Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
N-body problem wikipedia , lookup
Fluid dynamics wikipedia , lookup
Centripetal force wikipedia , lookup
Renormalization group wikipedia , lookup
Equations of motion wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Classical central-force problem wikipedia , lookup
Homework # 11 Extra credit Due Wednesday, April 16, 2008 ME 363 - Fluid Mechanics Spring Semester 2008 1] (a) Consider a liquid in a cylindrical container in which both the liquid and the container are rotating as a rigid body. The elevation difference h between the center of the liquid surface and the rim of the liquid surface is a function of angular velocity , fluid density , gravitational acceleration g, and radius R. Use the method of repeating variables to find a relationship between the parameters. (b) Consider now a transient process where the container was originally at rest and rotation started at time t = 0. After a certain period of time t = T the liquid will attain steady motion in which both the liquid and the container are rotating as a rigid body. At any time t = < T from the start of rotation, the fluid velocity V (r ) at distance r from the axis of rotation was found to be a function of time , angular velocity , liquid density , and liquid viscosity . Use the method of repeating variables to find a relationship between these parameters. If, in two separate experiments, honey and water are rotated in the same container at the same angular velocity, determine from the obtained dimensional analysis results which of the two liquids will attain steady motion faster. (a) Solution parameters. Assumptions We are to use dimensional analysis to find the functional relationship between the given 1 The given parameters are the only relevant ones in the problem. Analysis The step-by-step method of repeating variables is employed to obtain the nondimensional parameters (the s). Step 1 There are five parameters in this problem; n = 5, h f , , g , R List of relevant parameters: n5 (1) Step 2 The primary dimensions of each parameter are listed, h g R L t m L L t L 1 1 1 2 1 3 1 Step 3 As a first guess, j is set equal to 3, the number of primary dimensions represented in the problem (m, L, and t). j 3 Reduction: If this value of j is correct, the expected number of s is Number of expected s: k n j 53 2 Step 4 We need to choose three repeating parameters since j = 3. Following the guidelines outlined in this chapter, we elect not to pick the viscosity. We choose , , and R Repeating parameters: Step 5 The dependent is generated: 1 1 h a1 b1 R c1 L t 1 1 m L L a1 1 3 b1 1 c1 mass: m m 0 b1 b1 0 time: t t 0 a1 a1 0 0 1 3b1 c1 c1 1 length: b1 0 0 a1 L L L 0 1 3b1 Lc1 The dependent is thus 1: 1 h R The second Pi (the only independent in this problem) is generated: 2 2 g a2 b2 R c2 m m mass: t t time: length: b2 0 0 2 a2 t L L L 0 1 3b2 Lc2 L t 1 2 t m L L 1 a2 1 3 b2 1 c2 0 b2 b2 0 0 2 a2 a2 2 0 1 3b2 c2 c2 1 0 1 c2 which yields 2: 2 g 2R If we take 2 to the power –1/2 and recognize that R is the speed of the rim, we see that 2 can be modified into a Froude number, Modified 2: 2 Fr R gR Step 6 We write the final functional relationship as Relationship between s: h f Fr R (2) Discussion In the generation of the first , h and R have the same dimensions. Thus, we could have immediately written down the result, 1 = h/R. Notice that density does not appear in the result. Thus, density is not a relevant parameter after all. (b) Solution parameters. We are to use dimensional analysis to find the functional relationship between the given