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Fluid Mechanics (2nd Edition)
2nd Edition
ISBN: 9780134649290
Author: Russell C. Hibbeler
Publisher: PEARSON
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Question
Chapter 5, Problem 101P
To determine
The discharge of the water at point C.
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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 5 Solutions
Fluid Mechanics (2nd Edition)
Ch. 5 - Prob. 1FPCh. 5 - Oil is subjected to a pressure of 300 kPa at A,...Ch. 5 - Prob. 3FPCh. 5 - Water flows through the pipe at 8 m/s. Determine...Ch. 5 - The tank has a square base and is filled with...Ch. 5 - Prob. 6FPCh. 5 - Water flows from the reservoir through the...Ch. 5 - Crude oil flows through the 50-mm-diameter pipe...Ch. 5 - Water at A has a pressure of 400 kPa and a...Ch. 5 - Water from the reservoir flows through the...
Ch. 5 - Prob. 11FPCh. 5 - The jet engine takes in air and fuel having an...Ch. 5 - Determine the required average change in pressure...Ch. 5 - Prob. 2PCh. 5 - Prob. 3PCh. 5 - Air at 60°F flows through the horizontal tapered...Ch. 5 - Prob. 5PCh. 5 - The water in an open channel drainage canal flows...Ch. 5 - Water flows out of a faucet at A at 6 m/s....Ch. 5 - Water flows through the 30-mm-diameter pipe at...Ch. 5 - Water flows through the 30-mm-diameter pipe and is...Ch. 5 - Drainage under a canal is provided using a...Ch. 5 - Prob. 11PCh. 5 - Prob. 12PCh. 5 - A fountain is produced by water that flows up the...Ch. 5 - Prob. 14PCh. 5 - Air is drawn into the 200-mm-diameter cylinder...Ch. 5 - The level of mercury in the manometer has the...Ch. 5 - A fountain ejects water through the two nozzles A...Ch. 5 - Prob. 18PCh. 5 - Heavy rain has caused reservoir A to reach a...Ch. 5 - A fire hydrant supplies water under a pressure of...Ch. 5 - Determine the velocity of water through the pipe...Ch. 5 - The sewage siphon regulates the level of water in...Ch. 5 - If the manometer contains mercury, determine the...Ch. 5 - Prob. 24PCh. 5 - Prob. 25PCh. 5 - When the valve at A is opened, the initial...Ch. 5 - Prob. 27PCh. 5 - Prob. 28PCh. 5 - Air is pumped into the top of the tank so that the...Ch. 5 - Prob. 30PCh. 5 - Prob. 31PCh. 5 - A river has an average width of 5 m. Just after...Ch. 5 - A river has an average width of 5 m and flows with...Ch. 5 - Prob. 34PCh. 5 - Prob. 35PCh. 5 - Prob. 36PCh. 5 - Water flows through the transition at 0.3 m3/s,...Ch. 5 - If the water in piezometers A and B rises to hA =...Ch. 5 - Prob. 39PCh. 5 - Prob. 40PCh. 5 - Water flows through the pipe transition with a...Ch. 5 - Water from a faucet tapers from a diameter of 0.5...Ch. 5 - Prob. 43PCh. 5 - Prob. 44PCh. 5 - If the pressure at A is 325 kPa, and the velocity...Ch. 5 - If the pressure at A is 215 kPa, and the velocity...Ch. 5 - Prob. 47PCh. 5 - If the difference in the level of mercury within...Ch. 5 - Prob. 49PCh. 5 - Prob. 50PCh. 5 - Prob. 51PCh. 5 - If the pressure in the 6-in.-diameter pipe at A is...Ch. 5 - Prob. 53PCh. 5 - Prob. 54PCh. 5 - Prob. 56PCh. 5 - Prob. 57PCh. 5 - Prob. 58PCh. 5 - The solution is ejected from the 20-mm-diameter...Ch. 5 - Prob. 60PCh. 5 - Prob. 61PCh. 5 - Prob. 62PCh. 5 - Water from the large closed tank is to be drained...Ch. 5 - Prob. 64PCh. 5 - Carbon dioxide at 20°C passes through the...Ch. 5 - Prob. 66PCh. 5 - Prob. 67PCh. 5 - Determine the average velocity and the pressure in...Ch. 5 - Prob. 69PCh. 5 - Prob. 70PCh. 5 - Prob. 71PCh. 5 - Prob. 72PCh. 5 - Water at a pressure of 12 psi and a velocity of 5...Ch. 5 - Prob. 74PCh. 5 - Prob. 75PCh. 5 - The siphon spillway provides an automatic control...Ch. 5 - Prob. 77PCh. 5 - A piezometer and a manometer containing mercury...Ch. 5 - Water is drawn into the pump, such that the...Ch. 5 - Prob. 80PCh. 5 - Prob. 81PCh. 5 - Prob. 82PCh. 5 - Prob. 83PCh. 5 - A pump is used to deliver water from a large...Ch. 5 - A 6-hp pump with a 3-in.-diameter hose is used to...Ch. 5 - The pump is used with a 3-in.-diameter hose to...Ch. 5 - Solve Prob. 5–86 by including frictional head...Ch. 5 - The pump discharges water at B at 0.3 ft3/s. If...Ch. 5 - Prob. 89PCh. 5 - Draw the energy and hydraulic grade lines for the...Ch. 5 - The turbine removes energy from the water in the...Ch. 5 - Prob. 92PCh. 5 - Prob. 93PCh. 5 - Water in the reservoir flows through the...Ch. 5 - Prob. 95PCh. 5 - Determine the power delivered to the turbine if...Ch. 5 - The turbine at C draws a power of 90.5 hp. If the...Ch. 5 - Prob. 98PCh. 5 - Prob. 99PCh. 5 - Prob. 100PCh. 5 - The pump is connected to the 2-in.-diameter hose....Ch. 5 - Prob. 102PCh. 5 - Prob. 103PCh. 5 - Prob. 104PCh. 5 - Prob. 105PCh. 5 - Crude oil is pumped from a test separator at A to...Ch. 5 - Prob. 107PCh. 5 - Prob. 108PCh. 5 - Determine the power that the pump supplies to the...Ch. 5 - The pump delivers water at 120 ft3/min from the...Ch. 5 - Prob. 111PCh. 5 - Prob. 112P
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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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