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mechanical engineering

Questions and Answers of Mechanical Engineering

A solid circular cylinder of radius R rotates at angular velocity Ω in a viscous incompressible fluid which is at rest far from the cylinder, as in Fig. P4.82. Make simplifying assumptions and
The flow pattern in bearing lubrication can be illustrated by Fig. P4.83, where a viscous oil (ρ, μ) is forced into the gap h(x) between a fixed slipper block and a wall moving at velocity U.
Consider a viscous film of liquid draining uniformly down the side of a vertical rod of radius a, as in Fig. P4.84. At some distance down the rod the film will approach a terminal or fully developed
A flat plate of essentially infinite width and breadth oscillates sinusoidally in its own plane beneath a viscous fluid, as in Fig. P4.85 The fluid is at rest far above the plate. Making as many
SAE 10 oil at 20°C flows between parallel plates 8 mm apart, as in Fig. P4.86 A mercury manometer, with wall pressure taps 1 m apart, registers a 6-cm height, as shown. Estimate the flow rate of
SAE 30W oil at 20°C flows through the 9-cm-diameter pipe in Fig. P4.87 at an average velocity of 4.3 m/s(a) Verify that the flow is laminar.(b) Determine the volume flow rate in m3/h. (c)
The viscous oil in Fig P4.88 is set into steady motion by a concentric inner cylinder moving axially at velocity U inside a fixed outer cylinder. Assuming constant pressure and density and a purely
Modify Prob. 4.88 so that the outer cylinder also moves to the left at constant speed V. Find the velocity distribution υ z(r). For what ratio V/U will the wall shear stress be the same at both
SAE 10W oil at 20°C flows through a straight horizontal pipe. The pressure gradient is a constant 400 Pa/m. (a) What is the appropriate pipe diameter D in cm if the Reynolds number ReD of the flow
Consider 2-D incompressible steady Couette flow between parallel plates with the upper plate moving at speed V, as in Fig. 4.16a. Let the fluid be nonnewtonian, with stress given by a and c are
A tank of area Ao is draining in laminar flow through a pipe of diameter D and length L, as shown in Fig. P4.92. Neglecting the exit-jet kinetic energy and assuming the pipe flow is driven by the
A number of straight 25-cm-long microtubes, of diameter d, are bundled together into a “honeycomb” whose total cross-sectional area is 0.0006 m2. The pressure drop from entrance to exit is 1.5
For axial flow through a circular tube, the Reynolds number for transition to turbulence is approximately 2300 [see Eq. (6.2)], based upon the diameter and average velocity. If d = 5 cm and the fluid
In flow past a thin flat body such as an airfoil, transition to turbulence occurs at about Re = 1E6, based on the distance x from the leading edge of the wing. If an airplane flies at 450 mi/h at
When tested in water at 20°C flowing at 2 m/s, an 8-cm-diameter sphere has a measured drag of 5 N. What will be the velocity and drag force on a 1.5-m-diameter weather balloon moored in sea-level
An airplane has a chord length L = 1.2 m and flies at a Mach number of 0.7 in the standard atmosphere. If its Reynolds number, based on chord length, is 7E6, how high is it flying?
An automobile has a characteristic length and area of 8 ft and 60 ft2, respectively. When tested in sea-level standard air, it has the following measured drag force versus speed: V, mi/h: 20 40
SAE 10 oil at 20°C flows past an 8-cm-diameter sphere. At flow velocities of 1, 2, and 3 m/s, the measured sphere drag forces are 1.5, 5.3, and 11.2 N, respectively. Estimate the drag force if the
A body is dropped on the moon (g = 1.62 m/s2) with an initial velocity of 12 m/s. By using option 2 variables, Eq. (5.11), the ground impact occurs at ** t = 0.34 and S ** = 0.84. Estimate (a) the
The Morton number Mo, used to correlate bubble-dynamics studies, is a dimensionless combination of acceleration of gravity g, viscosity μ, density ρ, and surface tension coefficient Y. If
The acceleration number, Ac, sometimes used in compressible-flow theory, is a dimensionless combination of acceleration of gravity g, viscosity μ, density ρ, and bulk modulus B. If Ac is
Determine the dimension {MLTΘ} of the following quantities: All quantities have their standard meanings; for example, ρ is density, etc.
For a particle moving in a circle, its centripetal acceleration takes the form a = fcn (V, R), where V is its velocity and R the radius of its path. By pure dimensional reasoning, rewrite this
The Stokes number, St, used in particle-dynamics studies, is a dimensionless combination of five variables: acceleration of gravity g, viscosity μ, density ρ, particle velocity U, and
The speed of propagation C of a capillary wave in deep water is known to be a function only of density ρ, wavelength λ, and surface tension Y. Find the proper functional relationship,
In flow past a flat plate, the boundary layer thickness δ varies with distance x, free stream velocity U, viscosity μ, and density ρ. Find the dimensionless parameters for this problem
The wall shear stress τw in a boundary layer is assumed to be a function of stream velocity U, boundary layer thickness δ, local turbulence velocity u′, density ρ, and local
Convection heat-transfer data are often reported as a heat-transfer coefficient h, defined by Q = hAΔT Where Q = heat flow, J/s A = surface area, m2 ΔT = temperature difference,
The pressure drop per unit length Δp/L in a porous, rotating duct (Really! See Ref. 35) depends upon average velocity V, density ρ, viscosity μ, duct height h, wall injection velocity
Under laminar conditions, the volume flow Q through a small triangular-section pore of side length b and length L is a function of viscosity μ, pressure drop per unit length Δp/L, and
The period of oscillation T of a water surface wave is assumed to be a function of density ρ, wavelength λ, depth h, gravity g, and surface tension Y. Rewrite this relationship in
We can extend Prob. 5.18 to the case of laminar duct flow of a non-Newtonian fluid, for which the simplest relation for stress versus strain-rate is the power-law approximation: r = c (dθ /
In Example 5.1 we used the pi theorem to develop Eq. (5.2) from Eq. (5.1). Instead of merely listing the primary dimensions of each variable, some workers list the powers of each primary dimension
The angular velocity Ω of a windmill is a function of windmill diameter D, wind velocity V, air density ρ, windmill height H as compared to atmospheric boundary layer height L, and the
The period T of vibration of a beam is a function of its length L, area moment of inertia I, modulus of elasticity E, density ρ, and Poisson’s ratio σ. Rewrite this relation in
The lift force F on a missile is a function of its length L, velocity V, diameter D, angle of attack α, density ρ, viscosity μ, and speed of sound a of the air. Write out the
When a viscous fluid is confined between two long concentric cylinders as in Fig. 4.17, the torque per unit length T′ required to turn the inner cylinder at angular velocity Ω is a
A pendulum has an oscillation period T which is assumed to depend upon its length L, bob mass m, angle of swing θ, and the acceleration of gravity. A pendulum 1 m long, with a bob mass of 200 g,
In studying sand transport by ocean waves, A. Shields in 1936 postulated that the bottom shear stress τ required to move particles depends upon gravity g, particle size d and density ρp,
A simply supported beam of diameter D, length L, and modulus of elasticity E is subjected to a fluid cross flow of velocity V, density ρ, and viscosity μ. Its center deflection δ is
When fluid in a pipe is accelerated linearly from rest, it begins as laminar flow and then undergoes transition to turbulence at a time ttr which depends upon the pipe diameter D, fluid acceleration
The heat-transfer rate per unit area q to a body from a fluid in natural or gravitational convection is a function of the temperature difference ΔT, gravity g, body length L, and three fluid
The wall shear stress τw for flow in a narrow annular gap between a fixed and a rotating cylinder is a function of density ρ, viscosity μ, angular velocity Ω, outer radius R, and
A spar buoy (see Prob. 2.113) has a period T of vertical (heave) oscillation which depends upon the waterline cross-sectional area A, buoy mass m, and fluid specific weight . How does the
A weir is an obstruction in a channel flow which can be calibrated to measure the flow rate, as in Fig. P5.32The volume flow Q varies with gravity g, weir width b into the paper, and upstream water
To good approximation, the thermal conductivity k of a gas (see Ref. 8 of Chap. 1) depends only on the density ρ, mean free path ℓ, gas constant R, and absolute temperature T. For air at
The torque M required to turn the cone-plate viscometer in Fig. P5.35 depends upon the radius R, rotation rate Ω, fluid viscosity μ, and cone angle θ. Rewrite this relation in
The rate of heat loss, Qloss through a window is a function of the temperature difference ΔT, the surface area A, and the R resistance value of the window (in units of
The pressure difference Δp across an explosion or blast wave is a function of the distance r from the blast center, time t, speed of sound a of the medium, and total energy E in the blast.
The size d of droplets produced by a liquid spray nozzle is thought to depend upon the nozzle diameter D, jet velocity U, and the properties of the liquid ρ, μ, and Y. Rewrite this relation
In turbulent flow past a flat surface, the velocity u near the wall varies approximately logarithmically with distance y from the wall and also depends upon viscosity μ, density ρ, and wall
Reconsider the slanted-plate surface tension problem (see Fig. C1.1) as an exercise in dimensional analysis, let the capillary rise h vary only with fluid properties, bottom width, gravity, and the
A certain axial-flow turbine has an output torque M which is proportional to the volume flow rate Q and also depends upon the density ρ, rotor diameter D, and rotation rate Ω. How does the
Non-dimensionalize the thermal energy partial differential equation (4.75) and its boundary conditions (4.62), (4.63), and (4.70) by defining dimensionless temperature T* = T/To, where To is the
The differential equation of salt conservation for flowing seawater is where κ is a (constant) coefficient of diffusion,With typical units of square meters per second, and S is the salinity in
The differential energy equation for incompressible two-dimensional flow through a “Darcy-type” porous medium is approximately Where σ is the permeability of the porous medium, all other
A model differential equation, for chemical reaction dynamics in a plug reactor, is as follows:Where φ is the velocity potential and a is the (variable) speed of sound of the gas.
The differential equation for compressible in viscid flow of a gas in the xy plane isWhere φ is the velocity potential and a is the (variable) speed of sound of the gas. Non-dimensionalize this
The differential equation for small-amplitude vibrations y(x, t) of a simple beam is given byWhere ρ = beam material densityA = cross-sectional areaI = area moment of inertiaE =
A smooth steel (SG = 7.86) sphere is immersed in a stream of ethanol at 20°C moving at 1.5 m/s. Estimate its drag in N from Fig. 5.3a. What stream velocity would quadruple its drag? Take D = 2.5 cm.
The sphere in Prob. 5.48 is dropped in gasoline at 20°C. Ignoring its acceleration phase, what will its terminal (constant) fall velocity be, from Fig. 5.3a?
When a microorganism moves in a viscous fluid, inertia (fluid density) has a negligible influence on the organism’s drag force. These are called creeping flows. The only important parameters are
A ship is towing a sonar array which approximates a submerged cylinder 1 ft in diameter and 30 ft long with its axis normal to the direction of tow. If the tow speed is 12 kn (1 kn = 1.69 ft/s),
A 1-in-diameter telephone wire is mounted in air at 20°C and has a natural vibration frequency of 12 Hz. What wind velocity in ft/s will cause the wire to sing? At this condition what will the
Vortex shedding can be used to design a vortex flow meter (Fig. 6.34). A blunt rod stretched across the pipe sheds vortices whose frequency is read by the sensor downstream. Suppose the pipe diameter
A fishnet is made of 1-mm-diameter strings knotted into 2 × 2 cm squares. Estimate the horsepower required to tow 300 ft2 of this netting at 3 kn in seawater at 20°C. The net plane is normal to the
The radio antenna on a car begins to vibrate wildly at 8 Hz when the car is driven at 45 mi/h over a rutted road which approximates a sine wave of amplitude 2 cm and wavelength λ = 2.5 m. The
Flow past a long cylinder of square cross-section results in more drag than the comparable round cylinder. Here are data taken in a water tunnel for a square cylinder of side length b = 2 cm: V,
The simply supported 1040 carbon steel rod of Fig P5.57 is subjected to a cross flow stream of air at 20°C and 1 atm, for what stream velocity U will the rod center deflection be approximately 1
For the steel rod of Prob. 5.57, at what airstreams velocity U will the rod begin to vibrate laterally in resonance in its first mode (a half sine wave)? (Hint: Consult a vibration text under
Modify Prob. 5.55 as follows. If the circular antenna is steel, with L = 60 cm, and the car speed is 45 mi/h, what rod diameter would cause the natural vibration frequency to equal the shedding
A prototype water pump has an impeller diameter of 2 ft and is designed to pump 12 ft3/s at 750 r/min. A 1-ft-diameter model pump is tested in 20°C air at 1800 r/min, and Reynolds-number effects are
If viscosity is neglected, typical pump flow results are shown in Fig. P5.61 for a model pump tested in water. The pressure rise decreases and the power required increases with the dimensionless flow
Modify Prob. 5.61 so that the rotation speed is unknown but D = 12 cm and Q = 25 m3/h. What is the maximum rotation speed for which the power will not exceed 300 W? What will the pressure rise be for
The pressure drop per unit length Δp/L in smooth pipe flow is known to be a function only of the average velocity V, diameter D, and fluid properties ρ and μ. The following data were
The natural frequency ω of vibration of a mass M attached to a rod, as in Fig. P5.64 depends only upon M and the stiffness EI and length L of the rod. Tests with a 2-kg mass attached to a 1040
In turbulent flow near a flat wall, the local velocity u varies only with distance y from the wall, wall shear stress τw, and fluid properties ρ and μ. The following data were taken in
A torpedo 8 m below the surface in 20°C seawater cavitates at a speed of 21 m/s when atmospheric pressure is 101 kPa. If Reynolds-number and Froude-number effects are negligible, at what speed will
A student needs to measure the drag on a prototype of characteristic length dp moving at velocity Up in air at sea-level conditions. He constructs a model of characteristic length dm, such that the
Consider viscous flow over a very small object. Analysis of the equations of motion shows that the inertial terms are much smaller than viscous and pressure terms. Fluid density drops out, and these
A simple flow-measurement device for streams and channels is a notch, of angle α, cut into the side of a dam, as shown in Fig. P5.69. The volume flow Q depends only on α, the acceleration
A diamond-shaped body, of characteristic length 9 in, has the following measured drag forces when placed in a wind tunnel at sea-level standard conditions: V, ft/s: 30 38 48 56 61 F, lbf:
The pressure drop in a venturi meter (Fig P3.165) varies only with the fluid density, pipe approach velocity, and diameter ratio of the meter. A model venturi meter tested in water at 20°C shows a
A one-fifteenth-scale model of a parachute has a drag of 450 lbf when tested at 20 ft/s in a water tunnel. If Reynolds-number effects are negligible, estimate the terminal fall velocity at 5000-ft
The power P generated by a certain windmill design depends upon its diameter D, the air density ρ, the wind velocity V, the rotation rate Ω, and the number of blades n. (a) Write this
A one-tenth-scale model of a supersonic wing tested at 700 m/s in air at 20°C and 1 atm shows a pitching moment of 0.25 kN•m. If Reynolds-number effects are negligible, what will the pitching
A one-twelfth-scale model of an airplane is to be tested at 20°C in a pressurized wind tunnel. The prototype is to fly at 240 m/s at 10-km standard altitude. What should the tunnel pressure be in
A 2-ft-long model of a ship is tested in a freshwater tow tank. The measured drag may be split into “friction” drag (Reynolds scaling) and “wave” drag (Froude scaling). The model data are as
A dam spillway is to be tested by using Froude scaling with a one-thirtieth-scale model. The model flow has an average velocity of 0.6 m/s and a volume flow of 0.05 m3/s. What will the velocity and
A prototype spillway has a characteristic velocity of 3 m/s and a characteristic length of 10 m. A small model is constructed by using Froude scaling. What is the minimum scale ratio of the model
An East Coast estuary has a tidal period of 12.42 h (the semidiurnal lunar tide) and tidal currents of approximately 80 cm/s. If a one-five-hundredth-scale model is constructed with tides driven by a
A prototype ship is 35 m long and designed to cruise at 11 m/s (about 21 kn). Its drag is to be simulated by a 1-m-long model pulled in a tow tank. For Froude scaling find (a) the tow speed, (b) the
An airplane, of overall length 55 ft, is designed to fly at 680 m/s at 8000-m standard altitude. A one-thirtieth-scale model is to be tested in a pressurized helium wind tunnel at 20°C. What is the
A prototype ship is 400 ft long and has a wetted area of 30,000 ft2. A one-eightieth scale model is tested in a tow tank according to Froude scaling at speeds of 1.3, 2.0, and 2.7 kn (1 kn = 1.689
A one-fortieth-scale model of a ship’s propeller is tested in a tow tank at 1200 r/min and exhibits a power output of 1.4 ft•lbf/s. According to Froude scaling laws, what should the revolutions
A prototype ocean-platform piling is expected to encounter currents of 150 cm/s and waves of 12-s period and 3-m height. If a one-fifteenth-scale model is tested in a wave channel, what current
Solve Prob. 5.49, using the modified sphere-drag plot of Fig. 5.11.
Solve Prob. 5.50, using the modified sphere-drag plot of Fig. 5.11.
In Prob. 5.62 it was difficult to solve for Ω because it appeared in both power and flow coefficients. Rescale the problem, using the data of Fig. P5.61, to make a plot of dimensionless power
Modify Prob. 5.62 as follows: Let Ω = 32 r/s and Q = 24 m3/h for a geometrically similar pump what is the maximum diameter if the power is not to exceed 340 w solve this problem by rescaling the

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