Joseph - Helmholtz decomposition coupling rotational to irrotational flow of a viscous fluid.pdf

Joseph - Helmholtz decomposition coupling rotational to irrotational flow of a viscous fluid.pdf
Helmholtz decomposition coupling rotational
to irrotational flow of a viscous fluid
Daniel D. Joseph*
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455; and Department of Mechanical and Aerospace
Engineering, University of California, Irvine, CA 92697
Contributed by Daniel D. Joseph, July 10, 2006
In this work, I present the form of the Navier–Stokes equations
implied by the Helmholtz decomposition in which the relation of
the irrotational and rotational velocity fields is made explicit. The
idea of self-equilibration of irrotational viscous stresses is introduced. The decomposition is constructed by first selecting the
irrotational flow compatible with the flow boundaries and other
prescribed conditions. The rotational component of velocity is then
the difference between the solution of the Navier–Stokes equations and the selected irrotational flow. To satisfy the boundary
conditions, the irrotational field is required, and it depends on the
viscosity. Five unknown fields are determined by the decomposed
form of the Navier–Stokes equations for an incompressible fluid:
the rotational component of velocity, the pressure, and the harmonic potential. These five fields may be readily identified in
analytic solutions available in the literature. It is clear from these
exact solutions that potential flow of a viscous fluid is required to
satisfy prescribed conditions, like the no-slip condition at the
boundary of a solid or continuity conditions across a two-fluid
boundary. It can be said that equations governing the Helmholtz
decomposition describe the modification of irrotational flow due
to vorticity, but the analysis shows the two fields are coupled and
cannot be completely determined independently.
conditions. The boundary conditions for the irrotational flows have
a heavy weight in all this. Simple examples of unique decomposition, taken from hydrodynamics, will be presented later.
The decomposition of the velocity into rotational and irrotational
parts holds at each and every point and varies from point to point
in the flow domain. Various possibilities for the balance of these
parts at a fixed point and the distribution of these balances from
point to point can be considered as follows:
(i) The flow is purely irrotational or purely rotational. These two
possibilities do occur as approximations but are not typical.
(ii) Typically, the flow is mixed with rotational and irrotational
components at each point.
Navier–Stokes Equations for the Decomposition
To study solutions of the Navier–Stokes equations, it is convenient
to express the Navier–Stokes equations for an incompressible fluid,
␳
T
再
⭸␷
⭸␾ ␳
⫹ⵜ ␳
⫹ 兩ⵜ ␾ 兩 2 ⫹ p
⭸t
⭸t
2
divu ⫽ div␷ ⫹ ⵜ 2␾ ,
[2]
⫹␳
curlu ⫽ curl␷.
[3]
and
ⵜ 2␾ ⫽ 0.
[4]
Since ␾ is harmonic, we have from Eqs. 1 and 4 that
ⵜ 2 u ⫽ ⵜ 2 ␷.
[5]
The irrotational part of u is on the null space of the Laplacian, but
in special cases, like plane shear flow, ⵜ2␷ ⫽ 0, but curl␷ ⫽ 0.
Unique decompositions are generated by solutions of the
Navier–Stokes equation (Eq. 6) in decomposed form (Eq. 7) where
the irrotational flows satisfy Eqs. 1 and 3–5 and certain boundary
14272–14277 兩 PNAS 兩 September 26, 2006 兩 vol. 103 兩 no. 39
再
⭸
⭸␷i
⭸␾ ␳
⫹
⫹ 兩ⵜ␾兩2 ⫹ p
␳
⭸t
⭸xi
⭸t
2
␳
divu ⫽ div␷ ⫽ 0
[7]
or
[1]
This decomposition leads to the theory of the vector potential,
which is not followed here. The decomposition is unique on
unbounded domains without boundaries, and explicit formulas for
the scalar and vector potentials are well known. A framework for
embedding the study of potential flows of viscous fluids, in which
no special flow assumptions are made, is suggested by this decomposition (Eq. 1). If the fields are solenoidal, then
冎
⫹ ␳ div关 ␷ 丢 ⵜ ␾ ⫹ ⵜ ␾ 丢 ␷ ⫹ ␷ 丢 ␷兴 ⫽ ␮ ⵜ 2␷,
u ⫽ ␷ ⫹ ⵜ␾ ,
where
[6]
in terms of the rotational and irrotational fields implied by the
Helmholtz decomposition,
potential flow 兩 vorticity 兩 Navier–Stokes 兩 self-equilibration 兩 dissipation
Helmholtz Decomposition
he Helmholtz decomposition theorem says that every smooth
vector field u, defined everywhere in space and vanishing at
infinity together with its first derivatives can be decomposed into a
rotational part ␷ and an irrotational part ⵜ␾.
du
⫽ ⫺ⵜp ⫹ ␮ ⵜ 2u,
dt
␳
冉
冊
冎
⭸
⭸␾
⭸␾
␷
⫹ ␷i
⫹ ␷ i␷ j ⫽ ␮ ⵜ 2␷ i ,
⭸xj j ⭸xi
⭸x j
satisfying Eq. 4.
To solve this problem in a domain ⍀, say, when the velocity u ⫽
V is prescribed on ⭸⍀, we would need to compute a solenoidal field
␷ satisfying Eq. 7 and a harmonic function ␾ satisfying ⵜ2␾ ⫽ 0 such
that ␷ ⫹ ⵜ␾ ⫽ V on ⭸⍀. Since this system of five equations in five
unknowns is just the decomposed form of the four equations in four
unknowns that defines the Navier–Stokes system for u, it should be
possible to study this problem using exactly the same mathematical
tools that are used to study the Navier–Stokes equations.
In the Navier–Stokes theory for incompressible fluid, the solutions are decomposed into a space of gradients and its complement,
which is a space of solenoidal vectors. The gradient space is not, in
general, solenoidal because the pressure is not solenoidal. If it were
Conflict of interest statement: No conflicts declared.
Freely available online through the PNAS open access option.
*E-mail: [email protected]
© 2006 by The National Academy of Sciences of the USA
www.pnas.org兾cgi兾doi兾10.1073兾pnas.0605792103
冋
ⵜ2 ␳
册
⭸␾ ␳
⭸2
⫹ 兩ⵜ␾兩2 ⫹ p ⫽ ⫺␳
⭸t 2
⭸x i⭸x j
冉
␷j
冊
⭸␾
⭸␾
⫹ ␷i
⫹ ␷ i␷ j .
⭸x i
⭸x j
In fact, there may be hidden irrotational terms on the right-hand
side of this equation.
The boundary condition for solutions of Eq. 6 is u ⫺ a ⫽ 0 on
⭸⍀, where a is solenoidal field such that a ⫽ V on ⭸⍀; hence
␷ ⫺ a ⫹ ⵜ ␾ ⫽ 0.
[8a]
The decomposition depends on the selection of the harmonic
function ␾; the traditional boundary condition n䡠a ⫽ n䡠ⵜ␾ on ⭸⍀
together with a Dirichlet condition at infinity when the region of
flow is unbounded and a prescription of the value of the circulation
in doubly connected regions, gives rise to a unique ␾. Then the
rotational flow must satisfy
␷䡠n ⫽ 0
and
es䡠共 ␷ ⫺ a兲 ⫹ es䡠ⵜ ␾ ⫽ 0
[8b]
on ⭸⍀. The ␷ determined in this way is rotational and satisfies Eq.
3. However, ␷ may contain other harmonic components.
Purely rotational flows ␷ have no harmonic components. Purely
rotational velocities can be identified in the exact solutions exhibited in the examples where the parts of the solution which are
harmonic and the parts that are not are identified by inspection. Eq.
18, in which the purely irrotational flow is identified by selecting a
parameter ␣, is a good example. We have a certain freedom in
selecting the harmonic functions used for the decomposition. It is
possible to formulate problems of potential flow depending on a
parameter, say for streaming flow around bodies which would give
rise to the rotational flow in Eq. 18 when the body is a sphere.
However, in the general case in which explicit solutions are absent
and the potential flow is computed numerically, we have at present
no way to identify a purely irrotational flow. The examples show the
purely rotational flows exist in special cases. It remains to be seen
whether this concept makes sense in a general theory.
What is the value added to solutions of the Navier–Stokes
equations (Eq. 6) by solving them in the Helmholtz decomposed
form† (Eq. 7)? Certainly it is not easier to solve for five rather than
four fields; if you cannot solve Eq. 6 then you certainly cannot solve
Eq. 7. However, if the decomposed solution could be extracted
from solutions of Eq. 6 or computed directly, then the form of
the irrotational solution that is determined through coupling with
the rotational solution and the changes in its distribution as the
Reynolds number changes would be revealed. There is nothing
approximate about this; it is the correct description of the role of
irrotational solutions in the theory of the Navier–Stokes equations,
and it looks different and is different than the topic ‘‘potential flow
of an inviscid fluid,’’ which we all learned in school.
The form (Eq. 7) of the Navier–Stokes equations may be well
suited to the study of boundary layers of vorticity with irrotational
flow of viscous fluid outside. I conjecture that in such layers ␷ ⫽
0, while ␷ is relatively small in the irrotational viscous flow outside.
Rotational and irrotational flows are coupled in the mixed inertial
term on the left of Eq. 7. The irrotational flow does not vanish in
the boundary layers, and the rotational flow, although small,
probably will not be zero in the irrotational viscous flow outside.
This feature is also in Prandtl’s theory of boundary layers, but that
theory is not rigorous, and the irrotational part is, so to say, inserted
†A
cultured lady asked a famous conductor of Baroque music whether J.S. Bach was still
composing. The conductor replied, ‘‘No madame, he is decomposing.’’
Joseph
by hand and is not coupled to the rotational flow at the boundary.
The coupling terms are of considerable interest, and they should
play strongly in the region of small vorticity at the edge of the
boundary layer. The action of irrotational flow in the exact boundary layer solution of Hiemenz (1911) for steady two-dimensional
(2D) flow toward a ‘‘stagnation point’’ at a rigid body (2) and Hamel
(1917) flow (2, 3) in diverging and converging channels in the
Helmholtz decomposed form at the end of this work.
Effects on boundary layers on rigid solids arising from the
viscosity of the fluid in the irrotational flow outside have been
considered without the decomposition by Wang and Joseph (4, 5)
and Padrino and Joseph (6).
Self-Equilibration of the Irrotational Viscous Stress
The stress in a Newtonian incompressible fluid is given by
T ⫽ ⫺p1 ⫹ ␮ 共ⵜu ⫹ ⵜuT兲
⫽ ⫺p1 ⫹ ␮ 共ⵜ ␷ ⫹ ⵜ ␷T兲 ⫹ 2 ␮ ⵜ 丢 ⵜ ␾ .
Most flows have an irrotational viscous stress. The term ␮ⵜ2␷ in
Eq. 7 arises from the rotational part of the viscous stress.
The irrotational viscous stress ␶I ⫽ 2␮ⵜ R ⵜ␾ does not give rise
to a force density term in Eq. 7. The divergence of ␶I vanishes on
each and every point in the domain V of flow. Even though an
irrotational viscous stress exists, it does not produce a net force to
drive motions. Moreover,
冕
div ␶ IdV ⫽
冖
n䡠␶ IdS ⫽ 0.
[9]
The traction vectors n䡠␶I have no net resultant on each and every
closed surface in the domain V of flow. We say that the irrotational
viscous stresses, which do not drive motions, are self-equilibrated.
Irrotational viscous stresses are not equilibrated at boundaries, and
they may produce forces there.
Dissipation Function for the Decomposed Motion
The dissipation function evaluated on the decomposed field (Eq. 1)
sorts out into rotational, mixed, and irrotational terms given by
⌽⫽
冕
2␮DijDijdV ⫽
V
⫹ 4␮
冕
D ij关 ␷兴
冕
2 ␮ D ij关 ␷兴D ij关 ␷兴dV
⭸ 2␾
dV ⫹ 2 ␮
⭸x i⭸x j
冕
⭸ 2␾ ⭸ 2␾
dV.
⭸x i⭸x j ⭸x i⭸x j
[10]
Most flows have an irrotational viscous dissipation. In regions V⬘
where ␷ is small, we have approximately that
T ⫽ ⫺p1 ⫹ 2 ␮ ⵜ 丢 ⵜ ␾ ,
[11]
and
⌽ ⫽ 2␮
冕
V⬘
⭸ 2␾ ⭸ 2␾
dV.
⭸xi⭸xj ⭸xi⭸xj
[12]
Eq. 12 has been widely used to study viscous effects in irrotational flows since Stokes in 1851 (in ref. 7, ‘‘historical notes’’).
Irrotational Flow and Boundary Conditions
How is the irrotational flow determined? It frequently happens that
the rotational motion cannot satisfy the boundary conditions; this
well known problem is associated with difficulties in forming
boundary conditions for the vorticity. The potential ␾ is a harmonic
function, which can be selected so that the values of the sum of the
PNAS 兩 September 26, 2006 兩 vol. 103 兩 no. 39 兩 14273
APPLIED
MATHEMATICS
solenoidal, then ⵜ2p ⫽ 0, but ⵜ2p ⫽ ⫺div␳u䡠ⵜu satisfies Poisson’s
equation.
It is not true that only the pressure is found on the gradient space.
Indeed Eq. 7 gives rise to a Poisson’s equation for the Bernoulli
function, not just the pressure.
rotational and irrotational fields can be chosen to balance prescribed conditions at the boundary. The allowed irrotational fields
can be selected from harmonic functions that enter into the purely
irrotational solution of the same problem on the same domain. A
very important property of potential flow arises from the fact that
irrotational viscous stresses do not give rise to irrotational viscous
forces in the equations of motion (Eq. 7). The interior values of the
rotational velocity are coupled to the irrotational motion through
Bernoulli terms evaluated on the potential and inertial terms that
couple the irrotational and rotational fields. The dependence of the
irrotational field on viscosity can be generated by the boundary
conditions.
Poiseuille Flow
A simple example that serves as a paradigm for the relation of the
irrotational and rotational components of velocity in all the solutions of the Navier–Stokes equations is plane Poiseuille flow
u⫽
冋
⫺
␷⫽
册
P⬘
共b 2 ⫺ y 2兲, 0, 0 ,
2␮
冋
册
P⬘ 2
y , 0, 0 ,
2␮
curlu ⫽
ⵜ␾ ⫽
冋
⫺
冋
0, 0, ⫺
册
P⬘y
,
␮
册
P⬘ 2
b , 0, 0 .
2␮
[13]
The irrotational flow is a constrained field and cannot satisfy the
no-slip boundary condition. To satisfy the no-slip condition, we add
the irrotational flow ⭸␾兾⭸x ⫽ ⫺P⬘b2兾(2␮). The irrotational component is for uniform and unidirectional flow chosen so that u ⫽
0 at the boundary.
u ⫽ e␪ u共r兲,
A⫽⫺
u ⫽ e␪⍀ ab at r ⫽ b,
a 2⍀ a ⫺ b 2⍀ 2
,
b2 ⫺ a2
␷ ⫽ e␪ Ar,
B⫽
u共r兲 ⫽ Ar ⫹
共⍀ a ⫺ ⍀ b兲a 2b 2
,
b2 ⫺ a2
B
,
r
[14]
1 ⭸␾ B
⫽ .
r ⭸␪
r
The irrotational flow with ␷ ⫽ 0 is an exact solution of the
Navier–Stokes equations with no-slip at boundaries.
Stokes Flow Around a Sphere of Radius a in a Uniform
Stream U (2, 3)
The Helmholtz decomposition of the solution of the problem of
slow streaming motion over a stationary sphere is given in spherical
polar coordinates by
u ⫽ 共u r, u ␪兲 ⫽
冋
冋
冉
␷r ⫹
冊
⭸␾
1 ⭸␾
, ␷␪ ⫹
,
⭸r
r ⭸␪
冉 冊册
冉 冊册
3a 1 a
⫹
ur ⫽ U 1 ⫺
2r 2 r
cos ␪ ,
[15]
3 a
U cos ␪ ,
2 r
冋
␾⫽U r⫺
3 a
␮ U cos ␪ .
2 r2
[17]
册
␣ a3
cos ␪ .
4 r2
[18]
The normal component of velocity vanishes when ␣ ⫽ ⫺2, and the
tangential component vanishes when ␣ ⫽ 4. In the present case, to
satisfy the no-slip condition, we take ␣ ⫽ 1. Both the rotational and
irrotational components of velocity are required to satisfy the
no-slip condition.
Streaming Motion Past an Ellipsoid (8)
The problem of the steady translation of an ellipsoid in a viscous
liquid was solved by Lamb in 1932 (8) in terms of the gravitationalpotential ⍀ of the solid and another harmonic function ␹ corresponding to the case in which ⵜ2 ␹ ⫽ 0, finite at infinity with ␹ ⫽
1 for the internal space. Citing Lamb (8):
If the fluid be streaming past the ellipsoid, regarded as fixed, with
the general velocity U in the direction of x, we assume
u⫽A
冉
冊
⭸␹
⭸ 2⍀
⫺ ␹ ⫹ U,
2 ⫹B x
⭸x
⭸x
w⫽A
sin ␪ ,
册
␷␪ ⫽
3 a
U sin ␪ ,
4 r
14274 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0605792103
ⵜ2u ⫽ 2B
⭸ 2␹
,
⭸x2
␷⫽A
⭸␹
⭸ 2⍀
⫹ Bx ,
⭸x⭸y
⭸y
⭸␹
⭸ 2⍀
⫹ Bx .
⭸x⭸z
⭸z
ⵜ2␷ ⫽ 2B
[16]
⭸ 2␹
,
⭸x⭸y
ⵜ2w ⫽ 2B
⭸ 2␹
.
⭸x⭸z
We note next that ⵜ2␷ ⫽ ⵜ2u. It follows that the rotational velocity
is associated with ␹ and is given by the terms proportional to B. The
irrotational velocity is given by ⵜ␾ ⫽ ⵜ(A⭸⍀兾⭸x). The vorticity is
given by ␻ ⫽ ⫺ey(2B⭸␹兾⭸z) ⫹ ez(2B⭸␹兾⭸y). The vorticity is determined by x times the harmonic function ⭸␹兾⭸x.
Hadamard–Rybyshinsky Solution for Streaming Flow Past
a Liquid Sphere
As in the flow around a solid sphere, this problem is posed in
spherical coordinates with a stream function and potential
function related by Eqs. 15–17. The stream function is given by
␺ ⫽ f共r兲sin2␪ ,
[19]
A
⫹ Br ⫹ Cr2 ⫹ Dr4 .
r
[20]
The irrotational of Eq. 19 is the part that satisfies ⵜ 2␺ p ⫽ 0.
From this, it follows that
冉
冊
A
⫹ Cr2 sin2␪ ,
r
[21]
␺v ⫽ 共Br ⫹ Dr4兲sin2␪ .
[22]
␺p ⫽
3
3a
curlu ⫽ U 0, 0, ⫺ 2 sin ␪ ,
2r
␷r ⫽ ⫺
p ⫺ p⬁ ⫽ ⫺
The potential for flow over a sphere is
f共r兲 ⫺
3
3a 1 a
⫺
u␪ ⫽ ⫺U 1 ⫺
4r
4 r
冋
册
1 a3
cos ␪ ,
4 r2
These satisfy the equation of continuity, in virtue of the relations
ⵜ2⍀ ⫽ 0, ⵜ2␹ ⫽ 0; and they evidently make u ⫽ U, ␷ ⫽ 0, w ⫽
0 at infinity. Again, they make
Flow Between Rotating Cylinders
u ⫽ e␪⍀ aa at r ⫽ a,
冋
␾⫽U r⫺
The potential ␾ corresponding to ␺ p is given by Eq. 17.
The solution of this problem is determined by continuity
conditions at r ⫽ a. The inner solution for r ⬍ a is designated
by u៮ , ␷៮ , ␾៮ , ␺៮ , p៮ , ␮៮ , ␳៮ .
Joseph
The normal component of velocity
␷r ⫹
⭸␾
⭸␾៮
⫽ ␷៮ r ⫹
⭸r
⭸r
[23]
is continuous at r ⫽ a. The normal stress balance is approximated by a static balance in which the jump of pressure is
balanced by surface tension so large that the drop is approximately spherical. The shear stress
Lamb (8) calculated an exact solution of the problem of the
viscous decay of free gravity waves as a free surface problem of
this type. He decomposes the solution into a stream function ␺
and potential function ␾, and the solution is given by
u⫽
⭸␾ ⭸␺
⫹
,
⭸x
⭸y
␷⫽
⭸␾ ⭸␺
⫺
,
⭸y
⭸x
p
⭸␾
⫺ g y,
⫽⫺
␳
⭸t
[29]
where
␮e␪䡠关ⵜ ␷ ⫹ ⵜ ␷ ⫹ 2ⵜ 丢 ⵜ ␾ 兴䡠er
T
The coefficients A, B, C, D are determined by the condition that
u ⫽ Uex as r 3 ⬁ and the continuity conditions (Eqs. 23 and 24)
find that when r ⱕ a,
៮f共r兲 ⫽
␮ 1U 4
共r ⫺ a2r2兲,
␮ ⫹ ␮៮ 4 a2
[25]
where the r 2 term is associated with irrotational flow and when
r ⱖ a,
f共r兲 ⫽
冉
冊
1
␮៮ 1
a3
U共r2 ⫺ ar兲 ⫹
U
⫺ ar ,
2
␮៮ ⫹ ␮ 4
r
[26]
where the r 2 and a 2兾r terms are irrotational.
Axisymmetric Steady Flow Around a Spherical Gas Bubble
at Finite Reynolds Numbers
This problem is like the Hadamard–Rybyshinski problem, with
the internal motion of the gas neglected, but inertia cannot be
neglected. The coupling conditions reduce to
␷r ⫹
⭸␾
⫽ 0,
⭸r
e␪䡠关ⵜ␷ ⫹ ⵜ ␷T ⫹ 2ⵜ 丢 ⵜ ␾ 兴䡠er ⫽ 0.
[28]
There is no flow across the interface at r ⫽ a, and the shear stress
vanishes there.
The equations of motion are the r and ␪ components of Eq. 1
with time derivative zero. Since Levich in 1947 (in ref. 7,
‘‘historical notes’’) it has been assumed that at moderately large
Reynolds numbers, the flow in the liquid is almost purely
irrotational with a small vorticity layer where ␷ ⫽ 0 in the liquid
near r ⫽ a. The details of the flow in the vorticity layer, the
thickness of the layer, and the presence and variation of viscous
pressure contribution all are unknown.
It may be assumed that the irrotational flow in the liquid
outside the sphere can be expressed as a series of spherical
harmonics. The problem then is to determine the participation
coefficients of the different harmonics, the pressure distribution,
and the rotational velocity ␷ satisfying the continuity conditions
(Eqs. 27 and 28). The determination of the participation coefficients may be less efficient than a purely numerical simulation
of Laplace’s equation outside a sphere subject to boundary
conditions on the sphere, which are coupled to the rotational
flow. This important problem has not yet been solved.
Viscous Decay of Free Gravity Waves (8, 9)
Flows that depend on only two space variables such as plane
f lows or axisymmetric f lows admit a stream function. Such
f lows may be decomposed into a stream function and potential
function.
Joseph
[30]
This decomposition is a Helmholtz decomposition; it can be said
that Lamb solved this problem in the Helmholtz formulation.
Wang and Joseph (9) constructed a purely irrotational solution of this problem, which is in very good agreement with the
exact solution. The potential in the Lamb solution is not the same
as the potential in the purely irrotational solution because they
satisfy different boundary conditions. It is worth noting that
viscous potential flow rather than inviscid potential flow is
required to satisfy boundary conditions. The common idea that
the viscosity should be put to zero to satisfy boundary conditions
is deeply flawed. It is also worth noting that the viscous
component of the pressure does not arise in the boundary layer
for vorticity in the exact solution; the pressure is given by Eq. 29.
Oseen Flow (8, 10)
Steady-streaming flow of velocity U of an incompressible fluid
over a solid sphere of radius a, which is symmetric about the x
axis and satisfies
U
[27]
and
⭸␺
⫽ ␯ ⵜ 2␺ .
⭸t
ⵜ2␾ ⫽ 0,
[24]
1
⭸u
⫽ ⫺ ⵜp ⫹ ␯ ⵜ 2u,
⭸x
␳
[31]
where divu ⫽ 0 and U ⫽ 2k ␯ (for convenience). The inertial
terms in this approximation are linearized but not zero. The
equations of motion in decomposed form are
冋
ⵜ U
册
⭸␾ p
⭸␷
⫹
⫽ ␯ ⵜ 2 ␷.
⫹U
⭸x ␳
⭸x
[32]
Since div␷ ⫽ 0 and ␾ is harmonic, ⵜ 2p ⫽ 0 and U(⭸curl␷兾⭸x) ⫽
␯ⵜ2curl␷. The rotational velocity is determined by a function ␹,
␷⫽
1
ⵜ ␹ ⫺ ex ␹ ,
2k
␹⫽
a0
exp关⫺kr共1 ⫺ cos ␪ 兲兴.
r
[33]
The potential ␾ is governed by the Laplace equation ⵜ2␾ ⫽ 0,
for which the solution is given by
␾ ⫽ Ux ⫹
冉冊
b0
⭸ 1
⫹ b1
⫹···
r
⭸x r
⫽ Ur cos ␪ ⫹
b0
⫺1
⫹ b 1 2 cos ␪ ⫹ · · ·,
r
r
where Ux is the uniform velocity term. Using these, the velocity
u ⫽ (u r, u ␪) is expressed as
ur ⫽
1 ⭸␹
⭸␾
⫹
⫺ cos ␪␹ ,
⭸r 2k ⭸r
u␪ ⫽
1 ⭸␾
1 1 ⭸␹
⫹
⫹ sin ␪␹ .
r ⭸␪
2k r ⭸ ␪
PNAS 兩 September 26, 2006 兩 vol. 103 兩 no. 39 兩 14275
APPLIED
MATHEMATICS
⫽ ␮៮ e␪䡠关ⵜ ␷៮ ⫹ ⵜ ␷៮ T ⫹ 2ⵜ 丢 ⵜ ␾៮ 兴䡠er.
These should be zero at the sphere surface (r ⫽ a) to give
b0 ⫽ ⫺
3Ua
3a ␯
a0
⫽⫺
⫽⫺
,
2k
4k
2
a0 ⫽
3Ua
,
2
b1 ⫽
Ua
.
4
The higher-order terms in the potential vanish.
To summarize, the decomposition of the velocity into rotational and irrotational components is
u ⫽ 共u r , u ␪兲 ⫽
冉
␷r ⫹
⭸␾
,
⭸r
␷␪ ⫹
冊
1 ⭸␹
⫺ ␹ cos ␪ ,
2k ⭸r
␷␪ ⫽
1 ⭸␹
⫹ ␹ sin ␪ ,
2kr ⭸ ␪
[35]
w ⫽ ⫺共a ⫹ b兲z,
[41]
u2共X, Y, t兲 ⫽ ⫺
␻0kykx
Y ⫹ ␣ 2Y 3 ⫹ ␤ 2YX 2 ⫹ · · ·
k2
[42]
The vorticity vanishes at (X, Y) ⫽ (0, 0) and appears first at second
order
␻共X, Y, t兲 ⫽ ⫺␻ 0k x k y XY exp共⫺␯ k 2t兲 ⫹ · · ·
[43]
The Helmholtz decomposition of the local solution near (X,
Y) ⫽ (0, 0) is
⭸␾
⫹ ␷ 2,
⭸Y
[44]
⭸␾ ⭸␾
␻ 0k yk x
,
⫽
共X, ⫺Y兲,
⭸X ⭸Y
k2
[45]
u1 ⫽
⭸␾
⫹ ␷ 1,
⭸X
u2 ⫽
where
冉
冊
and
共␷1, ␷2兲 ⫽ 共␣1X 3 ⫹ ␤1XY2 ⫹ · · ·, ␣2Y3 ⫹ ␤2YX 2 ⫹ · · ·兲.
[46]
This solution is generally valid in eddy systems, which are
segregated into quadrants near the stagnation point as in Fig. 1.
Hiemenz 1911 Boundary Layer Solution for 2D Flow Toward
a ‘‘Stagnation Point’’ at a Rigid Boundary (2)
Stagnation points on solid bodies are very important because the
pressures at such points can be very high. But stagnation point
flow cannot persist all the way to the boundary because of the
no-slip condition. Hiemenz looked for a boundary layer solution
of the Navier–Stokes equations vanishing at y ⫽ 0, which tends
to stagnation point flow for large y expressed as
u ⫽ 共u, ␷ 兲,
Fig. 1. (After Taylor, ref. 11): Streamlines for system of eddies dying down
under the action of viscosity. The streamlines are isovorticity lines; the vorticity
vanishes on the border of the cells.
[40]
␻0 ky kx
X ⫹ ␣1X 3 ⫹ ␤1XY2 ⫹ · · ·
k2
u1共X, Y, t兲 ⫽
[37]
where a and b are unknown constants relating to the flow field.
Irrotational stagnation points in the plane are saddle points; centers
are stagnation points around which the fluid rotates. Saddle points
and centers are embedded in vortex arrays.
Taylor vortex flow (TVF) is a 2D (x, z) array of counter-rotating
vortices (Fig. 1) whose vorticity decays in time due to viscous
diffusion (⭸t␻ ⫽ ␯ⵜ2␻). TVF is an exact solution of the nonlinear
time-dependent, incompressible Navier–Stokes equations (Eq. 11).
The instantaneous velocity components are
14276 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0605792103
[39]
␻ (x, y, t) is the vorticity, and ␻0 is the initial maximum vorticity,
k x and k y are the wave numbers in the x and y directions, and
k 2 ⫽ k x2 ⫹ k 2y .
The vorticity vanishes at saddle points (X, Y). After expanding
the solution in power series centered on a generic stagnation point,
we find that
[36]
at leading order. The velocity components along the principal
axes (x, y, z) of the tensor a ij are
␷ ⫽ by,
ⵜ2␺ ⫽ ⫺k 2␺ ⫽ ␻ ⫽ ⭸ yu 1 ⫺ ⭸ xu 2 .
[34]
where the tensor aij is symmetric. At the stagnation point, ⵜ␾ ⫽ 0,
hence, ai ⫽ 0; and since ⵜ2␾ ⫽ 0 everywhere, we have aij ⫽ 0. The
velocity is
⭸␾
⫽ aij xj
⭸xi
␻0
cos共k xx兲cos共k yy兲exp共⫺␯ k 2t兲,
k2
and
Flows Near Internal Stagnation Points in Viscous
Incompressible Fluids
The fluid velocity relative to uniform motion or rest vanishes at
points of stagnation. These points occur frequently even in turbulent flow. When the flow is purely irrotational, the velocity potential
␾ can be expanded near the origin as a Taylor series
u ⫽ ax,
where
␺⫽
3Ua Ua3
⫺ 2 cos ␪ .
␾ ⫽ Ux ⫺
4kr
4r
1
␾ ⫽ ␾0 ⫹ ai xi ⫹ aij xi xj ⫹ O共xi xi3兲,
2
ky
cos共k x x兲sin共k yy兲exp共⫺␯ k 2t兲,
k2
[38]
kx
2
⫺⭸ x␺ ⫽ u 2共x, y, t兲 ⫽ ␻ 0 2 sin共k xx兲cos共k yy兲exp共⫺␯ k t兲,
k
1 ⭸␾
,
r ⭸␪
curlu ⫽ curl␷,
␷r ⫽
⭸y␺ ⫽ u1共x, y, t兲 ⫽ ⫺␻ 0
3
␻ 共u兲ez ⫽ curlu.
[47]
The motion in the outer region is irrotational flow near a
stagnation point at a plane boundary. The flow in the irrotational
region is described by the stream function, ␺ ⫽ kxy, where x and
y are rectlinear coordinates parallel and normal to the boundary
(see Fig. 2), with the corresponding velocity distribution
u ⫽ kx,
␷ ⫽ ⫺k y.
[48]
Joseph
Fig. 4. Hamel source flow. (a) Irrotational source flow. (b) Asymmetric
rotational flow. (c) Symmetric rotational flow.
Steady 2D flow toward a ‘‘stagnation point’’ at a rigid boundary.
k is a positive constant, which, in the case of a stagnation point on
a body fixed in a stream, must be proportional to the speed of the
stream.
The next step is to determine the distribution of vorticity in the
thin layer near the boundary from the equation
u
冉
冊
⭸␻
⭸ 2␻ ⭸ 2␻
⭸␻
⫹␷
⫽␯
⫹ 2 ,
⭸x
⭸y
⭸x2
⭸y
[49]
together with boundary conditions that u ⫽ 0 and ␷ ⫽ 0 at y ⫽
0 and that the flow tends to the form of Eq. 48 at the outer edge
of the layer. Hiemenz found such a solution in the form
␺ ⫽ xf共 y兲,
[50]
corresponding to u ⫽ xf ⬘( y), ␷ ⫽ ⫺f( y), and ␻ ⫽ ⭸␷兾⭸x ⫺
⭸u兾⭸y ⫽ ⫺xf ⬙( y), where f( y) is an unknown function and primes
denote differentiation with respect to y satisfying ⫺f ⬘f ⬙ ⫹ ff ⵮ ⫹
␯ f iv ⫽ 0, and the boundary conditions f ⫽ f ⬘ ⫽ 0 at y ⫽ 0, f 3
ky as y 3 ⬁. Heimenz (1911) showed that this system could be
computed numerically and that it had a boundary layer structure
in the limit of small ␯ (Fig. 2).
Alternatively, we may decompose the solution relative to a
stagnation point in the whole space. To satisfy the no-slip
condition, the x component of stagnation point flow on y ⫽ 0
must be put to zero by an equal and opposite rotational velocity.
For the decomposed motion, we have
冉
u ⫽ 共u, ␷ 兲 ␷ x ⫹
⭸␾
⭸␾
, ␷ ⫹
⭸x y
⭸y
冊
[51]
⫽ 共 ␷ x ⫹ kx, ␷ y ⫺ k y兲.
Instead of Eq. 49, we have the vorticity equation in the Helmholtz decomposed form
共␷x ⫹ kx兲
⭸␻共␷兲
⭸ ␻ 共 ␷兲
⫹ 共 ␷ y ⫺ k y兲
⫽ ␯ ⵜ 2␻ 共 ␷兲.
⭸x
⭸y
[52]
The solution of this problem is given by ␺ ⫽ xF( y), where F( y) ⫽
f ⫺ ky, F(0) ⫽ 0 and F⬘(0) ⫽ ⫺k.
Hamel Flow in Diverging and Converging Channels (2, 3)
The problem is to determine the steady flow between two plane
walls meeting at angle ␣ shown in Fig. 3a. Batchelor’s discussion
of this problem is framed in terms of the solutions u ⫽ ( ␷ r, ␷ z,
␷ ␸) ⫽ ( ␷ [r, ␸ ], 0, 0) of the Navier–Stokes equations (Eq. 6). The
continuity equation ⭸r ␷ 兾⭸r ⫽ 0 shows that
␷ ⫽ ␷˜共␸兲兾r.
[53]
The function ␷˜ (␸) is determined by an involved but straightforward
nonlinear analysis leading to the cartoons shown in Figs. 3 and 4.
I next consider the Helmholtz decomposition (Eq. 1) of Hamel
flow. Eq. 53 shows that the only irrotational flow allowed in this
decomposition is source or sink flow ␸ ⫽ C log r, where C is to be
determined from the condition that ␷ˆ(␸) ⫹ C ⫽ 0 at ␸ ⫽ ⫾ ␣. The
rotational field ␷ˆ(␸) and constant C can be uniquely determined by
an analysis like that given by Batchelor.
The Helmholtz decomposition gives rise to an exact theory of
potential flow in the frame of the Navier–Stokes equations in which
rotational and irrotational fields are tightly coupled and both fields
depend on viscosity. This kind of theory leads to boundary layers
of vorticity in asymptotic limits, but the fields are always coupled.
The exact theory is different than purely irrotational theories of the
effect of viscosity, which can lead to excellent but always approximate results.
Fig. 3. Hamel sink flow. (a) Flow channel. (b) Irrotational sink flow. (c) Sink
flow with rotational boundary layer.
I thank T. Funada for his help with these calculations, H. Weinberger for
discussions of conceptual issues, and G. I. Barenblatt who helped me with
this paper and with my work on viscous potential flow over many years.
This work was supported by National Science Foundation Chemical and
Transport Systems Grant 0302837.
1. Joseph DD (2006) Int J Multiphase Flow 32:886–887.
2. Batchelor GK (1967) An Introduction to Fluid Dynamics (Cambridge Univ
Press, Cambridge, UK).
3. Landau LD, Lifshitz EM (1987) Fluid Mechanics (Pergamon, Oxford), 2nd Ed.
4. Wang J, Joseph DD (2006) J Fluid Mech 557:145–165.
5. Wang J, Joseph DD (2006) J Fluid Mech 557:167–190.
6. Padrino JC, Joseph DD (2006) J Fluid Mech 557:191–223.
7. Joseph DD (2006) Int J Multiphase Flow 32:285–310.
8. Lamb H (1932) Hydrodynamics (Cambridge Univ Press, Cambridge, UK), 6th
Ed; reprinted (1945) (Dover, New York).
9. Wang J, Joseph DD (2006) J Fluid Mech 559:461–472.
10. Milne-Thomson LM (1968) Theoretical Hydrodynamics (Macmillan, New
York).
11. Taylor GI (1923) Philos Mag XLVI:671–674.
Joseph
PNAS 兩 September 26, 2006 兩 vol. 103 兩 no. 39 兩 14277
APPLIED
MATHEMATICS
Fig. 2.
␻共u兲 ⫽ ␻ 共 ␷ 兲,
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