Questions and Answers of Fundamentals Momentum Heat

When a doublet is added to a uniform stream so that the source part of the doublet faces the stream, a cylinder flow results. Plot the streamlines when the doublet is reversed so that the sink faces
A source of strength 1.5 m2/s at the origin is combined with a uniform stream moving at 9 m/s in the x direction. For the half-body that results, finda. The stagnation pointb. The body
A line vortex of strength K at (x, y) (0, α) is combined with opposite strength vortex at (0, α). Plot the streamline pattern and find the velocity that each vortex induces on the other vortex.
The stream function for an in-compressible, two dimensional flow field is ψ = 3x2y + y For this flow field, sketch several streamlines.
Sketch the streamlines and potential lines of the flow due to a line source of at (a, 0) plus an equivalent sink at (a, 0).
Calculate the total lift force on the Arctic hut shown below as a function of the location of the opening. The lift force results from the difference between the inside pressure and the outside
Determine the pressure gradient at the stagnation point of Problem 10.10(a).Data From Problem 10.10(a)For the velocity potentials given below, find the stream function and sketch the
In Problem 10.12, how far upstream does the flow from the source reach?Data From Problem 10.12For the case of a source at the origin with a uniform free stream plot the streamline ψ = 0.
For the case of a source at the origin with a uniform free stream plot the streamline ψ = 0.
The stream function for an in-compressible, two dimensional flow field is ψ = 2r3 sin 3θFor this flow field, plot several streamlines for 0 ≤ θ ≤ π/3.
Consider the process shown in the figure. A bulk gas stream containing 0.10 mole% of carbon monoxide (CO) gas, 2.0 mole% O2gas, and 97.9 mole% of CO2gas flows over a flat catalytic surface of length
For the velocity potentials given below, find the stream function and sketch the streamlines a. b.c. Зху?] 3 Ф%3D, 3. ху Ф — 0
At what point on the surface of the circular cylinder in a potential flow does the pressure equal the free-stream pressure?
In Problem 10.7, explain how one could obtain ∂vθ /∂θ at the stagnation point, using only r and ∂vr /∂r.Data From Problem 10.7For the flow about a cylinder, find the velocity variation
For the flow about a cylinder, find the velocity variation along the streamline leading to the stagnation point. What is the velocity derivative ∂vr/∂r at the stagnation point?
Make an analytical model of a tornado using an irrotational vortex (with velocity inversely proportional to distance from the center) outside a central core (with velocity directly proportional to
The velocity potential for a given two-dimensional flow field is ϕ = (5/3) x3 - 5xy2Show that the continuity equation is satisfied and determine the corresponding stream function.
In polar coordinates, the continuity equation for steady in-compressible flow becomes Derive equations (10-10), using this relation. 1 dve = 0 r д0 : (rv,) + - r дr 1 д Ur Ә Vө r д0 дr
Find the stream function for a flow with a uniform free stream velocity v∞.The free-stream velocity intersects the x axis at an angle α.
Determine the fluid rotation at a point in polar coordinates, using the method illustrated in Figure 10.1. УА B B Ду t + At Дr х
In polar coordinates, show that 1 ГО(rv) ди; ez дө Уху- дr
Consider steady, continuous, in-compressible, fully developed laminar flow of a Newtonian fluid in an infinitely long round pipe of diameter D inclined at an angle a. The fluid is not open to the
A wide moving belt passes through a container of a viscous liquid. The belt moves vertically upward with constant velocity vw, as illustrated in the figure. Because of viscous forces, the belt picks
Two immiscible fluids are flowing down between two flat, infinitely long flat parallel plates as shown in the figure below. The plate on the left is moving down with a velocity of vA, and the plate
Beginning with the appropriate form of the Navier€“ Stokes equations, develop an equation in the appropriate coordinate system to describe the velocity of a fluid that is flowing in the
Determine the velocity profile in a fluid situated between two coaxial rotating cylinders. Let the inner cylinder have radius R1, and angular velocity Ω1; let the outer cylinder have radius R2 and
For the flow described in Problem 8.13, obtain the differential equation of motion if vθ = f(r, t).
Using the Navier–Stokes equations in Appendix E, solve Problem 8.13.
Using the Navier–Stokes equations, find the differential equation for a radial flow in which vz = vθ = 0, and vr = f (r). Using continuity, show that the solution to the equation does not involve
Using the Navier–Stokes equations as given in Appendix E, work Problems 8.17 and 8.18.
Obtain the equations for one-dimensional inviscid, unsteady, compressible flow.
Obtain the equations for a one-dimensional steady, viscous, compressible flow in the x direction from the Navier– Stokes equations. (These equations, together with an equation of state and the
Using the laws for the addition of vectors and equation (9-19), show that in the absence of gravity,a. The fluid acceleration, pressure force, and viscous force all lie in the same planeb. In the
In polar coordinates, the continuity equation isShow thata. if vÎ¸ = 0, then vr = F(Î¸)/rb. if vr = 0, then vÎ¸ = f(r) |1 д 1 дуд (rv,) +- r д0 гдr
Derive equation (2-3) from equation (9-27). Dv = pg – VP Dt
Write equations (9-17) in component form for Cartesian coordinates. ду +V. (μV υ.) дх + V. OP Dvx = P8x P Dt Əx
In a velocity field where v = 400[(y/L)2ex + (x/L)2ey] fps, determine the pressure gradient at the point (L, 2L). The y axis is vertical, the density is 64.4 Ibm/ft3 and the flow may be considered
The atmospheric density may be approximated by the relation ρ = ρ0 exp(-y/β), where β = 22,000 ft. Determine the rate at which the density changes with respect to body falling at v fps. If v =
Does the velocity distribution in Example 2 satisfy continuity?
Using the Navier€“Stokes equations and the continuity equation, obtain an expression for the velocity profile between two flat, parallel plates.Continuity equation, дv0 ду дих дх
For flow at very low speeds and with large viscosity (the so-called creeping flows) such as occur in lubrication, it is possible to delete the inertia terms, Dv/Dt from the Navier- Stokes equation.
Find Dv/Dt in polar coordinates by taking the derivative of the velocity. (v = vr (r, θ, t)er + vθ (r, θ, t)eθ. Remember that the unit vectors have derivatives.)
In an in-compressible flow, the volume of the fluid is constant. Using the continuity equation, ∇ • v = 0, show that the fluid volume change is zero.
In Cartesian coordinates, show thatmay be written (v €¢ ˆ‡). What is the physical meaning of the term (v €¢ ˆ‡)? д д д + Vz д ду dy дх
Apply the law of conservation of mass to an element in a polar coordinate system and obtain the continuity equation for a steady, two-dimensional, in-compressible flow.
You have been asked to calculate the density of an in-compressible Newtonian fluid in steady flow that is flowing continuously at 250°F along a 2500-ft pipe with a constant diameter of 4 in. and a
Benzene, which is an in-compressible Newtonian fluid, flows steadily and continuously at 150°F through a 3000-ft pipe with a constant diameter of 4 in. with a volumetric flow rate of 3.5 ft3/s.
A Newtonian fluid in continuous, in-compressible laminar flow is moving steadily through a very long 700-m, horizontal pipe. The inside radius is 0.25 m for the entire length, and the pressure drop
Benzene flows steadily and continuously at 100°F through a 3000-ft horizontal pipe with a constant diameter of 4 in. The pressure drop across the pipe under these conditions is 300 lbf/ft2. Assuming
Determine the maximum film velocity in Problem 8-15.Data From Problem 8.15A viscous film drains uniformly down the side of a vertical rod of radius R. At some distance down the rod, the film
A viscous film drains uniformly down the side of a vertical rod of radius R. At some distance down the rod, the film approaches a terminal or fully developed flow such that the film thickness, h, is
Oil is supplied at the center of two long plates. The volumetric flow rate per unit length is Q and the plates remain a constant distance, b, apart. Determine the vertical force per unit length as a
The device in the schematic diagram on the next page is a viscosity pump. It consists of a rotating drum inside of a stationary case. The case and the drum are concentric. Fluid enters at A, flows
A continuous belt passes upward through a chemical bath at velocity v0 and picks up a film of liquid of thickness h, density, ρ, and viscosity μ. Gravity tends to make the liquid drain down, but
Derive the equation of motion for a one-dimensional, inviscid, unsteady compressible flow in a pipe of constant cross sectional area neglect gravity.
Fluid flows between two parallel plates, a distance h apart. The upper plate moves at velocity, v0; the lower plate is stationary. For what value of pressure gradient will the shear stress at the
Determine the velocity profile for fluid flowing between two parallel plates separated by a distance 2h. The pressure drop is constant.
Two immiscible fluids of different density and viscosity are flowing between two parallel plates. Express the boundary conditions at the interface between the two fluids.
The viscosity of heavy liquids, such as oils, is frequently measured with a device that consists of a rotating cylinder inside a large cylinder. The annular region between these cylinders is filled
A thin rod of diameter d is pulled at constant velocity through a pipe of diameter D. If the wire is at the center of the pipe, find the drag per unit length of wire. The fluid filling the space
Derive the expressions for the velocity distribution and for the pressure drop for a Newtonian fluid in fully developed laminar flow in the annular space between two horizontal, concentric pipes.
A common type of viscosimeter for liquids consists of a relatively large reservoir with a very slender outlet tube, the rate of outflow being determined by timing the fall in the surface level. If
A 0.635-cm hydraulic line suddenly ruptures 8 m from a reservoir with a gage pressure of 207 kPa. Compare the laminar and inviscid flow rates from the ruptured line in cubic meters per second.
A 40-km-long pipeline delivers petroleum at a rate of 4000 barrels per day. The resulting pressure drop is 3.45 × 106 Pa. If a parallel line of the same size is laid along the last 18 km of the
Express equation (8-9) in terms of the flow rate and the pipe diameter. If the pipe diameter is doubled at constant pressure drop, what percentage change will occur in the flow rate?Equation (8-9)
The figure below shows the geometry of a rheological experiment. A fluid lies between R0and Ri, where Ri= 16.00 mm and R0= 17.00 mm. The gap between the two cylinders can be modeled as two parallel
A thin coating is to be applied to both sides of a piece of thin plastic that is being mechanically transported. The plastic is 4.5- µm-thick and 0.0254-meters-wide and is very long. We want to coat
A Newtonian oil with a density of 60 lbm/ft3, viscosity of 0.206 Ã— 10-3Ibm/ft-s and kinematic viscosity of 0.342 Ã— 10-5ft2/s undergoes steady shear between a horizontal
The rate of shear work per unit volume is given by the product τv. For a parabolic velocity profile in a circular tube (see Example 4.2), determine the distance from the wall at which the shear work
What pressure drop per foot of tube is caused by the shear stress in Problem 7.17?Data From Problem 7.17For water flowing in a 0.1-in.-diameter tube, the velocity distribution is parabolic (see
For water flowing in a 0.1-in.-diameter tube, the velocity distribution is parabolic (see Example 4.2). If the average velocity is 2 fps, determine the magnitude of the shear stress at the tube wall.
The conical pivot shown in the figure has angular velocity Ï‰ and rests on an oil film of uniform thickness h. Determine the frictional moment as a function of the angle Î±,
A cross-flow molecular filtration device equipped with a mesoporous membrane is used to separate the enzyme lysozyme from a fermentation broth, as shown in the figure (right column). Water at
If the ram and auto rack in the previous problem together have a mass of 680 kg, estimate the maximum sinking speed of the ram and rack when gravity and viscous friction are the only forces acting.
An auto lift consists of 36.02-cm-diameter ram that slides in a 36.04-cm-diameter cylinder. The annular region is filled with oil having a kinematic viscosity of 0.00037 m2/s and a specific gravity
Two ships are traveling parallel to each other and are connected by flexible hoses. Fluid is transferred from one ship to the other for processing and then returned. If the fluid is flowing at 100
Gasoline from an under-storage storage tank leaked down onto an impermeable clay barrier and collected into a liquid pool. A simplified picture of the situation is provided in the figure below.
If the speed of the shaft is doubled in Problem 7.11, what will be the percentage increase in the heat transferred from the bearing? Assume that the bearing remains at constant temperature.
A flat steel plate of 2.0 m length and 2.0 m width initially contains a very thin coating of light hydrocarbon lubricating oil (species A) used in a manufacturing process. An engineer is considering
An automobile crankshaft is 3.175 cm in diameter. A bearing on the shaft is 3.183 cm in diameter and 2.8 cm long. The bearing is lubricated with SAE 30 oil at a temperature of 365 K. Assuming that
Repeat the preceding problem for air.Data From Problem 7.9According to the Hagen–Poiseuille laminar flow model, the volumetric flow rate is inversely proportional to the viscosity. What percentage
A well-mixed open pond contains wastewater that is contaminated with a dilute concentration of dissolved methylene chloride. The pond is rectangular with dimensions of 500 m by 100 m, as shown in the
According to the Hagen–Poiseuille laminar flow model, the volumetric flow rate is inversely proportional to the viscosity. What percentage change in volumetric flow rate occurs in a laminar flow as
At what temperature is the kinematic viscosity of glycerin the same as the kinematic viscosity of helium?
What is the percentage change in the viscosity of water when the water temperature rises from 60 to 120°F?
Calculate the viscosity of oxygen at 350 K and compare with the value given in Appendix I.
Estimate the viscosity of nitrogen at 175 K using equation (7-10). 28 · 183 1.200 · 10- µ = 2.6693 · 10-6. Pa · s (3.681)²(1.175)
Using a cylindrical element, show that Stokes€™s viscosity relation yields the following shear stress components: 1д0- Trо— Төr — и|r- дr rдө 1д0. дvв + поб) д. Теz —
Show that the axial strain rate in a one-dimensional flow, vx = vx (x), is given by ∂vx/∂x. What is the rate of volume change? Generalize for a three-dimensional element, and determine the rate
For a two-dimensional, in-compressible flow with velocity vx = vx(y), sketch a three-dimensional fluid element and illustrate the magnitude, direction, and surface of action of each stress component.
Sketch the deformation of a fluid element for the following cases:a. ∂vx/∂y is much larger than ∂vy/∂xb. ∂vy/∂x is much larger than ∂vx/∂y
In the development of the approximate solution for solving the laminar concentration boundary layer formed by fluid flow over a flat plate, it is necessary to assume a concentration profile. Equation
You have been asked to do an analysis of a pump that will transfer a fluid. The inlet to a pump has a diameter of 0.35 m, and the pressure to the pump is 2500 kg/m-s2. The outlet from the pump has a
Air flows steadily through a turbine that produces 3.5 105ft-lbf/s of work. Using the data below at the inlet and outlet, where the inlet is 10 feet below the outlet, please calculate the heat
You have been asked to do an analysis of a steady state pump that will transfer Aniline, an organic material, from one area to another in a chemical plant. Please determine the temperature change
In using the von KÃ¡rmÃ¡n approximate method for analyzing the turbulent boundary layer over a flat plate, the following velocity and concentration profiles were
Butyl alcohol, a Newtonian fluid, is being pumped at steady state with the density of 50 Ibm/ft3. viscosity of 1.80 × 10-3 Ibm/ft s, heat capacity of 0.58 Btu/lbm °F, and kinematic
A client has asked you to find the pressure change in a pumping station. The outlet from the pump is 20 ft above the inlet. A Newtonian fluid is being pumped at steady state. At the inlet to the pump
The tank in the previous problem feeds two lines, a 4-cm pipe that exits 10 m below the water level in the tank and a second line, also 4 cm in diameter runs from the tank to a station 20 m below the
Consider a 4-cm pipe that runs between a tank open to the atmosphere and a station open to the atmosphere 10 m below the water surface in the tank. Assuming friction less flow, what will be the mass

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