Incompressible fluid flows steadily through a plane diverging channel. At the inlet, of height H, the flow is uniform with magnitude V1. At the outlet, of height 2H, the velocity profile is
where y is measured from the channel centerline. Express Vm in terms of V1.
Trending nowThis is a popular solution!
Chapter 4 Solutions
Fox and McDonald's Introduction to Fluid Mechanics
Additional Engineering Textbook Solutions
Engineering Mechanics: Statics
INTERNATIONAL EDITION---Engineering Mechanics: Statics, 14th edition (SI unit)
Thinking Like an Engineer: An Active Learning Approach (4th Edition)
Introduction to Heat Transfer
Mechanics of Materials
Engineering Mechanics: Statics & Dynamics (14th Edition)
- 4. Consider a wide liquid film of constant thickness h flowing steadily due to gravity down an inclined plane at angle 0. AS shown in below figure. The atmosphere exerts constant pressure and negligible shear on the free surface. Show that the velocity distribution is given by pg sine u y(2h-y) 2µ and that the volume flow rate per unit width is Q pgh3 sin 0 /3µ. u(y)arrow_forwardB/ Two components of velocity in an incompressible fluid flow are given by v=z³y³ determine the third component. u= x² - y³ and Do the velocity field U = 5xi + (3y + ty2)j represent physically possible flow?arrow_forward2. Water flows in a two-dimensional channel of width W and depth D as shown in the diagram. The hypothetical velocity profile for the water is y? 1- D² 4x2 V (x,y)=|V, 1 where V, is the velocity at the water surface midway between the channel walls. The coordinate system is as shown; x is measured from the center plane of the channel and y downward from the water surface. Find the discharge in the channel in terms of V, D, and W.arrow_forward
- In a tapered horizontal pipeline, the seawater's speed is 3.75 m/s and the gauge pressure is 21 kPa at the first point. Find the gauge pressure at a second point in the line if the cross-sectional area at the second point is thrice that at the first.arrow_forwardIn the pipe below, an incompressible fluid flows. If the radius at A₁ is twice the at A2, V₁ V2. 2 V2 4 V2 V2/2 V2/4 A₁ V₁ A₂ 2 V₂arrow_forwardA venturi meter is used to measure the flow speed of a fluid in a pipe. The meter is connected between two sections of the pipe (the figure); the cross-sectional area A of the entrance and exit of the meter matches the pipe's cross-sectional area. Between the entrance and exit, the fluid flows from the pipe with speed V and then through a narrow ''throat'' of cross-sectional area a with speed v. A manometer connects the wider portion of the meter to the narrower portion. The change in the fluid's speed is accompanied by a change Δp in the fluid's pressure, which causes a height difference h of the liquid in the two arms of the manometer. (Here Δp means pressure in the throat minus pressure in the pipe.) Let A equal 5·a. Suppose the pressure p1 at A is 2.1 atm. Compute the values of (a) the speed V at A and (b) the speed v at a that make the pressure p2 at a equal to zero. (c) Compute the corresponding volume flow rate if the diameter at A is 4.0 cm.The phenomenon that occurs at a when…arrow_forward
- An incompressible fluid flows between two infinite stationary parallel plates. The velocity profile is given by u =umax (Ay² + By + C), where A, B, and C are constants and y is measured upward from the lower plate. The total gap width is h. Use appropriate boundary conditions to express the constants in terms of h. Develop an expression for volume flow rate per unit depth and evaluate the ratio V/umax.arrow_forward3. Consider the one-dimensional, incompressible flow through the circular channel shown. The velocity at section 1 is given by U=Uo+ U₁ sin(t), where Uo= 20 m/s, U₁ = 2 m/s, and co-0.3 rad/s. The channel dimensions are L- 1 m, R₁-0.4 m, and R₂-0.2 m. Determine the particle acceleration at the channel exit. Plot the results as a function of time over a complete cycle. On the same plot, show the acceleration at the channel exit if the channel is constant area, rather than convergent, and explain the difference between the curves. T R₁ ✓ X1 L X2arrow_forwardA venturi meter is used to measure the flow speed of a fluid in a pipe. The meter is connected between two sections of the pipe (the figure); the cross-sectional area A of the entrance and exit of the meter matches the pipe's cross-sectional area. Between the entrance and exit, the fluid flows from the pipe with speed Vand then through a narrow "throat" of cross-sectional area a with speed v. A manometer connects the wider portion of the meter to the narrower portion. The change in the fluid's speed is accompanied by a change Ap in the fluid's pressure, which causes a height difference h of the liquid in the two arms of the manometer. (Here Ap means pressure in the throat minus pressure in the pipe.) Suppose that the fluid is fresh water, that the cross- sectional areas are 93 cm2 in the pipe and 31 cm² in the throat, and that the pressure is 53 kPa in the pipe and 44 kPa in the throat. What is the rate of water flow in cubic meters per second? Meter Meter entrance Venturi meter exit A…arrow_forward
- A brand of toothpaste is made using a mixture of white and red paste. During the manufacturing process, both pastes are pumped through a pipe, with the red paste lying in the center of the pipe, as shown in Figure Q2. The pipe has an internal radius, R2, and the segment of red paste has a radius R1. The flow is found to follow the Poiseuille equation, i.e. 1 dp u(r) = -- 4µdx (R²2-²) Where u is the viscosity of the paste and dp dx is the pressure gradient. The total flow of toothpaste through the pipe is Qtot, and the flowrate of the red paste is Qred. The toothpaste has a density of p = 1200 kg/m3, and a viscosity of 0.03 Pa.s. The pipe has a radius R2 = 5 cm and a length L = 2 m. a) Find an expression for the flowrate of the red toothpaste. You answer should be in terms of µ, R1, R2, dp dx) b) The flow in the pipe becomes turbulent when the Reynolds number reaches 2000 (where Re is calculated using the mean velocity across the pipe). What is the maximum pressure drop along the pipe…arrow_forwardA brand of toothpaste is made using a mixture of white and red paste. During the manufacturing process, both pastes are pumped through a pipe, with the red paste lying in the centre of the pipe, as shown in Figure Q2. The pipe has an internal radius, R₂, and the segment of red paste has a radius R₁. The flow is found to follow the Poiseuille equation, i.e. u(r) = 1 dp - (R² - r²) dp Where u is the viscosity of the paste and μ is the pressure gradient. The total flow of toothpaste through the pipe is Qtot, and the flowrate of the red paste is Qred. dx 4μ dx The toothpaste has a density of p = 1200 kg/m³, and a viscosity of 0.03 Pa.s. The pipe has a radius R₂ = 5 cm and a length L = 2 m. R₁ a) Find an expression for the flowrate of the red toothpaste. You answer should be in terms of µ, R₁, R₂, P) b) The flow in the pipe becomes turbulent when the Reynolds number reaches 2000 (where Re is calculated using the mean velocity across the pipe). What is the maximum pressure drop along the…arrow_forwardA pipe of diameter 10cm conveying 200 liters/s of water has bend of angle (θ = 90⁰) and θ = 0⁰ through horizontal plane. Find theresultant force exerted on the bend if the pressure at the inlet and outlet of the bend are 15N/cm2 and 10 N/cm2respectivelyarrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY