EXAMPLES
  1. f(z) = U z (Φ = U x, Ψ = U y)

    w = f′(z) = U = u − i v
    so
    u = U,     v = 0
    This is a uniform flow in the x direction.
    To find the streamlines, use
    f(z) = U (x + i y),
    to obtain
    Ψ = U y,
    so the equation for the streamlines is
    U y = c,
    which can be drawn for different values of c as
    fluid_uniform.jpg
  2. f(z) = m ln z
    Express z in polar form as
    z = r ei θ
    it follows
    f(z) = m (ln r + i θ)
    from which the streamline is
    Ψ = m θ = const.
    This flow is called a source flow if m is positive, and it is called a sink flow if m is negative.
    fluid_source.jpg
    The velocity is obtained as
    vr = ∂Φ
    ∂r
    		 = m
    r
    		 .
  3. f(z) = m ln (z − a) − m ln (z + a)
    If a source exists at z = a with a strength of m, and a sink exists at z = −a with a strength of −m, the streamlines should look like:
    fluid_double.jpg
    It can be shown that the streamlines are part of circles. Set
    z − a = r1 ei θ1,     z + a = r2 ei θ2,
    		(1)
    then it follows

    f(z)
    =
    m ln ⎛
    ⎝     		z − a
    z + a
    		⎞
    ⎠
    =
    m ln ⎛
    ⎝     		r1
    r2
    		 ei (θ1 − θ2)⎞
    ⎠
    =
    m ln  r1
    r2
    		 + m i (θ1 − θ2),
    		(2)
    so the equation for the streamlines is

    (θ1 − θ2) = const.
    		(3)
    From the figures below, the loci of θ1 − θ2 = const form a circle.
    fluid_source_and_sink.jpg
    Let 2 a → 0 while 2 a m remains a constant in the above as
    f(z)
    =
    m ln ⎛
    ⎝     		1 − a/z
    1 + a/z
    		⎞
    ⎠
    =
    m ⎛
    ⎝     		− a
    z
    		 − 1
    2
    		 ⎛
    ⎝     		a
    z
    		⎞
    ⎠     		2

     
    		 − 1
    3
    		 ⎛
    ⎝     		a
    z
    		⎞
    ⎠     		3

     
    		 − …⎞
    ⎠     		−m ⎛
    ⎝     		a
    z
    		 − 1
    2
    		 ⎛
    ⎝     		a
    z
    		⎞
    ⎠     		2

     
    		 + 1
    3
    		 ⎛
    ⎝     		a
    z
    		⎞
    ⎠     		3

     
    		 − …⎞
    ⎠
     ∼ 
    − 2 a m
    z
    		 − 2
    3
    		 ⎛
    ⎝     		a m
    z
    		⎞
    ⎠     		3

     
    		 − 2
    5
    		 ⎛
    ⎝     		a m
    z
    		⎞
    ⎠     		5

     
    		 − …
     ∼ 
    − 2 a m
    z
    		 .
    		(4)
    fluid_couple.jpg
  4. f(z) = U z + m  ln z
    Use the polar form, z = r ei θ, so that
    f(z)
    =
    U r ( cos θ+ i sin θ) + m ( ln r + i θ)
    =
    (U r cos θ+ m ln r) + i( U r sin θ+ m θ),
    		(5)
    thus,

    Ψ = U r sin θ+ m θ
    The streamlines can be found by solving
    U r sin θ+ m θ = c     (const.)
    Several solutions are possible for the above equation.
    1. c = 0
      U r sin θ+ m θ = 0 can be satisfied for θ = 0 ( the positive x axis).
    2. c = m π
      U r sin θ+ m θ = m π can be satisfied for θ = π (the negative x axis). Besides this trivial solution, one can also solve for r as

      r = m (π− θ)
      U sin θ
      		 .
      As θ→ π, r → m/U. Also as θ→ 0, r sin θ→ m π/U. For other values of θ, one can numerically compute r as
      fluid_solid1.jpg
    3. For other values of c, the stream line should look like
      fluid_solid2.jpg
    fluid_1.jpg
  5. f(z) = U z + m ln (z + a) − m ln (z − a)
    This is called the Rankine oval.
    (details will be discussed in class. Will be filled in later)
  6. f(z) = U z + [(μ)/(z)]
    With the polar form, z = r ei θ,

    f(z) = U r ei θ + μ
    r
    		 e− i θ = ⎛
    ⎝     		U r + μ
    r
    		⎞
    ⎠     		cos θ+ i⎛
    ⎝     		U r − μ
    r
    		⎞
    ⎠     		sin θ,
    so
    Ψ = (U r − μ
    r
    		 ) sin θ
    and the equation for the streamlines is
    (U r − μ
    r
    		 ) sin θ = c     (const).
    1. c = 0
      If sin θ = 0, it follows θ = 0, π. Otherwise, U r − μ/r = 0 from which
      r =   ⎛
      √
      μ
      U
       
      		 .
      This is a circle with μ/U as the radius.
      fluid_circle1.jpg
    2. For other values of c, the graph of (r, θ) may look like
      fluid_circle2.jpg
    As is seen, this represents a uniform flow around a circular cylinder.


A sample code in Mathematica to visualize the flow is shown below:
fluid_example.jpg


File translated from TEX by TTH, version 4.03.
On 18 Jan 2021, 22:07.