Modified Form of the Helmholtz Vorticity Equation and its Solution
for Spherical Flow Within a Droplet in Uniform
or Counterflow Streams
SIAVASH H. SOHRAB
Robert McCormick School of Engineering and Applied Science
Department of Mechanical Engineering
Northwestern University, Evanston, Illinois 60208
UNITED STATES OF AMERICA
Abstract
:

The scale

invariant form of conservation equations in reactive fields are described. The modified
form of the
Helmholtz
vorticity equation is solved to determine flo
w field within a droplet that is located either
in a uniform stream or at the stagnation

point of axi

symmetric counterflow. For the former case, the classical
solution of
Hill
spherical vortex is recovered. The latter case results in a spherical flow pro
duced by two semi

spherical ring vortices. For both cases, the stream functions representing two concentric embedded spherical
flows are also determined as product solutions.
Key
–
Words:

Helmholtz vorticity equation. Hill vortex. Turbulent dissipat
ion. Dynamo theory.
1 Introduction
The universality of turbulent phenomena from
stochastic quantum fields to classical hydrodynamic
fields resulted in recent introduction of a scale

invariant model of statistical mechanics and its
application to the fie
ld of thermodynamics [4]. The
implications of the model to the study of transport
phenomena and invariant forms of conservation
equations have also been addressed [5, 6]. In the
present study, the modified form of the
Helmholtz
vorticity equation is solv
ed for the classical problem
of flow within a droplet that is located either in a
uniform stream or at the stagnation point of axi

symmetric counterflow.
2 A Scale

Invariant Model of Statistical
Mechanics
Following the classical methods [1

3], the invaria
nt
definitions of the density
, and the velocity of
atom
u
,
element
v
, and
system
w
at the scale
are given
as [4, 5]
ρ n m m f du
u
=
v
(1)
1
m f d
v u u
w
=
v
(2)
The invariant definit
ions of the peculiar
and the
diffusion velocities have been introduced as [4]
V'
=
u
†
v
V
=
v
=
V'
(3)
3 Invariant Form of the Conservation
Equations for Chemically Reactive
Fields
Following the classical methods [1

3], the scale

invariant forms of mass, thermal energy, linear and
angular momentum conservation equations [5, 6] at
scale
are given as
β
β β β
ρ
ρ
t
v
(4)
β
β β
ε
ε 0
t
v
(5)
β
β β
0
t
p
p v
(6)
β
β β
0
t
π
π v
(7)
where
=
h
,
p
=
v
,
and
=
are the
volumetric density
of thermal
energy
,
linear and
angular
momentum
of the field, respectively and
ω v
is the vorticity. Also,
is the
chemical reaction rate and h
is the absolute enthalpy
[5].
2
The local velocity
v
in (8)

(11) is expressed as the
sum of con
vective
w
=
v
> and diffusive velocities
[5]
v
=
w
+
V
g
g
D ln( )
V
(8a)
v
=
w
+
V
tg
tg
ln( )
V
(8a)
v
=
w
+
V
hg
hg
ln( )
V p
c
v
=
w
+
V
rhg
rhg
ln( )
V
π
d
where (
V
g
,
V
tg
,
V
hg
,
V
rhg
) are respectively the
diffusive,
the thermo

diffusive, the linear hydro

diffusive, and the angular hydro

diffusive
velocities.
For unity Schmidt and
Prandtl numbers Sc
= Pr
=
/D
=
/
= 1, one may express
tg g t
V V V
(9a)
hg g h
V V V
(9b)
rhg g rh
V V V
(9a)
that involve the thermal
V
t
, the line
ar (translational)
hydrodynamic
V
h
and the angular (rotational)
hydrodynamic
V
rh
diffusion velocities defined as [6]
t
ln(h )
V
(10a)
h
ln( )
V v
(10b)
rh
ln( )
V
ω
(10c)
Since for an ideal gas h
= c
p
T
, when c
p
is constant
and T = T
, Eq.(3.6a) reduces to the
Fourier
law of
heat conduction
t
h
κ Τ
q V
(11)
where
and
=
/(
c
p
) are the thermal
conductivity and diffusivit
y. Similarly, (10b) may be
identified as the shear stress associated with
diffusional flux of linear momentum and expressed
by the generalized
Newton
law of viscosity [5]
ij
β β jβ ijβh jβ i
ρ μ/
τ v V v x
Finally, (10c) may be identified as
the torsional stress
induced by diffusional flux of angular momentum
and expressed as
ijr
β β jβ ijβrh β jβ i
ρ μ/
τ ω V ω x
(13)
Substitutions from (8a)

(8d) into (4)

(7),
neglecting cross

diffusion terms and assuming
constant transport coefficien
ts with unity
Prandtl
and
Schmidt
numbers Sc
= Pr
= 1, result in [6]
2
ρ
ρ D ρ
t
+w
(14)
2
ρ
h
ρ D ρ
t
+w
2
h
ρ h h 0
t
+w
(15)
2
ρ
ρ D ρ
t
v +w
2
0
t
v
+w v v
(16)
2
ρ
ρ D ρ
t
ω +w
2
0
t
ω
+w
ω ω w ω
(17)
The above forms of the conservation equations
perhaps help to better reveal the coupling between
the gravitational versus the inertial contributions to
total energy and momentum densities of the field.
Substitutions from (14) into (15)

(17) result in
scale

invariant forms of conservation equations in
chemically reactive fields [6]
2
ρ
ρ D ρ
t
+w
(18)
2
p
T
T T h/( c )
t
+w
(19)
2
/
t
v
+w v v v
(20)
2
t
ω ω
+w
ω ω ω w
(21)
3
Equation (21) is the modified form of the
Helmholtz
vorticity equation for chemically reactive flow
fields. The last two terms of (21) respectively
correspond to vorticity generation b
y vortex

stretching and chemical reactions. Hence,
(
/
) represents generation
< 0
(annihilation
> 0) of angular momentum
accompanied by release (absorption) of thermal
energy associated with exothermic (endothermic)
chemical reactions. As an
example, the latter source
term may be used to describe the change of angular
momentum of a ballet dancer. In this case, the loss
of mass due to chemical reactions in the body of a
spinning dancer that brings the arms inward, thus
doing work against cent
rifugal forces, results in an
increase in the dancer's angular momentum.
4 Solution of the Modified Helmholtz
Vorticity Equation for Flow Inside a
Droplet in a Uniform Stream
For flow within a droplet located in a uniform
stream the non

dimensional stea
dy modified
Helmholtz vorticity equation (21) without reactions
= 0 in axi

symmetric cylindrical coordinate
reduces to
r
r z
w
w w
r z r
ω
ω ω
2 2
2 2 2
1
r r r r z
ω
ω ω ω
(22)
where
=
'
/(U
2
/
) is the d
imensionless azimuthal
vorticity and dimensionless
convective
velocity
components are given by
w
z
=
1
,
w
r
=
(23)
where
w
z
=
w'
z
/U,
w
r
=
w'
r
/U, and U is the uniform
stream velocity. An exact solution of (22)

(23) may
be expressed as
= 5r/R
2
(24)
where R = R'/(
/U) and R' is the droplet radius. The
solution (24) corresponds to the axial and the radial
velocity components within the droplet (
v
z
=
v'
z
/U ,
v
r
=
v'
r
/U) given by
v
z
=
[1
2(r/R)
2
(z/R)
2
] ,
v
r
= rz/
R
2
(25)
with the associated dimensionless stream function
= (1/2) r
2
[1
(r/R)
2
(z/R)
2
]
(26)
where
=
'
/(
2
/U).
Therefore, the exact solution
of the modified Helmholtz vorticity equation (22)
agrees with the classical sol
ution of Hill [7, 8]. One
notes however that even though the final solutions
(24)

(26) are
identical to the classical results, the
mathematical model such as the nature of the
convective velocity field (23) and the vorticity
con
servation equation (22) are different. Some of
the streamlines calculated from Eq.(26) using
Mathematica [9] are shown in Fig.1.
1
0.5
0
0.5
1
1
0.5
0
0.5
1
Fig.1 Hill spherical vortex for flow within a
droplet in a uniform stream.
Near the center of the spherical
flow (Fig.1) i.e. for
small r
0 and z
0, the velocity field (25) reduces
to
v
z
= v
z
= w
z
= 1 , v
r
= v
r
= w
r
= 0 (27)
where subscript
refers to the scale [4]. The local
velocity (2
7) is similar to the outer convective
velocity field (23) except that it is in the opposite
direction. Therefore, in view of the scale

invariant
form of (22), one may arrive at a cascade of
concentric
Hill
spherical vortices that are embedded
within each
other with alternating sense of rotation.
This is because when the inner spherical
Hill
vortex
is small enough, it will experience a locally uniform
external flow field (27) that is produced by the outer
spherical
Hill
vortex.
4
Equation (21) is the modifi
ed form of the
Helmholtz
vorticity equation for chemically reactive flow
fields. The last two terms of (21) respectively
correspond to vorticity generation by vortex

stretching and chemical reactions. Hence,
(
/
) represents generation
< 0
(annih
ilation
> 0) of angular momentum
accompanied by release (absorption) of thermal
energy associated with exothermic (endothermic)
chemical reactions. As an example, the latter source
term may be used to describe the change of angular
momentum of a ballet
dancer. In this case, the loss
of mass due to chemical reactions in the body of a
spinning dancer that brings the arms inward, thus
doing work against centrifugal forces, results in an
increase in the dancer's angular momentum.
4 Solution of the Modifi
ed Helmholtz
Vorticity Equation for Flow Inside a
Droplet in a Uniform Stream
For flow within a droplet located in a uniform
stream the non

dimensional steady modified
Helmholtz vorticity equation (21) without reactions
= 0 in axi

symmetric cylindrical coordinate
reduces to
r
r z
w
w w
r z r
ω
ω ω
2 2
2 2 2
1
r r r r z
ω
ω ω ω
(22)
where
=
'
/(U
2
/
) is the dimensionless azimuthal
vorticity and dimensionless
convective
ve
locity
components are given by
w
z
=
1
,
w
r
=
(23)
where
w
z
=
w'
z
/U,
w
r
=
w'
r
/U, and U is the uniform
stream velocity. An exact solution of (22)

(23) may
be expressed as
= 5r/R
2
(24)
where R = R'/(
/U) and R' is the
droplet radius. The
solution (24) corresponds to the axial and the radial
velocity components within the droplet (
v
z
=
v'
z
/U ,
v
r
=
v'
r
/U) given by
v
z
=
[1
2(r/R)
2
(z/R)
2
] ,
v
r
= rz/R
2
(25)
with the associated dimensionless stream functi
on
= (1/2) r
2
[1
(r/R)
2
(z/R)
2
]
(26)
where
=
'
/(
2
/U).
Therefore, the exact solution
of the modified Helmholtz vorticity equation (22)
agrees with the classical solution of Hill [7, 8]. One
notes however that even though the fi
nal solutions
(24)

(26) are identical to the classical results, the
mathematical model such as the nature of the
convective velocity field (23) and the vorticity
conservation equation (22) are different. Some of
the streamlines calculated from Eq.(26) usi
ng
Mathematica [9] are shown in Fig.1.
1
0.5
0
0.5
1
1
0.5
0
0.5
1
Fig.1 Hill spherical vortex for flow within a
droplet in a uniform stream.
Near the center of the spherical flow (Fig.1) i.e. for
small r
0 and z
0, the velocity field (25) reduces
to
v
z
= v
z
= w
z
= 1 , v
r
= v
r
= w
r
= 0 (27)
where subscript
refers to the scale [4]. The local
velocity (27) is similar to the outer convective
velocity field (23) except that it is in the opposite
direction. Therefore, in view of the scale

invariant
form of (22), one may arrive at a cascade of
concentric
Hill
spherical vortices that are embedded
within each other with alternating sense of rotation.
This is because when the inner spherical
Hill
vo
rtex
is small enough, it will experience a locally uniform
external flow field (27) that is produced by the outer
spherical
Hill
vortex.
5
1
0.5
0
0.5
1
1
0.5
0
0.5
1
Fig.2 Streamlines of concentric spherical Hill
vortices calculated from Eq.(36).
In view of the linear
ity of the governing
equations, one can show that the streamline for two
embedded concentric
Hill
spherical vortices may be
presented as product solutions. To show this, first
the modified
Helmholtz
vorticity equation (22) is
written as
L
(
) = 0
(28)
where the linear operator
L
is defined as
r
r z
w
= w w
r z r
L
2 2
2 2 2
1 1
r r r r z
(29)
Next, the azimuthal vorticity is expressed in terms
of stream function as
2 2
2 2
1 1
( )
r r r r z
J
(30)
wher
e a second linear operator
J
is defined as
2 2
2 2
1 1
r r r r z
J
(31)
such that (28) becomes
L
[
J
(
] = 0
(32)
Let us now consider the flow field within two
concentric droplets that are located in a uniform
gaseous stream. The droplets are supposed to be
composed of different immiscible fluids. For the
inner droplet of radius R
1
= 1
and the outer droplet
of radius R
2
=
2
, the corresponding stream
functions from (26) become
1
= (1/2) r
2
[1
r
2
z
2
]
,
2
= (1/4) r
2
[2
r
2
z
2
]
(33)
Although (27) and (23) are different, since w
r
= 0
and
z
w (/z) 0
, the operator
L
becomes
identical for the outer and the inner flow fields.
Hence, applying the vorticity equation (32) to the
stream functions in (33) gives
L
[
J
(
] =
L
[
J
(
]
=
(34)
that in view of the linearity of the o
perators leads to
the product solution
L
[
J
(
]
=
L
[
J
(
] =
(35)
Therefore, for flow within two concentric droplets
in a uniform flow the stream function is expressed
by the product solution obtained from (35) and (33)
as
=
=
= (1/8) r
4
[1
r
2
z
2
] [2
r
2
z
2
] (36)
Some of the streamlines for two concentric
embedded
Hill
spherical vortices calculated from
(36) are shown in Fig.2. It is noted that as the radius
of the outer droplet R
2
is increase
d, the streamlines
within the outer sphere (Fig.2) become increasingly
similar to the streamlines for
Stokes
flow over a
sphere in accordance with the classical results [8].
5 Solution of the Modified Helmholtz
Vorticity Equation for Flow Inside a
Droplet
at Stagnation

Point of Axi

symmetric Counterflow
Following the classical solution of
Hill
[7, 8], the
spherical flow generated in a very small droplet that
is located at the stagnation point of an axi

symmetric gaseous counterflow is considered [10].
Th
e convective velocity of the counterflow outside
of the droplet is given by [2]
w'
z
=
2
z' ,
w'
r
=
r'
(37)
6
where
is the counterflow velocity gradient. With
the definitions of dimensionless velocity and
coordinates
(
v
r
,
v
z
,
w
r
,
w
z
) =
(
v'
r
,
v'
z
,
w'
r
,
w'
z
)/
,
r
=
r
'
/
/
,
z
=
z'
/
/
(38)
the dimensionless azimuthal vorticity is obtained
from (22) as
= 14 r
z/R
2
(39)
where
=
'
/
,
R
=
R'
/
/
, and R' is the droplet
radius. The spherical flow within the droplet is
describ
e
d by the dimens
ionless stream function
z
r
2
[
1
(r/R)
2
(z/R)
2
]
(40)
where
'
The corresponding
components of the axial and the radial velocity are
v
z
2z
[1
2 (r/R)
2
(z/R)
2
]
,
v
r
r
[1
(r/R)
2
3(z/
R)
2
] (41)
Some of the streamlines calculated from Eq.(40) are
shown in Fig.3.
It is interesting to note that even if there were no
droplet at the stagnation point, it is expected that a
small spherical region of flow recirculation like that
shown in Fig.3 (or like an ellipsoidal body of
revolution) will form around the stagnation point.
Therefore, for fluids with finite viscosity, the critical
singularity located at the stagnation point will be
avoided by the global flow through the formatio
n of
such a closed region of secondary flow. The radius
of such a secondary flow region is given by
R
' =
/
and hence depends on the viscosity and
the rate of strain. It is very interesting to compare the
spherical flow in Fig.3 of the present study with
Fig.2
b of the previous investigation [11] involving
secondary flows produced in the vicinity of
stagnation

point of axi

symmetric counter

rotating
counterflows. Clearly, in the latter study [11] the two
toroidal vortices formin
g an e
llipsoidal body of
revolution also possess azimuthal rotational velocity
induced by the rotating outer counterflow jets.
Indeed, it is the rotation of the jets that causes a
reduction in the strain rate
leading to an increase in
the radius of the recirc
ulation zone
R
' =
/
,
thereby facilitating its observation. The application of
the modified
Helmholtz
vorticity equation (21) to
describe flow fields within rotating droplets will be
considered in a future investigation.
In the vicinity of the stagnation po
int,
r 0
and
z 0
, the local velocity field (41) reduces to
v'
z
= v'
z
= w'
z
= 2
z' ,
v'
r
= v'
r
= w'
r
=
r'
(42)
that is identical in for
m but opposite in direction to
the outer convective velocity field in (37).
Therefore, as was noted earlier [10], because of the
scale

invariant nature of the conservation equations,
one expects a cascade of embedded concentric
spherical flows at ever sma
ller scales to form around
the stagnation point. Following the reasoning and the
procedures similar to those described in (28)

(36), it
can be shown that for two concentric droplets located
at the stagnation

point of a counterflow with the radii
R
1
= 1 an
d R
2
=
2
and with the respective stream
functions obtained from (40) as
z
r
2
[
1
r
2
z
2
]
,
z
r
2
[
2
r
2
z
2
]
(43)
one arrives at the product solution given by
=
= (1/2) z
2
r
4
[1
r
2
z
2
] [2
r
2
z
2
]
(44)
Some of the streamlines for flow within two
concentric droplets that are located at the stagnation
point of a counterflow calculated from Eq.(44) are
shown in Fig.4.
1
0.5
0
0.5
1
1
0.5
0
0.5
1
Fig.3 Spherical flow within a d
roplet at
stagnation point of a counterflow.
7
1
0.5
0
0.5
1
1
0.5
0
0.5
1
Fig.4 Steamlines of concentric embedded
spherical flows calculated from Eq.(44).
Examination of Fig.4 shows that generation of many
concentric spherical flows is accompanied by the
formation of
many new local counterflow regions. It
is expected that in the vicinity of each new stagnation
point smaller secondary spherical flows will be
generated. For example, one would expect small
spherical flows to form in the vicinity of the poles of
the inn
er sphere in Fig.4. Since strained flow fields
are common features of turbulent flows, the
generation of cascades of toroidal vortices at each
stagnation point is expected to play an important role
in turbulent dissipation process.
6 Concluding Remarks
In addition to their astrophysical significance, the
results will help the understanding of vortex
dynamics in turbulent fields as well as
motion/evaporation/combustion of droplets in
turbulent spray combustion. As an example of the
former class of proble
ms, it is known that the
direction of magnetic polarization of volcanic rocks
alternate every few million years with no known
mechanism to account for this behavior.
Examination of Figs.2 and 4 suggests that successive
generation and evolution of embedded
spherical
flows with alternating sense of rotation within the
molten core of a dynamo such as the earth [12] may
possibly account for such periodic change in the
direction of polarization. An example of the
implications of the results to the latter class
of
problems is that the generation of cascades of
embedded spherical vortices within locally strained
flows (Fig.4) could be identified as one possible
mechanism of turbulent dissipation.
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T
hermodynamics
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Williams, F. A.,
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Sohrab, S. H., Hydrodynamics of spherical
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Lin, T. H., and Sohrab, S. H., Influence of
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