(See Fluids in the News article titled A Sailing Ship

Looking at Eq. 6.37, what are the units of stream function? a) \(\mathrm{LT^{-1}}\) b) \(\mathrm{L^2T^{-1}}\) c) \(\mathrm{MLT^{-1}}\) d) \(\mathrm{T^{-1}}\)

Two tanks filled with water are connected by two straight circular pipes that have diameters \(D_1\) and \(D_2\), as shown in the figure. The water level in the left tank is twice that of the right tank. If the flow through the connection pipes is laminar and can be approximated by the fully developed Poiseuille solution, then the flow through pipe 2 will have the same velocity as the flow through pipe 1 when: a) the diameter of pipe 2 is less than the diameter of pipe 1. b) the diameter of pipe 2 is greater than the diameter of pipe 1. c) the diameter of pipe 2 is equal to the diameter of pipe 1.

A uniform flow moving to the left (in the negative x-axis direction) is imposed on the doublet shown in Fig. 6.23. For the resulting flow field, a) there are two stagnation points, one above and one below the doublet. b) there is one stagnation point to the left of the doublet. c) there is one stagnation point to the right of the doublet. d) there are no stagnation points anywhere in the flow.

For steady, laminar flow between fixed parallel plates, the velocity profile is parabolic in shape, as shown in Fig. 6.31. If the shear stress is given by \(\tau=\mu \partial u/ \partial y\), what is shear stress at the midpoint between the two plates? a) maximum shear b) negative shear c) zero shear d) equal to the wall shear

The velocity in a certain flow field is given by the equation \(\mathbf{V}=y z \hat{\mathbf{i}}+x^{2} z \hat{\mathbf{j}}+x \hat{\mathbf{k}}\) Determine the expressions for the three rectangular components of acceleration.

The velocity in a certain two-dimensional flow field is given by the equation \(\mathbf{V}=2 x t \hat{\mathbf{i}}-2 y t \hat{\mathbf{j}}\) where the velocity is in ft/s when x, y, and t are in feet and seconds, respectively. Determine expressions for the local and convective components of acceleration in the x and y directions. What is the magnitude and direction of the velocity and the acceleration at the point x = y = 2 ft at the time t = 0?

The velocity in a certain flow field is given by the equation \(\mathbf{V}=x \hat{\mathbf{i}}+x^{2} z \hat{\mathbf{j}}+y z \hat{\mathbf{k}}\) Determine the expressions for the three rectangular components of acceleration.

The three components of velocity in a flow field are given by \(u = x^2 + y^2 + z^2\) \(v = xy + yz + z^2\) \(w = -3xz - z^ 2/ 2 +4\) (a) Determine the volumetric dilatation rate and interpret the results. (b) Determine an expression for the rotation vector. Is this an irrotational flow field?

Determine the vorticity field for the following velocity vector: \(\mathbf{V}=\left(x^{2}-y^{2}\right) \hat{\mathbf{i}}-2 x y \hat{\mathbf{j}}\)

Determine an expression for the vorticity of the flow field described by \(\mathbf{V}=-x y^{3} \hat{\mathbf{i}}+y^{4} \hat{\mathbf{j}}\) Is the flow irrotational?

A two-dimensional flow field described by \(\mathbf{V}=\left(2 x^{2} y+x\right) \hat{\mathbf{i}}+\left(2 x y^{2}+y+1\right) \hat{\mathbf{j}}\) where the velocity is in m/s when x and y are in meters. Determine the angular rotation of a fluid element located at x = 0.5 m, y = 1.0 m.

For a certain incompressible, two-dimensional flow field the velocity component in the y direction is given by the equation \(v = 3xy + x^2y\) Determine the velocity component in the x direction so that the volumetric dilatation rate is zero.

An incompressible viscous fluid is placed between two large parallel plates as shown in Fig. P6.9. The bottom plate is fixed and the upper plate moves with a constant velocity, U. For these conditions the velocity distribution between the plates is linear and can be expressed as \(u=U \frac{y}{b}\) Determine: (a) the volumetric dilatation rate, (b) the rotation vector, (c) the vorticity, and (d) the rate of angular deformation.

A viscous fluid is contained in the space between concentric cylinders. The inner wall is fixed, and the outer wall rotates with an angular velocity \(\omega\). (See Fig. P6.10a and Video V6.3.) Assume that the velocity distribution in the gap is linear as illustrated in Fig. P6.10b. For the small rectangular element shown in Fig. P6.10b, determine the rate of change of the right angle \(\gamma\) due to the fluid motion. Express your answer in terms of \(r_o, r_i\), and \(\omega\).

For incompressible fluids the volumetric dilatation rate must be zero; that is, \(\nabla \cdot \mathbf{V}=0\). For what combination of constants, a, b, c, and e can the velocity components u = ax + by v = cx + ey w = 0 be used to describe an incompressible flow field?

For a certain incompressible flow field it is suggested that the velocity components are given by the equations \(u=2xy \quad v=-x^2y \quad w=0\) Is this a physically possible flow field? Explain.

The velocity components of an incompressible, two dimensional velocity field are given by the equations \(u=y^2 -x(1+x)\) v = y(2x + 1) Show that the flow is irrotational and satisfies conservation of mass.

For each of the following stream functions, with units of \(\mathrm{m^2/s}\), determine the magnitude and the angle the velocity vector makes with the x axis at x = 1 m, y = 2 m. Locate any stagnation points in the flow field. (a) \(\psi=xy\) (b) \(\psi=-2x^2+y\)

The stream function for an incompressible, two dimensional flow field is \(\psi=ay - by^3\) where a and b are constants. Is this an irrotational flow? Explain.

The stream function for an incompressible, two-dimensional flow field is \(\psi=ay^2 - bx\) where a and b are constants. Is this an irrotational flow? Explain.

The velocity components in an incompressible, two dimensional flow field are given by the equations \(u=x^2\) \(v=-2xy+x\) Determine, if possible, the corresponding stream function.

The velocity components of an incompressible, two dimensional velocity field are given by the equations u = 2xy \(v=x^2-y^2\) Show that the flow is irrotational and satisfies conservation of mass.

For a certain two-dimensional flow field u = 0 v = V (a) What are the corresponding radial and tangential velocity components? (b) Determine the corresponding stream function expressed in Cartesian coordinates and in cylindrical polar coordinates.

Some velocity measurements in a three-dimensional incompressible flow field indicate that \(u=6xy^2\) and \(v=-4y^2z\). There is some conflicting data for the velocity component in the z direction. One set of data indicates that \(w=4yz^2\) and the other set indicates that \(w= 4yz^2 - 6y^2 z\). Which set do you think is correct? Explain.

A two-dimensional, incompressible flow is given by u = -y and v = x. Show that the streamline passing through the point x = 10 and y = 0 is a circle centered at the origin.

In a certain steady, two-dimensional flow field the fluid density varies linearly with respect to the coordinate x: that is, \(\rho =Ax\) where A is a constant. If the x component of velocity u is given by the equation u = y, determine an expression for v.

In a two-dimensional, incompressible flow field, the x component of velocity is given by the equation u = 2x. (a) Determine the corresponding equation for the y component of velocity if v = 0 along the x axis. (b) For this flow field, what is the magnitude of the average velocity of the fluid crossing the surface OA of Fig. P6.23? Assume that the velocities are in feet per second when x and y are in feet.

The radial velocity component in an incompressible, two-dimensional flow field \((v_z = 0)\) is \(v_r=2r+3r^2 \sin \theta\) Determine the corresponding tangential velocity component, \(v_\theta\), required to satisfy conservation of mass.

The stream function for an incompressible flow field is given by the equation \(\psi=3x^2y-y^3\) where the stream function has the units of \(\mathrm{m}^{2} / \mathrm{s}\) with x and y in meters. (a) Sketch the streamline(s) passing through the origin. (b) Determine the rate of flow across the straight path AB shown in Fig. P6.25.

The streamlines in a certain incompressible, two-dimensional flow field are all concentric circles so that \(v_r=0\). Determine the stream function for (a) \(v_\theta= Ar\) and for (b) \(v_\theta= Ar^{-1}\), where A is a constant.

It is proposed that a two-dimensional, incompressible flow field be described by the velocity components u = Ay v = Bx where A and B are both positive constants. (a) Will the continuity equation be satisfied? (b) Is the flow irrotational? (c) Determine the equation for the streamlines and show a sketch of the streamline that passes through the origin. Indicate the direction of flow along this streamline.

The stream function for an incompressible, two dimensional flow field is \(\psi=3x^2y+y\) For this flow field, plot several streamlines.

Consider the incompressible, two-dimensional flow of a nonviscous fluid between the boundaries shown in Fig. P6.29. The velocity potential for this flow field is \(\phi=x^2-y^2\) (a) Determine the corresponding stream function. (b) What is the relationship between the discharge, q (per unit width normal to plane of paper) passing between the walls and the coordinates \(x_i , y_i\) of any point on the curved wall? Neglect body forces.

A fluid with a density of \(2000~\mathrm{kg/m^3}\) flows steadily between two flat plates as shown in Fig. P6.30. The bottom plate is fixed and the top one moves at a constant speed in the x direction. The velocity \(V=0.20~y ~\mathbf{\hat i}~\mathrm{m/s}\) is where y is in meters. The acceleration of gravity is \(g=-9.8~\mathbf{\hat j}~\mathrm{m/s^2}\). The only nonzero shear stresses, \(\tau_{yx}=\tau_{xy}\), are constant throughout the flow with a value of \(5~\mathrm{N/m^2}\). The normal stress at the origin (x = y = 0) is \(\sigma_{xx}=- 100\) kPa. Use the x and y components of the equations of motion (Eqs. 6.50a and b) to determine the normal stress throughout the fluid. Assume that \(\sigma_{xx} = \sigma_{yy}\).

A fluid with a density of \(2~ \mathrm{slug/ft^3}\) flows steadily between two stationary flat plates as shown in Fig. P6.31. The velocity is \(\boldsymbol{V}=0.5\left[1-(y / h)^{2}\right] \hat{\mathbf{i}} \ \mathrm{ft} / \mathrm{s}\) where y and h are in feet. The only nonzero shear stresses, \(\tau_{yx}=\tau_{xy}\), are given by \(\tau_{yx}=-4.0~y~\mathrm{lb/ft^2}\) and the acceleration of gravity is negligible. The normal stress at the origin (x = y = 0) is \(\sigma_{xx}=-10~ \mathrm{lb/ft^2}\). Use the x and y components of the equations of motion (Eqs. 6.50a and b) to determine the normal stress throughout the fluid. Assume that \(\sigma_{xx}=\sigma_{yy}\).

Given the stream function for a flow as \(\psi=4x^2-4y^2\), show that the Bernoulli equation can be applied between any two points in the flow field.

A two-dimensional flow field for a nonviscous, incompressible fluid is described by the velocity components \(u = U_0+ 2y \) v = 0 where \(U_0\) is a constant. If the pressure at the origin (Fig. P6.33) is \(p_0\), determine an expression for the pressure at (a) point A, and (b) point B. Explain clearly how you obtained your answer. Assume that the units are consistent and body forces may be neglected.

In a certain two-dimensional flow field, the velocity is constant with components u = -4 ft/s and v = -2 ft/s. Determine the corresponding stream function and velocity potential for this flow field. Sketch the equipotential line \(\phi=0\) which passes through the origin of the coordinate system.

The stream function for a given two-dimensional flow field is \(\psi=5x^2y-(5/3)y^3\) Determine the corresponding velocity potential.

A certain flow field is described by the stream function \(\psi=A \ \theta+B \ r \sin\theta\) where A and B are positive constants. Determine the corresponding velocity potential and locate any stagnation points in this flow field.

It is known that the velocity distribution for two dimensional flow of a viscous fluid between wide parallel plates (Fig. P6.37) is parabolic; that is, \(u=U_{c}\left[1-\left(\frac{y}{h}\right)^{2}\right]\) with v = 0. Determine, if possible, the corresponding stream function and velocity potential.

The velocity potential for a certain inviscid flow field is \(\phi=-(3x^2y-y^3)\) where \(\phi\) has the units of \(\mathrm{ft^2 /s}\) when x and y are in feet. Determine the pressure difference (in psi) between the points (1, 2) and (4, 4), where the coordinates are in feet, if the fluid is water and elevation changes are negligible.

The velocity potential for a flow is given by \(\phi=\frac{a}{2}\left(x^{2}-y^{2}\right)\) where a is a constant. Determine the corresponding stream function and sketch the flow pattern.

The stream function for a two-dimensional, nonviscous, incompressible flow field is given by the expression \(\psi=-2(x-y)\) where the stream function has the units of \(\mathrm{ft^2 /s}\) with x and y in feet. (a) Is the continuity equation satisfied? (b) Is the flow field irrotational? If, so, determine the corresponding velocity potential. (c) Determine the pressure gradient in the horizontal x direction at the point x = 2 ft, y = 2 ft.

The velocity potential for a certain inviscid, incompressible flow field is given by the equation \(\phi=2 x^{2} y-\left(\frac{2}{3}\right) y^{3}\) where \(\phi\) has the units of \(\mathrm{m^2 /s}\) when x and y are in meters. Determine the pressure at the point x = 2 m, y = 2 m if the pressure at x = 1 m, y = 1 m is 200 kPa. Elevation changes can be neglected, and the fluid is water.

A steady, uniform, incompressible, inviscid, two-dimensional flow makes an angle of \(30^\circ\) with the horizontal x axis. (a) Determine the velocity potential and the stream function for this flow. (b) Determine an expression for the pressure gradient in the vertical y direction. What is the physical interpretation of this result?

The streamlines for an incompressible, inviscid, two dimensional flow field are all concentric circles, and the velocity varies directly with the distance from the common center of the streamlines; that is \(v_ \theta=Kr\) where K is a constant. (a) For this rotational flow, determine, if possible, the stream function. (b) Can the pressure difference between the origin and any other point be determined from the Bernoulli equation? Explain.

The velocity potential \(\phi=-k(x^2-y^2) \quad(k=\text {constant})\) may be used to represent the flow against an infinite plane boundary, as illustrated in Fig. P6.44. For flow in the vicinity of a stagnation point, it is frequently assumed that the pressure gradient along the surface is of the form \(\frac{\partial p}{\partial x}=Ax\) where A is a constant. Use the given velocity potential to show that this is true.

In a certain steady, two-dimensional flow field the fluid may be assumed to be ideal and the weight of the fluid (specific weight = \(50~\mathrm{lb/ft^3}\)) is the only body force. The x component of velocity is known to be u = 6x, which gives the velocity in ft/s when x is measured in feet, and the y component of velocity is known to be a function of only y. The y axis is vertical, and at the origin the velocity is zero. (a) Determine the y component of velocity so that the continuity equation is satisfied. (b) Can the difference in pressures between the points x = 1 ft, y = 1 ft and x = 1 ft, y = 4 ft be determined from the Bernoulli equation? If so, determine the value in \(\mathrm{lb/ft^2 }\). If not, explain why not.

Water is flowing between wedge-shaped walls into a small opening as shown in Fig. P6.46. The velocity potential with units \(m^2 /s\) for this flow is \(\phi=-2 \ln r\) with r in meters. Determine the pressure differential between points A and B.

A certain flow field is described by the velocity potential \(\phi = A \ln r + Br \cos \theta\) where A and B are positive constants. Determine the corresponding stream function and locate any stagnation points in this flow field.

The velocity potential for a given two-dimensional flow field is \(\phi = \left(\frac{5}{3} \right)x^3-5xy^2\) Show that the continuity equation is satisfied and determine the corresponding stream function.

It is suggested that the velocity potential for the incompressible, nonviscous, two-dimensional flow along the wall shown in Fig. P6.49 is \(\phi=r^{4 / 3} \cos \frac{4}{3} \theta\) Is this a suitable velocity potential for flow along the wall? Explain.

As illustrated in Fig. P6.50, a tornado can be approximated by a free vortex of strength \(\Gamma\) for \(r > R_c\), where \(R_c\) is the radius of the core. Velocity measurements at points A and B indicate that \(V_A = 125\) ft/s and \(V_B = 60\) ft/s. Determine the distance from point A to the center of the tornado. Why can the free vortex model not be used to approximate the tornado throughout the flow field \((r \geq 0)\)?

The streamlines in a particular two-dimensional flow field are all concentric circles, as shown in Fig. P6.51. The velocity is given by the equation \(v_\theta = \omega r\) where \(\omega\) is the angular velocity of the rotating mass of fluid. Determine the circulation around the path ABCD.

The motion of a liquid in an open tank is that of a combined vortex consisting of a forced vortex for \(0 \leq r \leq 2\) ft and a free vortex for r > 2 ft. The velocity profile and the corresponding shape of the free surface are shown in Fig. P6.52. The free surface at the center of the tank is a depth h below the free surface at \(r=\infty\). Determine the value of h. Note that \(h=h_{\text {forced}}+h_{\text{free}}\), where \(h_{\text {forced}}\) and \(h_{\text{free}}\) are the corresponding depths for the forced vortex and the free vortex, respectively. (See Section 2.12.2 for further discussion of the forced vortex.)

A source of strength m is located a distance \(\ell\) from a vertical solid wall as shown in Fig. P6.53. The velocity potential for this incompressible, irrotational flow is given by \(\phi=\frac{m}{4 \pi}\left\{\ln \left[(x-\ell)^{2}+y^{2}\right]+\ln \left[(x+\ell)^{2}+y^{2}\right]\right\}\) (a) Show that there is no flow through the wall. (b) Determine the velocity distribution along the wall. (c) Determine the pressure distribution along the wall, assuming \(p = p_0\) far from the source. Neglect the effect of the fluid weight on the pressure.

Water flows through a two-dimensional diffuser having a \(20^\circ\) expansion angle as shown in Fig. P6.54. Assume that the flow in the diffuser can be treated as a radial flow emanating from a source at the origin O. (a) If the velocity at the entrance is 20 m/s, determine an expression for the pressure gradient along the diffuser walls. (b) What is the pressure rise between the entrance and exit?

When water discharges from a tank through an opening in its bottom, a vortex may form with a curved surface profile, as shown in Fig. P6.55 and Video V6.4. Assume that the velocity distribution in the vortex is the same as that for a free vortex. At the same time the water is being discharged from the tank at point A, it is desired to discharge a small quantity of water through the pipe B. As the discharge through A is increased, the strength of the vortex, as indicated by its circulation, is increased. Determine the maximum strength that the vortex can have in order that no air is sucked in at B. Express your answer in terms of the circulation. Assume that the fluid level in the tank at a large distance from the opening at A remains constant and viscous effects are negligible.

Water flows over a flat surface at 4 ft/s, as shown in Fig. P6.56. A pump draws off water through a narrow slit at a volume rate of \(0.1 ~ \mathrm{ft^3 /s}\) per foot length of the slit. Assume that the fluid is incompressible and inviscid and can be represented by the combination of a uniform flow and a sink. Locate the stagnation point on the wall (point A) and determine the equation for the stagnation streamline. How far above the surface, H, must the fluid be so that it does not get sucked into the slit?

Two sources, one of strength m and the other with strength 3m, are located on the x axis as shown in Fig. P6.57. Determine the location of the stagnation point in the flow produced by these sources.

The velocity potential for a spiral vortex flow is given by \(\phi=(\Gamma/2 \pi) \theta - (m/2 \pi) \ln r\), where \(\Gamma\) and m are constants. Show that the angle, \(\alpha\), between the velocity vector and the radial direction is constant throughout the flow field (see Fig. P6.58).

For a free vortex (see Video V6.4) determine an expression for the pressure gradient (a) along a streamline, and (b) normal to a streamline. Assume that the streamline is in a horizontal plane, and express your answer in terms of the circulation.

(See Fluids in the News article titled “Some Hurricane Facts,” Section 6.5.3.) Consider a category five hurricane that has a maximum wind speed of 160 mph at the eye wall, 10 mi from the center of the hurricane. If the flow in the hurricane outside of the hurricane’s eye is approximated as a free vortex, determine the wind speeds at locations 20 mi, 30 mi, and 40 mi from the center of the storm.

A plane flow field is developed by the addition of a free vortex and a uniform stream in the positive x direction. If the vortex is located at the origin, determine the pressure variation in this flow field. Neglect the effect of the fluid weight. Express your answer in terms of the uniform velocity, U, the strength of the vortex, \(\Gamma\), and the pressure, \(p_0\), far from the origin.

Potential flow against a flat plate (Fig. P6.62a) can be described with the stream function \(\psi=Axy\) where A is a constant. This type of flow is commonly called a stagnation point flow since it can be used to describe the flow in the vicinity of the stagnation point at O. By adding a source of strength m at O, stagnation point flow against a flat plate with a “bump” is obtained as illustrated in Fig. P6.62b. Determine the relationship between the bump height, h, the constant, A, and the source strength, m.

Show on a plot several “bumps” that can be generated by combining stagnation point flow against a flat plate and a source as described in Problem 6.62.

Consider a uniform flow in the positive x direction combined with a free vortex located at the origin of the coordinate system. The streamline \(\psi=0\) passes through the point x = 4, y = 0. Determine the equation of this streamline.

The combination of a uniform flow and a source can be used to describe flow around a streamlined body called a half-body. (See Video V6.5.) Assume that a certain body has the shape of a half-body with a thickness of 0.5 m. If this body is placed in an airstream moving at 15 m/s, what source strength is required to simulate flow around the body?

A vehicle windshield is to be shaped as a portion of a half-body with the dimensions shown in Fig. P6.66. (a) Make a scale drawing of the windshield shape. (b) For a free-stream velocity of 55 mph, determine the velocity of the air at points A and B.

A body having the general shape of a half-body is placed in a stream of fluid. At a great distance upstream the velocity is U as shown in Fig. P6.67. Show how a measurement of the differential pressure between the stagnation point and point A can be used to predict the free-stream velocity, U. Express the pressure differential in terms of U and fluid density. Neglect body forces and assume that the fluid is nonviscous and incompressible.

One end of a pond has a shoreline that resembles a half-body as shown in Fig. P6.68. A vertical porous pipe is located near the end of the pond so that water can be pumped out. When water is pumped at the rate of \(0.08 ~ \mathrm{m^3 /s}\) through a 3-m-long pipe, what will be the velocity at point A? Hint: Consider the flow inside a half-body. (See Video V6.5.)

Consider two sources having equal strengths located along the x axis at x = 0 and x = 2 m, and a sink located on the y axis at y = 2 m. Determine the magnitude and direction of the fluid velocity at x = 5 m and y = 0 due to this combination if the flowrate from each of the sources is \(0.5~ \mathrm{m^3 /s}\) per m and the flowrate into the sink is \(1.0~ \mathrm{m^3 /s}\) per m.

Consider a uniform flow with velocity U in the positive x direction combined with two free vortices of equal strength located along the y-axis. Let one vortex located at y = a be a clockwise vortex \((\psi=K \ln r)\) and the other at y = -a be a counterclockwise vortex, where K is a positive constant. It can be shown by plotting streamlines that for Ua/K < 2 the streamline \(\psi=0\) forms a closed contour, as shown in Fig. P6.70. Thus, this combination can be used to represent flow around a family of bodies (called Kelvin ovals). Show, with the aid of a graph, how the dimensionless height, H/a, varies with the parameter Ua/K in the range 0.3 < Ua/K < 1.75.

A Rankine oval is formed by combining a source–sink pair, each having a strength of \(36~ \mathrm{ft^2 /s}\) and separated by a distance of 12 ft along the x axis, with a uniform velocity of 10 ft/s (in the positive x direction). Determine the length and thickness of the oval.

Make use of Eqs. 6.107 and 6.109 to construct a table showing how \(\ell/a\), h/a, \(\ell/h\) and for Rankine ovals depend on the parameter \(\pi Ua/m\). Plot \(\ell/h\) versus \(\pi Ua/m\) and describe how this plot could be used to obtain the required values of m and a for a Rankine oval having a specific value of \(\ell\) and h when placed in a uniform fluid stream of velocity, U.

An ideal fluid flows past an infinitely long, semicircular “hump” located along a plane boundary, as shown in Fig. P6.73. Far from the hump the velocity field is uniform, and the pressure is \(p_0\). (a) Determine expressions for the maximum and minimum values of the pressure along the hump, and indicate where these points are located. Express your answer in terms of \(\rho, U\), and \(p_0\). (b) If the solid surface is the \(\psi=0\) streamline, determine the equation of the streamline passing through the point \(\theta = \pi/2, r=2a\).

Assume that the flow around the long circular cylinder of Fig. P6.74 is nonviscous and incompressible. Two pressures, \(p_1\), and \(p_2\), are measured on the surface of the cylinder, as illustrated. It is proposed that the free-stream velocity, U, can be related to the pressure difference \(\Delta p=p_1 -p_2\) by the equation \(U=C \sqrt{\frac{\Delta p}{\rho}}\) Where \(\rho\) is the fluid density. Determine the value of the constant C. Neglect body forces.

Water flows around a 6-ft-diameter bridge pier with a velocity of 12 ft/s. Estimate the force (per unit length) that the water exerts on the pier. Assume that the flow can be approximated as an ideal fluid flow around the front half of the cylinder, but due to flow separation (see Video V6.8), the average pressure on the rear half is constant and approximately equal to 1/2 the pressure at point A (see Fig. P6.75).

Consider the steady potential flow around the circular cylinder shown in Fig. 6.26. On a plot show the variation of the magnitude of the dimensionless fluid velocity, V/U, along the positive y axis. At what distance, y/a (along the y axis), is the velocity within 1% of the free-stream velocity?

The velocity potential for a cylinder (Fig. P6.77) rotating in a uniform stream of fluid is \(\phi=U r\left(1+\frac{a^{2}}{r^{2}}\right) \cos \theta+\frac{\Gamma}{2 \pi} \theta\) Where \(\Gamma\) is the circulation. For what value of the circulation will the stagnation point be located at: (a) point A; (b) point B?

Show that for a rotating cylinder in a uniform flow, the following pressure ratio equation is true. \(\frac{p_{\text {top }}-p_{\text {bottom }}}{p_{\text {stagnation }}}=\frac{8 q}{U}\) Here U is the velocity of the uniform flow and q is the surface speed of the rotating cylinder.

(See Fluids in the News article titled “A Sailing Ship without Sails,” Section 6.6.3.) Determine the magnitude of the total force developed by the two rotating cylinders on the Flettner “rotorship” due to the Magnus effect. Assume a wind-speed relative to the ship of (a) 10 mph and (b) 30 mph. Each cylinder has a diameter of 9 ft, a length of 50 ft, and rotates at 750 rev/min. Use Eq. 6.124 and calculate the circulation by assuming the air sticks to the rotating cylinders. Note: This calculated force is at right angles to the direction of the wind and it is the component of this force in the direction of motion of the ship that gives the propulsive thrust. Also, due to viscous effects, the actual propulsive thrust will be smaller than that calculated from Eq. 6.124 which is based on inviscid flow theory.

A long porous pipe runs parallel to a horizontal plane surface as shown in Fig. P6.80. The longitudinal axis of the pipe is perpendicular to the plane of the paper. Water flows radially from the pipe at a rate of \(0.5~\pi~ \mathrm{ft^3/s}\) per foot of pipe. Determine the difference in pressure (in \(\mathrm{lb} / \mathrm{ft}^{2}\)) between point B and point A. The flow from the pipe may be approximated by a two-dimensional source. Hint: To develop the stream function or velocity potential for this type of flow, place (symmetrically) another equal source on the other side of the wall. With this combination there is no flow across the x axis, and this axis can be replaced with a solid boundary. This technique is called the method of images.

Typical inviscid flow solutions for flow around bodies indicate that the fluid flows smoothly around the body, even for blunt bodies as shown in Video V6.11. However, experience reveals that due to the presence of viscosity, the main flow may actually separate from the body, creating a wake behind the body. As discussed in a later section (Section 9.2.6), whether or not separation takes place depends on the pressure gradient along the surface of the body, as calculated by inviscid flow theory. If the pressure decreases in the direction of flow (a favorable pressure gradient), no separation will occur. However, if the pressure increases in the direction of flow (an adverse pressure gradient), separation may occur. For the circular cylinder of Fig. P6.81 placed in a uniform stream with velocity, U, determine an expression for the pressure gradient in the direction of flow on the surface of the cylinder. For what range of values for the angle \(\theta\) will an adverse pressure gradient occur?

The velocity field for a two-dimensional source or sink flow is a solution of the Euler equations of motion for inviscid flow. Show that this flow field is also a solution of the Navier–Stokes equations of motion for viscous flow.

The stream function for a certain incompressible, two dimensional flow field is \(\psi=3r^3 \sin 2 \theta + 2 \theta\) Where \(\psi\) is in \(\mathrm{ft^2/s}\) when r is in feet and \(\theta\) in radians. Determine the shearing stress, \(\tau_{r \theta}\), at the point \(r = 2 ft, \theta= \pi/3\) radians if fluid is water.

Determine the shearing stress for an incompressible Newtonian fluid with a velocity distribution of \(\mathbf V=(3xy^2-4x^3) \mathbf{\hat i}+(12x^2y - y^3)\mathbf{\hat j}\).

The two-dimensional velocity field for an incompressible Newtonian fluid is described by the relationship \(\mathbf V=(12xy^2-6x^3) \mathbf{\hat i}+(18x^2y - 4y^3)\mathbf{\hat j}\) where the velocity has units of m/s when x and y are in meters. Determine the stresses \(\sigma_{xx}, \sigma_{yy}\), and \(\tau_{xy}\) at the point x = 0.5 m, y = 1.0 m if pressure at this point is 6 kPa and the fluid is glycerin at \(\mathrm{20~^\circ C}\). Show these stresses on a sketch.

The velocity of a fluid particle moving along a horizontal streamline that coincides with the x axis in a plane, two dimensional, incompressible flow field was experimentally found to be described by the equation \(u=x^2\). Along this streamline determine an expression for (a) the rate of change of the v component of velocity with respect to y, (b) the acceleration of the particle, and (c) the pressure gradient in the x direction. The fluid is Newtonian.

Oil (SAE 30) at \(\mathrm{15.6 ~^\circ C}\) flows steadily between fixed, horizontal, parallel plates. The pressure drop per unit length along the channel is 30 kPa/m, and the distance between the plates is 4 mm. The flow is laminar. Determine: (a) the volume rate of flow (per meter of width), (b) the magnitude and direction of the shearing stress acting on the bottom plate, and (c) the velocity along the centerline of the channel.

Two fixed, horizontal, parallel plates are spaced 0.4 in. apart. A viscous liquid \(\left(\mu=8 \times 10^{-3} \ \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}, S G=0.9\right)\) flows between the plates with a mean velocity of 0.5 ft/s. The flow is laminar. Determine the pressure drop per unit length in the direction of flow. What is the maximum velocity in the channel?

A viscous, incompressible fluid flows between the two infinite, vertical, parallel plates of Fig. P6.89. Determine, by use of the Navier–Stokes equations, an expression for the pressure gradient in the direction of flow. Express your answer in terms of the mean velocity. Assume that the flow is laminar, steady, and uniform.

A fluid of density \(\rho\) flows steadily downward between the two vertical, infinite, parallel plates shown in the figure for Problem 6.89. The flow is fully developed and laminar. Make use of the Navier–Stokes equation to determine the relationship between the discharge and the other parameters involved, for the case in which the change in pressure along the channel is zero.

(See Fluids in the News article titled “10 Tons on 8 psi,” Section 6.9.1.) A massive, precisely machined, 6-ft-diameter granite sphere rests on a 4-ft-diameter cylindrical pedestal as shown in Fig. P6.91. When the pump is turned on and the water pressure within the pedestal reaches 8 psi, the sphere rises off the pedestal, creating a 0.005-in. gap through which the water flows. The sphere can then be rotated about any axis with minimal friction. (a) Estimate the pump flowrate, \(Q_0\), required to accomplish this. Assume the flow in the gap between the sphere and the pedestal is essentially viscous flow between fixed, parallel plates. (b) Describe what would happen if the pump flowrate were increased to \(2Q_0\).

Two horizontal, infinite, parallel plates are spaced a distance b apart. A viscous liquid is contained between the plates. The bottom plate is fixed, and the upper plate moves parallel to the bottom plate with a velocity U. Because of the no-slip boundary condition (see Video V6.12), the liquid motion is caused by the liquid being dragged along by the moving boundary. There is no pressure gradient in the direction of flow. Note that this is a so-called simple Couette flow discussed in Section 6.9.2. (a) Start with the Navier–Stokes equations and determine the velocity distribution between the plates. (b) Determine an expression for the flowrate passing between the plates (for a unit width). Express your answer in terms of b and U.

A layer of viscous liquid of constant thickness (no velocity perpendicular to plate) flows steadily down an infinite, inclined plane. Determine, by means of the Navier–Stokes equations, the relationship between the thickness of the layer and the discharge per unit width. The flow is laminar, and assume air resistance is negligible so that the shearing stress at the free surface is zero.

An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates as is shown in Fig. P6.94. The two plates move in opposite directions with constant velocities, \(U_1\) and \(U_2\), as shown. The pressure gradient in the x direction is zero, and the only body force is due to the fluid weight. Use the Navier–Stokes equations to derive an expression for the velocity distribution between the plates. Assume laminar flow.

Two immiscible, incompressible, viscous fluids having the same densities but different viscosities are contained between two infinite, horizontal, parallel plates (Fig. P6.95). The bottom plate is fixed, and the upper plate moves with a constant velocity U. Determine the velocity at the interface. Express your answer in terms of \(U, \mu_1\), and \(\mu_2\). The motion of the fluid is caused entirely by the movement of the upper plate; that is, there is no pressure gradient in the x direction. The fluid velocity and shearing stress are continuous across the interface between the two fluids. Assume laminar flow.

The viscous, incompressible flow between the parallel plates shown in Fig. P6.96 is caused by both the motion of the bottom plate and a pressure gradient, \(\partial p/\partial x\). As noted in Section 6.9.2, an important dimensionless parameter for this type of problem is \(P=-(b^2/2 ~\mu U)(\partial p / \partial x)\) where \(\mu\) is the fluid viscosity. Make a plot of the dimensionless velocity distribution (similar to that shown in Fig. 6.32b) for P = 3. For this case where does the maximum velocity occur?

A viscous fluid (specific weight = \(80~\mathrm{ lb/ft^3}\); viscosity = \(0.03~\mathrm{ lb \cdot s/ft^2}\)) is contained between two infinite, horizontal parallel plates as shown in Fig. P6.97. The fluid moves between the plates under the action of a pressure gradient, and the upper plate moves with a velocity U while the bottom plate is fixed. A U-tube manometer connected between two points along the bottom indicates a differential reading of 0.1 in. If the upper plate moves with a velocity of 0.02 ft/s, at what distance from the bottom plate does the maximum velocity in the gap between the two plates occur? Assume laminar flow.

An infinitely long, solid, vertical cylinder of radius R is located in an infinite mass of an incompressible fluid. Start with the Navier–Stokes equation in the \(\theta\) direction and derive an expression for the velocity distribution for the steady flow case in which the cylinder is rotating about a fixed axis with a constant angular velocity \(\omega\). You need not consider body forces. Assume that the flow is axisymmetric and the fluid is at rest at infinity

A vertical shaft passes through a bearing and is lubricated with an oil having a viscosity of \(0.2~ \mathrm{N \cdot s/m^2}\) as shown in Fig. P6.99. Assume that the flow characteristics in the gap between the shaft and bearing are the same as those for laminar flow between infinite parallel plates with zero pressure gradient in the direction of flow. Estimate the torque required to overcome viscous resistance when the shaft is turning at 80 rev/min.

A viscous fluid is contained between two long concentric cylinders. The geometry of the system is such that the flow between the cylinders is approximately the same as the laminar flow between two infinite parallel plates. (a) Determine an expression for the torque required to rotate the outer cylinder with an angular velocity \(\omega\). The inner cylinder is fixed. Express your answer in terms of the geometry of the system, the viscosity of the fluid, and the angular velocity. (b) For a small, rectangular element located at the fixed wall, determine an expression for the rate of angular deformation of this element. (See Video V6.3 and Fig. P6.10.)

Oil (SAE 30) flows between parallel plates spaced 5 mm apart. The bottom plate is fixed, but the upper plate moves with a velocity of 0.2 m/s in the positive x direction. The pressure gradient is 60 kPa/m, and it is negative. Compute the velocity at various points across the channel and show the results on a plot. Assume laminar flow.

Ethyl alcohol flows through a horizontal tube having a diameter of 10 mm. If the mean velocity is 0.15 m/s, what is the pressure drop per unit length along the tube? What is the velocity at a distance of 2 mm from the tube axis?

A simple flow system to be used for steady-flow tests consists of a constant head tank connected to a length of 4-mm diameter tubing as shown in Fig. P6.103. The liquid has a viscosity of \(0.015~ \mathrm{N \cdot s/m^2}\) a density of \(1200~ \mathrm{kg/m^3}\), and discharges into the atmosphere with a mean velocity of 2 m/s. (a) Verify that the flow will be laminar. (b) The flow is fully developed in the last 3 m of the tube. What is the pressure at the pressure gage? (c) What is the magnitude of the wall shearing stress, \(\tau_{rz}\), in the fully developed region?

(a) Show that for Poiseuille flow in a tube of radius R the magnitude of the wall shearing stress, \(\tau_{rz}\), can be obtained from the relationship \(\left|\left(\tau_{r z}\right)_{\text {wall }}\right|=\frac{4 \mu Q}{\pi R^{3}}\) for a Newtonian fluid of viscosity \(\mu\). The volume rate of flow is Q. (b) Determine the magnitude of the wall shearing stress for a fluid having a viscosity of \(0.004~\mathrm{ N \cdot s/m^2}\) flowing with an average velocity of 130 mm/s in a 2-mm-diameter tube.

An infinitely long, solid, vertical cylinder of radius R is located in an infinite mass of an incompressible fluid. Start with the Navier–Stokes equation in the \(\theta\) direction and derive an expression for the velocity distribution for the steady-flow case in which the cylinder is rotating about a fixed axis with a constant angular velocity \(\omega\). You need not consider body forces. Assume that the flow is axisymmetric and the fluid is at rest at infinity.

As is shown by Eq. 6.150, the pressure gradient for laminar flow through a tube of constant radius is given by the expression \(\frac{\partial p}{\partial z}=-\frac{8 \mu Q}{\pi R^{4}}\) For a tube whose radius is changing very gradually, such as the one illustrated in Fig. P6.106, it is expected that this equation can be used to approximate the pressure change along the tube if the actual radius, R(z), is used at each cross section. The following measurements were obtained along a particular tube. Compare the pressure drop over the length \(\ell\) for this nonuniform tube with one having the constant radius \(R_o\). Hint: To solve this problem you will need to numerically integrate the equation for the pressure gradient given previously.

A liquid (viscosity = \(0.002~ \mathrm{N \cdot s/m^2}\); density = \(1000~ \mathrm{kg/m^3}\)) is forced through the circular tube shown in Fig. P6.107. A differential manometer is connected to the tube as shown to measure the pressure drop along the tube. When the differential reading, \(\Delta h\), is 9 mm, what is the mean velocity in the tube?

An incompressible Newtonian fluid flows steadily between two infinitely long, concentric cylinders as shown in Fig. P6.108. The outer cylinder is fixed, but the inner cylinder moves with a longitudinal velocity \(V_0\) as shown. The pressure gradient in the axial direction is \(- \Delta p/ \ell\). For what value of \(V_0\) will the drag on the inner cylinder be zero? Assume that the flow is laminar, axisymmetric, and fully developed.

A viscous fluid is contained between two infinitely long, vertical, concentric cylinders. The outer cylinder has a radius \(r_0\) and rotates with an angular velocity \(\omega\). . The inner cylinder is fixed and has a radius \(r_i\). Make use of the Navier–Stokes equations to obtain an exact solution for the velocity distribution in the gap. Assume that the flow in the gap is axisymmetric (neither velocity nor pressure are functions of angular position \(\theta\) within the gap) and that there are no velocity components other than the tangential component. The only body force is the weight.

For flow between concentric cylinders, with the outer cylinder rotating at an angular velocity \(\omega\) and the inner cylinder fixed, it is commonly assumed that the tangential velocity \(v_\theta\) distribution in the gap between the cylinders is linear. Based on the exact solution to this problem (see Problem 6.109) the velocity distribution in the gap is not linear. For an outer cylinder with radius \(r_o=2.00\) in. and an inner cylinder with radius \(r_i=1.80\) in., show, with the aid of a plot, how the dimensionless velocity distribution, \(v_\theta / r_o \omega\),varies with the dimensionless radial position, \(r/r_o\), for the exact and approximate solutions.

A viscous liquid \(\left(\mu=0.012~ \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}, \rho=1.79~ \mathrm{slugs} / \mathrm{ft}^{3}\right)\) flows through the annular space between two horizontal, fixed, concentric cylinders. If the radius of the inner cylinder is 1.5 in. and the radius of the outer cylinder is 2.5 in., what is the pressure drop along the axis of the annulus per foot when the volume flowrate is \(0.14 ~\mathrm{ft}^{3} / \mathrm{s}\)?

Show how Eq. 6.155 is obtained.

A wire of diameter d is stretched along the centerline of a pipe of diameter D. For a given pressure drop per unit length of pipe, by how much does the presence of the wire reduce the flowrate if (a) d/D = 0.1; (b) d/D = 0.01?

Obtain a photograph/image of a situation in which CFD has been used to solve a fluid flow problem. Print this photo and write a brief paragraph that describes the situation involved.

What sometimes appear at first glance to be simple fluid flows can contain subtle, complex fluid mechanics. One such example is the stirring of tea leaves in a teacup. Obtain information about “Einstein’s tea leaves” and investigate some of the complex fluid motions interacting with the leaves. Summarize your findings in a brief report.

Computational fluid dynamics (CFD) has moved from a research tool to a design tool for engineering. Initially, much of the work in CFD was focused in the aerospace industry, but now has expanded into other areas. Obtain information on what other industries (e.g., automotive) make use of CFD in their engineering design. Summarize your findings in a brief report.