Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 1, Problem 21P
A man goes to a traditional market to buy a steak for diner. He finds a 12-oz steak (1 Ibm = 16 oz) for $3.15. He then goes to the adjacent international market and finds a 320-g steak of identical quality for $3.30. Which steak is the better buy?
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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: Fundamentals and Applications
Ch. 1 - What is a fluid? How does it differ from a solid?...Ch. 1 - Define internal, external, and open-channel flows.Ch. 1 - Define incompressible flow and in compressible...Ch. 1 - Consider the flow of air over the wings of an...Ch. 1 - What is forced flow? How does it differ from...Ch. 1 - How is the Mach number of a flow defined? That...Ch. 1 - When an airplane is flying at a constant speed...Ch. 1 - Consider the flow of air at a Mach number of 0.12....Ch. 1 - What is the no-slip condition? What causes it?Ch. 1 - What is a boundary layer? What causes a boundary...
Ch. 1 - What is a steady-flow process?Ch. 1 - Define stress, normal stress, shear stress, and...Ch. 1 - What are system, surroundings, and boundary?Ch. 1 - When analyzing the acceleration of gases as they...Ch. 1 - When is a system a closed system, and when is it a...Ch. 1 - You are to understand how a reciprocating air...Ch. 1 - What is the difference between pound-mass and...Ch. 1 - In a news ankle, is stated that a recently...Ch. 1 - Explain why the light-year has the dimension of...Ch. 1 - What is the net force acting on a car cruising at...Ch. 1 - A man goes to a traditional market to buy a steak...Ch. 1 - What is the weight, in N, of an object with a mass...Ch. 1 - What is the weight of a 1-kg substance in N,...Ch. 1 - Determine the mass and the weight of the air...Ch. 1 - A 3-kW resistance heater a water beater...Ch. 1 - A195-Ibm astronaut took his bathroom scale (a...Ch. 1 - The acceleration of high-speed aircraft sometimes...Ch. 1 - A 10-kg rock is thrown upward with a force of 280...Ch. 1 - The value of the gravitational acceleration g...Ch. 1 - At 45° latitude: the gravitational acceleration as...Ch. 1 - 1-32 The gravitational constant g is 9.807m/s2 at...Ch. 1 - On average, an adult person breathes in about 7.0...Ch. 1 - While solving a problem, a person ends up with...Ch. 1 - An airplane flies horizontally at 70m/s . Its...Ch. 1 - If the airplane of Prob. 1-35 weighs 17 lbf,...Ch. 1 - The boom of a fire truck raises a fireman (and his...Ch. 1 - A 6-kg plastic tank that has a volume of 0.18m3 is...Ch. 1 - Water at 15°C from a garden hose fills a 1.5 L...Ch. 1 - A forklift raises a 90.5 kg crate 1.80 m. (a)...Ch. 1 - The gas tank of a car is filled with a nozzle that...Ch. 1 - A pool of volume V (in m3) is to filled with water...Ch. 1 - Based on unit considerations alone, show that the...Ch. 1 - What is the importance of modeling in engineering?...Ch. 1 - What is the difference between the analytical and...Ch. 1 - When modeling an engineering process, how is the...Ch. 1 - What is the difference between precision and...Ch. 1 - How do the differential equations in the study of...Ch. 1 - What is the value of the engineering software...Ch. 1 - The weight of bodies may change somewhat from one...Ch. 1 - The reactive force developed by a jet to push an...Ch. 1 - An important design consideration in two-phase...Ch. 1 - Consider the flow of air through a wind turbine...Ch. 1 - A tank is filled with oil whose density is =850...Ch. 1 - If mass, heat, and work are not allowed to cross...Ch. 1 - The speed of an aircraft is given to be 260 m/s in...Ch. 1 - One J/kg is equal to (a) 1kPam3 (b) 1kNm/kg (c)...Ch. 1 - Which is a unit for power? (a) Btu (b) kwh (c)...Ch. 1 - The speed of an aircraft is given to be 950 km/h....Ch. 1 - The weight of a 10-kg mass at sea level is (a)...Ch. 1 - The weight of a 1 -Ibm mass is (a) 1Ibmft/s2 (b)...Ch. 1 - A hydroelectric power plant operates at its rated...Ch. 1 - Write an essay on the various mass- and...Ch. 1 - Search the Internet to find out how to properly...Ch. 1 - Another unit is kgf, which is a force unit used...Ch. 1 - Discuss why pressure tests of pressurized tanks...
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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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