Fox and McDonald's Introduction to Fluid Mechanics
9th Edition
ISBN: 9781118912652
Author: Philip J. Pritchard, John W. Mitchell
Publisher: WILEY
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Textbook Question
Chapter 4, Problem 49P
A conical enlargement in a vertical pipeline is 5 ft long and enlarges the pipe diameter from 12 in. to 24 in. Calculate the magnitude and direction of the vertical force on this enlargement when 10 cfs of water flow upward through the line and the pressure at the smaller end of the enlargement is 30 psi.
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Problem 2 (55 pts). We now consider the FEM solution of Problem 1.(a) [5pts] Briefly describe the 4 steps necessary to obtain the approximate solution of thatBVP using the Galerkin FEM. Use the minimum amount of math necessary to supportyour explanations.(b) [20pts] Derive the weak form of the BVP.(c) [10pts] Assuming a mesh of two equal elements and linear shape functions, sketch byhand how you expect the FEM solution to look like. Also sketch the analytical solutionfor comparison. In your sketch, identify the nodal degrees of freedom that the FEMsolution seeks to find.(d) [10pts] By analogy with the elastic rod problem and heat conduction problem considered in class, write down the stiffness matrix and force vector for each of the twoelements considered in (c).(e) [10pts] Assemble the global system of equations, and verbally explain how to solve it.
An aluminum rod of length L = 1m has mass density ρ = 2700 kgm3 andYoung’s modulus E = 70GPa. The rod is fixed at both ends. The exactnatural eigenfrequencies of the rod are ωexactn =πnLqEρfor n=1,2,3,. . . .1. What is the minimum number of linear elements necessary todetermine the fundamental frequency ω1 of the system? Discretizethe rod in that many elements of equal length, assemble the globalsystem of equations KU = ω2MU, and find the fundamentalfrequency ω1. Compute the relative error e1 = (ω1 − ωexact1)/ωexact1.Sketch the fundamental mode of vibration.
Chapter 4 Solutions
Fox and McDonald's Introduction to Fluid Mechanics
Ch. 4 - An ice-cube tray containing 250 mL of freshwater...Ch. 4 - A hot air balloon with an initial volume of 2600...Ch. 4 - A fully loaded Boeing 777-200 jet transport...Ch. 4 - On the Milford Trek in New Zealand, there is a...Ch. 4 - A high school experiment consists of a block of...Ch. 4 - For a small particle of styrofoam (density = 19.2...Ch. 4 - Air at 20C and an absolute pressure of 101.3 kpa...Ch. 4 - A block of copper of mass 5 kg is heated to 90C...Ch. 4 - The average rate of heat loss from a person to the...Ch. 4 - The velocity field in the region shown is given by...
Ch. 4 - The area shown shaded is in a flow where the...Ch. 4 - Obtain an expression for the kinetic energy flux,...Ch. 4 - A 0.3 m by 0.5 m rectangular air duct carries a...Ch. 4 - Across a shock wave in a gas flow there is a great...Ch. 4 - Water flows in a pipeline composed of 75-mm and...Ch. 4 - The velocity distribution for laminar flow in a...Ch. 4 - A farmer is spraying a liquid through 10 nozzles,...Ch. 4 - A university laboratory that generates 15 m3/s of...Ch. 4 - Hydrogen is being pumped through a pipe system...Ch. 4 - Calculate the mean velocities for these...Ch. 4 - If the velocity profile in a passage of width 2R...Ch. 4 - Fluid with 1040 kg/m3 density is flowing steadily...Ch. 4 - A rice farmer needs to fill a 150 m 400 m field...Ch. 4 - In your kitchen, the sink is 60 cm by 45.7 cm. by...Ch. 4 - Fluid passes through this set of thin closely...Ch. 4 - A pipeline 0.3 m in diameter divides at a Y into...Ch. 4 - A manifold pipe of 3 in. diameter has four...Ch. 4 - You are trying to pump storm water out of your...Ch. 4 - In the incompressible flow through the device...Ch. 4 - Water enters a wide, flat channel of height 2h...Ch. 4 - Find the average efflux velocity V if the flow...Ch. 4 - Find V for this mushroom cap on a pipeline. P4.32Ch. 4 - Incompressible fluid flows steadily through a...Ch. 4 - A two-dimensional reducing bend has a linear...Ch. 4 - Water enters a two-dimensional, square channel of...Ch. 4 - Viscous liquid from a circular tank. D = 300 mm in...Ch. 4 - A rectangular tank used to supply water for a...Ch. 4 - A cylindrical tank, 0.3 m in diameter, drains...Ch. 4 - Air enters a tank through an area of 0.018 m2 with...Ch. 4 - A cylindrical tank, of diameter D = 50 mm, drains...Ch. 4 - A conical flask contains water to height H = 36.8...Ch. 4 - Water flows steadily past a porous flat plate....Ch. 4 - A tank of fixed volume contains brine with initial...Ch. 4 - A conical funnel of half-angle = 30 drains...Ch. 4 - Evaluate the net rate of flux of momentum out...Ch. 4 - Water flows steadily through a pipe of length L...Ch. 4 - Evaluate the net momentum flux through the bend of...Ch. 4 - Evaluate the net momentum flux through the channel...Ch. 4 - A conical enlargement in a vertical pipeline is 5...Ch. 4 - A 100-mm nozzle is bolted (with 6 bolts) to the...Ch. 4 - The projectile partially fills the end of the 0.3...Ch. 4 - Considering that in the fully developed region of...Ch. 4 - A jet of water issuing from a stationary nozzle at...Ch. 4 - A circular cylinder inserted across a stream of...Ch. 4 - A 6-in.-diameter horizontal pipeline bends through...Ch. 4 - The axes of the pipes are in a vertical plane. The...Ch. 4 - Water flows through a tee in a horizontal pipe...Ch. 4 - In a laboratory experiment, the water flow rate is...Ch. 4 - A gate is 1 m wide and 1.2 m tall and hinged at...Ch. 4 - Water flows steadily through a fire hose and...Ch. 4 - Two types of gasoline are blended by passing them...Ch. 4 - A circular cylinder inserted across a stream of...Ch. 4 - The pressure difference results from head loss...Ch. 4 - Obtain expressions for the rate of change in mass...Ch. 4 - Water is flowing steadily through the 180 elbow...Ch. 4 - Water flows steadily through the nozzle shown,...Ch. 4 - The pump, suction pipe, discharge pipe, and nozzle...Ch. 4 - The passage is 1.2 m wide normal to the paper....Ch. 4 - If the two-dimensional flow rate through this...Ch. 4 - Assume the bend of Problem 4.35 is a segment of a...Ch. 4 - A flat plate orifice of 50 mm diameter is located...Ch. 4 - At rated thrust, a liquid-fueled rocket motor...Ch. 4 - Flow from the end of a two-dimensional open...Ch. 4 - Calculate the magnitude and direction of the...Ch. 4 - This water jet of 50 mm diameter moving at 30 m/s...Ch. 4 - If the splitter is removed from the plate of...Ch. 4 - Consider flow through the sudden expansion shown....Ch. 4 - A conical spray head is shown. The fluid is water...Ch. 4 - A curved nozzle assembly that discharges to the...Ch. 4 - The pump maintains a pressure of 10 psi at the...Ch. 4 - A motorboat moves up a river at a speed of 9 m/s...Ch. 4 - A 30 reducing elbow is shown. The fluid is water....Ch. 4 - A monotube boiler consists of a 6 m length of...Ch. 4 - Water is discharged at a flow rate of 0.3m3/s from...Ch. 4 - A nozzle for a spray system is designed to produce...Ch. 4 - The horizontal velocity in the wake behind an...Ch. 4 - An incompressible fluid flows steadily in the...Ch. 4 - Consider the incompressible flow of fluid in a...Ch. 4 - Air at standard conditions flows along a flat...Ch. 4 - Gases leaving the propulsion nozzle of a rocket...Ch. 4 - Two large tanks containing water have small...Ch. 4 - Students are playing around with a water hose....Ch. 4 - A 2-kg disk is constrained horizontally but is...Ch. 4 - A stream of water from a 50-mm-diameter nozzle...Ch. 4 - A plane nozzle discharges vertically 1200 L/s per...Ch. 4 - In ancient Egypt, circular vessels filled with...Ch. 4 - Incompressible fluid of negligible viscosity is...Ch. 4 - The narrow gap between two closely spaced circular...Ch. 4 - Design a clepsydra (Egyptian water clock), which...Ch. 4 - Water from a stationary nozzle impinges on a...Ch. 4 - A freshwater jet boat takes in water through side...Ch. 4 - The Canadair CL-215T amphibious aircraft is...Ch. 4 - Water, in a 100-mm-diameter jet with speed of 30...Ch. 4 - Consider a series of turning vanes struck by a...Ch. 4 - A steady jet of water is used to propel a small...Ch. 4 - The cart of Problem 4.105 is accelerated by a jet...Ch. 4 - A vane/slider assembly moves under the influence...Ch. 4 - A cart is propelled by a liquid jet issuing...Ch. 4 - For the vane/slider problem of Problem 4.107, find...Ch. 4 - If the cart of Problem 4.105 is released at t = 0,...Ch. 4 - The wheeled cart shown rolls with negligible...Ch. 4 - A rocket sled is to be slowed from an initial...Ch. 4 - Starting from rest, the cart shown is propelled by...Ch. 4 - Solve Problem 4.107 if the vane and slider ride on...Ch. 4 - For the vane/slider problem of Problem 4.114, plot...Ch. 4 - A rectangular block of mass M, with vertical...Ch. 4 - A vertical jet of water impinges on a horizontal...Ch. 4 - A rocket sled traveling on a horizontal track is...Ch. 4 - A rocket sled accelerates from rest on a level...Ch. 4 - A rocket sled with initial mass of 900 kg is to be...Ch. 4 - A rocket sled with initial mass of 3 metric tons,...Ch. 4 - A home-made solid propellant rocket has an initial...Ch. 4 - Neglecting air resistance, what speed would a...Ch. 4 - The moving tank shown is to be slowed by lowering...Ch. 4 - The 90 reducing elbow of Example 4.6 discharges to...Ch. 4 - Crude oil (SG = 0:95) from a tanker dock flows...Ch. 4 - The simplified lawn sprinkler shown rotates in the...Ch. 4 - Calculate the torque about the pipes centerline in...Ch. 4 - A fire truck is equipped with a 66 ft long...Ch. 4 - Calculate the torque exerted on the flange joint...Ch. 4 - Consider the sprinkler of Problem 4.130 again....Ch. 4 - A small lawn sprinkler is shown. The sprinkler...Ch. 4 - When a garden hose is used to fill a bucket, water...Ch. 4 - A pipe branches symmetrically into two legs of...Ch. 4 - Compressed air is stored in a pressure bottle with...Ch. 4 - A turbine is supplied with 0.6 m3/s of water from...Ch. 4 - Air is drawn from the atmosphere into a...Ch. 4 - At high speeds the compressor and turbine of the...Ch. 4 - Transverse thrusters are used to make large ships...Ch. 4 - All major harbors are equipped with fire boats for...Ch. 4 - A pump draws water from a reservoir through a...Ch. 4 - Liquid flowing at high speed in a wide, horizontal...
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- Problem 1 (65 pts, suggested time 50 mins). An elastic string of constant line tension1T is pinned at x = 0 and x = L. A constant distributed vertical force per unit length p(with units N/m) is applied to the string. Under this force, the string deflects by an amountv(x) from its undeformed (horizontal) state, as shown in the figure below.The PDE describing mechanical equilibrium for the string isddx Tdvdx− p = 0 . (1)(a) [5pts] Identify the BCs for the string and identify their type (essential/natural). Writedown the strong-form BVP for the string, including PDE and BCs.(b) [10pts] Find the analytical solution of the BVP in (a). Compute the exact deflectionof the midpoint v(L/2).(c) [15pts] Derive the weak-form BVP.(d) [5pts] What is the minimum number of linear elements necessary to compute the deflection of the midpoint?(e) [15pts] Write down the element stiffness matrix and the element force vector for eachelement.arrow_forwardProblem 1 (35 pts). An elastic string of constant line tension1 T is pinned at x = 0 andx = L. A constant distributed vertical force per unit length p (with units N/m) is appliedto the string. Under this force, the string deflects by an amount v(x) from its undeformed(horizontal) state, as shown in the figure below.Force equilibrium in the string requires thatdfdx − p = 0 , (1)where f(x) is the internal vertical force in the string, which is given byf = Tdvdx . (2)(a) [10pts] Write down the BVP (strong form) that the string deflection v(x) must satisfy.(b) [2pts] What order is the governing PDE in the BVP of (a)?(c) [3pts] Identify the type (essential/natural) of each boundary condition in (a).(d) [20pts] Find the analytical solution of the BVP in (a).arrow_forwardProblem 2 (25 pts, (suggested time 15 mins). An elastic string of line tension T andmass per unit length µ is pinned at x = 0 and x = L. The string is free to vibrate, and itsfirst vibration mode is shown below.In order to find the frequency of the first mode (or fundamental frequency), the string isdiscretized into a certain number of linear elements. The stiffness and mass matrices of thei-th element are, respectivelyESMi =TLi1 −1−1 1 EMMi =Liµ62 11 2 . (2)(a) [5pts] What is the minimum number of linear elements necessary to compute the fundamental frequency of the vibrating string?(b) [20pts] Assemble the global eigenvalue problem and find the fundamental frequency ofvibration of the stringarrow_forward
- I need part all parts please in detail (including f)arrow_forwardProblem 3 (10 pts, suggested time 5 mins). In class we considered the mutiphysics problem of thermal stresses in a rod. When using linear shape functions, we found that the stress in the rod is affected by unphysical oscillations like in the following plot E*(ux-a*T) 35000 30000 25000 20000 15000 10000 5000 -5000 -10000 0 Line Graph: E*(ux-a*T) MULT 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Arc length (a) [10pts] What is the origin of this issue and how can we fix it?arrow_forwardanswer the questions and explain all of it in words. Ignore where it says screencast and in class explanationarrow_forward
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