Fluid Mechanics
8th Edition
ISBN: 9780073398273
Author: Frank M. White
Publisher: McGraw-Hill Education
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Chapter 3, Problem 3.37P
To determine
The formula for the time required to reduce excess water to
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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.
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.
Problem 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).
Chapter 3 Solutions
Fluid Mechanics
Ch. 3 - Prob. 3.1PCh. 3 - Consider the angular momentum relation in the form...Ch. 3 - For steady low-Reynolds-number (laminar) flow...Ch. 3 - Water at 20°C flows through a long elliptical duct...Ch. 3 - Water at 20°C flows through a 5-in-diameter smooth...Ch. 3 - Water fills a cylindrical tank to depth h. The...Ch. 3 - A spherical tank, of diameter 35 cm, is leaking...Ch. 3 - Three pipes steadily deliver water at 20°C to a...Ch. 3 - A laboratory test tank contains seawater of...Ch. 3 - Water flowing through an 8-cm-diameter pipe enters...
Ch. 3 - Water flows from a faucet into a sink at 3 U.S....Ch. 3 - The pipe flow in Fig, P3.12 fills a cylindrical...Ch. 3 - The cylindrical container in Fig. P3.13 is 20 cm...Ch. 3 - The open tank in Fig. F3.14 contains water at 20°C...Ch. 3 - Water, assumed incompressible, flows steadily...Ch. 3 - P3.16 An incompressible fluid flows past an...Ch. 3 - Incompressible steady flow in the inlet between...Ch. 3 - Gasoline enters section 1 in Fig, P3.18 at 0.5...Ch. 3 - Water from a storm drain flows over an outfall...Ch. 3 - Oil (SG = 0.89) enters at section 1 in Fig, P3.20...Ch. 3 - Prob. 3.21PCh. 3 - Prob. 3.22PCh. 3 - Prob. 3.23PCh. 3 - Prob. 3.24PCh. 3 - Prob. 3.25PCh. 3 - A thin layer of liquid, draining from an inclined...Ch. 3 - Prob. 3.27PCh. 3 - Prob. 3.28PCh. 3 - Prob. 3.29PCh. 3 - Prob. 3.30PCh. 3 - Prob. 3.31PCh. 3 - Prob. 3.32PCh. 3 - In some wind tunnels the test section is...Ch. 3 - A rocket motor is operati ng steadily, as shown in...Ch. 3 - In contrast to the liquid rocket in Fig. P3.34,...Ch. 3 - The jet pump in Fig. P3.36 injects water at U1 =...Ch. 3 - Prob. 3.37PCh. 3 - Prob. 3.38PCh. 3 - A wedge splits a sheet of 20°C water, as shown in...Ch. 3 - The water jet in Fig, P3,40 strikes normal to a...Ch. 3 - P3.41 In Fig. P3.41 the vane turns the water jet...Ch. 3 - Prob. 3.42PCh. 3 - P3.43 Water at 20°C flows through a 5-cm-diameter...Ch. 3 - P3.44 When a uniform stream flows past an immersed...Ch. 3 - Water enters and leaves the 6-cm-diameter pipe...Ch. 3 - When a jet strikes an inclined fixed plate, as in...Ch. 3 - A liquid jet of velocity Vjand diameter Djstrikes...Ch. 3 - The small boat in Fig. P3.48 is driven at a steady...Ch. 3 - The horizontal nozzle in Fig. P3.49 has D1 = 12 in...Ch. 3 - Prob. 3.50PCh. 3 - P3.51 A liquid jet of velocity Vj and area Aj...Ch. 3 - A large commercial power washer delivers 21...Ch. 3 - Prob. 3.53PCh. 3 - For the pipe-flow-reducing section of Fig. P3.54,...Ch. 3 - In Fig. P3.55 the jet strikes a vane that moves to...Ch. 3 - Prob. 3.56PCh. 3 - Prob. 3.57PCh. 3 - Prob. 3.58PCh. 3 - Prob. 3.59PCh. 3 - Prob. 3.60PCh. 3 - Prob. 3.61PCh. 3 - P3.62 Water at 20°C exits to the standard...Ch. 3 - Water flows steadily through the box in Fig....Ch. 3 - The 6-cm-diameter 20°C water jet in Fig. P3.64...Ch. 3 - Prob. 3.65PCh. 3 - Prob. 3.66PCh. 3 - Prob. 3.67PCh. 3 - Prob. 3.68PCh. 3 - P3.69 A uniform rectangular plate, 40 cm long and...Ch. 3 - Prob. 3.70PCh. 3 - Prob. 3.71PCh. 3 - When immersed in a uniform stream, a thick...Ch. 3 - P3.73 A pump in a tank of water at 20°C directs a...Ch. 3 - P3.74 Water at 20°C flows down through a vertical,...Ch. 3 - Prob. 3.75PCh. 3 - Prob. 3.76PCh. 3 - Prob. 3.77PCh. 3 - Prob. 3.78PCh. 3 - P3.79 The Saturn V rocket in the chapter opener...Ch. 3 - Prob. 3.80PCh. 3 - Prob. 3.81PCh. 3 - Prob. 3.82PCh. 3 - Prob. 3.83PCh. 3 - Air at 20°C and 1 atm flows in a 25-cm-diameter...Ch. 3 - Prob. 3.85PCh. 3 - Prob. 3.86PCh. 3 - Prob. 3.87PCh. 3 - Prob. 3.88PCh. 3 - Prob. 3.89PCh. 3 - Prob. 3.90PCh. 3 - Prob. 3.91PCh. 3 - Prob. 3.92PCh. 3 - Prob. 3.93PCh. 3 - A water jet 3 in in diameter strikes a concrete...Ch. 3 - P3.95 A tall water tank discharges through a...Ch. 3 - Prob. 3.96PCh. 3 - Prob. 3.97PCh. 3 - Prob. 3.98PCh. 3 - Prob. 3.99PCh. 3 - Prob. 3.100PCh. 3 - Prob. 3.101PCh. 3 - Prob. 3.102PCh. 3 - Suppose that the solid-propellant rocket of Prob....Ch. 3 - A rocket is attached to a rigid horizontal rod...Ch. 3 - Extend Prob. P3.104 to the case where the rocket...Ch. 3 - Actual airflow past a parachute creates a variable...Ch. 3 - Prob. 3.107PCh. 3 - Prob. 3.108PCh. 3 - Prob. 3.109PCh. 3 - Prob. 3.110PCh. 3 - Prob. 3.111PCh. 3 - A jet of alcohol strikes the vertical plate in...Ch. 3 - Prob. 3.113PCh. 3 - Prob. 3.114PCh. 3 - Prob. 3.115PCh. 3 - P3.116 For the container of Fig. P3.116 use...Ch. 3 - Water at 20°C, in the pressurized tank of Fig....Ch. 3 - P3.118 Bernoulli's 1738 treatise Hydrodynamica...Ch. 3 - Prob. 3.119PCh. 3 - Prob. 3.120PCh. 3 - Prob. 3.121PCh. 3 - Prob. 3.122PCh. 3 - The air-cushion vehicle in Fig, P3.123 brings in...Ch. 3 - Prob. 3.124PCh. 3 - Prob. 3.125PCh. 3 - Prob. 3.126PCh. 3 - Prob. 3.127PCh. 3 - Prob. 3.128PCh. 3 - Prob. 3.129PCh. 3 -
P3.130 In Fig. P3.130 the fluid is gasoline at...Ch. 3 - Prob. 3.131PCh. 3 - Prob. 3.132PCh. 3 - Prob. 3.133PCh. 3 - Prob. 3.134PCh. 3 - Prob. 3.135PCh. 3 - Air, assumed frictionless, flows through a tube,...Ch. 3 - In Fig. P3.137 the piston drives water at 20°C....Ch. 3 - Prob. 3.138PCh. 3 - Prob. 3.139PCh. 3 - Prob. 3.140PCh. 3 - Prob. 3.141PCh. 3 - Prob. 3.142PCh. 3 - Prob. 3.143PCh. 3 - Prob. 3.144PCh. 3 - Prob. 3.145PCh. 3 - The pump in Fig. P3.146 draws gasoline at 20°C...Ch. 3 - The very large water tank in Fig. P3.147 is...Ch. 3 - Prob. 3.148PCh. 3 - P3.149 The horizontal lawn sprinkler in Fig....Ch. 3 - Prob. 3.150PCh. 3 - Prob. 3.151PCh. 3 - Prob. 3.152PCh. 3 - Prob. 3.153PCh. 3 - Prob. 3.154PCh. 3 - Prob. 3.155PCh. 3 - Prob. 3.156PCh. 3 - Prob. 3.157PCh. 3 - Prob. 3.158PCh. 3 - Prob. 3.159PCh. 3 - Prob. 3.160PCh. 3 - Prob. 3.161PCh. 3 - The waterwheel in Fig. P3.162 is being driven at...Ch. 3 - Prob. 3.163PCh. 3 - Prob. 3.164PCh. 3 - Prob. 3.165PCh. 3 - A power plant on a river, as in Fig. P3.166, must...Ch. 3 - Prob. 3.167PCh. 3 - Prob. 3.168PCh. 3 - P3.169 When the pump in Fig. P3.169 draws 220 m3/h...Ch. 3 - Prob. 3.170PCh. 3 - P3.171 Consider a turbine extracting energy from a...Ch. 3 - Prob. 3.172PCh. 3 - Prob. 3.173PCh. 3 - Prob. 3.174PCh. 3 - Prob. 3.175PCh. 3 - Prob. 3.176PCh. 3 - Prob. 3.177PCh. 3 - Prob. 3.178PCh. 3 - Prob. 3.179PCh. 3 - Prob. 3.180PCh. 3 - Prob. 3.181PCh. 3 - Prob. 3.182PCh. 3 - Prob. 3.183PCh. 3 - The large turbine in Fig. P3.184 diverts the river...Ch. 3 - Prob. 3.185PCh. 3 - Prob. 3.1WPCh. 3 - Prob. 3.2WPCh. 3 - Prob. 3.3WPCh. 3 - Prob. 3.4WPCh. 3 - W3.5 Consider a long sewer pipe, half full of...Ch. 3 - Put a table tennis ball in a funnel, and attach...Ch. 3 - How does a siphon work? Are there any limitations...Ch. 3 - Prob. 3.1FEEPCh. 3 - Prob. 3.2FEEPCh. 3 - In Fig, FE3.1 water exits from a nozzle into...Ch. 3 - Prob. 3.4FEEPCh. 3 - Prob. 3.5FEEPCh. 3 - FE3.6 A fireboat pump delivers water to a...Ch. 3 - A fireboat pump delivers water to a vertical...Ch. 3 - Prob. 3.8FEEPCh. 3 - Water flowing in a smooth 6-cm-diameter pipe...Ch. 3 - Prob. 3.10FEEPCh. 3 - In a certain industrial process, oil of density ...Ch. 3 - Prob. 3.2CPCh. 3 - Prob. 3.3CPCh. 3 - Prob. 3.4CPCh. 3 - Prob. 3.5CPCh. 3 - Prob. 3.1DP
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- Problem 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_forwardI 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_forward
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