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engineering
engineering fluid mechanics

Questions and Answers of Engineering Fluid Mechanics

A viscous fluid is contained between two infinitely long, vertical, concentric cylinders. The outer cylinder has a radius \(r_{o}\) and rotates with an angular velocity \(\omega\). The inner cylinder
A double-pipe heat exchanger consists of two concentric tubes with one fluid flowing in the central tube and the other flowing in the annulus between the tubes. In a particular exchanger, cold
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
Look through several professional society magazines such as Civil Engineering, Mechanical Engineering, or ASEE Prism. Count the advertisements for CFD software and calculate an "ad density" in
Consider the Navier-Stokes equations as written in Eq. 6.158. Write the physical meaning of each term in the equation; for example, "The term \(ho \mathbf{g}\) is the gravity force (or weight) per
Find out what CFD software packages your college or university has. How many "seats" of each? How much does your institution spend annually on CFD software?
Obtain a photograph/image of a situation in which a fluid is flowing. Print this photo and draw in some lines to represent how you think some streamlines may look. Write a brief paragraph to describe
The surface velocity of a river is measured at several locations \(x\) and can be reasonably represented by\[ V=V_{0}+\Delta V\left(1-e^{-a x}\right) \]where \(V_{0}, \Delta V\), and \(a\) are
The velocity field of a flow is given by \(\mathbf{V}=2 x^{2} t \hat{\mathbf{i}}+[4 y(t-1)\) \(\left.+2 x^{2} t\right] \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}\), where \(x\) and \(y\) are in meters
A two-dimensional velocity field is given by \(u=1+y\) and \(v=1\). Determine the equation of the streamline that passes through the origin. On a graph, plot this streamline.
Streamlines are given in Cartesian coordinates by the equation.\[ \psi=U\left(y-\frac{y}{x^{2}+y^{2}}\right) . \quad x^{2}+y^{2} \geq 1 \]Plot the streamlines for \(\psi=0, \pm 60.0625\), and
A flow can be visualized by plotting the velocity field as velocity vectors at representative locations in the flow as shown in Fig. E4.1. Consider the velocity field given in polar coordinates by
A car accelerates from rest to a final constant velocity \(V_{f}\) and a police officer records the following velocities at various locations \(x\) along the highway:Find a mathematical Eulerian
The components of a velocity field are given by \(u=x+y\), \(v=x y^{3}+16\), and \(w=0\). Determine the location of any stagnation points \((\mathbf{V}=0)\) in the flow field.
A two-dimensional, unsteady velocity field is given by\[ u=5 x(1+t) \quad \text { and } \quad v=5 y(-1+t) \]where \(u\) is the \(x\)-velocity component and \(v\) the \(y\)-velocity component. Find
The velocity field of a flow is given by \(u=-V_{0} y /\left(x^{2}+\right.\) \(\left.y^{2}\right)^{1 / 2}\) and \(v=V_{0} x /\left(x^{2}+y^{2}\right)^{1 / 2}\), where \(V_{0}\) is a constant. Where
A velocity field is given by \(\mathbf{V}=x \hat{\mathbf{i}}+x(x-1)(y+1) \hat{\mathbf{j}}\), where \(u\) and \(v\) are in \(\mathrm{ft} / \mathrm{s}\) and \(x\) and \(y\) are in feet. Plot the
From time \(t=0\) to \(t=5 \mathrm{hr}\) radioactive steam is released from a nuclear power plant accident located at \(x=-1\) mile and \(y=3\) miles. The following wind conditions are expected:
The \(x\) and \(y\) components of a velocity field are given by \(u=x^{2} y\) and \(v=-x y^{2}\). Determine the equation for the streamlines of this flow and compare it with those in Example 4.2. Is
In addition to the customary horizontal velocity components of the air in the atmosphere (the "wind"), there often are vertical air currents (thermals) caused by buoyant effects due to uneven heating
A test car is traveling along a level road at \(88 \mathrm{~km} / \mathrm{hr}\). In order to study the acceleration characteristics of a newly installed engine, the car accelerates at its maximum
For any steady flow the streamlines and streaklines are the same. For most unsteady flows this is not true. However, there are unsteady flows for which the streamlines and streaklines are the same.
A tornado has the following velocity components in polar coordinates:\[ V_{r}=-\frac{C_{1}}{r} \quad \text { and } \quad V_{\theta}=-\frac{C_{2}}{r} \]Note that the air is spiraling inward. Find an
The Wide World of Fluids article titled "Follow those particles,". Two photographs of four particles in a flow past a sphere are superposed as shown in Fig. P4.18. The time interval between the
Air flows steadily through a circular, constant-diameter duct. The air is perfectly inviscid, so the velocity profile is flat across each flow area. However, the air density decreases as the air
A constant-density fluid flows in the converging, twodimensional channel shown in Fig. P4.20. The width perpendicular to the paper is quite large compared to the channel height. The velocity in the
Pathlines and streaklines provide ways to visualize flows. Another technique would be to instantly inject a line of dye across streamlines and observe how this line moves as time increases. For
Classify the following flows as one-, two-, or three-dimensional. Sketch a few streamlines for each.(a) Rainwater flow down a wide driveway(b) Flow in a straight horizontal pipe(c) Flow in a straight
The velocity components of \(u\) and \(v\) of a two-dimensional flow are given by\[ u=a x+\frac{b x}{x^{2} y^{2}} \quad \text { and } \quad v=a y+\frac{b y}{x^{2} y^{2}} \]where \(a\) and \(b\) are
Air is delivered through a constant-diameter duct by a fan. The air is inviscid, so the fluid velocity profile is "flat" across each cross section. During the fan start-up, the following velocities
Water flows through a constant diameter pipe with a uniform velocity given by \(\mathbf{V}=(8 / t+5) \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}\), where \(t\) is in seconds. Determine the acceleration
The velocity of air in the diverging pipe shown in Fig. P4.26 is given by \(V_{1}=4 t \mathrm{ft} / \mathrm{s}\) and \(V_{2}=2 t \mathrm{ft} / \mathrm{s}\), where \(t\) is in seconds. (a) Determine
A certain flow field has the velocity vector\(\mathbf{V}=\frac{-2 x y z}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{i}+\frac{\left(x^{2}-y^{2}\right) z}{\left(x^{2}+y^{2}\right)^{2}}
Determine the \(x\)-component of the acceleration, \(a_{x}\), along the centerline \((y=0)\) for the flow of Problem 4.20. Can you determine the acceleration vector at a location not on the
The velocity of the water in the pipe shown in Fig. P4.29 is given by \(V_{1}=0.50 t \mathrm{~m} / \mathrm{s}\) and \(V_{2}=1.0 t \mathrm{~m} / \mathrm{s}\), where \(t\) is in seconds. Determine the
A shock wave is a very thin layer (thickness \(=\ell\) ) in a highspeed (supersonic) gas flow across which the flow properties (velocity, density, pressure, etc.) change from state (1) to state (2)
Estimate the average acceleration of water as it travels through the nozzle on your garden hose. List all assumptions and show all calculations.
A stream of water from the faucet strikes the bottom of the sink. Estimate the maximum acceleration experienced by the water particles. List all assumptions and show calculations.
As a valve is opened, water flows through the diffuser shown in Fig. P4.33 at an increasing flowrate so that the velocity along the centerline is given by \(\mathbf{V}=\hat{u}
The fluid velocity along the \(x\) axis shown in Fig. P4.34 changes from \(6 \mathrm{~m} / \mathrm{s}\) at point \(A\) to \(18 \mathrm{~m} / \mathrm{s}\) at point \(B\). It is also known that the
A fluid flows along the \(x\) axis with a velocity given by \(\mathbf{V}=(x / t) \hat{\mathbf{i}}\), where \(x\) is in feet and \(t\) in seconds. (a) Plot the speed for \(0 \leq x \leq 10
A constant-density fluid flows through a converging section having an area \(A\) given by\[ A=\frac{A_{0}}{1+(x / \ell)} \]where \(A_{0}\) is the area at \(x=0\). Determine the velocity and
A hydraulic jump is a rather sudden change in depth of a liquid layer as it flows in an open channel as shown in Fig. P4.37 and Video V10.12. In a relatively short distance (thickness \(=\ell\) ) the
A fluid particle flowing along a stagnation streamline, as shown in Video V4.9 and Fig. P4.38, slows down as it approaches the stagnation point. Measurements of the dye flow in the video indicate
A nozzle is designed to accelerate the fluid from \(V_{1}\) to \(V_{2}\) in a linear fashion. That is, \(V=a x+b\), where \(a\) and \(b\) are constants. If the flow is constant with \(V_{1}=10
An incompressible fluid flows through the converging duct shown in Fig. P4.40a with velocity \(V_{0}\) at the entrance. Measurements indicate that the actual velocity of the fluid near the wall of
Air flows steadily through a variable area pipe with a velocity of \(\mathbf{V}=u(x) \hat{\mathbf{i}} \mathrm{ft} / \mathrm{s}\), where the approximate measured values of \(u(x)\) are given in the
As is indicated in Fig. P4.42, the speed of exhaust in a car's exhaust pipe varies in time and distance because of the periodic nature of the engine's operation and the damping effect with distance
Water flows down the face of the dam shown in Fig. P4.43. The face of the dam consists of two circular arcs with radii of 10 and \(20 \mathrm{ft}\) as shown. If the speed of the water along
Water flows over the crest of a dam with speed \(V\) as shown in Fig. P4.44. Determine the speed if the magnitude of the normal acceleration at point (1) is to equal the acceleration of gravity,
Water flows under the sluice gate shown in Fig. P4.45. If \(V_{1}=3 \mathrm{~m} / \mathrm{s}\), what is the normal acceleration at point (1)?Figure P4.45 Sluice gate R = 0.12 m (1) V = 3 m/s
A fluid flows past a sphere with an upstream velocity of \(V_{0}=40 \mathrm{~m} / \mathrm{s}\) as shown in Fig. P4.46. From a more advanced theory it is found that the speed of the fluid along the
Assume that the streamlines for the wingtip vortices from an airplane Fig. P4.15 can be approximated by circles of radius \(r\) and that the speed is \(V=K / r\), where \(K\) is a constant. Determine
The velocity components for steady flow through the nozzle shown in Fig. P4.48 are \(u=-V_{0} x / \ell\) and \(v=V_{0}[1+(y / \ell)]\), where \(V_{0}\) and \(\ell\) are constants. Determine the ratio
Water flows through the curved hose shown in Fig. P4.49 with an increasing speed of \(V=10 t \mathrm{ft} / \mathrm{s}\), where \(t\) is in seconds. For \(t=2 \mathrm{~s}\) determine(a) the component
Water flows though the slit at the bottom of a two-dimensional water trough as shown in Fig. P4.50. Throughout most of the trough the flow is approximately radial (along rays from \(O\) ) with a
Air flows from a pipe into the region between two parallel circular disks as shown in Fig. P4.51. The fluid velocity in the gap between the disks is closely approximated by \(V=V_{0} R / r\), where
Air flows into a pipe from the region between a circular disk and a cone as shown in Fig. P4.52. The fluid velocity in the gap between the disk and the cone is closely approximated by \(V=V_{0} R^{2}
Fluid flows through a pipe with a velocity of \(2.0 \mathrm{ft} / \mathrm{s}\) and is being heated, so the fluid temperature \(T\) at axial position \(x\) increases at a steady rate of \(30.0^{\circ}
A gas flows along the \(x\) axis with a speed of \(V=5 x \mathrm{~m} / \mathrm{s}\) and a pressure of \(p=10 x^{2} \mathrm{~N} / \mathrm{m}^{2}\), where \(x\) is in meters.(a) Determine the time rate
Assume the temperature of the exhaust in an exhaust pipe can be approximated by \(T=T_{0}\left(1+a e^{-b x}\right)[1+c \cos (\omega t)]\), where \(T_{0}=100^{\circ} \mathrm{C}, a=3, b=0.03
A bicyclist leaves from her home at 9 A.M. and rides to a beach \(40 \mathrm{mi}\) away. Because of a breeze off the ocean, the temperature at the beach remains \(60^{\circ} \mathrm{F}\) throughout
The following pressures for the air flow in Problem 4.24 were measured:Find the local rate of change of pressure \(\partial p / \partial t\) and the convective rate of change of pressure \(V \partial
Obtain a photograph/image of a situation in which a fluid is flowing. Print this photo and draw a control volume through which the fluid flows. Write a brief paragraph that describes how the fluid
In the region just downstream of a sluice gate, the water may develop a reverse flow region as is indicated in Fig. P4.59 and Video V10.9. The velocity profile is assumed to consist of two uniform
At time \(t=0\) the valve on an initially empty (perfect vacuum, \(ho=0)\) tank is opened and air rushes in. If the tank has a volume of \(\forall_{0}\) and the density of air within the tank
From calculus, one obtains the following formula (Leibnitz rule) for the time derivative of an integral that contains time in both the integrand and the limits of the integration:\[ \frac{d}{d t}
Air enters an elbow with a uniform speed of \(10 \mathrm{~m} / \mathrm{s}\) as shown in Fig. P4.62. At the exit of the elbow, the velocity profile is not uniform. In fact, there is a region of
A layer of oil flows down a vertical plate as shown in Fig. P4.63 with a velocity of \(\mathbf{V}=\left(V_{0} / h^{2}\right)\left(2 h x-x^{2}\right) \hat{\mathbf{j}}\) where \(V_{0}\) and \(h\) are
Figure P4.64 shows a fixed control volume. It has a volume \(V_{0}=1.0 \mathrm{ft}^{3}\), a flow area \(A=1.0 \mathrm{ft}^{2}\), and a length \(\ell_{0}=1.0 \mathrm{ft}\). Position \(x\) represents
Find \(D V / D t\) for the system in Problem 4.64.Problem 4.64Figure P4.64 shows a fixed control volume. It has a volume \(V_{0}=1.0 \mathrm{ft}^{3}\), a flow area \(A=1.0 \mathrm{ft}^{2}\), and a
Water enters a 5 -ft-wide, 1 -ft-deep channel as shown in Fig. P4.66. Across the inlet the water velocity is \(6 \mathrm{ft} / \mathrm{s}\) in the center portion of the channel and \(1 \mathrm{ft} /
Figure P4.67 illustrates a system and fixed control volume at time \(t\) and the system at a short time \(\delta t\) later. The system temperature is \(T=100^{\circ} \mathrm{F}\) at time \(t\) and
The wind blows across a field with an approximate velocity profile as shown in Fig. P4.68. Use Eq. 4.16 with the parameter \(b\) equal to the velocity to determine the momentum flowrate across the
Water flows from a nozzle with a speed of \(V=10 \mathrm{~m} / \mathrm{s}\) and is collected in a container that moves toward the nozzle with a speed of \(V_{c v}=2 \mathrm{~m} / \mathrm{s}\) as
The Wide World of Fluids article titled "Giraffe's blood pressure,".(a) Determine the change in hydrostatic pressure in a giraffe's head as it lowers its head from eating leaves \(6 \mathrm{~m}\)
What would be the barometric pressure reading, in \(\mathrm{mm} \mathrm{Hg}\), at an elevation of \(4 \mathrm{~km}\) in the U.S. standard atmosphere?
What pressure gradient along the streamline, \(d p / d s\), is required to accelerate water upward in a vertical pipe at a rate of \(30 \mathrm{ft} / \mathrm{s}^{2}\) ? What is the answer if the flow
The Bernoulli equation is valid for steady, inviscid, incompressible flows with constant acceleration of gravity. Consider flow on a planet where the acceleration of gravity varies with height so
2013 Indianapolis 500 champion Tony Kanaan holds his hand out of his IndyCar while driving through still air with standard atmospheric conditions.(a) For safety, the pit lane speed limit is \(60
Water flows steadily from a large open tank and discharges into the atmosphere though a 3-in.-diameter pipe as shown in Fig. P3.89. Determine the diameter, \(d\), in the narrowed section of the pipe
Water flows from a large tank as shown in Fig. P3.90. Atmospheric pressure is \(14.5 \mathrm{psia}\), and the vapor pressure is 1.60 psia. If viscous effects are neglected, at what height, \(h\),
Water flows into the sink shown in Fig. P3.91 and Video V5.1 at a rate of \(2 \mathrm{gal} / \mathrm{min}\). If the drain is closed, the water will eventually flow through the overflow drain holes
What pressure, \(p_{1}\), is needed to produce a flowrate of 0.09 \(\mathrm{ft}^{3} / \mathrm{s}\) from the tank shown in Fig. P3.92?Figure P3.92 Air P1 T Gasoline 2.0 ft Salt water SG = 1.1 0.06-ft
The vent on the tank shown in Fig. P3.93 is closed and the tank pressurized to increase the flowrate. What pressure, \(p_{1}\), is needed to produce twice the flowrate of that when the vent is
A sump pump is submerged in \(60^{\circ} \mathrm{F}\) ordinary water that vaporizes at a pressure of \(0.256 \mathrm{psia}\). The pump inlet has an inside diameter of \(2.067 \mathrm{in}\). and is
Water is siphoned from a large tank and discharges into the atmosphere through a 2-in.-diameter tube as shown in Fig. P3.95. The end of the tube is \(3 \mathrm{ft}\) below the tank bottom, and
Water flows in the system shown in Fig. P3.96. Assume frictionless flow.(a) Calculate the rate \(Q\) at which water must be added at the inlet to maintain the 16-ft height.(b) Calculate the height
Water flows steadily from the pipe shown in Fig. P3.97 with negligible viscous effects. Determine the maximum flowrate if the water is not to flow from the open vertical tube at \(A\).Figure P3.97 3
JP-4 fuel \((S G=0.77)\) flows through the Venturi meter shown in Fig. P3.98 with a velocity of \(15 \mathrm{ft} / \mathrm{s}\) in the 6 -in. pipe. If viscous effects are negligible, determine the
Water, considered an inviscid, incompressible fluid, flows steadily as shown in Fig. P3.99. Determine \(h\).Figure P3.99 Q = 4 ft/s h Air Water 0.5-ft diameter 1-ft diameter 3 ft
Determine the flowrate through the submerged orifice shown in Fig. P3.100 if the contraction coefficient is \(C_{c}=0.63\).Figure P3.100 4 ft 3-in. diameter 2 ft 6 ft
The water clock (clepsydra) shown in Fig. P3.101 is an ancient device for measuring time by the falling water level in a large glass container. The water slowly drains out through a small hole in the
A long water trough of triangular cross section is formed from two planks as is shown in Fig. P3.102. A gap of 0.1 in. remains at the junction of the two planks. If the water depth initially was \(2
Pop (with the same properties as water) flows from a 4-in.-diameter pop container that contains three holes as shown in Fig. P3.103 (see Video 3.9). The diameter of each fluid stream is \(0.15
A spherical tank of diameter \(D\) has a drain hole of diameter \(d\) at its bottom. A vent at the top of the tank maintains atmospheric pressure at the liquid surface within the tank. The flow is
A small hole develops in the bottom of the stationary rowboat shown in Fig. P3.105. Estimate the amount of time it will take for the boat to sink. List all assumptions and show all
When the drain plug is pulled, water flows from a hole in the bottom of a large, open cylindrical tank. Show that if viscous effects are negligible and if the flow is assumed to be quasisteady, then
Someone siphoned \(15 \mathrm{gal}\) of gasoline from a gas tank in the middle of the night. The gas tank measures 12 in. wide \(\times 24\) in. long \(\times 18\) in. high and was full when the
The surface area, \(A\), of the pond shown in Fig. P3. 108 varies with the water depth, \(h\), as shown in the table. At time \(t=0\) a valve is opened and the pond is allowed to drain through a pipe

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