課程名稱︰中等流體力學
課程性質︰工科應力組必修
課程教師︰蔡武廷
開課學院：工
開課系所︰工科海洋
考試日期（年月日）︰2013 04 12
考試時限（分鐘）：180 分
是否需發放獎勵金：是 謝謝
（如未明確表示，則不予發放）
試題 :
1. A three-dimensional velocity field is given by:
_ _ _ _
v(x, y, z, t) = (-xt + y + z)i + (x + y + 2z +t)j + [x + 2y + z(t+1)]k
the density ρ is only dependent on time, i.e. ρ = ρ(t)
a.[5%] Is this a Lagrangian or Eulerian description of the flow field? Why?
b.[10%] What are the 'local','advective' and 'total' acceleration of the flow
field?
c.[10%] Calculate the rates of rotation and the rates of shearing on the x-y,
y-z and z-x planes.
d.[5%] Is the flow rotational or irrotational?
e.[5%] Is thw flow compressible or incompressible?
f.[15%] What is the general differential equation govering conservation of
mass for a flow field? Write down the equation using both 'vector'
and 'tensor' notaion and explain the physical meaning of each them.
g.[10%] Find the density field ρ(t) such that the conversation of mass is
satisfied.
2.[20%] Express the following tensor terms is Cratesian coordinate system.
(a) ui∂uk/∂xi (b) ∂(ρui)/∂xi (c)∂σij/∂xi
(d)[∂uj/∂xk - ∂uk/∂xj]
3.[10%] The temperature of a themometer that drift down a river at 10 km/day
shows an increase of 0.2°C/day. A thermometer anchored at a spot in
the river shows a decrete of 0.2°C/day. What is the temperature
gradient along the river?
4. Consider a steady and viscous flow u = u(y), p = p(x) and v = w = 0 between
two parallel stationary non-slip walls located at y = 0 and y = h respecti-
vely. The density of the fluid is ρ, the dynamic viscosity is μ, and the
gravity acceleration is g.
(a)[10%] Write down the general three-dimensional Navier-Stokes equations,
and then simplify the equations for u in this case.
(b)[5%] Show that if p(x) = 2x + 3, then u(y) = C1 + C2 + y^2/μ, Where C1
and C2 are constants.
(c)[5%] Apply suitable boundary conditions to determine C1 and C2.
(d)[5%] Calculate the shear stress τxy anywhere in the fluid.
(e)[5%] What is the stress τo at the lower wall?
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