Fluid Dynamics: Examples< Potential flow>
１． Consider a two-dimentional steady state potential flow of an inviscid incompresiive fluid,
whose complex potential is given by

W = U 0 (z +a2).
zHere, z is the comlex, which is expressed byz = x + iy = re iθ ,and a is a possitive constant. Answer the following questions.
(1) Calculate the real and imaginary parts of W , and then obtain the velocity potential

φ (r,θ ) and stream function ϕ (r,θ ) ，respectively.
(2) Using the result of question (1), obtain the radial velocity componet ur (r,θ ) andcircumferential velocity component uθ (r,θ ) .
(3) Using the result of question (2)，obtain the velocity componet in the x –direcitonux (r,θ ) and elocity componet in the y –direciton, u y (r,θ ) ．

(4) Using the result of question (2)，explian the flow field at r = a ．
(5) Using the result of question (3)，explian the flow field far from the origin. r = a ．
(6) Draw the flow field given by the potentail, W ．
２． Consider the following complex velocity potential, W , which is given by

W = α ln z .
Here, m and z are a complex constant and variable, respectively, given by

α = a − ib (a ≥ 0, b ≥ 0) ,,z = x + iy = re iθAnd the following equations give Q and Γ ;,Q=!
∫ (u ⋅ n) dsCΓ = !∫ u ⋅ d s
C,where C is a closed loop oriented counterclockwise around the origin. Answer thefollowing questions.
(1) Calculate the real and imaginary parts of W , and then obtain the velocity potential
φ (r,θ ) and stream function ψ (r,θ ) .

1(2) Using the result of question (1), obtain the radial velocity component ur (r,θ ) and thecircumferential velocity component uθ (r,θ ) ．
(3) Draw the streamlines and evaluate Q and Γ in case of b = 0 ．
(4) Draw the streamlines and evaluate Q and Γ in case of a = 0 ．
３． Consider a two-dimensional steady state potential flow of an inviscid incompressible fluid
around a corner as shown in Fig.1, whose complex velocity potential, W , is given by
W = A zα .
Here, A and z are the complex constant and variable, respectively, given by

A = A e iβ ， z = x + iy = re iθ ，and α is a positive constant and β is a constant of −π < β < π . Answer the followingquestions.
(1) Calculate the real and imaginary parts of W , and then obtain the velocity potential
φ (r,θ ) and stream function ψ (r,θ ) .
(2) Using the result of question (1), obtain the radial velocity component ur (r,θ ) and thecircumferential velocity component uθ (r,θ ) . Then obtain the velocity component inthe x -direction, ux (x, y) and the velocity component in the y -direction, u y (x, y) ．
(3) Using the result of question (2)，determine β from a boundary condition in terms ofuθ (r,0) ．
(4) Using the results of questions (2) and (3)，determine α from a boundary condition in3
terms of uθ (r, π ) ．
2(5) Using the results of questions (2), (3) and (4), obtain the absolute value of the flow
velocity at the origin.

Fig.1

2

<Channel flow>
1. As shown in Fig. 2, the two types of fluid A and B flow in the x direction between the paralleldp= F0 (< 0) without mixing with each other. It isdx
assumed that the flow is sufficiently developed, and the width for the fluids A flow is “ a ” and thatplates with a constant pressure gradient,

for the fluid B is “ b ”. In addition, the viscosity coefficients of fluid A and fluid B are given by µ A
and µ B , respectively.

And it is also assumed that the adhesive condition is satisfied on the wall.

Answer the following questions.
(1) Show the velocity conditions on the wall.
(2) Show the conditions that hold at the boundary between fluids A and B.
(3) Find U ( y) .
(4) Using the results of Question (3), show a schematic diagram of the flow velocity distribution in
the case of µ A > µ B .
(4) Write down what kind of phenomenon will occur when F0 (< 0) becomes smaller from this state.

Fig. 2
2. Consider a concentric double circular pipe with an inner radius “ a ” , an outer radius,“ b (b > a) ”and a length unit, as shown in Fig. 3. The gap between the pipes is filled with a fluid
having a viscosity coefficient, µ , the outer pipe is mechanically fixed, and the inner pipe is rotatedat an angular velocity, ω . Here the flow is laminar and the adhesive condition is satisfied on thewall. Answer the following questions.
(1) Show the conditions that hold on the wall.
(2) Find the flow velocity distribution.
(3) Find the torque required to rotate the inner tube.
(4) Find the torque required to fix the outer pipe.
Fig.3
3