Fluid Mechanics
8th Edition
ISBN: 9780073398273
Author: Frank M. White
Publisher: McGraw-Hill Education
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Chapter 1, Problem 1.84P
In the early 1900s, the British chemist Sir Cyril Hinshelwood quipped that fluid dynamics study was divided into "workers who observed things they could not explain and workers who explained things they could not observe." To what historic situation was he referring?
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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 1 Solutions
Fluid Mechanics
Ch. 1 - Prob. 1.1PCh. 1 - Table A.6 lists the density of the standard...Ch. 1 - For the triangular element in Fig, P1.3,show that...Ch. 1 - Sand, and other granular materials, appear to...Ch. 1 - The mean free path of a gas, l, is defined as the...Ch. 1 - Henri Darcy, a French engineer, proposed that the...Ch. 1 - Convert the following inappropriate quantities...Ch. 1 - Suppose we know little about the strength of...Ch. 1 - A hemispherical container, 26 inches in diameter,...Ch. 1 - The Stokes-Oseen formula [33] for drag force F on...
Ch. 1 - P1.11 In English Engineering units, the specific...Ch. 1 - For low-speed (laminar) steady flow through a...Ch. 1 - The efficiency ? of a pump is defined as the...Ch. 1 - Figure P1.14 shows the flow of water over a dam....Ch. 1 - The height H that fluid rises in a liquid...Ch. 1 - Algebraic equations such as Bernoulli's relation,...Ch. 1 - The Hazen-Williams hydraulics formula for volume...Ch. 1 - For small particles at low velocities, the first...Ch. 1 - In his study of the circular hydraulic jump formed...Ch. 1 - Books on porous media and atomization claim that...Ch. 1 - Aeronautical engineers measure the pitching moment...Ch. 1 - Prob. 1.22PCh. 1 - During World War II, Sir Geoffrey Taylor, a...Ch. 1 - Air, assumed to be an ideal gas with k = 1.40,...Ch. 1 - On a summer day in Narragansett, Rhode Island, the...Ch. 1 - When we in the United States say a car's tire is...Ch. 1 - Prob. 1.27PCh. 1 - Wet atmospheric air at 100 percent relative...Ch. 1 - Prob. 1.29PCh. 1 - P1.30 Repeat Prob. 1.29 if the tank is filled with...Ch. 1 - Prob. 1.31PCh. 1 - Prob. 1.32PCh. 1 - A tank contai as 9 kg of CO2at 20°C and 2.0 MPa....Ch. 1 - Consider steam at the following state near the...Ch. 1 - In Table A.4, most common gases (air, nitrogen,...Ch. 1 - Prob. 1.36PCh. 1 - A near-ideal gas has a molecular weight of 44 and...Ch. 1 - In Fig. 1.7, if the fluid is glycerin at 20°C and...Ch. 1 - Prob. 1.39PCh. 1 - Glycerin at 20°C fills the space between a hollow...Ch. 1 - An aluminum cylinder weighing 30 N, 6 cm in...Ch. 1 - Prob. 1.42PCh. 1 - Prob. 1.43PCh. 1 - One type of viscometer is simply a long capillary...Ch. 1 - A block of weight W slides down an inclined plane...Ch. 1 - A simple and popular model for two nonnewtonian...Ch. 1 - Data for the apparent viscosity of average human...Ch. 1 - A thin plate is separated from two fixed plates by...Ch. 1 - An amazing number of commercial and laboratory...Ch. 1 - Prob. 1.50PCh. 1 - Prob. 1.51PCh. 1 - The belt in Fig. P1.52 moves at a steady velocity...Ch. 1 - A solid tune of angle 2 , base r0, and density...Ch. 1 - A disk of radius R rotates at an angular velocity ...Ch. 1 - A block of weight W is being pulled over a table...Ch. 1 - The device in Fig. P1.56 is called a cone-plate...Ch. 1 - Extend the steady flow between a fixed lower plate...Ch. 1 - The laminar pipe flow example of Prob. 1.12 can be...Ch. 1 - A solid cylinder of diameter D, length L, and...Ch. 1 - Prob. 1.60PCh. 1 - Prob. 1.61PCh. 1 - P1.62 The hydrogen bubbles that produced the...Ch. 1 - Derive Eq. (1.33) by making a force balance on the...Ch. 1 - Pressure in a water container can be measured by...Ch. 1 - The system in Fig. P1.65 is used to calculate the...Ch. 1 - Prob. 1.66PCh. 1 - Prob. 1.67PCh. 1 - Prob. 1.68PCh. 1 - A solid cylindrical needle of diameter d, length...Ch. 1 - Derive an expression for the capillary height...Ch. 1 - A soap bubble of diameter D1coalesces with another...Ch. 1 - Early mountaineers boiled water to estimate their...Ch. 1 - A small submersible moves al velocity V, in fresh...Ch. 1 - Oil, with a vapor pressure of 20 kPa, is delivered...Ch. 1 - An airplane flies at 555 mi/h. At what altitude in...Ch. 1 - Prob. 1.76PCh. 1 - Prob. 1.77PCh. 1 - P1.78 Sir Isaac Newton measured the speed of sound...Ch. 1 - Prob. 1.79PCh. 1 - Prob. 1.80PCh. 1 - Use Eq. (1.39) to find and sketch the streamlines...Ch. 1 - P1.82 A velocity field is given by u = V cos, v =...Ch. 1 - Prob. 1.83PCh. 1 - In the early 1900s, the British chemist Sir Cyril...Ch. 1 - Prob. 1.85PCh. 1 - A right circular cylinder volume v is to be...Ch. 1 - The absolute viscosity of a fluid is primarily a...Ch. 1 - Prob. 1.2FEEPCh. 1 - Helium has a molecular weight of 4.003. What is...Ch. 1 - An oil has a kinematic viscosity of 1.25 E-4 m2/s...Ch. 1 - Prob. 1.5FEEPCh. 1 - Prob. 1.6FEEPCh. 1 - FE1.7 Two parallel plates, one moving at 4 m/s...Ch. 1 - Prob. 1.8FEEPCh. 1 - A certain water flow at 20°C has a critical...Ch. 1 - Prob. 1.10FEEPCh. 1 - Sometimes we can develop equations and solve...Ch. 1 - When a person ice skates, the surface of the ice...Ch. 1 - Two thin flat plates, tilted at an angle a, are...Ch. 1 - Oil of viscosity and density drains steadily...Ch. 1 - Prob. 1.5CPCh. 1 - Prob. 1.6CPCh. 1 - Prob. 1.7CPCh. 1 -
C1.8 A mechanical device that uses the rotating...Ch. 1 - Prob. 1.9CPCh. 1 - A popular gravity-driven instrument is the...Ch. 1 - Mott [Ref. 49, p. 38] discusses a simple...Ch. 1 - A solid aluminum disk (SG = 2.7) is 2 in in...
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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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