Questions and Answers of Introduction To Fluid Mechanics

A simple but effective anemometer to measure wind speed can be made from a thin plate hinged to deflect in the wind. Consider a thin plate made from brass that is \(20 \mathrm{~mm}\) high and \(10
It is proposed to build a pyramidal building with a square base with sides of \(160 \mathrm{ft}\), which has the same volume as the Willis Tower. Calculate the maximum drag force on this building. Do
A circular disk is hung in an air stream from a pivoted strut as shown. In a wind-tunnel experiment, performed in air at \(15 \mathrm{~m} / \mathrm{s}\) with a \(25-\mathrm{mm}\) diameter disk,
An F-4 aircraft is slowed after landing by dual parachutes deployed from the rear. Each parachute is \(12 \mathrm{ft}\) in diameter. The F-4 weighs 32,000 lbf and lands at 160 knots. Estimate the
A 180-hp sports car of frontal area \(1.72 \mathrm{~m}^{2}\), with a drag coefficient of 0.31 , requires \(17 \mathrm{hp}\) to cruise at \(100 \mathrm{~km} / \mathrm{h}\). At what speed does
An object falls in air down a long vertical chute. The speed of the object is constant at \(3 \mathrm{~m} / \mathrm{s}\). The flow pattern around the object is shown. The static pressure is uniform
An object of mass \(m\), with cross-sectional area equal to half the size of the chute, falls down a mail chute. The motion is steady. The wake area is \(\frac{3}{4}\) the size of the chute at its
A light plane tows an advertising banner over a football stadium on a Saturday afternoon. The banner is \(4 \mathrm{ft}\) tall and \(45 \mathrm{ft}\) long. According to Hoerner [16], the drag
Consider small oil droplets \((\mathrm{SG}=0.85)\) rising in water. Develop a relation for calculating terminal speed of a droplet (in \(\mathrm{m} / \mathrm{s}\) ) as a function of droplet diameter
Compute the terminal speed of a 3-mm-diameter spherical raindrop in standard air.
A tennis ball with a mass of \(57 \mathrm{~g}\) and diameter of \(64 \mathrm{~mm}\) is dropped in standard sea level air. Calculate the terminal velocity of the ball. Assuming as an approximation
A cast-iron "12-pounder" cannonball rolls off the deck of a ship and falls into the ocean at a location where the depth is \(1000 \mathrm{~m}\). Estimate the time that elapses before the cannonball
A rectangular airfoil of \(40 \mathrm{ft}\) span and \(6 \mathrm{ft}\) chord has lift and drag coefficients of 0.5 and 0.04 , respectively, at an angle of attack of \(6^{\circ}\). Calculate the drag
An air bubble, \(0.3 \mathrm{in}\). in diameter, is released from the regulator of a scuba diver swimming \(100 \mathrm{ft}\) below the sea surface where the water temperature is \(86^{\circ}
Why is it possible to kick a football farther in a spiral motion than in an end-over-end tumbling motion?
If \(C_{L}=1.0\) and \(C_{D}=0.05\) for an airfoil, then find the span needed for a rectangular wing of \(10 \mathrm{~m}\) chord to lift \(3560 \mathrm{kN}\) at a take-off speed of \(282 \mathrm{~km}
A barge weighing \(8820 \mathrm{kN}\) that is \(10 \mathrm{~m}\) wide, \(30 \mathrm{~m}\) long, and \(7 \mathrm{~m}\) tall has come free from its tug boat in the Mississippi River. It is in a section
While walking across campus one windy day, an engineering student speculates about using an umbrella as a "sail" to propel a bicycle along the sidewalk. Develop an algebraic expression for the speed
The NACA 23015 airfoil is to move at \(180 \mathrm{mph}\) through standard sea level air. Determine the minimum drag, drag at optimum \(L / D\) and drag at point of maximum lift. Calculate the lift
Wiffle \(^{\mathrm{TM}}\) balls made from light plastic with numerous holes are used to practice baseball and golf. Explain the purpose of the holes and why they work. Explain how you could test your
The "shot tower," used to produce spherical lead shot, has been recognized as a mechanical engineering landmark. In a shot tower, molten lead is dropped from a high tower; as the lead solidifies,
A model airfoil of chord 6 in. and span 30 in. is placed in a wind tunnel with an air flow of \(100 \mathrm{ft} / \mathrm{s}\) at \(70^{\circ} \mathrm{F}\). It is mounted on a cylindrical support rod
How do cab-mounted wind deflectors for tractor-trailer trucks work? Explain using diagrams of the flow pattern around the truck and pressure distribution on the surface of the truck.
The U.S. Air Force F-16 fighter aircraft has wing planform area \(A=300 \mathrm{ft}^{2}\); it can achieve a maximum lift coefficient of \(C_{L}=1.6\). When fully loaded, its weight is \(26,000
A light airplane has 35 -ft effective wingspan and 5.5 - \(\mathrm{ft}\) chord. It was originally designed to use a conventional (NACA 23015) airfoil section. With this airfoil, its cruising speed on
Jim Hall's Chaparral \(2 \mathrm{~F}\) sports-racing cars in the 1960s pioneered use of airfoils mounted above the rear suspension to enhance stability and improve braking performance. The airfoil
Some cars come with a "spoiler," a wing section mounted on the rear of the vehicle that salespeople sometimes claim significantly increases traction of the tires at highway speeds. Investigate the
Roadside signs tend to oscillate in a twisting motion when a strong wind blows. Discuss the phenomena that must occur to cause this behavior.
A class demonstration showed that lift is present when a cylinder rotates in an air stream. A string wrapped around a paper cylinder and pulled causes the cylinder to spin and move forward
A baseball pitcher throws a ball at \(80 \mathrm{mph}\). Home plate is \(60 \mathrm{ft}\) away from the pitcher's mound. What spin should be placed on the ball for maximum horizontal deviation from a
Consider incompressible flow in a circular channel. Derive general expressions for Reynolds number in terms of (a) volume flow rate and tube diameter and (b) mass flow rate and tube diameter. The
What is the maximum flow rate of air that may occur at laminar condition in a 4-in.-diameter pipe at an absolute pressure of \(30 \mathrm{psia}\) and \(100^{\circ} \mathrm{F}\) ? If the pressure is
For flow in circular tubes, transition to turbulence usually occurs around \(R e \approx 2300\). Investigate the circumstances under which the flows of (a) standard air and (b) water at \(15^{\circ}
Air flows at \(100^{\circ} \mathrm{F}\) in a pipe system in which the diameter increases in two stages from 2 in. to 3 in. to 4 in. Each section is \(6 \mathrm{ft}\) long. The initial flow rate is
An incompressible fluid flows between two infinite stationary parallel plates. The velocity profile is given by \(u=u_{\max }\left(A y^{2}+\right.\) \(B y+C)\), where \(A, B\), and \(C\) are
Oil is confined in a 4-in.-diameter cylinder by a piston having a radial clearance of \(0.001 \mathrm{in}\). and a length of \(2 \mathrm{in}\). A steady force of \(4500 \mathrm{lbf}\) is applied to
Viscous oil flows steadily between parallel plates. The flow is fully developed and laminar. The pressure gradient is \(1.25 \mathrm{kPa} / \mathrm{m}\) and the channel half-width is \(h=1.5
Calculate \(\alpha\) for the flow in this two-dimensional passage if \(q\) is \(1.5 \mathrm{~m}^{3} / \mathrm{s} \cdot \mathrm{m}\). 3 m/s 0.6 m >Paraboals P8.8
The velocity profile in a two-dimensional open channel may be approximated by the parabola shown. Calculate the flow rate and the kinetic energy coefficient \(\alpha\). 10 ft 4 ft/s 2 ft/s P8.9 A 8 ft
A large mass is supported by a piston of diameter \(D=4 \mathrm{in}\). and length \(L=4\) in. The piston sits in a cylinder closed at the bottom, and the gap \(a=0.001 \mathrm{in}\). between the
A hydraulic jack supports a load of \(9000 \mathrm{~kg}\). The following data are given:Estimate the rate of leakage of hydraulic fluid past the piston, assuming the fluid is SAE 30 oil at
The basic component of a pressure gage tester consists of a piston-cylinder apparatus as shown. The piston, \(6 \mathrm{~mm}\) in diameter, is loaded to develop a pressure of known magnitude. The
When a horizontal laminar flow occurs between two parallel plates of infinite extent \(0.3 \mathrm{~m}\) apart, the velocity at the midpoint between the plates is \(2.7 \mathrm{~m} / \mathrm{s}\).
In a laminar flow of water of \(0.007 \mathrm{~m}^{3} / \mathrm{s}\) between parallel plates spaced \(75 \mathrm{~mm}\) apart, the measured shearing stress at the pipe wall is \(47.9 \mathrm{~Pa}\).
Consider the simple power-law model for a non-Newtonian fluid given by Eq. 2.16. Extend the analysis of Section 8.2 to show that the velocity profile for fully developed laminar flow of a powerlaw
A sealed journal bearing is formed from concentric cylinders. The inner and outer radii are 25 and \(26 \mathrm{~mm}\), the journal length is \(100 \mathrm{~mm}\), and it turns at \(2800
Using the profile of Problem 8.15, show that the flow rate for fully developed laminar flow of a power-law fluid between stationary parallel plates may be written as\[Q=\left(\frac{h}{k} \frac{\Delta
In a laminar flow between parallel plates spaced 12 in. apart, the shear stress at the wall is \(1.0 \mathrm{psf}\) and the fluid viscosity \(0.002 \mathrm{lb} \mathrm{s} / \mathrm{ft}^{2}\). What is
A fluid of specific gravity 0.90 flows at a Reynolds number of 1500 between parallel plates spaced \(0.3 \mathrm{~m}\) apart. The velocity \(50 \mathrm{~mm}\) from the wall is \(3 \mathrm{~m} /
Two immiscible fluids are contained between infinite parallel plates. The plates are separated by distance \(2 h\), and the two fluid layers are of equal thickness \(h\); the dynamic viscosity of the
The record-read head for a computer disk-drive memory storage system rides above the spinning disk on a very thin film of air (the film thickness is \(0.25 \mu \mathrm{m}\) ). The head location is
Consider steady, incompressible, and fully developed laminar flow of a viscous liquid down an incline with no pressure gradient. The velocity profile was derived in Example 5.9. Plot the velocity
In a flow of air between parallel plates spaced \(0.03 \mathrm{~m}\) apart, the centerline velocity is \(1.2 \mathrm{~m} / \mathrm{s}\) and that \(5 \mathrm{~mm}\) from the pipe wall is \(0.8
Two immiscible fluids of equal density are flowing down a surface inclined at a \(60^{\circ}\) angle. The two fluid layers are of equal thickness \(h=10 \mathrm{~mm}\); the kinematic viscosity of the
Consider fully developed flow between parallel plates with the upper plate moving at \(U=5 \mathrm{ft} / \mathrm{s}\). The spacing between the plates is \(a=0.1 \mathrm{in}\). Determine the flow rate
The velocity profile for fully developed flow of castor oil at \(20^{\circ} \mathrm{C}\) between parallel plates with the upper plate moving is given by Eq. 8.8. Assume \(U=1.5 \mathrm{~m} /
Free-surface waves begin to form on a laminar liquid film flowing down an inclined surface whenever the Reynolds number, based on mass flow per unit width of film, is larger than about 33. Estimate
A viscous-shear pump is made from a stationary housing with a close-fitting rotating drum inside. The clearance is small compared with the diameter of the drum, so flow in the annular space may be
The efficiency of the viscous-shear pump of Fig. P8.29 is given by\[\eta=6 q \frac{(1-2 q)}{(4-6 q)}\]where \(q=Q / a b R \omega\) is a dimensionless flow rate, \(Q\) is the flow rate at pressure
An inventor proposes to make a "viscous timer" by placing a weighted cylinder inside a slightly larger cylinder containing viscous liquid, creating a narrow annular gap close to the wall. Analyze the
A continuous belt, passing upward through a chemical bath at speed \(U_{0}\), picks up a liquid film of thickness \(h\), density \(ho\), and viscosity \(\mu\). Gravity tends to make the liquid drain
A wet paint film of uniform thickness, \(\delta\), is painted on a vertical wall. The wet paint can be approximated as a Bingham fluid with a yield stress, \(\tau_{y}\), and density, \(ho\). Derive
When dealing with the lubrication of bearings, the governing equation describing pressure is the Reynolds equation, generally written in one dimension as\[\frac{d}{d x}\left(\frac{h^{3}}{\mu} \frac{d
Consider first water and then SAE 10W lubricating oil flowing at \(40^{\circ} \mathrm{C}\) in a 6-mm-diameter tube. Determine the maximum flow rate and the corresponding pressure gradient, \(\partial
Using for the viscosity of water, find the viscosity at \(-20^{\circ} \mathrm{C}\) and \(120^{\circ} \mathrm{C}\). Plot the viscosity over this range. Find the maximum laminar flow rate \((\mathrm{L}
Consider fully developed laminar flow in the annulus between two concentric pipes. The outer pipe is stationary, and the inner pipe moves in the \(x\) direction with speed \(V\). Assume the axial
Carbon dioxide flows in a 50-mm-diameter pipe at a velocity of \(1.5 \mathrm{~m} / \mathrm{s}\), temperature \(66^{\circ} \mathrm{C}\), and absolute pressure \(50 \mathrm{kPa}\). Is the flow laminar
Consider fully developed laminar flow in a circular pipe. Use a cylindrical control volume as shown. Indicate the forces acting on the control volume. Using the momentum equation, develop an
What is the largest diameter of pipeline that may be used to carry \(100 \mathrm{gpm}\) of jet fuel (JP-4) at \(59^{\circ} \mathrm{F}\) if the flow is to be laminar?
Consider fully developed laminar flow in the annular space formed by the two concentric cylinders shown in the diagram for Problem 8.36, but with pressure gradient, \(\partial p / \partial x\), and
Consider fully developed pressure-driven flow in a cylindrical tube of radius, \(R\), and length, \(L=10 \mathrm{~mm}\), with flow generated by an applied pressure gradient, \(\Delta p\). Tests are
In the laminar flow of an oil of viscosity \(1 \mathrm{~Pa} \cdot \mathrm{s}\), the velocity at the center of a \(0.3 \mathrm{~m}\) pipe is \(4.5 \mathrm{~m} / \mathrm{s}\) and the velocity
In a laminar flow of \(0.007 \mathrm{~m}^{3} / \mathrm{s}\) in a 75 -mm-diameter pipeline the shearing stress at the pipe wall is known to be \(47.9 \mathrm{~Pa}\). Calculate the viscosity of the
Consider blood flow in an artery. Blood is non-Newtonian; the shear stress versus shear rate is described by the Casson relationship:\[\begin{cases}\sqrt{\tau}=\sqrt{\tau_{c}}+\sqrt{\mu \frac{d u}{d
The classic Poiseuille flow (Eq. 8.12), is for no-slip conditions at the walls. If the fluid is a gas, and when the mean free path, \(l\) (the average distance a molecule travels before collision
For pressure-driven, steady, fully developed laminar flow of an incompressible fluid through a straight channel of length \(L\), we can define the hydraulic resistance as \(R_{\text {hyd }}=\Delta p
In a laminar flow in a 12-in.-diameter pipe the shear stress at the wall is \(1.0 \mathrm{psf}\) and the fluid viscosity \(0.002 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}\). Calculate the
A fluid of specific gravity 0.90 flows at a Reynolds number of 1500 in a 0.3-m-diameter pipeline. The velocity \(50 \mathrm{~mm}\) from the wall is \(3 \mathrm{~m} / \mathrm{s}\). Calculate the flow
In a food industry plant, two immiscible fluids are pumped through a tube such that fluid \(1\left(\mu_{1}=0.5 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)\) forms an inner core
A horizontal pipe carries fluid in fully developed turbulent flow. The static pressure difference measured between two sections is \(750 \mathrm{psi}\). The distance between the sections is \(15
Kerosene is pumped through a smooth tube with inside diameter \(D=30 \mathrm{~mm}\) at close to the critical Reynolds number. The flow is unstable and fluctuates between laminar and turbulent states,
In a flow of water in a 0.3-m-diameter pipe, the centerline velocity is \(6 \mathrm{~m} / \mathrm{s}\) and that \(50 \mathrm{~mm}\) from the pipe wall is \(5.2 \mathrm{~m} / \mathrm{s}\). Assuming
A liquid drug, with the viscosity and density of water, is to be administered through a hypodermic needle. The inside diameter of the needle is \(0.25 \mathrm{~mm}\) and its length is \(50
Laufer [5] measured the following data for mean velocity in fully developed turbulent pipe flow at \(R e_{U}=50,000\) :In addition, Laufer measured the following data for mean velocity in fully
Equation 8.23 gives the power-law velocity profile exponent, \(n\), as a function of centerline Reynolds number, \(R e_{U}\), for fully developed turbulent flow in smooth pipes. Equation 8.24 relates
Consider fully developed laminar flow of water between stationary parallel plates. The maximum flow speed, plate spacing, and width are \(20 \mathrm{ft} / \mathrm{s}, 0.075\mathrm{in}\). and \(1.25
Consider fully developed laminar flow in a circular tube. Evaluate the kinetic energy coefficient for this flow.
Show that the kinetic energy coefficient, \(\alpha\), for the "power law" turbulent velocity profile of Eq. 8.22 is given by Eq. 8.27. Plot \(\alpha\) as a function of \(R e_{\bar{V}}\), for \(R
If the turbulent velocity profile in a pipe \(0.6 \mathrm{~m}\) in diameter may be approximated by \(v=3.56 y^{1 / 7}\), where \(v\) is in \(\mathrm{m} / \mathrm{s}\) and \(y\) is in \(\mathrm{m}\),
Water flows in a horizontal constant-area pipe; the pipe diameter is \(75 \mathrm{~mm}\) and the average flow speed is \(5 \mathrm{~m} / \mathrm{s}\). At the pipe inlet, the gage pressure is \(275
For a given volume flow rate and piping system, will the pressure loss be greater for hot water or cold water? Explain.
Consider the pipe flow from the water tower of Example 8.7. To increase delivery, the pipe length is reduced from \(600 \mathrm{ft}\) to \(450 \mathrm{ft}\) (the flow is still fully turbulent and
At the inlet to a constant-diameter section of the Alaskan pipeline, the pressure is \(8.5 \mathrm{MPa}\) and the elevation is \(45 \mathrm{~m}\); at the outlet the elevation is \(115 \mathrm{~m}\).
When oil (kinematic viscosity \(1 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}\), specific gravity 0.92) flows at a mean velocity of \(1.5 \mathrm{~m} / \mathrm{s}\) through a 50 -mm-diameter
When fluid of specific weight \(50 \mathrm{lb} / \mathrm{ft}^{3}\) flows in a 6-in.diameter pipeline, the frictional stress between fluid and pipe is \(0.5 \mathrm{psf}\). Calculate the head lost per
If the head lost in 30 -m-diameter of \(75-\mathrm{mm}\)-diameter pipe is \(7.6 \mathrm{~m}\) for a given flow rate of water, what is the total drag force exerted by the water on this length of pipe?
Water flows at \(10 \mathrm{~L} / \mathrm{min}\) through a horizontal \(15-\mathrm{mm}\) diameter tube. The pressure drop along a \(20-\mathrm{m}\) length of tube is \(85 \mathrm{kPa}\). Calculate
Laufer [5] measured the following data for mean velocity near the wall in fully developed turbulent pipe flow at \(R e_{U}=50,000(U=9.8 \mathrm{ft} / \mathrm{s}\) and \(R=4.86\) in. \()\) in air:Plot
Water is pumped at the rate of \(0.075 \mathrm{~m}^{3} / \mathrm{s}\) from a reservoir \(20 \mathrm{~m}\) above a pump to a free discharge \(35 \mathrm{~m}\) above the pump. The pressure on the
Just downstream from the nozzle tip the velocity distribution is as shown. Calculate the flow rate past section 1, the kinetic energy coefficient \(\alpha\), and the momentum flux. Assume water is

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