Flow in a Corner

    The flow of a fluid in the corner 0≤θ≤π/3 is described by the field
        FormBox[RowBox[{Overscript[v, ⇀], =, RowBox[{U (x^2 - y^2, -2x y), Cell[]}]}], TraditionalForm] defined for 0≤θ≤π/3
where U is a parameter describing the speed of the flow. The potential is
        ϕ(x, y) = U/3 (x^3 - 3 x y^2),
and here is a graph of the potential.

Off[Plot3D :: "plnc"] ;            ...      (* Turn warnings back on *)On[Plot3D :: "gval"] ;

[Graphics:../HTMLFiles/index_196.gif]

Now we plot the vector field together with some representative equipotential curves.

Needs["Graphics`PlotField`"] Needs["Graphics`ImplicitPlot`"] U = 1/10 ; v[ ... lotRange {{0, 1}, {0, 1}}}], ]}], ;}],         }]

[Graphics:../HTMLFiles/index_198.gif]

    When we work in the corner 0≤θ≤π/4, we have
        FormBox[RowBox[{Overscript[v, ⇀], =, RowBox[{U (x^3 - 3x y^2, y^3 - 3x^2y), Cell[]}]}], TraditionalForm]
and the potential is
        ϕ(x, y) = U/4 (x^4 - 6x^2y^2 + y^4).
Its graph is

Off[Plot3D :: "plnc"] ;            ... nbsp;     (* Turn on warnings *)On[Plot3D :: "gval"] ;

[Graphics:../HTMLFiles/index_203.gif]

Now we plot the vector field together with some representative equipotential curves.

Needs["Graphics`PlotField`"] Needs["Graphics`ImplicitPlot`"] U = 1/10 ; v[ ... n, PlotRange {{0, 1}, {0, 1}}] ;         

[Graphics:../HTMLFiles/index_205.gif]

    Although we have described how these flows vary as the parameter U changes, we have not yet described how the vary as the domain varies. In particular, let us consider the flow in the domain 0≤θ≤Θ, where Θ lies in the range 0≤Θ≤π. The cases Θ = π/3 and Θ = π/4 have already been discussed.

    To begin, the potential ϕ(x, y) in the domain 0≤θ≤Θ is given by the function
            ϕ(x, y) = ( U Θ)/π (x^2 + y^2)^π /(2Θ) cos (π/Θ cos^(-1) (( x)/(x^2 + y^2)^(1/2))).
The right side is the real part of the complex number ( U Θ)/π (x + i y)^(π / Θ). This expression can be simplified when Θ = π/n for some integer n. Indeed, if n is an integer, then
            cos ( n cos^(-1) ξ) = T_n(ξ)    
where T_n is the Chebyshev polynomial. The first five Chebyshev polynomials are:

Do[Print[Subscript["T", n], "(ξ) = ", ChebyshevT[n, ξ]], {n, 1, 5}]

T_1(ξ) = ξ

T_2(ξ) =  -1 + 2 ξ^2

T_3(ξ) =  -3 ξ + 4 ξ^3

T_4(ξ) = 1 - 8 ξ^2 + 8 ξ^4

T_5(ξ) = 5 ξ - 20 ξ^3 + 16 ξ^5

Thus, the potential in the region 0≤θ≤π/n is
        ϕ(x, y) = ( U)/n (x^2 + y^2)^(n/2) T_n (( x)/(x^2 + y^2)^(1/2)).
We can compare this result to the values previously given for the potentials in 0≤θ≤π/3 and 0≤θ≤π/4:

Clear[U] ; ϕ[x_, y_] := U/3 (x^2 + y^2)^(3/2) ChebyshevT[3, x/(x^2 + y^2)^(1/2)] Print[&q ... ;, η], ξ], D[ϕ[ξ, η], η]} /. {ξx, ηy}]] ;

ϕ(x,y) = 1/3 U (x^3 - 3 x y^2)

⇀    v    (x,y) =  {U (x^2 - y^2), -2 U x y}

Clear[U] ; ϕ[x_, y_] := U/4 (x^2 + y^2)^(4/2) ChebyshevT[4, x/(x^2 + y^2)^(1/2)] Print[&q ... ;, η], ξ], D[ϕ[ξ, η], η]} /. {ξx, ηy}]] ;

ϕ(x,y) = 1/4 U (x^4 - 6 x^2 y^2 + y^4)

⇀    v    (x,y) =  {U x (x^2 - 3 y^2), U y (-3 x^2 + y^2)}

    Returning to the general case, the velocity field is given by Overscript[v, ⇀] = ∇ϕ so that
        Overscript[v, ⇀] = U (x^2 + y^2)^(π/(2 Θ) - 1) (x cos (π/Θcos^(-1) ( ... 60;/Θcos^(-1) (x/(x^2 + y^2)^(1/2))) - x sin (π/Θcos^(-1) (x/(x^2 + y^2)^(1/2)))) .
Indeed, we have

Clear[U, Θ] ϕ[x_, y_, Θ_] := (U Θ)/π (x^2 + y^2)^π/(2 Θ) Co ... 640; Print[   v    (x,y) = , Assuming[x∈Reals && y≥0, FullSimplify[v[x, y]]]]

⇀    v    (x,y) =  {U (x^2 + y^2)^(-1 + π/(2 Θ)) (x Cos[(π ArcCos ... π ArcCos[x/(x^2 + y^2)^(1/2)])/Θ] - x Sin[(π ArcCos[x/(x^2 + y^2)^(1/2)])/Θ])}

We can then plot both the vector field and the potential for a range of values. Here is an animation showing the different fields for Θ<π/2.

Needs["Graphics`PlotField`"] Needs["Graphics`ImplicitPlot`"] Θ_min =  ... #62754; {{0, 1}, {0, 1}}}], ]}]}], ,, {Θ, Θ_min, Θ_max, Θ_step}}], ]}], ;}]}]

Flow in an Acute Corner

We can even allow Θ>π/2.

Needs["Graphics`PlotField`"] Needs["Graphics`ImplicitPlot`"] Θ_min =  ... 371;PlotRange {{-1, 1}, {0, 2}}],  {Θ, Θ_min, Θ_max, Θ_step}]

Flow in an Obtuse Corner