Questions and Answers of Introduction To Fluid Mechanics

A crude approximation for the \(x\) component of velocity in an incompressible laminar boundary layer is a linear variation from \(u=0\) at the surface \((y=0)\) to the freestream velocity, \(U\), at
A useful approximation for the \(x\) component of velocity in an incompressible laminar boundary layer is a parabolic variation from \(u=0\) at the surface \((y=0)\) to the freestream velocity,
A useful approximation for the \(x\) component of velocity in an incompressible laminar boundary layer is a cubic variation from \(u=0\) at the surface \((y=0)\) to the freestream velocity, \(U\), at
For a flow in the \(x y\) plane, the \(x\) component of velocity is given by \(u=A x^{2} y^{2}\), where \(A=0.3 \mathrm{~m}^{-3} \cdot \mathrm{s}^{-1}\), and \(x\) and \(y\) are measured in meters.
Consider a water stream from a jet of an oscillating lawn sprinkler. Describe the corresponding pathline and streakline.
Which of the following sets of equations represent possible incompressible flow cases?(a) \(V_{r}=U \cos \theta ; V_{\theta}=-U \sin \theta\)(b) \(V_{r}=-q / 2 \pi r ; V_{\theta}=K / 2 \pi r\)(c)
For an incompressible flow in the \(r \theta\) plane, the \(r\) component of velocity is given as \(V_{r}=U \cos \theta\).(a) Determine a possible \(\theta\) component of velocity.(b) How many
A viscous liquid is sheared between two parallel disks of radius \(R\), one of which rotates while the other is fixed. The velocity field is purely tangential, and the velocity varies linearly with
A velocity field in cylindrical coordinates is given as \(\vec{V}=\hat{\mathrm{e}}_{r} A / r+\hat{\mathrm{e}}_{\theta} B / r\), where \(A\) and \(B\) are constants with dimensions of \(\mathrm{m}^{2}
Determine the family of stream functions \(\psi\) that will yield the velocity field \(\vec{V}=2 y(2 x+1) \hat{i}+\left[x(x+1)-2 y^{2}\right] \hat{j}\).
The stream function for a certain incompressible flow field is given by the expression \(\psi=-U r \sin \theta+q \theta / 2 \pi\). Obtain an expression for the velocity field. Find the stagnation
Determine the stream functions for the following flow fields. For the value of \(\psi=2\), plot the streamline in the region between \(x=-1\) and \(x=1\).(a) \(u=4 ; v=3\)(b) \(u=4 y ; v=0\)(c) \(u=4
Determine the stream function for the steady incompressible flow between parallel plates. The velocity profile is parabolic and given by \(u=u_{c}+a y^{2}\), where \(u_{c}\) is the centerline
An incompressible frictionless flow field is specified by the stream function \(\psi=-5 A x-2 A y\), where \(A=2 \mathrm{~m} / \mathrm{s}\), and \(x\) and \(y\) are coordinates in meters.(a) Sketch
A parabolic velocity profile was used to model flow in a laminar incompressible boundary layer in Problem 5.11. Derive the stream function for this flow field. Locate streamlines at one-quarter and
A flow field is characterized by the stream function \(\psi=3 x^{2} y-y^{3}\). Demonstrate that the flow field represents a twodimensional incompressible flow. Show that the magnitude of the velocity
A flow field is characterized by the stream function \(\psi=x y\). Plot sufficient streamlines to represent the flow field. Determine the location of any stagnation points. Give at least two possible
A cubic velocity profile was used to model flow in a laminar incompressible boundary layer in Problem 5.12. Derive the stream function for this flow field. Locate streamlines at one-quarter and
A flow field is characterized by the stream function\[\psi=\frac{1}{2 \pi}\left(\tan ^{-1} \frac{y-a}{x}-\tan ^{-1} \frac{y+a}{x}\right)-\frac{1}{2 \pi} \ln \sqrt{x^{2}+y^{2}}\]Locate the stagnation
In a parallel one-dimensional flow in the positive \(x\) direction, the velocity varies linearly from zero at \(y=0\) to \(30 \mathrm{~m} / \mathrm{s}\) at \(y=1.5 \mathrm{~m}\). Determine an
Consider the flow field given by \(\vec{V}=x y^{2} \hat{i}-\frac{1}{3} y^{3} \hat{j}+x y \hat{k}\). Determine (a) the number of dimensions of the flow, (b) if it is a possible incompressible flow,
Consider the flow field given by \(\vec{V}=a x^{2} y \hat{i}-b y \hat{j}+c z^{2} \hat{k}\), where \(a=2 \mathrm{~m}^{-2} \cdot \mathrm{s}^{-1}, b=2 \mathrm{~s}^{-1}\), and \(c=1 \mathrm{~m}^{-1}
The velocity field within a laminar boundary layer is approximated by the expression\[\vec{V}=\frac{A U y}{x^{1 / 2}} \hat{i}+\frac{A U y^{2}}{4 x^{3 / 2}} \hat{j}\]In this expression, \(A=141
A velocity field is given by \(\vec{V}=10 t \hat{i}-\frac{10}{t^{3}} \hat{j}\). Show that the flow field is a two-dimensional flow and determine the acceleration as a function of time.
The \(y\) component of velocity in a two-dimensional, incompressible flow field is given by \(v=-A x y\), where \(v\) is in \(\mathrm{m} / \mathrm{s}, x\) and \(y\) are in meters, and \(A\) is a
A \(4 \mathrm{~m}\) diameter tank is filled with water and then rotated at a rate of \(\omega=2 \pi\left(1-\mathrm{e}^{-t}\right) \mathrm{rad} / \mathrm{s}\). At the tank walls, viscosity prevents
An incompressible liquid with negligible viscosity flows steadily through a horizontal pipe of constant diameter. In a porous section of length \(L=0.3 \mathrm{~m}\), liquid is removed at a constant
Sketch the following flow fields and derive general expressions for the acceleration:(a) \(u=2 x y ; v=-x^{2} y\)(b) \(u=y-x+x^{2} ; v=x+y-2 x y\)(c) \(u=x^{2} t+2 y ; v=2 x-y t^{2}\)(d)
Consider the low-speed flow of air between parallel disks as shown. Assume that the flow is incompressible and inviscid, and that the velocity is purely radial and uniform at any section. The flow
As part of a pollution study, a model concentration \(c\) as a function of position \(x\) has been developed,\[c(x)=A\left(e^{-x / 2 a}-e^{-x / a}\right)\]where \(A=3 \times 10^{-5} \mathrm{ppm}\)
As an aircraft flies through a cold front, an onboard instrument indicates that ambient temperature drops at the rate of \(0.7^{\circ} \mathrm{F} / \mathrm{min}\).Other instruments show an air speed
Wave flow of an incompressible fluid into a solid surface follows a sinusoidal pattern. Flow is axisymmetric about the \(z\) axis, which is normal to the surface. The \(z\) component of the flow
A steady, two-dimensional velocity field is given by \(\vec{V}=A x \hat{i}-A y \hat{j}\), where \(A=1 \mathrm{~s}^{-1}\). Show that the streamlines for this flow are rectangular hyperbolas, \(x
A velocity field is represented by the expression \(\vec{V}=(A x-B)\) \(\hat{i}+C y \hat{j}+D t \hat{k}\) where \(A=0.2 \mathrm{~s}^{-1}, B=0.6 \mathrm{~m} \cdot \mathrm{s}^{-1}, D=5 \mathrm{~m}
A parabolic approximate velocity profile was used in Problem 5.11 to model flow in a laminar incompressible boundary layer on a flat plate. For this profile, find the \(x\) component of acceleration,
A cubic approximate velocity profile was used in Problem 5.12 to model flow in a laminar incompressible boundary layer on a flat plate. For this profile, obtain an expression for the \(x\) and \(y\)
The velocity field for steady inviscid flow from left to right over a circular cylinder, of radius \(R\) is given by\[\vec{V}=U \cos \theta\left[1-\left(\frac{R}{r}\right)^{2}\right] \hat{e}_{r}-U
Consider the incompressible flow of a fluid through a nozzle as shown. The area of the nozzle is given by \(A=A_{0}(1-b x)\) and the inlet velocity varies according to \(U=U_{0}(0.5+0.5 \cos \omega
Consider the one-dimensional, incompressible flow through the circular channel shown. The velocity at section (1) is given by \(U=U_{0}+U_{1} \sin \omega t\), where \(U_{0}=20 \mathrm{~m} /
Expand \((\vec{V} \cdot abla) \vec{V}\) in cylindrical coordinates by direct substitution of the velocity vector to obtain the convective acceleration of a fluid particle. Verify the results given in
Determine the velocity potential for(a) a flow field characterized by the stream function \(\psi=3 x^{2} y-y^{3}\).(b) a flow field characterized by the stream function \(\psi=x y\).
Determine whether the following flow fields are irrotational.(a) \(u=2 x y ; v=-x^{2} y\)(b) \(u=y-x+x^{2} ; v=x+y-2 x y\)(c) \(u=x^{2} t+2 y ; v=2 x-y t^{2}\)(d) \(u=-x^{2}-y^{2}-x y t ;
The velocity profile for steady flow between parallel is parabolic and given by \(u=u_{c}+a y^{2}\), where \(u_{c}\) is the centerline velocity and \(y\) is the distance measured from the centerline.
Consider the velocity field for flow in a rectangular "corner," \(\vec{V}=A x \hat{i}-A y \hat{i}\), with \(A=0.3 \mathrm{~s}^{-1}\), as in Example 5.8. Evaluate the circulation about the unit square
Consider the two-dimensional flow field in which \(u=A x^{2}\) and \(v=B x y\), where \(A=1 / 2 \mathrm{ft}^{-1} \cdot \mathrm{s}^{-1}, B=-1 \mathrm{ft}^{-1} \cdot \mathrm{s}^{-1}\), and the
Consider a flow field represented by the stream function \(\psi=3 x^{5} y-10 x^{3} y^{3}+3 x y^{5}\). Is this a possible two-dimensional incompressible flow? Is the flow irrotational?
Fluid passes through the set of thin, closely space blades at a velocity of \(3 \mathrm{~m} / \mathrm{s}\). Determine the circulation for the flow. P5.56 D= 0.6 m 30 V
A two-dimensional flow field is characterized as \(u=A x^{2}\) and \(u=B x y\) where \(A=\frac{1}{2} \mathrm{~m}^{-1} \mathrm{~s}^{-1}\) and \(B=-1 \mathrm{~m}^{-1} \mathrm{~s}^{-1}\), and \(x\) and
A flow field is represented by the stream function \(\psi=x^{4}-2 x^{3} y+2 x y^{3}-y^{4}\). Is this a possible two-dimensional flow? Is the flow irrotational?
Consider a velocity field for motion parallel to the \(x\) axis with constant shear. The shear rate is \(d u / d y=A\), where \(A=0.1 \mathrm{~s}^{-1}\). Obtain an expression for the velocity field,
Consider the flow field represented by the stream function \(\psi=A x y+A y^{2}\), where \(A=1 \mathrm{~s}^{-1}\). Show that this represents a possible incompressible flow field. Evaluate the
Consider the velocity field given by \(\vec{V}=A x^{2} \hat{i}+B x y \hat{j}\), where \(A=1 \mathrm{ft}^{-1} \cdot \mathrm{s}^{-1}, B=-2 \mathrm{ft}^{-1} \cdot \mathrm{s}^{-1}\), and the coordinates
Consider again the viscometric flow of Example 5.7. Evaluate the average rate of rotation of a pair of perpendicular line segments oriented at \(\pm 45^{\circ}\) from the \(x\) axis. Show that this
The velocity field near the core of a tornado can be approximated as\[\vec{V}=-\frac{q}{2 \pi r} \hat{e}_{r}+\frac{K}{2 \pi r} \hat{e}_{\theta}\]Is this an irrotational flow field? Obtain the stream
A velocity field is given by \(\vec{V}=2 \hat{i}-4 x \hat{j} \mathrm{~m} / \mathrm{s}\). Determine an equation for the streamline. Calculate the vorticity of the flow.
Consider the pressure-driven flow between stationary parallel plates separated by distance \(2 b\). Coordinate \(y\) is measured from the channel centerline. The velocity field is given by \(u=\)
Consider a steady, laminar, fully developed, incompressible flow between two infinite plates, as shown. The flow is due to the motion of the left plate as well a pressure gradient that is applied in
Assume the liquid film in Example 5.9 is not isothermal, but instead has the following distribution:\[T(y)=T_{0}+\left(T_{w}-T_{0}\right)\left(1-\frac{y}{h}\right)\]where \(T_{0}\) and \(T_{w}\) are,
Consider a steady, laminar, fully developed incompressible flow between two infinite parallel plates as shown. The flow is due to a pressure gradient applied in the \(x\) direction. Given that
Consider a steady, laminar, fully developed incompressible flow between two infinite parallel plates separated by a distance \(2 h\) as shown. The top plate moves with a velocity \(V_{0}\). Derive an
A linear velocity profile was used to model flow in a laminar incompressible boundary layer in Problem 5.10. Express the rotation of a fluid particle. Locate the maximum rate of rotation. Express the
A cylinder of radius \(r_{\mathrm{i}}\) rotates at a speed \(\omega\) coaxially inside a fixed cylinder of radius \(r_{0}\). A viscous fluid fills the space between the two cylinders. Determine the
The velocity profile for fully developed laminar flow in a circular tube is \(u=u_{\max }\left[1-(r / R)^{2}\right]\). Obtain an expression for the shear force per unit volume in the \(x\) direction
Assume the liquid film in Example 5.9 is horizontal (i.e., \(\theta=0^{\circ}\) ) and that the flow is driven by a constant shear stress on the top surface \((y=h), \tau_{y x}=C\). Assume that the
The common thermal polymerase chain reaction (PCR) process requires the cycling of reagents through three distinct temperatures for denaturation \(\left(90-94^{\circ} \mathrm{C}\right)\), annealing
A tank contains water \(\left(20^{\circ} \mathrm{C}\right)\) at an initial depth \(y_{0}=1 \mathrm{~m}\). The tank diameter is \(D=250 \mathrm{~mm}\) and a tube of diameter \(d=3 \mathrm{~mm}\) and
Use Excel to generate the solution of Eq. 5.31 for \(m=1\) shown in Fig. 5.18 To do so, you need to learn how to perform linear algebra in Excel. For example, for \(N=4\) you will end up with the
For a small spherical particle of styrofoam (density \(=16 \mathrm{~kg} / \mathrm{m}^{3}\) ) with a diameter of \(5 \mathrm{~mm}\) falling in air, the drag is given by \(F_{\mathrm{D}}=3 \pi \mu V
Following the steps to convert the differential equation Eq. 5.31 (for \(m=1\) ) into a difference equation (for example, Eq. 5.37 for \(N=4\) ), solve\[\frac{d u}{d x}+u=2 x^{2}+x \quad 0 \leq x
Use Excel to generate the progression to an iterative solution Eq. 5.31 for \(m=2\), as illustrated in Fig. 5.21Data From Equation 5.31Data From Fig. 5.21 du +um=0; 0x1; u(0)=1 dx
Use Excel to generate the solutions of Eq. 5.31 for \(m=-1\), with \(u(0)=3\), using 4 and 16 points over the interval from \(x=0\) to \(x=3\), with sufficient iterations, and compare to the exact
An incompressible frictionless flow field is given by \(\vec{V}=(A x+B y) \hat{i}+(B x-A y) \hat{j}\), where \(A=2 \mathrm{~s}^{-1}\) and \(B=2 s^{-1}\), and the coordinates are measured in meters.
A velocity field in a fluid with density of \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) is given by \(\vec{V}=(-A x+B y) t \hat{i}+(A y+B x) t \hat{j}\), where \(A=2 \mathrm{~s}^{-2}\) and \(B=1
The \(x\) component of velocity in an incompressible flow field is given by \(u=A x\), where \(A=2 \mathrm{~s}^{-1}\) and the coordinates are measured in meters. The pressure at point \((x,
Consider the flow field with the velocity given by \(\vec{V}=3 \hat{i}+5 t \hat{j}+\) \(8 t^{2} \hat{k}\), where the velocity is in \(\mathrm{m} / \mathrm{s}\) and \(t\) is in seconds. The fluid
Consider the flow field with the velocity given by \(\vec{V}=4 y \hat{i}+3 x \hat{j}\), where the velocity is in \(\mathrm{ft} / \mathrm{s}\) and the coordinates are in feet. The fluid density is
The velocity field for a plane source located distance \(h=1 \mathrm{~m}\) above an infinite wall aligned along the \(x\) axis is given by\[\begin{aligned} \vec{V}= & \frac{q}{2
In a two-dimensional frictionless, incompressible \(\left(ho=1500 \mathrm{~kg} / \mathrm{m}^{3}\right)\) flow, the velocity field in meters per second is given by \(\vec{V}=(A x+B y) \hat{i}+(B x-A
Consider a two-dimensional incompressible flow flowing downward against a plate. The velocity is given by \(\vec{V}=A x \hat{i}-A y \hat{j}\), where \(A=2 \mathrm{~s}^{-1}\) and \(x\) and \(y\) are
An incompressible liquid with a density of \(900 \mathrm{~kg} / \mathrm{m}^{3}\) and negligible viscosity flows steadily through a horizontal pipe of constant diameter. In a porous section of length
Consider a flow of water in pipe. What is the pressure gradient required to accelerate the water at \(20 \mathrm{ft} / \mathrm{s}^{2}\) if the pipe is(a) horizontal,(b) vertical with the water
The velocity field for a plane vortex sink is given by \(\vec{V}=(-q / 2 \pi r) \hat{e}_{r}+(K / 2 \pi r) \hat{e}_{\theta}\), where \(q=2 \mathrm{~m}^{3} / \mathrm{s} / \mathrm{m}\) and \(K=1\)
An incompressible liquid with negligible viscosity and density \(ho=1.75 \mathrm{slug} / \mathrm{ft}^{3}\) flows steadily through a horizontal pipe. The pipe cross-section area linearly varies from
Consider water flowing in a circular section of a twodimensional channel. Assume the velocity is uniform across the channel at \(12 \mathrm{~m} / \mathrm{s}\). The pressure is \(120 \mathrm{kPa}\) at
Consider a tornado as air moving in a circular pattern in the horizontal plane. If the wind speed is \(200 \mathrm{mph}\) and the diameter of the tornado is \(200 \mathrm{ft}\), determine the radial
A nozzle for an incompressible, inviscid fluid of density \(ho=1000 \mathrm{~kg} / \mathrm{m}^{3}\) consists of a horizontal converging section of pipe. At the inlet the diameter is \(D_{i}=100
A diffuser for an incompressible, inviscid fluid of density \(ho=1000 \mathrm{~kg} / \mathrm{m}^{3}\) consists of a horizontal diverging section of pipe. At the inlet the diameter is \(D_{i}=0.25
A liquid layer separates two plane surfaces as shown. The lower surface is stationary; the upper surface moves downward at constant speed \(V\). The moving surface has width \(w\), perpendicular to
Consider Problem 6.15 with the nozzle directed upward. Assuming that the flow is uniform at each section, derive and plot the acceleration of a fluid particle for an inlet speed of \(V_{i}=2
Consider Problem 6.16 with the diffuser directed upward. Assuming that the flow is uniform at each section, derive and plot the acceleration of a fluid particle for an inlet speed of \(V_{i}=12
A rectangular computer chip floats on a thin layer of air, \(h=0.5 \mathrm{~mm}\) thick, above a porous surface. The chip width is \(b=40 \mathrm{~mm}\), as shown. Its length, \(L\), is very long in
Heavy weights can be moved with relative ease on air cushions by using a load pallet as shown. Air is supplied from the plenum through porous surface \(A B\). It enters the gap vertically at uniform
The \(y\) component of velocity in a two-dimensional incompressible flow field is given by \(v=-A x y\), where \(v\) is in \(\mathrm{m} / \mathrm{s}\), the coordinates are measured in meters, and
The velocity field for a plane doublet is given in Table 6.2. Find an expression for the pressure gradient at any point \((r, \theta)\).Data From Table 6.2 Table 6.2 Elementary Plane Flows Uniform
To model the velocity distribution in the curved inlet section of a water channel, the radius of curvature of the streamlines is expressed as \(R=L R_{0} / 2 y\). As an approximation, assume the
Repeat Example 6.1, but with the somewhat more realistic assumption that the flow is similar to a free vortex (irrotational) profile, \(V_{\theta}=c / r\) (where \(c\) is a constant), as shown in
Using the analyses of Example 6.1 and Problem 6.25, plot the discrepancy (percent) between the flow rates obtained from assuming uniform flow and the free vortex (irrotational) profile as a function
The \(x\) component of velocity in a two-dimensional incompressible flow field is given by \(u=-\Lambda\left(x^{2}-y^{2}\right) /\left(x^{2}+y^{2}\right)^{2}\), where \(u\) is in \(\mathrm{m} /
Water flows at a speed of \(25 \mathrm{ft} / \mathrm{s}\). Calculate the dynamic pressure of this flow. Express your answer in inches of mercury.
Plot the speed of air versus the dynamic pressure (in millimeters of mercury), up to a dynamic pressure of \(250 \mathrm{~mm} \mathrm{Hg}\).

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