Applied Fluid Mechanics (7th Edition)
7th Edition
ISBN: 9780132558921
Author: Robert L. Mott, Joseph A. Untener
Publisher: PEARSON
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Chapter 2, Problem 2.14PP
Define a Newtonian fluid.
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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 2 Solutions
Applied Fluid Mechanics (7th Edition)
Ch. 2 - Define shear stress as it applies to a moving...Ch. 2 - Define velocity gradient.Ch. 2 - State the mathematical definition for dynamic...Ch. 2 - Which would have the greater dynamic viscosity, a...Ch. 2 - State the standard units for dynamic viscosity in...Ch. 2 - State the standard units for dynamic viscosity in...Ch. 2 - State the equivalent units for poise in terms of...Ch. 2 - Why are the units of poise and centipoise...Ch. 2 - State the mathematical definition for kinematic...Ch. 2 - State the standard units for kinematic viscosity...
Ch. 2 - State the standard units for kinematic viscosity...Ch. 2 - State the equivalent units for stoke in terms of...Ch. 2 - Why are the units of stoke and centistoke...Ch. 2 - Define a Newtonian fluid.Ch. 2 - Define a non-Newtonian fluid.Ch. 2 - Give five examples of Newtonian fluids.Ch. 2 - Give four examples of the types of fluids that are...Ch. 2 - Appendix D iS gives dynamic viscosity for a...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D iS gives dynamic viscosity for a...Ch. 2 - Appendix D iS gives dynamic viscosity for a...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - Appendix D gives dynamic viscosity for a variety...Ch. 2 - If you want to choose a fluid that exhibits a...Ch. 2 - Which type of viscosity measurement method uses...Ch. 2 - In the rotating-drum viscometer, describe how the...Ch. 2 - In the rotating-drum viscometer, describe how the...Ch. 2 - What measurements must be taken to determine...Ch. 2 - Define the term terminal velocity as it applies to...Ch. 2 - What measurements must be taken to determine...Ch. 2 - Describe the basic features of the Saybolt...Ch. 2 - Are the results of the Saybolt viscometer tests...Ch. 2 - Does the Saybolt viscometer produce data related...Ch. 2 - Which type of viscometer is prescribed by SAE for...Ch. 2 - Describe the difference between an SAE 20 oil and...Ch. 2 - What grades of SAE oil are suitable for...Ch. 2 - What grades of SAE oil are suitable for...Ch. 2 - If you were asked to check the viscosity of an oil...Ch. 2 - If you were asked to check the viscosity of an oil...Ch. 2 - Prob. 2.53PPCh. 2 - The viscosity of a lubricating oil is given as 500...Ch. 2 - Using the data from Table 2.5. report the minimum,...Ch. 2 - Convert a dynamic viscosity measurement of 4500 cP...Ch. 2 - Convert a kinematic viscosity measurement of 5.6...Ch. 2 - The viscosity of an oil is given as 80 SUS at...Ch. 2 - Convert a viscosity measurement of 6.5x103 Pa.s...Ch. 2 - An oil container indicates that it has a viscosity...Ch. 2 - In a falling-ball viscometer, a steel ball 1.6 mm...Ch. 2 - A capillary tube viscometer similar to that shown...Ch. 2 - In a falling-ball viscometer, a steel ball with a...Ch. 2 - A capillary type viscometer similar to that shown...Ch. 2 - A fluid has a kinematic viscosity of 15.0 mm2/s at...Ch. 2 - A fluid has a kinematic viscosity of 55.3 mm2/s at...Ch. 2 - A fluid has a kinematic viscosity of 188 mm2/s at...Ch. 2 - A fluid has a kinematic viscosity of 244 mm2/s at...Ch. 2 - A fluid has a kinematic viscosity of 153mm2/s at...Ch. 2 - A fluid has a kinematic viscosity of 205mm2/s at...Ch. 2 - An oil is tested using a Saybolt viscometer and...Ch. 2 - An oil is tested using a Saybolt viscometer and...Ch. 2 - Prob. 2.73PPCh. 2 - Prob. 2.74PPCh. 2 - An oil is tested using a Saybolt viscometer and...Ch. 2 - Prob. 2.76PPCh. 2 - Convert all of the kinematic viscosity data in...Ch. 2 - Use a spreadsheet to display the values of...
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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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