Questions and Answers of Introduction To Fluid Mechanics

Water flows in a pipeline. At a point in the line where the diameter is 7 in., the velocity is \(12 \mathrm{fps}\) and the pressure is \(50 \mathrm{psi}\). At a point \(40 \mathrm{ft}\) away the
In a pipe \(0.3 \mathrm{~m}\) in diameter, \(0.3 \mathrm{~m}^{3} / \mathrm{s}\) of water are pumped up a hill. On the hilltop (elevation 48), the line reduces to \(0.2 \mathrm{~m}\) diameter. If the
A jet of air from a nozzle is blown at right angles against a wall in which two pressure taps are located. A manometer connected to the tap directly in front of the jet shows a head of \(25
The inlet contraction and test section of a laboratory wind tunnel are shown. The air speed in the test section is \(U=50 \mathrm{~m} / \mathrm{s}\). A totalhead tube pointed upstream indicates that
Maintenance work on high-pressure hydraulic systems requires special precautions. A small leak can result in a high-speed jet of hydraulic fluid that can penetrate the skin and cause serious injury
An open-circuit wind tunnel draws in air from the atmosphere through a well-contoured nozzle. In the test section, where the flow is straight and nearly uniform, a static pressure tap is drilled into
Water is flowing. Calculate \(H(\mathrm{~m})\) and \(p(\mathrm{kPa})\). 175 mm- -125 mm d -100 mm d Hg (13.57) P6.36 75 mm d H
If each gauge shows the same reading for a flow rate of \(1.00 \mathrm{cfs}\), what is the diameter of the constriction? 3-in.-dia. El. 100 - El. 90 P6.37 El. 108 3-in.-dia.
Derive a relation between \(A_{1}\) and \(A_{2}\) so that for a flow rate of \(0.28 \mathrm{~m}^{3} / \mathrm{s}\) the static pressure will be the same at sections (1) and (2). Also calculate the
Water flows steadily up the vertical 1-in.-diameter pipe and out the nozzle, which is 0.5 in. in diameter, discharging to atmospheric pressure. The stream velocity at the nozzle exit must be \(30
Your car runs out of gas unexpectedly and you siphon gas from another car. The height difference for the siphon is \(1 \mathrm{ft}\). The hose diameter is 0.5 in. What is your gasoline flow rate?
A tank at a pressure of \(50 \mathrm{kPa}\) gage gets a pinhole rupture and benzene shoots into the air. Ignoring losses, to what height will the benzene rise?
The water flow rate through the siphon is \(5 \mathrm{~L} / \mathrm{s}\), its temperature is \(20^{\circ} \mathrm{C}\), and the pipe diameter is \(25 \mathrm{~mm}\). Compute the maximum allowable
Water flows from a very large tank through a \(5 \mathrm{~cm}\) diameter tube. The dark liquid in the manometer is mercury. Estimate the velocity in the pipe and the rate of discharge from the tank.
Consider frictionless, incompressible flow of air over the wing of an airplane flying at \(200 \mathrm{~km} / \mathrm{hr}\). The air approaching the wing is at \(65 \mathrm{kPa}\) and \(-10^{\circ}
A closed tank contains water with air above it. The air is maintained at a gage pressure of \(150 \mathrm{kPa}\) and \(3 \mathrm{~m}\) below the water surface a nozzle discharges into the atmosphere.
Water jets upward through a 3-in.-diameter nozzle under a head of \(10 \mathrm{ft}\). At what height \(h\) will the liquid stand in the pitot tube? What is the cross-sectional area of the jet at
Calculate the rate of flow through this pipeline and the pressures at \(A, B, C\), and \(D\). Sketch the EL and HGL showing vertical distances. 5 ft B 20 ft 6 in. dl Water A P6.47 12 ft 2 in. d 8 ft
A mercury barometer is carried in a car on a day when there is no wind. The temperature is \(20^{\circ} \mathrm{C}\) and the corrected barometer height is \(761 \mathrm{~mm}\) of mercury. One window
A racing car travels at \(235 \mathrm{mph}\) along a straightaway. The team engineer wishes to locate an air inlet on the body of the car to obtain cooling air for the driver's suit. The plan is to
The velocity field for a plane source at a distance \(h\) above an infinite wall aligned along the \(x\) axis was given in Problem 6.6. Using the data from that problem, plot the pressure
A smoothly contoured nozzle, with outlet diameter \(d=20 \mathrm{~mm}\), is coupled to a straight pipe by means of flanges. Water flows in the pipe, of diameter \(D=50 \mathrm{~mm}\), and the nozzle
Water flows steadily through a 3.25-in.-diameter pipe and discharges through a 1.25-in.-diameter nozzle to atmospheric pressure. The flow rate is \(24.5 \mathrm{gpm}\). Calculate the minimum static
A flow nozzle is a device for measuring the flow rate in a pipe. This particular nozzle is to be used to measure low-speed air flow for which compressibility may be neglected. During operation, the
The head of water on a \(50 \mathrm{~mm}\) diameter smooth nozzle is \(3 \mathrm{~m}\). If the nozzle is directed upward at angles of(a) \(30^{\circ}\),(b) \(45^{\circ}\),(c) \(60^{\circ}\),(d)
Water flows from one reservoir in a \(200-\mathrm{mm}\) pipe, while water flows from a second reservoir in a \(150-\mathrm{mm}\) pipe. The two pipes meet in a "tee" junction with a
Barometric pressure is 14.0 psia. What is the maximum flow rate that can be obtained by opening the valve? (a) if cavitation is not a consideration and (b) if cavitation needs to be prevented? CCI4
A spray system is shown in the diagram. Water is supplied at \(p=10 \mathrm{kPa}\) gage, through the flanged opening of area \(A=1900 \mathrm{~mm}^{2}\). The water leaves in a steady free jet at
Water flows out of a kitchen faucet of 1.25 -in.-diameter at the rate of \(0.1 \mathrm{~L} / \mathrm{s}\). The bottom of the sink is \(45 \mathrm{~cm}\) below the faucet outlet. Will the
A horizontal axisymmetric jet of air with 0.4-in.-diameter strikes a stationary vertical disk of \(7.5 \mathrm{in}\). diameter. The jet speed is \(225 \mathrm{ft} / \mathrm{s}\) at the nozzle exit. A
The water level in a large tank is maintained at height \(H\) above the surrounding level terrain. A rounded nozzle placed in the side of the tank discharges a horizontal jet. Neglecting friction,
Many recreation facilities use inflatable "bubble" structures. A tennis bubble to enclose four courts is shaped roughly as a circular semicylinder with a diameter of \(50 \mathrm{ft}\) and a length
Water flows at low speed through a circular tube with inside diameter of 2 in. A smoothly contoured body of \(1.5 \mathrm{in}\). diameter is held in the end of the tube where the water discharges to
Describe the pressure distribution on the exterior of a multistory building in a steady wind. Identify the locations of the maximum and minimum pressures on the outside of the building. Discuss the
An aspirator provides suction by using a stream of water flowing through a venturi. Analyze the shape and dimensions of such a device. Comment on any limitations on its use.
Carefully sketch the energy grade lines (EGL) and hydraulic grade lines (HGL) for the system shown in Fig. 6.6 if the pipe is horizontal (i.e., the outlet is at the base of the reservoir), and a
Carefully sketch the energy grade lines (EGL) and hydraulic grade lines (HGL) for the system shown in Fig. 6.6 if a pump adding energy to the fluid is located at point (2), such that flow is into the
Water is being pumped from the lower reservoir through a nozzle into the upper reservoir. If the vacuum gauge at \(A\) reads \(2.4 \mathrm{psi}\) vacuum,(a) find the flow velocity through the
The turbine extracts power from the water flowing from the reservoir. Find the horsepower extracted if the flow through the system is 1000 cfs. Draw the energy line and the hydraulic grade line. El.
Consider a two-dimensional fluid flow: \(u=a x+b y\) and \(v=\) \(c x+d y\), where (a,b, c, and d) are constant. If the flow is incompressible and irrotational, find the relationships among (a,b, c,
The velocity field for a two-dimensional flow is \(\vec{V}=\) \((A x-B y) t \hat{i}-(B x+A y) t \hat{j}\), where \(A=1 \mathrm{~s}^{-2} B=2 \mathrm{~s}^{-2}, t\) is in seconds, and the coordinates
A flow field is characterized by the stream function \(\psi=A x y\), where \(A=2 \mathrm{~s}^{-1}\) and the coordinates are measured in feet. Verify that the flow is irrotational and determine the
The flow field for a plane source at a distance \(h\) above an infinite wall aligned along the \(x\) axis is given by \[\begin{aligned}\vec{V} & =\frac{q}{2 \pi\left[x^{2}+(y-h)^{2}\right]}[x
The stream function of a flow field is \(\psi=A x^{2} y-B y^{3}\), where \(A=1 \mathrm{~m}^{-1} \cdot \mathrm{s}^{-1}, B=\frac{1}{3} \mathrm{~m}^{-1} \cdot \mathrm{s}^{-1}\), and the coordinates are
A flow field is characterized by the stream function\[\psi=2 y+\frac{1}{2 \pi}\left(\tan ^{-1} \frac{y-a}{x}-\tan ^{-1} \frac{y+a}{x}\right)\]Derive an expression for the location of the stagnation
A flow field is characterized by the stream function\[\psi=x y^{2}+B x^{3}\]What does the value of \(B\) need to be for the flow to be irrotational? For that value of \(B\), determine the velocity
The stream function of a flow field is \(\psi=A x^{3}-B x y^{2}\), where \(A=1 \mathrm{~m}^{-1} \cdot \mathrm{s}^{-1}\) and \(B=3 \mathrm{~m}^{-1} \cdot \mathrm{s}^{-1}\), and coordinates are
A flow field is represented by the stream function \(\psi=x^{5}-\) \(15 x^{4} y^{2}+15 x^{2} y^{4}-y^{6}\). Find the corresponding velocity field. Show that this flow field is irrotational and obtain
Consider the flow field represented by the potential function \(\phi=x^{5}-10 x^{3} y^{2}+5 x y^{4}-x^{2}+y^{2}\). Verify that this is an incompressible flow, and obtain the corresponding stream
Show by expanding and collecting real and imaginary terms that \(f=z^{6}\) (where \(z\) is the complex number \(z=x+i y\) ) leads to a valid velocity potential (the real part of \(f\) ) and a
Consider the flow field represented by the velocity potential \(\phi=A x+B x^{2}-B y^{2}\), where \(A=1 \mathrm{~m} \cdot \mathrm{s}^{-1}, B=1 \mathrm{~m}^{-1} \cdot \mathrm{s}^{-1}\), and the
An incompressible flow field is characterized by the stream function \(\psi=3 A x^{2} y-A y^{3}\), where \(A=1 \mathrm{~m}^{-1} \cdot \mathrm{s}^{-1}\). Show that this flow field is irrotational.
Consider an air flow over a flat wall with an upstream velocity of \(6 \mathrm{~m} / \mathrm{s}\). There is a narrow slit through which air is drawn in at a flow rate of \(0.2 \mathrm{~m}^{3} /
A source with a strength of \(q=3 \pi \mathrm{m}^{2} / \mathrm{s}\) and a sink with a strength of \(q=\pi \mathrm{m}^{2} / \mathrm{s}\) are located on the \(x\) axis at \(x=-1 \mathrm{~m}\) and \(x=1
The velocity distribution in a two-dimensional, steady, inviscid flow field in the \(x y\) plane is \(\vec{V}=(A x+B) \hat{i}+(C-A y) \hat{j}\), where \(A=3 \mathrm{~s}^{-1}, B=6 \mathrm{~m} /
Consider the flow past a circular cylinder, of radius \(a\), used in Example 6.11. Show that \(V_{r}=0\) along the lines \((r, \theta)=(r, \pm \pi / 2)\). Plot \(V_{\theta} / U\) versus radius for
The flow in a corner with an angle \(\alpha\) can be described in radial coordinates by the stream function as \(\psi=A r^{\frac{\pi}{\alpha}} \sin \frac{\pi \theta}{\alpha}\). Determine the velocity
Consider the two-dimensional flow against a flat plate that is characterized by the stream function \(\psi=A x y\). Superimpose a plane source of strength \(B\) placed at the origin. Determine the
A source and a sink with strengths of equal magnitude, \(q=3 \pi \mathrm{m}^{2} / \mathrm{s}\), are placed on the \(x\) axis at \(x=-a\) and \(a\), respectively. A uniform flow, with speed \(U=20
A flow field is formed by combining a uniform flow in the positive \(x\) direction, with \(U=10 \mathrm{~m} / \mathrm{s}\), and a counterclockwise vortex, with strength \(K=16 \pi \mathrm{m}^{2} /
Consider the flow field formed by combining a uniform flow in the positive \(x\) direction and a source located at the origin. Obtain expressions for the stream function, velocity potential, and
The slope of the free surface of a steady wave in one-dimensional flow in a shallow liquid layer is described by the equation\[\frac{\partial h}{\partial x}=-\frac{u}{g} \frac{\partial u}{\partial
One-dimensional unsteady flow in a thin liquid layer is described by the equation\[\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}=-g \frac{\partial h}{\partial x}\]Use a length scale,
In atmospheric studies the motion of the earth's atmosphere can sometimes be modeled with the equation\[\frac{D \vec{V}}{D t}+2 \vec{\Omega} \times \vec{V}=-\frac{1}{ho} abla p\]where \(\vec{V}\) is
Fluid fills the space between two parallel plates. The differential equation that describes the instantaneous fluid velocity for unsteady flow with the fluid moving parallel to the walls is\[ho
By using order of magnitude analysis, the continuity and Navier-Stokes equations can be simplified to the Prandtl boundarylayer equations. For steady, incompressible, and two-dimensional flow,
Consider a disk of radius \(R\) rotating in an incompressible fluid at a speed \(\omega\). The equations that describe the boundary layer on the disk are:\[\begin{aligned}&
An unsteady, two-dimensional, compressible, inviscid flow can be described by the equation\[\begin{aligned}\frac{\partial^{2} \psi}{\partial t^{2}}+\frac{\partial}{\partial t} &
Experiments show that the pressure drop for flow through an orifice plate of diameter \(d\) mounted in a length of pipe of diameter . You are asked to organize some experimental data. Obtain the
At very low speeds, the drag on an object is independent of fluid density. Thus the drag force, \(F\), on a small sphere is a function only of speed, \(V\), fluid viscosity, \(\mu\), and sphere
We saw in Chapter 3 that the buoyant force, \(F_{B}\), on a body submerged in a fluid is directly proportional to the specific weight of the fluid, \(\gamma\). Demonstrate this using dimensional
Assume that the velocity acquired by a body falling from rest (without resistance) depends on weight of body, acceleration due to gravity, and distance of fall. Prove by dimensional analysis that
Derive by dimensional analysis an expression for the local velocity in established pipe flow through a smooth pipe if this velocity depends only on mean velocity, pipe diameter, distance from pipe
The speed of shallow water waves in the ocean (e.g., seismic sea waves or tsunamis) depends only on the still water depth and the acceleration due to gravity. Derive an expression for wave speed.
The speed, \(V\), of a free-surface wave in shallow liquid is a function of depth, \(D\), density, \(ho\), gravity, \(g\), and surface tension, \(\sigma\). Use dimensional analysis to find the
The boundary-layer thickness, \(\delta\), on a smooth flat plate in an incompressible flow without pressure gradients depends on the freestream speed, \(U\), the fluid density, \(ho\), the fluid
The speed, \(V\), of a free-surface gravity wave in deep water is a function of wavelength, \(\lambda\), depth, \(D\), density, \(ho\), and acceleration of gravity, \(g\). Use dimensional analysis to
Derive an expression for the velocity of very small ripples on the surface of a liquid if this velocity depends only on ripple length and density and surface tension of the liquid.
Derive an expression for the axial thrust exerted by a propeller if the thrust depends only on forward speed, angular speed, size, and viscosity and density ofthe fluid. How would the expression
Derive an expression for drag force on a smooth submerged object moving through incompressible fluid if this force depends only on speed and size of object and viscosity and density of the fluid.
The energy released during an explosion, \(E\), is a function of the time after detonation \(t\), the blast radius \(R\) at time \(t\), and the ambient air pressure \(p\), and density \(ho\).
Measurements of the liquid height upstream from an obstruction placed in an open-channel flow can be used to determine volume flow rate. (Such obstructions, designed and calibrated to measure rate of
The load-carrying capacity, \(W\), of a journal bearing is known to depend on its diameter, \(D\), length, \(l\), and clearance, \(c\), in addition to its angular speed, \(\omega\), and lubricant
Derive an expression for the drag force on a smooth object moving through compressible fluid if this force depends only on speed and size of object, and viscosity, density, and modulus of elasticity
A circular disk of diameter \(d\) and of negligible thickness is rotated at a constant angular speed, \(\omega\), in a cylindrical casing filled with a liquid of viscosity \(\mu\) and density \(ho\).
Two cylinders are concentric, the outer one fixed and the inner one movable. A viscous incompressible fluid fills the gap between them. Derive an expression for the torque required to maintain
The time, \(t\), for oil to drain out of a viscosity calibration container depends on the fluid viscosity, \(\mu\), and density, \(ho\), the orifice diameter, \(d\), and gravity, \(g\). Use
You are asked to find a set of dimensionless parameters to organize data from a laboratory experiment, in which a tank is drained through an orifice from initial liquid level \(h_{0}\). The time,
A continuous belt moving vertically through a bath of viscous liquid drags a layer of liquid, of thickness \(h\), along with it. The volume flow rate of liquid, \(Q\), is assumed to depend on \(\mu,
Derive an expression for the frictional torque exerted on the journal of a bearing if this torque depends only on the diameters of journal and bearing, their axial lengths (these are the same),
Tests on the established flow of six different liquids in smooth pipes of various sizes yield the following data:Make a dimensional analysis of this problem and a plot of the resulting dimensionless
The power, \(\mathscr{P}\), required to drive a fan is believed to depend on fluid density, \(ho\), volume flow rate, \(Q\), impeller diameter, \(D\), and angular velocity, \(\omega\). Use
The sketch shows an air jet discharging vertically. Experiments show that a ball placed in the jet is suspended in a stable position. The equilibrium height of the ball in the jet is found to depend
The diameter, \(d\), of bubbles produced by a bubble-making toy depends on the soapy water viscosity, \(\mu\), density, \(ho\), and surface tension, \(\sigma\), the ring diameter, \(D\), and the
Choked-flow nozzles are often used to meter the flow of gases through piping systems. The mass flow rate of gas is thought to depend on nozzle area \(A\), pressure \(p\), and temperature \(T\)
A large tank of liquid under pressure is drained through a smoothly contoured nozzle of area \(A\). The mass flow rate is thought to depend on nozzle area, \(A\), liquid density, \(ho\), difference
Spin plays an important role in the flight trajectory of golf, ping-pong, and tennis balls. Therefore, it is important to know the rate at which spin decreases for a ball in flight. The aerodynamic
The power loss, \(\mathscr{P}\), in a journal bearing depends on length, \(l\), diameter, \(D\), and clearance, \(c\), of the bearing, in addition to its angular speed, \(\omega\). The lubricant
The thrust of a marine propeller is to be measured during "open-water" tests at a variety of angular speeds and forward speeds ("speeds of advance"). The thrust, \(F_{T}\), is thought to depend on
The rate \(d T / d t\) at which the temperature \(T\) at the center of a rice kernel falls during a food technology process is critical-too high a value leads to cracking of the kernel, and too low a

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