Fox And McDonald's Introduction To Fluid Mechanics 9th Edition Philip J. Pritchard, John W. Mitchell - Solutions

When a valve is closed suddenly in a pipe with flowing water, a water hammer pressure wave is set up. The very high pressures generated by such waves can damage the pipe. The maximum pressure, \(p_{\text {max }}\), generated by water hammer is a function of liquid density, \(ho\), initial flow
An airship is to operate at \(20 \mathrm{~m} / \mathrm{s}\) in air at standard conditions. A model is constructed to 1:20 scale and tested in a wind tunnel at the same air temperature to determine drag. What criterion should be considered to obtain dynamic similarity? If the model is tested at \(75
An airplane wing of \(3 \mathrm{~m}\) chord length moves through still air at \(15^{\circ} \mathrm{C}\) and \(101.3 \mathrm{kPa}\) at a speed of \(320 \mathrm{~km} / \mathrm{h}\). A 1:20 scale model of this wing is placed in a wind tunnel, and dynamic similarity between model and prototype is
A flat plate \(1.5 \mathrm{~m}\) long and \(0.3 \mathrm{~m}\) wide is towed at \(3 \mathrm{~m} / \mathrm{s}\) in a towing basin containing water at \(20^{\circ} \mathrm{C}\), and the drag force is observed to be \(14 \mathrm{~N}\). Calculate the dimensions of a similar plate that will yield
This 1:12 pump model using water at \(15^{\circ} \mathrm{C}\) simulates a prototype for pumping oil of specific gravity 0.90 . The input to the model is \(0.522 \mathrm{~kW}\). Calculate the viscosity of the oil and the prototype power for complete dynamic similarity between model and prototype. 3
An ocean-going vessel is to be powered by a rotating circular cylinder. Model tests are planned to estimate the power required to rotate the prototype cylinder. A dimensional analysis is needed to scale the power requirements from model test results to the prototype. List the parameters that should
On a cruise ship, passengers complain about the noise emanating from the ship's propellers (probably due to turbulent flow effects between the propeller and the ship). You have been hired to find out the source of this noise. You will study the flow pattern around the propellers and have decided to
A 1:3 scale model of a torpedo is tested in a wind tunnel to determine the drag force. The prototype operates in water, has \(533 \mathrm{~mm}\) diameter, and is \(6.7 \mathrm{~m}\) long. The desired operating speed of the prototype is \(28 \mathrm{~m} / \mathrm{s}\). To avoid compressibility
A flow rate of \(0.18 \mathrm{~m}^{3} / \mathrm{s}\) of water at \(20^{\circ} \mathrm{C}\) discharges from a \(0.3 \mathrm{~m}\) pipe through a \(0.15 \mathrm{~m}\) nozzle into the atmosphere. The axial force component exerted by water on the nozzle is \(3 \mathrm{kN}\). If frictional effects may
A force of \(9 \mathrm{~N}\) is required to tow a 1:50 ship model at \(4.8 \mathrm{~km} / \mathrm{h}\). Assuming the same water in towing basin and sea, calculate the corresponding speed and force in the prototype if the flow is dominated by:(a) density and gravity,(b) density and surface
An airplane wing, with chord length of \(1.5 \mathrm{~m}\) and span of \(9 \mathrm{~m}\), is designed to move through standard air at a speed of \(7.5 \mathrm{~m} / \mathrm{s}\). A 1:10 scale model of this wing is to be tested in a water tunnel. What speed is necessary in the water tunnel to
A water pump with impeller diameter of 24 in. is to be designed to move \(15 \mathrm{ft}^{3} / \mathrm{s}\) when running at \(750 \mathrm{rpm}\). Testing is performed on a 1:4 scale model running at \(2400 \mathrm{rpm}\) using air \(\left(68^{\circ} \mathrm{F}\right)\) as the fluid. For similar
A model hydrofoil is to be tested at 1:20 scale. The test speed is chosen to duplicate the Froude number corresponding to the 60-knot prototype speed. To model cavitation correctly, the cavitation number also must be duplicated. At what ambient pressure must the test be run? Water in the model test
A ship \(120 \mathrm{~m}\) long moves through freshwater at \(15^{\circ} \mathrm{C}\) at \(32 \mathrm{~km} / \mathrm{h}\). A 1:100 model of this ship is to be tested in a towing basin containing a liquid of specilic gravity 0.92 . What viscosity must this liquid have for both Reynolds' and Froude's
A 1:30 scale model of a cavitating overflow structure is to be tested in a vacuum tank wherein the pressure is maintained at 2.0 psia. The prototype liquid is water at \(70^{\circ} \mathrm{F}\). The barometric pressure on the prototype is \(14.5 \mathrm{psia}\). If the liquid to be used in the
In some speed ranges, vortices are shed from the rear of bluff cylinders placed across a flow. The vortices alternately leave the top and bottom of the cylinder, as shown, causing an alternating force normal to the freestream velocity. The vortex shedding frequency, \(f\), is thought to depend on
A 1:8 scale model of a tractor-trailer rig is tested in a pressurized wind tunnel. The rig width, height, and length are \(W=0.305 \mathrm{~m}\), \(H=0.476 \mathrm{~m}\), and \(L=2.48 \mathrm{~m}\), respectively. At wind speed \(V=75.0 \mathrm{~m} / \mathrm{s}\), the model drag force is \(F_{D}=128
On a cruise ship, passengers complain about the amount of smoke that becomes entrained behind the cylindrical smoke stack. You have been hired to study the flow pattern around the stack, and have decided to use a 1:15 scale model of the \(15-\mathrm{ft}\) smoke stack. What range of wind tunnel
When a sphere of \(0.25 \mathrm{~mm}\) diameter and specific gravity 5.54 is dropped in water at \(25^{\circ} \mathrm{C}\) it will attain a constant velocity of \(0.07 \mathrm{~m} / \mathrm{s}\). What specific gravity must a \(2.5 \mathrm{~mm}\) sphere have so that when it is dropped in crude oil
The flow about a \(150 \mathrm{~mm}\) artillery projectile which travels at \(600 \mathrm{~m} / \mathrm{s}\) through still air at \(30^{\circ} \mathrm{C}\) and absolute pressure \(101.4 \mathrm{kPa}\) is to be modeled in a high-speed wind tunnel with a 1:6 model. If the wind tunnel air has a
Your favorite professor likes mountain climbing, so there is always a possibility that the professor may fall into a crevasse in some glacier. If that happened today, and the professor was trapped in a slowly moving glacier, you are curious to know whether the professor would reappear at the
A 1:50-scale model of a submarine is to be tested in a towing tank under two conditions: motion at the free surface and motion far below the surface. The tests are performed in freshwater. On the surface, the submarine cruises at 24 knots. At what speed should the model be towed to ensure dynamic
Consider water flow around a circular cylinder, of diameter \(D\) and length \(l\). In addition to geometry, the drag force is known to depend on liquid speed, \(V\), density, \(ho\), and viscosity, \(\mu\). Express drag force, \(F_{D}\), in dimensionless form as a function of all relevant
A 1:10 scale model of a tractor-trailer rig is tested in a wind tunnel. The model frontal area is \(A_{m}=0.1 \mathrm{~m}^{2}\). When tested at \(V_{m}=75 \mathrm{~m} / \mathrm{s}\) in standard air, the measured drag force is \(F_{D}=350 \mathrm{~N}\). Evaluate the drag coefficient for the model
The power, \(\mathscr{P}\), required to drive a fan is assumed to depend on fluid density \(ho\), volume flow rate \(Q\), impeller diameter \(D\), and angular speed \(\omega\). If a fan with \(D_{1}=8\) in. delivers \(Q_{1}=15 \mathrm{ft}^{3} / \mathrm{s}\) of air at \(\omega_{1}=2500
Over a certain range of air speeds, \(V\), the lift, \(F_{L}\), produced by a model of a complete aircraft in a wind tunnel depends on the air speed, air density, \(ho\), and a characteristic length (the wing base chord length, \(c=150 \mathrm{~mm}\) ). The following experimental data is obtained
The pressure rise, \(\Delta p\), of a liquid flowing steadily through a centrifugal pump depends on pump diameter \(D\), angular speed of the rotor \(\omega\), volume flow rate \(Q\), and density \(ho\). The table gives data for the prototype and for a geometrically similar model pump. For
An axial-flow pump is required to deliver \(0.75 \mathrm{~m}^{3} / \mathrm{s}\) of water at a head of \(15 \mathrm{~J} / \mathrm{kg}\). The diameter of the rotor is \(0.25 \mathrm{~m}\), and it is to be driven at \(500 \mathrm{rpm}\). The prototype is to be modeled on a small test apparatus having
A model propeller \(1 \mathrm{~m}\) in diameter is tested in a wind tunnel. Air approaches the propeller at \(50 \mathrm{~m} / \mathrm{s}\) when it rotates at \(1800 \mathrm{rpm}\). The thrust and torque measured under these conditions are \(100 \mathrm{~N}\) and \(10 \mathrm{~N} \cdot
Consider Problem 7.38. Experience shows that for ship-size propellers, viscous effects on scaling are small. Also, when cavitation is not present, the nondimensional parameter containing pressure can be ignored. Assume that torque, \(T\), and power, \(\mathscr{P}\), depend on the same parameters as
Closed-circuit wind tunnels can produce higher speeds than open-circuit tunnels with the same power input because energy is recovered in the diffuser downstream from the test section. The kinetic energy ratio is a figure of merit defined as the ratio of the kinetic energy flux in the test section
A 1:16 model of a bus is tested in a wind tunnel in standard air. The model is \(152 \mathrm{~mm}\) wide, \(200 \mathrm{~mm}\) high, and \(762 \mathrm{~mm}\) long. The measured drag force at \(26.5 \mathrm{~m} / \mathrm{s}\) wind speed is \(6.09 \mathrm{~N}\). The longitudinal pressure gradient in
The propagation speed of small-amplitude surface waves in a region of uniform depth is given by\[c^{2}=\left(\frac{2 \pi \sigma}{\lambda ho}+\frac{g \lambda}{2 \pi}\right) \tanh \frac{2 \pi h}{\lambda}\]where \(h\) is depth of the undisturbed liquid and \(\lambda\) is wavelength. Explore the
What is the ratio between the viscosities of air and water at \(10^{\circ} \mathrm{C}\) ? What is the ratio between their kinematic viscosities at this temperature and standard barometric pressure?
A cylindrical can \(76 \mathrm{~mm}\) in diameter and \(152 \mathrm{~mm}\) high, weighing \(1.11 \mathrm{~N}\), contains water to a depth of \(76 \mathrm{~mm}\). When this can is placed in water, how deep will it sink?
If the 10-ft-long box is floating on the oil-water system, calculate how much the box and its contents must weigh. -8 ft- 2 ft 1 ft P3.71 Oil (SG= 0.80) Water
The mean free path \(\lambda\) of a molecule of gas is the average distance it travels before collision with another molecule. It is given by\[\lambda=C \frac{m}{ho d^{2}}\]where \(m\) and \(d\) are the molecule's mass and diameter, respectively, and \(ho\) is the gas density. What are the
A rectangular gate (width \(w=2 \mathrm{~m}\) ) is hinged as shown, with a stop on the lower edge. At what depth \(H\) will the gate tip? Water H 0.55 m 0.45 m P3.46 Hinge Stop
Gates in the Poe Lock at Sault Ste. Marie, Michigan, close a channel \(W=34 \mathrm{~m}\) wide, \(L=360 \mathrm{~m}\) long, and \(D=10 \mathrm{~m}\) deep. The geometry of one pair of gates is shown; each gate is hinged at the channel wall. When closed, the gate edges are forced together at the
A plane gate of uniform thickness holds back a depth of water as shown. Find the minimum weight needed to keep the gate closed. 0=30 L=3m Water w=2m P3.45
Obtain an expression for the kinetic energy flux, \(\int\left(V^{2} / 2\right)\) \(ho \vec{V} \cdot d \vec{A}\), through cross section (1) of the control volume shown. x CV Width = w P4.12 h V
The velocity distribution for laminar flow in a long circular tube of radius \(R\) is given by the one-dimensional expression,\[\vec{V}=u \hat{i}=u_{\max }\left[1-\left(\frac{r}{R}\right)^{2}\right] \hat{i}\]For this profile obtain expressions for the volume flow rate and the momentum flux through
Hydrogen is being pumped through a pipe system whose temperature is held at \(273 \mathrm{~K}\). At a section where the pipe diameter is \(10 \mathrm{~mm}\), the absolute pressure and average velocity are \(200 \mathrm{kPa}\) and \(30 \mathrm{~m} / \mathrm{s}\). Find all possible velocities and
Calculate the mean velocities for these two-dimensional velocity profiles If \(v_{c}=3 \mathrm{~m} / \mathrm{s}\). Parabola De Circle DE Parabola Equal Equal (a) (b) (c) (d) (e)
If the velocity profile in a passage of width \(2 R\) is given by the equation \(v / v_{c}=(y / R)^{1 / n}\), derive an expression for \(V / v_{c}\) in terms of \(n\) :(a) for a two-dimensional passage.(b) for a cylindrical pass age.
Fluid with \(1040 \mathrm{~kg} / \mathrm{m}^{3}\) density is flowing steadily through the rectangular box shown. Given \(A_{1}=0.046 \mathrm{~m}^{2}, A_{2}=0.009 \mathrm{~m}^{2}\), \(A_{3}=0.056 \mathrm{~m}^{2}, \vec{V}_{1}=3 \hat{i} \mathrm{~m} / \mathrm{s}\) and \(\vec{V}_{2}=6 \hat{j}
In your kitchen, the sink is \(60 \mathrm{~cm}\) by \(45.7 \mathrm{~cm}\). by \(30.5 \mathrm{~cm}\). deep. You are filling it with water at the rate of \(252 \times 10^{-6} \mathrm{~m}^{3} / \mathrm{s}\). How long will it take (in min) to half fill the sink? After this you turn off the faucet and
A pipeline \(0.3 \mathrm{~m}\) in diameter divides at a \(\mathrm{Y}\) into two branches \(200 \mathrm{~mm}\) and \(150 \mathrm{~mm}\) in diameter. If the flow rate in the main line is \(0.3 \mathrm{~m}^{3} / \mathrm{s}\) and the mean velocity in the \(200-\mathrm{mm}\) pipe is \(2.5 \mathrm{~m} /
A number of common substances areSome of these materials exhibit characteristics of both solid and fluid behavior under different conditions. Explain and give examples. Tar Sand "Silly Putty" Jello Modeling clay Wax Toothpaste Shaving cream
Give a word statement of each of the five basic conservation laws stated in Section 1.2, as they apply to a system.
The barrel of a bicycle tire pump becomes quite warm during use. Explain the mechanisms responsible for the temperature increase.
Very small particles moving in fluids are known to experience a drag force proportional to speed. Consider a particle of net weight \(W\) dropped in a fluid. The particle experiences a drag force, \(F_{D}=k V\), where \(V\) is the particle speed. Determine the time required for the particle to
In a combustion process, gasoline particles are to be dropped in air at \(200^{\circ} \mathrm{F}\). The particles must drop at least \(10 \mathrm{in}\). in \(1 \mathrm{~s}\). Find the diameter \(d\) of droplets required for this. (The drag on these particles is given by \(F_{D}=3 \pi \mu V d\),
In a pollution control experiment, minute solid particles (typical mass \(1 \times 10^{-13}\) slug) are dropped in air. The terminal speed of the particles is measured to be \(0.2 \mathrm{ft} / \mathrm{s}\). The drag of these particles is given by \(F_{D}=k V\), where \(V\) is the instantaneous
A rocket payload with a weight on earth of \(2000 \mathrm{lb}\) is landed on the moon where the acceleration due to the moon's gravity \(g_{m} \approx g_{n} / 6\). Find the mass of the payload on the earth and the moon and the payload's moon weight.
A cubic metre of air at \(101 \mathrm{kPa}\) absolute and \(15^{\circ} \mathrm{C}\) weighs \(12.0 \mathrm{~N}\). What is its specific volume? What is the specific volume if it is cooled to \(-10^{\circ} \mathrm{C}\) at constant pressure?
Calculate the specific weight, specific volume, and density of air at \(40^{\circ} \mathrm{F}\) and 50 psia. What are these values if the air is then compressed isentropically to \(100 \mathrm{psia}\) ?
For Problem 1.6, find the distance the particles travel before reaching 99 percent of terminal speed. Plot the distance traveled as a function of time.Data From Problem 1.6 1.16 A fluid occupying 3.2 m has a mass of 4 Mg. Calculate its den- sity and specific volume in SI, EE, and BG units.
A sky diver with a mass of \(70 \mathrm{~kg}\) jumps from an aircraft. The aerodynamic drag force acting on the sky diver is known to be \(F_{D}=k V^{2}\), where \(k=0.25 \mathrm{~N} \cdot \mathrm{s}^{2} / \mathrm{m}^{2}\). Determine the maximum speed of free fall for the sky diver and the speed
The English perfected the longbow as a weapon after the Medieval period. In the hands of a skilled archer, the longbow was reputed to be accurate at ranges to \(100 \mathrm{~m}\) or more. If the maximum altitude of an arrow is less than \(h=10 \mathrm{~m}\) while traveling to a target \(100
For each quantity listed, indicate dimensions using mass as a primary dimension, and give typical SI and English units:(a) Power(b) Pressure(c) Modulus of elasticity(d) Angular velocity(e) Energy(f) Moment of a force (g) Momentum (h) Shear stress (i) Strain (j) Angular momentum
The density of a sample of sea water is 1.99 slugs \(/ \mathrm{ft}^{3}\). What are the values in SI and EE units?
A pump is rated at \(50 \mathrm{hp}\). What is the rating in \(\mathrm{kW}\) and \(\mathrm{Btu} / \mathrm{hr}\) ?
A fluid occupying \(3.2 \mathrm{~m}^{3}\) has a mass of \(4 \mathrm{Mg}\). Calculate its density and specific volume in SI, EE, and BG units.
If a power plant is rated at \(2000 \mathrm{MW}\) output and operates (on average) at \(75 \%\) of rated power, how much energy (in J and ft.lbs) does it put out in a year?
For each quantity listed, indicate dimensions using force as a primary dimension, and give typical SI and English units:(a) Power(b) Pressure(c) Modulus of elasticity(d) Angular velocity(e) Energy(f) Momentum(g) Shear stress(h) Specific heat(i) Thermal expansion coefficient(j) Angular momentum
Derive the following conversion factors:(a) Convert a pressure of \(1 \mathrm{psi}\) to \(\mathrm{kPa}\).(b) Convert a volume of 1 liter to gallons.(c) Convert a viscosity of \(1 \mathrm{lbf} \cdot \mathrm{s} / \mathrm{ft}^{2}\) to \(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\).
Express the following in SI units:(a) \(5 \mathrm{acre.}/ \mathrm{ft}\)(b) \(150 \mathrm{in.}{ }^{3} / \mathrm{s}\)(c) \(3 \mathrm{gpm}\)(d) \(3 \mathrm{mph} / \mathrm{s}\)
Express the following in SI units:(a) \(100 \mathrm{cfm}\left(\mathrm{ft}^{3} / \mathrm{min}\right)\)(b) \(5 \mathrm{gal}\)(c) \(65 \mathrm{mph}\)(d) \(5.4 \mathrm{acres}\)
Express the following in \(\mathrm{BG}\) units:(a) \(50 \mathrm{~m}^{2}\)(b) \(250 \mathrm{cc}\)(c) \(100 \mathrm{~kW}\)(d) \(5 \mathrm{~kg} / \mathrm{m}^{2}\)
While you're waiting for the ribs to cook, you muse about the propane tank of your barbecue. You're curious about the volume of propane versus the actual tank size. Find the liquid propane volume when full (the weight of the propane is specified on the tank). Compare this to the tank volume (take
Derive the following conversion factors:(a) Convert a volume flow rate in cubic inches per minute to cubic millimeters per minute.(b) Convert a volume flow rate in cubic meters per second to gallons per minute (gpm).(c) Convert a volume flow rate in liters per minute to gpm.(d) Convert a volume
The kilogram force is commonly used in Europe as a unit of force. (As in the U.S. customary system, where \(1 \mathrm{lbf}\) is the force exerted by a mass of \(1 \mathrm{lbm}\) in standard gravity, \(1 \mathrm{kgf}\) is the force exerted by a mass of \(1 \mathrm{~kg}\) in standard gravity.)
From thermodynamics, we know that the coefficient of performance of an ideal air conditioner \(\left(C O P_{\text {ideal }}\right)\) is given by\[C O P_{\text {ideal }}=\frac{T_{L}}{T_{H}-T_{L}}\]where \(T_{L}\) and \(T_{H}\) are the room and outside temperatures (absolute). If an \(\mathrm{AC}\)
The maximum theoretical flow rate (slug/s) through a supersonic nozzle is\[\dot{m}_{\max }=2.38 \frac{A_{t} p_{0}}{\sqrt{T_{0}}}\]where \(A_{t}\left(\mathrm{ft}^{2}\right)\) is the nozzle throat area, \(p_{0}\) (psia) is the tank pressure, and \(T_{0}(\mathrm{R})\) is the tank temperature. Is this
A container weighs \(3.5 \mathrm{lbf}\) when empty. When filled with water at \(90^{\circ} \mathrm{F}\), the mass of the container and its contents is 2.5 slug. Find the weight of water in the container, and its volume in cubic feet.
A parameter that is often used in describing pump performance is the specific speed, \(N_{S_{c u}}\), given by\[N_{s_{c u}}=\frac{N(\mathrm{rpm})[Q(\mathrm{gpm})]^{1 / 2}}{[H(\mathrm{ft})]^{3 / 4}}\]
Calculate the density of standard air in a laboratory from the ideal gas equation of state. Estimate the experimental uncertainty in the air density calculated for standard conditions (29.9 in. of mercury and \(59^{\circ} \mathrm{F}\) ) if the uncertainty in measuring the barometer height is \(\pm
The mass of the standard American golf ball is \(1.62 \pm 0.01 \mathrm{oz}\) and its mean diameter is \(1.68 \pm 0.01\) in. Determine the density and specific gravity of the American golf ball. Estimate the uncertainties in the calculated values.
A can of pet food has the following internal dimensions: \(102 \mathrm{~mm}\) height and \(73 \mathrm{~mm}\) diameter (each \(\pm 1 \mathrm{~mm}\) at odds of 20 to 1). The label lists the mass of the contents as \(397 \mathrm{~g}\). Evaluate the magnitude and estimated uncertainty of the density of
The mass flow rate in a water flow system determined by collecting the discharge over a timed interval is \(0.2 \mathrm{~kg} / \mathrm{s}\). The scales used can be read to the nearest \(0.05 \mathrm{~kg}\) and the stopwatch is accurate to \(0.2 \mathrm{~s}\). Estimate the precision with which the
The mass flow rate of water in a tube is measured using a beaker to catch water during a timed interval. The nominal mass flow rate is \(100 \mathrm{~g} / \mathrm{s}\). Assume that mass is measured using a balance with a least count of \(1 \mathrm{~g}\) and a maximum capacity of \(1 \mathrm{~kg}\),
The mass of the standard British golf ball is \(45.9 \pm 0.3 \mathrm{~g}\) and its mean diameter is \(41.1 \pm 0.3 \mathrm{~mm}\). Determine the density and specific gravity of the British golf ball. Estimate the uncertainties in the calculated values.
The viscosity \(\mu\left(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)\) of water at temperature \(T(\mathrm{~K})\) can be computed from \(\mu=A 10^{B /(T-C)}\), where \(A=2.414 \times 10^{-5} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}, B=247.8 \mathrm{~K}\), and \(C=140 \mathrm{~K}\).
An enthusiast magazine publishes data from its road tests on the lateral acceleration capability of cars. The measurements are made using a 150 -ft-diameter skid pad. Assume the vehicle path deviates from the circle by \(\pm 2 \mathrm{ft}\) and that the vehicle speed is read from a fifthwheel
The height of a building may be estimated by measuring the horizontal distance to a point on the ground and the angle from this point to the top of the building. Assuming these measurements are \(L=100 \pm 0.5 \mathrm{ft}\) and \(\theta=30 \pm 0.2^{\circ}\), estimate the height \(H\) of the
An American golf ball is described in Problem 1.32 Assuming the measured mass and its uncertainty as given, determine the precision to which the diameter of the ball must be measured so the density of the ball may be estimated within an uncertainty of \(\pm 1\) percent.Data from problem 1.32 1.32
For the velocity fields given below, determine:(a) whether the flow field is one-, two-, or three-dimensional, and why.(b) whether the flow is steady or unsteady, and why.(The quantities \(a\) and \(b\) are constants.)\(1 \vec{V}=\left[(a x+t) e^{b y}\right] \hat{i}\)\(2 \vec{V}=(a x-b y)
For the velocity fields given below, determine:(a) whether the flow field is one-, two-, or three-dimensional, and why.(b) whether the flow is steady or unsteady, and why.(The quantities \(a\) and \(b\) are constants.)\(1 \vec{V}=\left[a y^{2} e^{-b t}\right] \hat{i}\)\(2 \vec{V}=a x^{2} \hat{i}+b
A viscous liquid is sheared between two parallel disks; the upper disk rotates and the lower one is fixed. The velocity field between the disks is given by \(\vec{V}=\hat{e}_{\theta} r \omega z / h\). (The origin of coordinates is located at the center of the lower disk; the upper disk is located
For the velocity field \(\vec{V}=A x^{2} y \hat{i}+B x y^{2} \hat{j}\), where \(A=2 \mathrm{~m}^{-2} \mathrm{~s}^{-1}\) and \(B=1 \mathrm{~m}^{-2} \mathrm{~s}^{-1}\), and the coordinates are measured in meters, obtain an equation for the flow streamlines. Plot several streamlines in the first
A fluid flow has the following velocity components: \(u=1 \mathrm{~m} / \mathrm{s}\) and \(v=2 x \mathrm{~m} / \mathrm{s}\). Find an equation for and sketch the streamlines of this flow.
When an incompressible, nonviscous fluid flows against a plate in a plane (two-dimensional) flow, an exact solution for the equations of motion for this flow is \(u=A x, v=-A y\), with \(A>0\) for the sketch shown. The coordinate origin is located at the stagnation point 0 , where the flow
For the free vortex flow the velocities are \(v_{t}=5 / r\) and \(v_{r}=0\). Assume that lengths are in feet or meters and times are in seconds. Plot the streamlines of this flow. How does the velocity vary with distance from the origin? What is the velocity at the origin \((0,0)\) ?
For the forced vortex flow the velocities are \(v_{t}=\omega r\) and \(v_{r}=0\). Plot the streamlines of this flow. How does the velocity vary with distance from the origin? What is the velocity at the origin \((0,0)\) ?
A velocity field is specified as \(\vec{V}=a x y \hat{i}+b y^{2} \hat{j}\), where \(a=\) \(2 \mathrm{~m}^{-1} \mathrm{~s}^{-1}, b=-6 \mathrm{~m}^{-1} \mathrm{~s}^{-1}\), and the coordinates are measured in meters. Is the flow field one-, two-, or three-dimensional? Why? Calculate the velocity
A velocity field is given by \(\vec{V}=a x^{3} \hat{i}+b x y^{3} \hat{j}\), where \(a=1 \mathrm{~m}^{-2} \mathrm{~s}^{-1}\) and \(b=1 \mathrm{~m}^{-3} \mathrm{~s}^{-1}\). Find the equation of the streamlines. Plot several streamlines in the first quadrant.
The velocity for a steady, incompressible flow in the \(x y\) plane is given by \(\vec{V}=\hat{i} A / x+\hat{j} A y / x^{2}\), where \(A=2 \mathrm{~m}^{2} / \mathrm{s}\), and the coordinates are measured in meters. Obtain an equation for the streamline that passes through the point \((x,
The flow field for an atmospheric flow is given by\[\vec{V}=-\frac{K y}{2 \pi\left(x^{2}+y^{2}\right)} \hat{i}+\frac{K x}{2 \pi\left(x^{2}+y^{2}\right)} \hat{j}\]where \(K=10^{5} \mathrm{~m}^{2} / \mathrm{s}\), and the \(x\) and \(y\) coordinates are parallel to the local latitude and longitude.
For the velocity field \(\vec{V}=A x \hat{i}-A y \hat{j}\), where \(A=2 \mathrm{~s}^{-1}\), which can be interpreted to represent flow in a corner, show that the parametric equations for particle motion are given by \(x_{p}=c_{1} e^{A t}\) and \(y_{p}=c_{2} e^{-A t}\). Obtain the equation for the

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Fox And McDonald's Introduction To Fluid Mechanics 9th Edition Philip J. Pritchard, John W. Mitchell - Solutions

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