課程名稱︰高等流體力學 課程性質︰必修(化工所核心課程) 課程教師︰趙玲 開課學院：工學院 開課系所︰化工所 考試日期（年月日）︰2014/11/18 考試時限（分鐘）：150分鐘 是否發放獎勵金：Yes 試題 : 1. (10pts) Please obtain the expansions of the following two expressions: (a) τ:τ (τ is a tensor) (b) ▽‧[▽×v] (v is a vector) 2. (10pts) Please derive the r component of [v‧▽w] in spherical coodinates. (Note: v and w are vectors and ▽ in different coordinates can be found in the appendices) 3. (10pts) Cylindrical coordinates are used in order to solve the flow problem outside a solid cylinder. The orientation of the cylinder, the defined cylindrical coordinates and the direction of the gravity field are shown in Figure 1. http://imgur.com/LYaWJcJ The size of the gravity accerlation rate is g0. Please derive the explicit correlation between the dynamic pressure (P) and the absolute pressure (p) in the fluid (Note that ▽P≡▽p-ρg) 4. (25pts) A solid plate inclined at an angle α relative to horizontal is being withdrawn from an incompressible Newtonian liquid at a tangential velocity U. The gas-liquid interface (corresponding to θ=0 in cylindrical coordinates) is assumed to be flat. That is, wetting phenomena and the tendency for the moving plate to disturb the interface are neglected. Assume Re＜＜1 and the shear stress at the gas-liquid interface is negligible. http://imgur.com/9w8WA0N (a) (7pts) Write the governing equation and the associated boundary conditions for the stream function Ψ. (b) (6pts) Guess the solution form of Ψ from the boundary conditions. (c) (12pts) Find Ψ, and the flow velocity Vr (r-dir velocity) and Vθ (θ-dir velocity). (Note: (d^2/dθ^2)+constant)(d^2/dθ^2)+constant)f=0 can be solved by using two steps: (d^2/dθ^2)+constant)g=0 and (d^2/dθ^2)+constant)f=g, if necessary.) 5. (20pts) A surface tension method is developed to pump a liquid through a microchannel (a tube with a micron-scale radius). At t = 0, a larger half- droplet with a radius of Rs0 and a smaller half-droplet with a radius of Rs0 were placed above the left and right ports of a fluid-filled microchannel. The fluid with an interfacial tension of γ started to flow due to the surface tension effect and the sizes of both of the droplets kept a hemispherical shape during the entire process. Another assumption is that the change rate of the droplet size was much slower than the fluid response and a quasi-steady state situation can apply. The microchannel has a circular cross- section with a diameter of d and the length is L. The vertical part of the channel connecting the ports has laminar flow and the fluid is incompressible and Newtonian. The atmosphere pressure is P0. Please answer the following questions. (Note that the volume of a sphere with a radius r is 4πr^3/3.) http://imgur.com/JNH9qT4 (a) Is the fluid flowing to the right or the left? Why? (b) Please derive Rs(t) (function of t) as a function of given parameters. (It does not need to be a clean expression as long as the equation only has Rs(t) and the given parameters.) (c) Please sketch Rs(t) as a fuction of time in the time region when the large droplet is much larger than the small droplet. 6. (25pts) Consider that an incompressible Newtonian fluid with a density of ρ and a velocity of μ is confined in the gap with a height of b between two parallel elastic disks with an average radius of R. We provided an external oscillating pressure from inlets at the center of each disk. The resulting pressure drop in the r-direction is given in Equation 1 and the fluid flows radially outward. The largest average volumetric flow rate sending from the inlets is Q. dP/dr = -500*cos(ωt)/(πbr) (Equation 1) http://imgur.com/t9NxPyX http://imgur.com/XRFq9LW (a) Assume that Vθ = Vz = 0 (no θ-dir and z-dir velocity) and the entrance and exit effects can be neglected. Please use the equation of continuity and equation of motion to derive the governing equation which can be used to solve Vr (Do simplify the equation to the extent as much as you can.) (b) If the values of all of the given parameters are shown in Table 1, please write down the estimated size of each term in the governing equation obtained in (a). (c) Please obtain the approximate solution for Vr. (You can leave the integration constant in the solution since the boundary and initial conditions are not given.) Appendices (3 pages): http://imgur.com/UtjTR7w http://imgur.com/OiaRQQ3 http://imgur.com/6CzWrx6 --

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