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Questions and Answers of Mechanical Engineering

An open water jet exits from a nozzle into sea-level air, as shown, and strikes a stagnation tube. If the centerline pressure at section (1) is 110 kPa and losses are neglected, estimate (a) the mass
A venturi meter, shown in Fig P3.165, is a carefully designed constriction whose pressure difference is a measure of the flow rate in a pipe. Using Bernoulli€™s equation for steady
A wind tunnel draws in sea-level standard air from the room and accelerates it into a 1-m by 1-m test section. A pressure transducer in the test section wall measures Δp = 45 mm water between
In Fig P3.167 the fluid is gasoline at 20°C at a weight flux of 120 N/s. Assuming no losses, estimate the gage pressure at section 1.
In Fig P3.168 both fluids are at 20°C. If V1 = 1.7 ft/s and losses are neglected, what should the manometer reading h ft be?
Once it has been started by sufficient suction, the siphon in Fig. P3.169 will run continuously as long as reservoir fluid is available. Using Bernoulli€™s equation with no losses, show (a)
If losses are neglected in Fig P3.170, for what water level h will the flow begin to form vapor cavities at the throat of the nozzle?
For the 40°C water flow in Fig P3.171, estimate the volume flow through the pipe, assuming no losses; then explain what is wrong with this seemingly innocent question. If the actual flow rate is
The 35°C water flow of Fig P3.172 discharges to sea-level standard atmosphere. Neglecting losses, for what nozzle diameter D will cavitation begin to occur? To avoid cavitation, should you increase
The horizontal wye fitting in Fig P3.173 splits the 20°C water flow rate equally, if Q1 = 5 ft3/s and p1 = 25 lbf/in2 (gage) and losses are neglected, estimate (a) p2, (b) p3, and (c) the vector
In Fig P3.174 the piston drives water at 20°C. Neglecting losses, estimate the exit velocity V2 ft/s. If D2 is further constricted, what is the maximum possible value of V2?
If the approach velocity is not too high, a hump in the bottom of a water channel causes a dip Δh in the water level, which can serve as a flow measurement. If, as shown in Fig P3.175, Δh
In the spillway flow of Fig P3.176, the flow is assumed uniform and hydrostatic at sections 1 and 2. If losses are neglected, compute (a) V2 and (b) the force per unit width of the water on the
For the water-channel flow of Fig P3.177, h1 = 1.5 m, H = 4 m, and V1 = 3 m/s. Neglecting losses and assuming uniform flow at sections 1 and 2, find the downstream depth h2, and show that two
For the water channel flow of Fig P3.178, h1 = 0.45 ft, H = 2.2 ft, and V1 = 16 ft/s. Neglecting losses and assuming uniform flow at sections 1 and 2, find the downstream depth h2. Show that two
A cylindrical tank of diameter D contains liquid to an initial height ho. At time t = 0 a small stopper of diameter d is removed from the bottom. Using Bernoulli€™s equation with no losses,
The large tank of incompressible liquid in Fig P3.180 is at rest when, at t = 0, the valve is opened to the atmosphere. Assuming h ≈ constant (negligible velocities and accelerations in the
Modify Prob. 3.180 as follows. Let the top of the tank be enclosed and under constant gage pressure po. Repeat the analysis to find V(t) in the pipe.
The incompressible-flow form of Bernoulli’s relation, Eq. (3.77), is accurate only for Mach numbers less than about 0.3. At higher speeds, variable density must be accounted for. The most common
The pump in Fig P3.183 draws gasoline at 20°C from a reservoir. Pumps are in big trouble if the liquid vaporizes (cavitates) before it enters the pump. (a) Neglecting losses and assuming a flow
For the system of Prob. 3.183, let the pump exhaust gasoline at 65 gal/min to the atmosphere through a 3-cm-diameter opening, with no cavitation, when x = 3 m, y = 2.5 m, and z = 2 m. If the friction
Water at 20°C flows through a vertical tapered pipe at 163 m3/h, the entrance diameter is 12 cm, and the pipe diameter reduces by 3 mm for every 2 meter rise in elevation. For frictionless flow,
An idealized velocity field is given by the formula V=4txi−2t2yj+4xzk Is this flow field steady or unsteady? Is it two- or three-dimensional? At the point (x, y, z) = (–1, +1, 0), compute
Flow through the converging nozzle in Fig. P4.2 can be approximated by the one-dimensional velocity distribution u ≈ Vo (1 + 2x/L) v ≈ 0 w ≈ 0(a) Find a general expression for the
A two-dimensional velocity field is given by in arbitrary units. V = (x2 – y2 + x)i – (2xy + y)j At (x, y) = (1, 2), compute (a) The accelerations ax and ay, (b) The velocity component in
Suppose that the temperature field T = 4x2 – 3y3, in arbitrary units, is associated with the velocity field of Prob. 4.3. Compute the rate of change dT/dt at (x, y) = (2, 1).
The velocity field near a stagnation point (see Example 1.10) may be written in the form u = Uox / L v = -Uoy/L Uo and L are constants (a) Show that the acceleration vector is purely radial. (b)
Assume that flow in the converging nozzle of Fig. P4.2 has the form V =Vo (1 + 2x/L) i. Compute (a) The fluid acceleration at x = L; and (b) The time required for a fluid particle to travel from
Consider a sphere of radius R immersed in a uniform stream Uo, as shown in Fig. P4.7. According to the theory of Chap. 8, the fluid velocity along streamline AB is given byv = ui = Uo (1+R3/x3) iFind
When a valve is opened, fluid flows in the expansion duct of Fig. P4.8 according to the approximationv = iU (1-x/2L) tanh Ut/LFind (a) the fluid acceleration at (x, t) = (L, L/U) and (b) the time for
An idealized incompressible flow has the proposed three-dimensional velocity distribution V = 4xy2i + f(y) j – zy2k Find the appropriate form of the function f(y) which satisfies the continuity
After discarding any constants of integration, determine the appropriate value of the unknown velocities u or v which satisfy the equation of two-dimensional incompressible continuity for: (a) u =
Derive Eq. (4.12b) for cylindrical coordinates by considering the flux of an incompressible fluid in and out of the elemental control volume in Fig. 4.2.
Spherical polar coordinates (r, θ, φ) are defined in Fig. P4.12. The cartesian transformations arex = r sinθ cosφy = r sinθ sinφz = r cosθDo not show that the
A two dimensional velocity field is given by u = - Ky/ x2 + y2 v = Kx/x2 + y2 Where K is constant does this field satisfy incompressible continuity? Transform these velocities to polar components
For incompressible polar-coordinate flow, what is the most general form of a purely circulatory motion, υθ = υθ (r, θ, t) and υ r = 0, which satisfies continuity?
What is the most general form of a purely radial polar-coordinate incompressible flow pattern, υ r = υ r(r, θ, t) and υθ = 0, which satisfies continuity?
After discarding any constants of integration, determine the appropriate value of the unknown velocities w or v which satisfy the equation of three-dimensional incompressible continuity for:(a)
A reasonable approximation for the two-dimensional incompressible laminar boundary layer on the flat surface in Fig P4.17 is u = U (2y/∂ - y2/∂2) for y Where ∂ ≈ Cx 1/2 , C =
A piston compresses gas in a cylinder by moving at constant speed V, as in Fig. P4.18. Let the gas density and length at t = 0 be ρo and Lo, respectively. Let the gas velocity vary linearly from
An incompressible flow field has the cylinder components υθ = Cr, υ z = K (R2 – r2), Υr = 0, where C and K are constants and r ≤ R, z ≤ L. Does this flow satisfy
A two-dimensional incompressible velocity field has u = K (1 – e–ay), for x ≤ L and 0 ≤ y ≤ ∞. What is the most general form of v(x, y) for which continuity is satisfied
Air flows under steady, approximately one-dimensional conditions through the conical nozzle in Fig. P4.21. If the speed of sound is approximately 340 m/s, what is the minimum nozzle-diameter ratio
A flow field in the x-y plane is described by u = Uo = constant, v = Vo = constant. Convert these velocities into plane polar coordinate velocities, vr and vθ.
A tank volume V contains gas at conditions (ρ0, p0, T0). At time t = 0 it is punctured by a small hole of area A. According to the theory of Chap. 9, the mass flow out of such a hole is
Reconsider Fig. P4.17 in the following general way, It is known that the boundary layer thickness δ(x) increases monotonically and that there is no slip at the wall (y = 0). Further, u(x, y)
An incompressible flow in polar coordinates is given byvr = Kcos (1-b/r2)vo = - Ksin θ (1 + br2)Does this field satisfy continuity? For consistency, what should the dimensions of constants K
Curvilinear, or streamline, coordinates are defined in Fig. P4.26, where n is normal to the streamline in the plane of the radius of curvature R. Show that Euler€™s frictionless momentum
A frictionless, incompressible steady-flow field is given by V = 2xyi – y2j in arbitrary units. Let the density be ρ0 = constant and neglect gravity. Find an expression for the pressure
If z is “up,” what are the conditions on constants a and b for which the velocity field u = ay, υ = bx, w = 0 is an exact solution to the continuity and Navier-Stokes equations for
Consider a steady, two-dimensional, incompressible flow of a Newtonian fluid with the velocity field u = –2xy, v = y2 – x2, and w = 0. (a) Does this flow satisfy conservation of mass? (b) Find
Show that the two-dimensional flow field of Example 1.10 is an exact solution to the incompressible Navier-Stokes equation. Neglecting gravity, compute the pressure field p(x, y) and relate it to the
According to potential theory (Chap.8) for the flow approaching a rounded two- dimensional body, as in Fig P4.31, the velocity approaching the stagnation point is given by u = U(1 €“
The answer to Prob. 4.14 is υθ = f(r) only. Do not reveal this to your friends if they are still working on Prob. 4.14. Show that this flow field is an exact solution to the Navier-Stokes
From Prob. 4.15 the purely radial polar-coordinate flow which satisfies continuity is υr = f (θ)/r, where f is an arbitrary function. Determine what particular forms of f(θ) satisfy
A proposed three-dimensional incompressible flow field has the following vector form: V = Kxi + Kyj – 2Kzk (a) Determine if this field is a valid solution to continuity and Navier-Stokes. (b)
From the Navier-Stokes equations for incompressible flow in polar coordinates (App. E for cylindrical coordinates), find the most general case of purely circulating motion υθ (r), vr =
A constant-thickness film of viscous liquid flows in laminar motion down a plate inclined at angle θ, as in Fig. P4.36. The velocity profile is u = Cy(2h €“ y) v = w = 0 Find the
A viscous liquid of constant density and viscosity falls due to gravity between two parallel plates a distance 2h apart, as in the figure. The flow is fully developed, that is, w = w(x) only. There
Reconsider the angular-momentum balance of Fig. 4.5 by adding a concentrated body couple Cz about the z axis [6]. Determine a relation between the body couple and shear stress for equilibrium. What
Problems involving viscous dissipation of energy are dependent on viscosity μ, thermal conductivity k, stream velocity Uo, and stream temperature to. Group these parameters into the dimensionless
Problems involving viscous dissipation of energy are dependent on viscosity μ, thermal conductivity k, stream velocity Uo, and stream temperature to. Group these parameters into the
The approximate velocity profile in Prob. 3.18 for steady laminar flow through a rectangular duct,u = umax [1 €“ (y/b) 2 ][1 €“ (z/h)2]Satisfies the no-slip condition and gave a
Suppose that we wish to analyze the rotating, partly-full cylinder of Fig. 2.23 as a spin-up problem, starting from rest and continuing until solid-body-rotation is achieved. What are the appropriate
For the draining liquid film of Fig P4.36, what are the appropriate boundary conditions (a) at the bottom y 0 and (b) at the surface y h?
Suppose that we wish to analyze the sudden pipe-expansion flow of Fig. P3.59, using the full continuity and Navier-Stokes equations, what are the proper boundary conditions to handle this problem?
Suppose that we wish to analyze the U-tube oscillation flow of Fig P3.96, using the full continuity and Navier-Stokes equations. What are the proper boundary conditions to handle this problem?
Fluid from a large reservoir at temperature to flows into a circular pipe of radius R. The pipe walls are wound with an electric-resistance coil which delivers heat to the fluid at a rate qw (energy
Given the incompressible flow V = 3yi + 2xj. Does this flow satisfy continuity? If so, find the stream function ψ(x, y) and plot a few streamlines, with arrows.
Consider the following two-dimensional incompressible flow, which clearly satisfies continuity: u = Uo = constant, v = Vo = constant Find the stream function ψ(r, θ) of this flow, that is,
Investigate the stream function ψ = K(x2 €“ y2), K = constant. Plot the streamlines in the full xy plane, find any stagnation points, and interpret what the flow could represent.
Investigate the polar-coordinate stream function ψ = Kr1/2sin 12 θ, K = constant. Plot the streamlines in the full xy plane, find any stagnation points, and interpret.
Investigate the polar-coordinate stream function ψ = Kr2/3sin (2θ/3), K = constant. Plot the streamlines in all except the bottom right quadrant, and interpret.
A two-dimensional, incompressible, frictionless fluid is guided by wedge-shaped walls into a small slot at the origin, as in Fig. P4.52 The width into the paper is b, and the volume flow rate is Q.
For the fully developed laminar-pipe-flow solution of Prob. 4.34, find the axisymmetric stream function ψ(r, z). Use this result to determine the average velocity V = Q/A in the pipe as a ratio
An incompressible stream function is defined by Ψ (x, y) = Y/L2 (3x2y - y3)Where U and L are (positive) constants, where in this chapter are the streamlines of this flow plotted? Use this
In spherical polar coordinates, as in Fig. P4.12, the flow is called axisymmetric if υθ ≡ 0 and ∂ /∂φ ≡ 0, so that υr = υr(r, θ) and υθ
Investigate the velocity potential φ = Kxy, K = constant. Sketch the potential lines in the full xy plane, find any stagnation points, and sketch in by eye the orthogonal streamlines. What could
A two-dimensional incompressible flow field is defined by the velocity components where V and L are constants, if they exist, find the stream function and velocity potential.
Consider the two-dimensional incompressible velocity potential φ = xy + x2 – y2. (a) Is it true that ∆2φ = 0, and, if so, what does this mean? (b) If it exists, find the stream
Liquid drains from a small hole in a tank, as shown in Fig. P4.60, such that the velocity field set up is given by υr ≈ 0, υz ≈ 0, υθ = ωR2/r, where z
Show that the linear Coquette flow between plates in Fig. 1.6 has a stream function but no velocity potential. Why is this so?
Find the two-dimensional velocity potential φ(r, θ) for the polar-coordinate flow pattern υ r = Q/r, υθ = K/r, where Q and K are constants.
A two-dimensional incompressible flow is defined by where K = constant. Is this flow irrotational? If so, find its velocity potential, sketch a few potential lines, and interpret the flow pattern.
A plane polar-coordinate velocity potential is defined by Find the stream function for this flow, sketch some streamlines and potential lines, and interpret the flow pattern.
A stream function for a plane, irrotational, polar-coordinate flow is ψ=Cθ−Klnr C and K=const Find the velocity potential for this flow. Sketch some streamlines and potential lines,
Find the stream function and plot some streamlines for the combination of a line source m at (x, y) = (0, +a) and an equal line source placed at (0, €“a).
Find the stream function and plot some streamlines for the combination of a counterclockwise line vortex K at (x, y) = (+a, 0) and an equal line vortex placed at (€“a, 0).
Take the limit of φ for the source-sink combination, Eq. (4.133), as strength m becomes large and distance a becomes small, so that (ma) = constant. What happens?
Find the stream function and plot some streamlines for the combination of a counterclockwise line vortex K at (x, y) = (+a, 0) and an opposite (clockwise) line vortex placed at (€“a, 0).
A coastal power plant takes in cooling water through a vertical perforated manifold, as in Fig. P4.72The total volume flow intake is 110 m3/s. Currents of 25 cm/s flow past the manifold, as shown.
A two-dimensional Rankine half body, 8 cm thick, is placed in a water tunnel at 20°C. The water pressure far upstream along the body centerline is 120 kPa. What is the nose radius of the
A small fish pond is approximated by a half-body shape, as shown in Fig. P4.74 Point O, which is 0.5 m from the left edge of the pond, is a water source delivering about 0.63 m3/s per meter of depth
Find the stream function and plot some streamlines for the combination of a line source 2m at (x, y) = (+a, 0) and a line source m at (€“a, 0). Are there any stagnation points in the flow
Air flows at 1.2 m/s along a flat wall when it meets a jet of air issuing from a slot at A. The jet volume flow is 0.4 m3/s per m of width into the paper, If the jet is approximated as a line source,
A tornado is simulated by a line sink m = €“1000 m2/s plus a line vortex K = +1600 m2/s. Find the angle between any streamline and a radial line, and show that it is independent of both r
We wish to study the flow due to a line source of strength m placed at position (x, y) = (0, +L), above the plane horizontal wall y = 0. Using Bernoulli€™s equation, find (a) the point(s)
Study the combined effect of the two viscous flows in Fig. 4.16. That is, find u(y) when the upper plate moves at speed V and there is also a constant pressure gradient (dp/dx). Is superposition
An oil film drains steadily down the side of a vertical wall, as shown. After an initial development at the top of the wall, the film becomes independent of z and of constant thickness. Assume that w
Modify the analysis of Fig. 4.17 to find the velocity vθ when the inner cylinder is fixed and the outer cylinder rotates at angular velocity Ω0. May this solution be added to Eq. (4.146)

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