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being withdrawn from an incompressible Newtonian liquid at a tangential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