Fluid dynamics phenomenon
In the science of fluid flow , Stokes' paradox is the phenomenon that there can be no creeping flow of a fluid around a disk in two dimensions; or, equivalently, the fact there is no non-trivial steady-state solution for the Stokes equations around an infinitely long cylinder. This is opposed to the 3-dimensional case, where Stokes' method provides a solution to the problem of flow around a sphere.[1] [2]
Stokes' paradox was resolved by Carl Wilhelm Oseen in 1910, by introducing the Oseen equations which improve upon the Stokes equations – by adding convective acceleration .
Derivation [ edit ]
The velocity vector
u
{\displaystyle \mathbf {u} }
of the fluid may be written in terms of the stream function
ψ
{\displaystyle \psi }
as
u
=
(
∂
ψ
∂
y
,
−
∂
ψ
∂
x
)
.
{\displaystyle \mathbf {u} =\left({\frac {\partial \psi }{\partial y}},-{\frac {\partial \psi }{\partial x}}\right).}
The stream function in a Stokes flow problem,
ψ
{\displaystyle \psi }
satisfies the biharmonic equation .[3] By regarding the
(
x
,
y
)
{\displaystyle (x,y)}
-plane as the complex plane , the problem may be dealt with using methods of complex analysis . In this approach,
ψ
{\displaystyle \psi }
is either the real or imaginary part of
z
¯
f
(
z
)
+
g
(
z
)
{\displaystyle {\bar {z}}f(z)+g(z)}
.[4]
Here
z
=
x
+
i
y
{\displaystyle z=x+iy}
, where
i
{\displaystyle i}
is the imaginary unit,
z
¯
=
x
−
i
y
{\displaystyle {\bar {z}}=x-iy}
, and
f
(
z
)
,
g
(
z
)
{\displaystyle f(z),g(z)}
are holomorphic functions outside of the disk. We will take the real part without loss of generality .
Now the function
u
{\displaystyle u}
, defined by
u
=
u
x
+
i
u
y
{\displaystyle u=u_{x}+iu_{y}}
is introduced.
u
{\displaystyle u}
can be written as
u
=
−
2
i
∂
ψ
∂
z
¯
{\displaystyle u=-2i{\frac {\partial \psi }{\partial {\bar {z}}}}}
, or
1
2
i
u
=
∂
ψ
∂
z
¯
{\displaystyle {\frac {1}{2}}iu={\frac {\partial \psi }{\partial {\bar {z}}}}}
(using the Wirtinger derivatives ).
This is calculated to be equal to
1
2
i
u
=
f
(
z
)
+
z
f
′
¯
(
z
)
+
g
′
¯
(
z
)
.
{\displaystyle {\frac {1}{2}}iu=f(z)+z{\bar {f\prime }}(z)+{\bar {g\prime }}(z).}
Without loss of generality, the disk may be assumed to be the unit disk , consisting of all complex numbers z of absolute value smaller or equal to 1.
The boundary conditions are:
lim
z
→
∞
u
=
1
,
{\displaystyle \lim _{z\to \infty }u=1,}
u
=
0
,
{\displaystyle u=0,}
whenever
|
z
|
=
1
{\displaystyle |z|=1}
,[1] [5]
and by representing the functions
f
,
g
{\displaystyle f,g}
as Laurent series :[6]
f
(
z
)
=
∑
n
=
−
∞
∞
f
n
z
n
,
g
(
z
)
=
∑
n
=
−
∞
∞
g
n
z
n
,
{\displaystyle f(z)=\sum _{n=-\infty }^{\infty }f_{n}z^{n},\quad g(z)=\sum _{n=-\infty }^{\infty }g_{n}z^{n},}
the first condition implies
f
n
=
0
,
g
n
=
0
{\displaystyle f_{n}=0,g_{n}=0}
for all
n
≥
2
{\displaystyle n\geq 2}
.
Using the polar form of
z
{\displaystyle z}
results in
z
n
=
r
n
e
i
n
θ
,
z
¯
n
=
r
n
e
−
i
n
θ
{\displaystyle z^{n}=r^{n}e^{in\theta },{\bar {z}}^{n}=r^{n}e^{-in\theta }}
.
After deriving the series form of u , substituting this into it along with
r
=
1
{\displaystyle r=1}
, and changing some indices, the second boundary condition translates to
∑
n
=
−
∞
∞
e
i
n
θ
(
f
n
+
(
2
−
n
)
f
¯
2
−
n
+
(
1
−
n
)
g
¯
1
−
n
)
=
0.
{\displaystyle \sum _{n=-\infty }^{\infty }e^{in\theta }\left(f_{n}+(2-n){\bar {f}}_{2-n}+(1-n){\bar {g}}_{1-n}\right)=0.}
Since the complex trigonometric functions
e
i
n
θ
{\displaystyle e^{in\theta }}
compose a linearly independent set, it follows that all coefficients in the series are zero.
Examining these conditions for every
n
{\displaystyle n}
after taking into account the condition at infinity shows that
f
{\displaystyle f}
and
g
{\displaystyle g}
are necessarily of the form
f
(
z
)
=
a
z
+
b
,
g
(
z
)
=
−
b
z
+
c
,
{\displaystyle f(z)=az+b,\quad g(z)=-bz+c,}
where
a
{\displaystyle a}
is an imaginary number (opposite to its own complex conjugate ), and
b
{\displaystyle b}
and
c
{\displaystyle c}
are complex numbers. Substituting this into
u
{\displaystyle u}
gives the result that
u
=
0
{\displaystyle u=0}
globally, compelling both
u
x
{\displaystyle u_{x}}
and
u
y
{\displaystyle u_{y}}
to be zero. Therefore, there can be no motion – the only solution is that the cylinder is at rest relative to all points of the fluid.
Resolution [ edit ]
The paradox is caused by the limited validity of Stokes' approximation, as explained in Oseen's criticism: the validity of Stokes' equations relies on Reynolds number being small, and this condition cannot hold for arbitrarily large distances
r
{\displaystyle r}
.[7] [2]
A correct solution for a cylinder was derived using Oseen's equations , and the same equations lead to an improved approximation of the drag force on a sphere .[8] [9]
Unsteady-state flow around a circular cylinder [ edit ]
On the contrary to Stokes' paradox , there exists the unsteady-state solution of the same problem which models a fluid flow moving around a circular cylinder with Reynolds number being small. This solution can be given by explicit formula in terms of vorticity of the flow's vector field.
Formula of the Stokes Flow around a circular cylinder [ edit ]
The vorticity of Stokes' flow is given by the following relation:[10]
w
k
(
t
,
r
)
=
W
|
k
|
,
|
k
|
−
1
−
1
[
e
−
λ
2
t
W
|
k
|
,
|
k
|
−
1
[
w
k
(
0
,
⋅
)
]
(
λ
)
]
(
t
,
r
)
.
{\displaystyle w_{k}(t,r)=W_{|k|,|k|-1}^{-1}\left[e^{-\lambda ^{2}t}W_{|k|,|k|-1}[w_{k}(0,\cdot )](\lambda )\right](t,r).}
Here
w
k
(
t
,
r
)
{\displaystyle w_{k}(t,r)}
- are the Fourier coefficients of the vorticity's expansion by polar angle which are defined on
(
r
0
,
∞
)
{\displaystyle (r_{0},\infty )}
,
r
0
{\displaystyle r_{0}}
- radius of the cylinder,
W
|
k
|
,
|
k
|
−
1
{\displaystyle W_{|k|,|k|-1}}
,
W
|
k
|
,
|
k
|
−
1
−
1
{\displaystyle W_{|k|,|k|-1}^{-1}}
are the direct and inverse special Weber's transforms,[11] and initial function for vorticity
w
k
(
0
,
r
)
{\displaystyle w_{k}(0,r)}
satisfies no-slip boundary condition.
Special Weber's transform has a non-trivial kernel, but from the no-slip condition follows orthogonality of the vorticity flow to the kernel.[10]
Derivation [ edit ]
Special Weber's transform [ edit ]
Special Weber's transform[11] is an important tool in solving problems of the hydrodynamics . It is defined for
k
∈
R
{\displaystyle k\in \mathbb {R} }
as
W
k
,
k
−
1
[
f
]
(
λ
)
=
∫
r
0
∞
J
k
(
λ
s
)
Y
k
−
1
(
λ
r
0
)
−
Y
k
(
λ
s
)
J
k
−
1
(
λ
r
0
)
J
k
−
1
2
(
λ
r
0
)
+
Y
k
−
1
2
(
λ
r
0
)
f
(
s
)
s
d
s
,
{\displaystyle W_{k,k-1}[f](\lambda )=\int _{r_{0}}^{\infty }{\frac {J_{k}(\lambda s)Y_{k-1}(\lambda r_{0})-Y_{k}(\lambda s)J_{k-1}(\lambda r_{0})}{\sqrt {J_{k-1}^{2}(\lambda r_{0})+Y_{k-1}^{2}(\lambda r_{0})}}}f(s)sds,}
where
J
k
{\displaystyle J_{k}}
,
Y
k
{\displaystyle Y_{k}}
are the
Bessel functions of the first and second kind
[12] respectively. For
k
>
1
{\displaystyle k>1}
it has a non-trivial kernel
[13] [10] which consists of the functions
C
/
r
k
∈
ker
(
W
k
,
k
−
1
)
{\displaystyle C/r^{k}\in \ker(W_{k,k-1})}
.
The inverse transform is given by the formula
W
k
,
k
−
1
−
1
[
f
^
]
(
r
)
=
∫
0
∞
J
k
(
λ
r
)
Y
k
−
1
(
λ
r
0
)
−
Y
k
(
λ
s
)
J
k
−
1
(
λ
r
0
)
J
k
−
1
2
(
λ
r
0
)
+
Y
k
−
1
2
(
λ
r
0
)
f
^
(
λ
)
λ
d
λ
.
{\displaystyle W_{k,k-1}^{-1}[{\hat {f}}](r)=\int _{0}^{\infty }{\frac {J_{k}(\lambda r)Y_{k-1}(\lambda r_{0})-Y_{k}(\lambda s)J_{k-1}(\lambda r_{0})}{\sqrt {J_{k-1}^{2}(\lambda r_{0})+Y_{k-1}^{2}(\lambda r_{0})}}}{\hat {f}}(\lambda )\lambda d\lambda .}
Due to non-triviality of the kernel, the inversion identity
f
(
r
)
=
W
k
,
k
−
1
−
1
[
W
k
,
k
−
1
[
f
]
]
(
r
)
{\displaystyle f(r)=W_{k,k-1}^{-1}\left[W_{k,k-1}[f]\right](r)}
is valid if
k
≤
1
{\displaystyle k\leq 1}
. Also it is valid in the case of
k
>
1
{\displaystyle k>1}
but only for functions, which are orthogonal to the kernel of
W
k
,
k
−
1
{\displaystyle W_{k,k-1}}
in
L
2
(
r
0
,
∞
)
{\displaystyle L_{2}(r_{0},\infty )}
with infinitesimal element
r
d
r
{\displaystyle rdr}
:
∫
r
0
∞
1
r
k
f
(
r
)
r
d
r
=
0
,
k
>
1.
{\displaystyle \int _{r_{0}}^{\infty }{\frac {1}{r^{k}}}f(r)rdr=0,~k>1.}
No-slip condition and Biot–Savart law [ edit ]
In exterior of the disc of radius
r
0
{\displaystyle r_{0}}
B
r
0
=
{
x
∈
R
2
,
|
x
|
>
r
0
}
{\displaystyle B_{r_{0}}=\{\mathbf {x} \in \mathbb {R} ^{2},~\vert \mathbf {x} \vert >r_{0}\}}
the Biot-Savart law
v
(
x
)
=
1
2
π
∫
B
r
0
(
x
−
y
)
⊥
|
x
−
y
|
2
w
(
y
)
d
y
+
v
∞
,
{\displaystyle \mathbf {v} (\mathbf {x} )={\frac {1}{2\pi }}\int _{B_{r_{0}}}{\frac {(\mathbf {x} -\mathbf {y} )^{\perp }}{\vert \mathbf {x} -\mathbf {y} \vert ^{2}}}w(\mathbf {y} )\operatorname {d\mathbf {y} } +\mathbf {v} _{\infty },}
restores the velocity field
v
(
x
)
{\displaystyle \mathbf {v} (\mathbf {x} )}
which is induced by the vorticity
w
(
x
)
{\displaystyle w(\mathbf {x} )}
with zero-circularity and given constant velocity
v
∞
{\displaystyle \mathbf {v} _{\infty }}
at infinity.
No-slip condition for
x
∈
S
r
0
=
{
x
∈
R
2
,
|
x
|
=
r
0
}
{\displaystyle \mathbf {x} \in S_{r_{0}}=\{\mathbf {x} \in \mathbb {R} ^{2},~\vert \mathbf {x} \vert =r_{0}\}}
1
2
π
∫
B
r
0
(
x
−
y
)
⊥
|
x
−
y
|
2
w
(
y
)
d
y
+
v
∞
=
0
{\displaystyle {\frac {1}{2\pi }}\int _{B_{r_{0}}}{\frac {(\mathbf {x} -\mathbf {y} )^{\perp }}{\vert \mathbf {x} -\mathbf {y} \vert ^{2}}}w(\mathbf {y} )\operatorname {d\mathbf {y} } +\mathbf {v} _{\infty }=0}
leads to the relations for
k
∈
Z
{\displaystyle k\in \mathbf {Z} }
:
∫
r
0
∞
r
−
|
k
|
+
1
w
k
(
r
)
d
r
=
d
k
,
{\displaystyle \int _{r_{0}}^{\infty }r^{-\vert k\vert +1}w_{k}(r)dr=d_{k},}
where
d
k
=
δ
|
k
|
,
1
(
v
∞
,
y
+
i
k
v
∞
,
x
)
,
{\displaystyle d_{k}=\delta _{\vert k\vert ,1}(v_{\infty ,y}+ikv_{\infty ,x}),}
δ
|
k
|
,
1
{\displaystyle \delta _{\vert k\vert ,1}}
is the
Kronecker delta ,
v
∞
,
x
{\displaystyle v_{\infty ,x}}
,
v
∞
,
y
{\displaystyle v_{\infty ,y}}
are the cartesian coordinates of
v
∞
{\displaystyle \mathbf {v} _{\infty }}
.
In particular, from the no-slip condition follows orthogonality the vorticity to the kernel of the Weber's transform
W
k
,
k
−
1
{\displaystyle W_{k,k-1}}
:
∫
r
0
∞
r
−
|
k
|
+
1
w
k
(
r
)
d
r
=
0
f
o
r
|
k
|
>
1.
{\displaystyle \int _{r_{0}}^{\infty }r^{-\vert k\vert +1}w_{k}(r)dr=0~for~|k|>1.}
Vorticity flow and its boundary condition [ edit ]
Vorticity
w
(
t
,
x
)
{\displaystyle w(t,\mathbf {x} )}
for Stokes flow satisfies to the vorticity equation
∂
w
(
t
,
x
)
∂
t
−
Δ
w
=
0
,
{\displaystyle {\frac {\partial w(t,\mathbf {x} )}{\partial t}}-\Delta w=0,}
or in terms of the Fourier coefficients in the expansion by polar angle
∂
w
k
(
t
,
r
)
∂
t
−
Δ
w
k
=
0
,
{\displaystyle {\frac {\partial w_{k}(t,r)}{\partial t}}-\Delta w_{k}=0,}
where
Δ
k
w
k
(
t
,
r
)
=
1
r
∂
∂
r
(
r
∂
∂
r
w
k
(
t
,
r
)
)
−
k
2
r
2
w
k
(
t
,
r
)
.
{\displaystyle \Delta _{k}w_{k}(t,r)={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial }{\partial r}}w_{k}(t,r)\right)-{\frac {k^{2}}{r^{2}}}w_{k}(t,r).}
From no-slip condition follows
d
d
t
∫
r
0
∞
r
−
|
k
|
+
1
w
k
(
t
,
r
)
d
r
=
0.
{\displaystyle {\frac {d}{dt}}\int _{r_{0}}^{\infty }r^{-\vert k\vert +1}w_{k}(t,r)dr=0.}
Finally, integrating by parts, we obtain the Robin boundary condition for the vorticity:
∫
r
0
∞
s
−
|
k
|
+
1
Δ
k
w
k
(
t
,
r
)
d
r
=
−
r
0
−
|
k
|
(
r
0
∂
w
k
(
t
,
r
)
∂
r
|
r
=
r
0
+
|
k
|
w
k
(
t
,
r
0
)
)
=
0.
{\displaystyle \int _{r_{0}}^{\infty }s^{-|k|+1}\Delta _{k}w_{k}(t,r)dr=-r_{0}^{-|k|}\left(r_{0}{\frac {\partial w_{k}(t,r)}{\partial r}}{\Big |}_{r=r_{0}}+|k|w_{k}(t,r_{0})\right)=0.}
Then the solution of the boundary-value problem can be expressed via Weber's integral above.
Formula for vorticity can give another explanation of the Stokes' Paradox. The functions
C
r
k
∈
k
e
r
(
Δ
k
)
,
k
>
1
{\displaystyle {\frac {C}{r^{k}}}\in ker(\Delta _{k}),~k>1}
belong to the kernel of
Δ
k
{\displaystyle \Delta _{k}}
and generate the stationary solutions of the vorticity equation with Robin-type boundary condition. From the arguments above any Stokes' vorticity flow with no-slip boundary condition must be orthogonal to the obtained stationary solutions. That is only possible for
w
≡
0
{\displaystyle w\equiv 0}
.
See also [ edit ]
References [ edit ]
^ a b Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 602–604 .
^ a b Van Dyke, Milton (1975). Perturbation Methods in Fluid Mechanics . Parabolic Press.
^ Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 602 .
^ Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics . CRC Press. ISBN 1584883472 .
^ Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 615 .
^ Sarason, Donald (1994). Notes on Complex Function Theory . Berkeley, California. {{cite book }}
: CS1 maint: location missing publisher (link )
^ Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 608–609 .
^ Lamb, Horace (1945). Hydrodynamics (Sixth ed.). New York: Dover Publications. pp. 609–616 .
^ Goldstein, Sydney (1965). Modern Developments in Fluid Dynamics . Dover Publications.
^ a b c Gorshkov, A.V. (2019). "Associated Weber–Orr Transform, Biot–Savart Law and Explicit Form of the Solution of 2D Stokes System in Exterior of the Disc". J. Math. Fluid Mech . 21 (41): 41. arXiv :1904.12495 . Bibcode :2019JMFM...21...41G . doi :10.1007/s00021-019-0445-2 . S2CID 199113540 .
^ a b Titchmarsh, E.C. (1946). Eigenfunction Expansions Associated With Second-Order Differential Equations, Part I . Clarendon Press, Oxford.
^ Watson, G.N. (1995). A Treatise on the Theory of Bessel Functions . Cambridge University Press.
^ Griffith, J.L. (1956). "A note on a generalisation of Weber's transform". J. Proc. Roy. Soc . 90 . New South Wales: 157–162.