualization is produced by a time exposure of air bubbles in water. Courtesy ONERA. Photograph by Werlé.

Problem 31P What is the definition of a timeline? How can timelines be produced in a water channel? Name an application where timelines are more useful than streaklines.

Problem 70P A two-dimensional fluid element of dimensions dx and dy translates and distorts as shown in Fig. P4?74 during the infinitesimal time period dt = t 2?t 1· The velocity components at point P at the initial time are u and v in the x- and y-directions, respectively. Show that the magnitude of the rate of rotation (angular velocity) about point P in the xy-plane is FIGURE P4?74

Problem 71P A two-dimensional fluid element of dimensions dx and dy translates and distorts as shown in Fig. P4?74 during the infinitesimal time period dt = t2 ?t1· The velocity components at point P at the initial time are u and v in the x- and y-directions, respectively. Consider the line segment PA in Fig. P4?74, and show that the magnitude of the linear strain rate in the x-direction is Fig:

Problem 106P Combine your results from Prob. 4-109 to form the axisymmetric strain rate tensor ?ij. Are the x- and r-axes principal axes? PROBLEM: For the axisymmetric Poiseuille flow of Prob. 4-108. calculate the linear strain rates in the x- and r-directions, and calculate the shear strain rate ?xr. The strain rate tensor in cylindrical coordinates (r, ?. x) and (ur,u?. u x). is

Problem 72P A two-dimensional fluid element of dimensions dxand dy translates and distorts as shown in Fig. P4?74 during the infinitesimal time period dt = t 2?t 1· The velocity components at point P at the initial time are u and v in the x- and y-directions, respectively. Show that the magnitude of the shear strain rate about point P in the xy-plane is

Problem 32P Consider a cross-sectional slice through an array of exchanger tubes (Fig. P4-33C). For each desired piece information, choose which kind of flow visualization plot (vector plot or contour plot) would be most appropriate, and explain why. (a) The location of maximum fluid speed is to be visualized. (b) Flow separation at the rear of the tubes is to be visualized. (c) The temperature field throughout the plane is to be visualized. (d) The distribution of the vorticity component normal to the plane is to be visualized. FIGURE P4?33C

Problem 1P What does the word kinematicsmean? Explain what the study of fluid kinematicsinvolves.

Problem 35P Consider the steady, incompressible, two-dimensional velocity field of Prob. 4?36. Generate a velocity vector plot in the upper-right quadrant from x = 0 to 5 and y = 0 to 6.

Problem 36P Consider the steady, incompressible, two-dimensional velocity field of Prob. 4?36. Generate a vector plot of the acceleration field in the upper-right quadrant from x = 0 to 5 and y = 0 to 6.

Problem 37P A steady, incompressible, two-dimensional velocity field is given by where the x- and y-coordinates are in m and the magnitude of velocity is in m/s. (a) Determine if there are any stagnation points in this flow field, and if so, where they are. (b) Sketch velocity vectors at several locations in the upper- right quadrant for x = 0 m to 4 m and y= 0 m to 4 m; qualitatively describe the flow field.

Problem 76P A cylindrical tank of radius rrim = 0.354 m rotates about its vertical axis (Fig. P4?78). The tank is partially filled with oil. The speed of the rim is 3.61 m/s in the counterclock wise direction (looking from the top), and the tank has been spinning long enough to be in solid-body rotation. For any fluid particle in the tank, calculate the magnitude of the component of vorticity in the vertical z-direction.

Problem 77P Consider a two-dimensional, incompressible flow field in which an initially square fluid particle moves and deforms. The fluid particle dimension is a at time t and is aligned with the x- and y-axes as sketched in Fig. P4?81. At some later time, the particle is still aligned with the x- and y-axes, but has deformed into a rectangle of horizontal length 2a. What is the vertical length of the rectangular fluid particle at this later time? FIGURE P4?81

Problem 78P Consider a two-dimensional, compressible flow field in which an initially square fluid particle moves and deforms. The fluid particle dimension is a at time t and is aligned with the x- and y-axes as sketched in Fig. P4?81. At some later time, the particle is still aligned with the x- and y-axes but has deformed into a rectangle of horizontal length 1.06a and vertical length 0.931a. (The particle’s dimension in the z-direction does not change since the flow is two-dimensional.) By what percentage has the density of the fluid particle increased or decreased?

Problem 112P In a steady, two-dimensional flow field in the xy-plane, the x-component of velocity is where a, b, c, and d are constants with appropriate dimensions. Generate a general expression for velocity component v such that the flow field is incompressible.

Problem 113P There are numerous occasions in which a fairly uniform free-stream flow encounters a long circular cylinder aligned normal to the flow (Fig. P11–58). Examples include air flowing around a car antenna, wind blowing against a flag pole or telephone pole, wind hitting electrical wires, and ocean currents impinging on the submerged round beams to support oil platforms. In all these cases, the flow at the rear of the cylinder is separated and unsteady, and usually turbulent. However, the flow in the front half of the cylinder is much more steady and predictable. In fact, except for a very thin boundary layer near the cylinder surface, the flow field may be approximated by the following steady, two-dimensional velocity components in the xy-or r?-plane: FIGURE P11–58 Is this flow field rotational or irrotational? Explain.

Problem 114P Consider the flow field of the Problem Below (flow over a circular cylinder). Consider only the front half of the flow (x < 0). There is one stagnation point in the front half of the flow field. Where is it? Give your answer in both cylindrical (r, ?)coordinates and Cartesian (x, y)coordinates. Problem: There are numerous occasions in which a fairly uniform free-stream flow encounters a long circular cylinder aligned normal to the flow (Fig. P11–58). Examples include air flowing around a car antenna, wind blowing against a flag pole or telephone pole, wind hitting electrical wires, and ocean currents impinging on the submerged round beams to support oil platforms. In all these cases, the flow at the rear of the cylinder is separated and unsteady, and usually turbulent. However, the flow in the front half of the cylinder is much more steady and predictable. In fact, except for a very thin boundary layer near the cylinder surface, the flow field may be approximated by the following steady, two-dimensional velocity components in the xy-or r?-plane: Is this flow field rotational or irrotational? Explain?

Problem 2P Consider steady flow of water through an axisymmetric garden hose nozzle (Fig. P11–2). Along the centerline of the nozzle, the water speed increases from uentrance toexit as sketched. Measurements reveal that the centerline water speed increases parabolically through the nozzle. Write an equation for centerline speed u(x), based on the parameters given here, from x = 0 to x = L.

Problem 3P Consider the following steady, two-dimensional velocity field: Is there a stagnation point in this flow field? If so, where is it?

Problem 4P A steady, two-dimensional velocity field is given by Calculate the location of the stagnation point.

Problem 38P Consider the steady, incompressible, two-dimensional velocity field of Prob. 4?39. (a) Calculate the material acceleration at the point (x = 2 m, y = 3 m). (b) Sketch the material acceleration vectors at the same array of x- and y-values as in Prob. 4?39.

Problem 40P The velocity field for a line vortexin the r?-plane (Fig. P4?42) is given by Where K is the line vortex strength. For the case with K= 1.5 m 2/s, plot a contour plot of velocity magnitude (speed). FIGURE P4?42 Specifically, draw curves of constant speed V = 0.5,1.0,1.5, 2.0, and 2.5 m/s. Be sure to label these speeds on your plot.

Problem 39P The velocity field for solid-body rotation in the r? plane (Fig. P4-41) is given by u r=0 u?=?r where ? is the magnitude of the angular velocity ( points in the z-direction). For the case with ? = 1.5 s -1, plot a contour plot of velocity magnitude (speed). Specifically, draw curves of constant speed V = 0.5, 1.0, 1.5, 2.0, and 2.5 m/s. Be sure to label these speeds on your plot. FIGURE P4?41

Problem 79P Consider the following steady, three-dimensional velocity field: Calculate the vorticity vector as a function of space (x, y, z).

Problem 80P Consider fully developed Couette flow-flow between two infinite parallel plates separated by distance h, with the top plate moving and the bottom plate stationary as illustrated in Fig. P11-43. The flow is steady, incompressible, and two-dimensional in the xy-plane. The velocity field is given by Is this flow rotational or irrotational? If it is rotational Caluclate the vorticity component in the z-direction. Do fluid particles in this flow rotate clockwise or counterclockwise FIGURE P11-43

Problem 81P For the Couette flow of Fig. P4?84, calculate the linear strain rates in the x- and y-directions, and calculate the shear strain rate ?xy.

Problem 115P Consider the upstream half (x < 0) of the flow field of Prob. 11–58 (flow over a circular cylinder). We introduce a parameter called the stream function ?. bwhich is constant along streamlinesin two-dimensional flows such as the one being considered here (Fig. P11–60) The velocity field of Prob. 11–58 corresponds to a stream function given by (a) Setting ?to a constant, generate an equation for a streamline. (Hint: Use the quadratic rule to solve for r as a function of ?.) (b) For the particular case in which V = 1.00 m/s and cylinder radius a = 10.0 cm, plot several streamlines in the upstream half of the flow (90°< ? < 270°). For consistency, plot in the range –0.4 m< x < 0 m, – 0.2 m < y < 0.2 m, with stream function values evenly spaced between –0.16 m2/s and –0.16 m2/s. FIGURE P11–60

Problem 116P Consider the flow field of Prob. 4?117 (flow over a circular cylinder). Calculate the two linear strain rates in the r?-plane; i.e., calculate ?rr and ???· Discuss whether fluid line segments stretch (or shrink) in this flow field.(Hint: The strain rate tensor in cylindrical coordinates is given in Prob. 4?109.) PROBLEM 4-117 There are numerous occasions in which a fairly uniform free-stream flow encounters a long circular cylinder aligned normal to the flow (Fig. P11–58). Examples include air flowing around a car antenna, wind blowing against a flag pole or telephone pole, wind hitting electrical wires, and ocean currents impinging on the submerged round beams to support oil platforms. In all these cases, the flow at the rear of the cylinder is separated and unsteady, and usually turbulent. However, the flow in the front half of the cylinder is much more steady and predictable. In fact, except for a very thin boundary layer near the cylinder surface, the flow field may be approximated by the following steady, two-dimensional velocity components in the xy-or ?-plane:Is this flow field rotational or irrotational? Explain. FIGURE P11–58 The strain rate tensor in cylindrical coordinates (r, ?. x) and (ur,u?. u x). is

Problem 117P Based on your results of Prob. 4-120. discuss the compressibility (or incompressibility) of this flow. PROBLEM 4-120: Consider the flow field of Prob. 4?117 (flow over a circular cylinder). Calculate the two linear strain rates in the r?-plane; i.e., calculate ?rr and???· Discuss whether fluid line segments stretch (or shrink) in this flow field.(Hint: The strain rate tensor in cylindrical coordinates is given in Prob. 4?109.) PROBLEM 4-177:There are numerous occasions in which a fairly uniform free-stream flow encounters a long circular cylinder aligned normal to the flow (Fig. P11–58). Examples include air flowing around a car antenna, wind blowing against a flag pole or telephone pole, wind hitting electrical wires, and ocean currents impinging on the submerged round beams to support oil platforms. In all these cases, the flow at the rear of the cylinder is separated and unsteady, and usually turbulent. However, the flow in the front half of the cylinder is much more steady and predictable. In fact, except for a very thin boundary layer near the cylinder surface, the flow field may be approximated by the following steady, two-dimensional velocity components in the xy-or r?-plane: Is this flow field rotational or irrotational? Explain. FIGURE P11–58 The strain rate tensor in cylindrical coordinates:

Problem 5P Consider the following steady, two-dimensional velocity field: Is there a stagnation point in this flow field? If so, where is it?

Problem 6P What is the Lagrangian descriptionof fluid motion?

Problem 7P Is the Lagrangian method of fluid flow analysis more similar to study of a system or a control volume? Explain.

Problem 41P The velocity field for a line source in the r? -plane (Fig. P4?43) is given by where m is the line source strength. For the case with m/(2?)= 1.5 m 2/s, plot a contour plot of velocity magnitude (speed). Specifically, draw curves of constant speed V = 0.5, 1.0,1.5, 2.0, and 2.5 m/s. Be sure to label these speeds on your plot. FIGURE P4?43

Problem 43P A very small circular cylinder of radius R i is rotating] at angular velocity ?i inside a much larger concentric cylinder of radius R o that is rotating at angular velocity?0. A liquid of density? and viscosity µ is confined between the two cylinders, as in Fig. P4?45. Gravitational and end effects can be neglected (the flow is two-dimensional into the page). If ?i = ?o and a long time has passed, generate an expression for the tangential velocity profile, u ? as a function of (al most)r, ?, R i, R o, ?, and µ, where ? = ?i, =?o. Also, calculate the torque exerted by the fluid on the inner cylinder and on the outer cylinder. FIGURE P4?45

Problem 45P Consider the same two concentric cylinders of Prob. 4?45. This time, however, the inner cylinder is rotating, but the outer cylinder is stationary. In the limit, as the outer cylinder is very large compared to the inner cylinder (imagine the inner cylinder spinning very fast while its radius gets very small), what kind of flow does this approximate? Explain. After a long time has passed, generate an expression for the tangential velocity profile, namely u ? as a function of (at most) r,?i, R i, R o, ?, and µ. Hint: Your answer may contain an (unknown) constant, which can be obtained by specifying a boundary condition at the inner cylinder surface.

Problem 82P Combine your results from Prob. 4?85 to form the two-dimensional strain rate tensor ?ij· Are the x- and y-axes principal axes? PROBLEM: FOr the Couette Flow of the figure, Calculate the linear strain rates in the x- and y-directions and calculate the shear strain rate ?zy.

Problem 83P A steady, three-dimensional velocity field is given by Calculate the vorticity vector as a function of space variables (x, y, z).

Problem 118 Consider the flow field of Prob. 4?117 (flow over a circular cylinder). Calculate ?r? the shear strain rate in the r?-plane. Discuss whether fluid particles in this flow deform with shear or not.(Hint: The strain rate tensor in cylindrical coordinates is given in Prob. 4?109.)

Problem 8P What is the Eulerian descriptionof fluid motion? How does it differ from the Lagrangian description?

Problem 9P A stationary probe is placed in a fluid flow and measures pressure and temperature as functions of time at one location in the flow (Fig. P 11-8C). Is this a Lagrangian or an Eulerian measurement? Explain.

Problem 49P Name and briefly describe the four fundamental types of motion or deformation of fluid particles.

Problem 10P A tiny neutrally buoyant electronic pressure probe is released into the inlet pipe of a water pump and transmits 2000 pressure readings per second as it passes through the pump. Is this a Lagrangian or an Eulerian measurement? Explain.

Problem 50P Explain the relationship between vorticity and rotationality.

Problem 51P Converging duct flow (Fig. P11–15) is modeled by the steady, two-dimensional velocity field of Prob. 11–15. Is this flow field rotational or irrotational? Show all your work.

Problem 85P A steady, three-dimensional velocity field is given by Calculate constants b and c such that the flow field is irrotational.

Problem 86P A steady, three-dimensional velocity field is given by Calculate constants a, b, and c such that the flow field is irrotational.

Problem 87P Briefly explain the purpose of the Reynolds transport theorem (RTT). Write the RTT for extensive property B as a "word equation." explaining each term in your own words.

Problem 84P A steady, two-dimensional velocity field is given by Calculate constant c such that the flow field is irrotational.