J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
1Ae28.
Secondary Flow for Slow Rotation of a Pervious Sphere with Source at its Centre in a Viscous Fluid Mukesh Awasthi1 .
Naveen Mani2 .
Amit Sharma3 .
Rahul Shukla4O Faculty of Mathematical and Statistical Sciences.
Shri Ramswaroop Memorial University.
India, mukeshfor6@gmail.
Department of Mathematics.
Chandigarh University.
India, naveenmani81@gmail.
Department of Mathematics.
Amity University Haryana.
India, amitsharma@gmail.
Department of Mathematical Sciences and Computing.
Walter Sisulu University.
South Africa, rshukla@wsu.
Abstract.
In this paper, the problem of secondary motion of fluid in region between two concentric spheres are studied.
Navier Stokes equations are employed to obtain the flow field, and the components of velocity for primary and secondary flow, stream function are obtained.
Further, two particular cases are deduced and discussed for the radius of cavity.
Finally.
Equi-pressure lines, streamlines, and vortex lines are constructed to visually demonstrate the impact of different factors on the flow Key words and Phrases: Primary flow, secondary flow, spherical cavity, pervious sphere, source, streamlines Introduction Datta .
investigated the steady slow viscous flow past a pervious sphere with a source at its center.
The source is of such a strength that it brings forth a linear part of the inertia terms neglected in Stokes flow for small Reynolds number Re.
Using the method of matched asymptotic expansion the problem was extended to include O(R.
terms in the case of a cylinder and a sphere by Sthapit & Datta (.
In 2011.
Datta and Singhal .
revisited the problem, incorporating the inner and outer expansions for the stream function as proposed by Proudman and Pearson .
to account for the effect of a source.
Their approach simplified the derivation of higher-order approximations, specifically the O(Re2 log R.
Subsequently.
Datta and Singhal .
extended their analysis to include the influence O Corresponding author 2020 Mathematics Subject Classification: 76D07, 76S05 Received: 27-12-2024, accepted: 22-05-2025.
of slip conditions.
They also investigated a related problem involving axial flow along a porous cylinder with a sink at its center .
ee Datta and Singhal .
Datta and Srivastava .
investigated the slow rotation of a sphere with a source at its center in a viscous fluid.
Bickley .
analyzed the secondary flow generated by a rotating sphere in a viscous medium.
Haberman .
addressed the secondary flow problem for a rotating sphere enclosed within a coaxially rotating spherical container.
Datta .
studied the steady rotation of a magnetized sphere in a viscous conducting fluid, while Datta .
explored the secondary flow around a magnetized sphere rotating in such a medium.
Ranger .
examined the axially symmetric flow past a rotating sphere induced by a uniform stream at infinity.
demonstrated that the leading terms of the flow represent a linear superposition of a primary Stokes flow past a non-rotating sphere and an antisymmetric secondary flow in the azimuthal plane caused by the sphereAos rotation.
In the paper Stokes flow around an axially symmetric rotating pervious body Srivastava .
has not attempted to satisfy the boundary conditions for the general case.
he has only reproduced the case of a sphere already covered in Datta & Srivastava .
In recent past years.
Some authors have studied and explored this kind of work for various medium by considering different parametrs.
Some of them are .
, 17, 18, 19, .
In this paper we study the analysis of the secondary flow for the slow rotation of a pervious sphere with a source at its center enclosed in a concentric spherical cavity at rest.
The analysis is done through a perturbation expansion in Reynolds number Re.
The first term corresponds to the azimuthal flow presented earlier by Datta & Srivastava .
and the second term provides the streamlines corresponding to the Stokes stream function in meridian flow.
Figure 1.
Figure for the rotation of pervious sphere with source at its center enclosing in concentric spherical cavity Formulation of the Problem Consider the slow steady rotation of a pervious sphere of radius a with constant angular velocity E carrying a source of strength Q at its center generating a radial flow field (Q/rA )rC and encompassed by a concentric pervious spherical cavity of radius b.
The region .
O rA O .
in between pervious sphere and cavity is filled with an incompressible viscous fluid of density A and kinematic viscosity , no fluid accumulates.
Now it is convenient to introduce dimensionless quantities ra = rA .
Eaq = qA .
AAEp = pA .
Here, denotes dimensional quantities, r space coordinate, q velocity and p The equations that govern the motion of an incompressible viscous fluid in the standard notation are:
Equation of continuity ON.
q = 0.
Navier-Stokes equation Re(.
ON).
= OeONp ON2 q, .
Re(((ON y .
ON.
) = OeONp Oe ON y (ON y .
, .
or using equation .
where Re = Ea2 / is Reynolds number.
The equations are to be solved under the following no slip boundary conditions:
q = sin iC at r = 1, .
q = 0 at r = 1/ = b/a, .
To solve the problem, we make use of the following perturbation scheme:
rC w.
, )iC Re.
, )rC v.
, )C), q.
, ) = Re r2 where s = Q/a is source parameter.
p = p0 Rep1 .
Inserting above representation in equation of motion .
and equating O.
and O(R.
terms, we obtain the following equations:
Primary flow:
s OC ON2 w Oe 2 2 = 3 .
, p0 = constant.
r OCr r sin Secondary Flow :
ON2 u Oe ON2 v Oe 2 OC .
OC u OCp1 r2 sin OCr r2 OCr s OC.
1 OCp1
OCr
r OC
2 OCu
Oe 3
r2 sin2 r2 OC
ON2 O
OC2
2 OC
cot OC
1 OC2
OCr2 r OCr r2 OC2 r OC Primary Flow As in Datta & Srivastva .
and Devi .
, we consider the primary motion which is induced in the fluid by slow rotation of pervious sphere with a source at its center.
The motion is governed by the equation .
The boundary conditions .
, .
for the azimuthal velocity component w are w.
, ) = sin , .
/, ) = 0.
The boundary condition .
suggests that solution of equation .
of the form w = f .
sin , .
where f .
is derived below and given by equation .
Now equation .
provides f AA .
Oe 2 f A .
Oe f .
= 0.
Solving the above differential equation, using boundary conditions .
, we get 2r2 Oe 2rs s2 Oe r2 e(Oe r )s 2 s2 Oe 2s 2 = r .
2 Oe 2s .
Oe e(Oe.
s2 Oe 2s .
For the case of the pervious sphere rotating an infinite expanse of fluid ( Ie .
, and the result above reduces to 2r2 1 Oe eOe r Oe 2rs s2 f .
) = .
2 Oe 2s 2 .
Oe eOes )) which conforms to the result obtained by Datta & Srivastava .
The couple on the sphere required to maintain the motion is obtained by integration of viscous stress in dimensionless form Eri .
, ) A Eri .
, ) AAE s.
(Oe.
s2 Oe 2s .
Oe .
) sin.
e(Oe.
s2 Oe 2s .
Oe .
2 Oe 2s .
OCw w Oe OCr = .
A .
Oe f .
)sin Thus, the moment of the required couple is given by(.
) Z A A M = Oe2Aa ErA A i sin2 d, .
where ErA A i is the tangential stress on the pervious sphere is given by .
Thus, on evaluating the integral .
, the dimensional moment of the required couple M A is comes out.
(Oe.
s2 Oe 2s .
Oe .
) M A = Oe8aEa3 (Oe.
s2 Oe 2s .
Oe .
2 Oe 2s .
) The non dimensional moment of the required couple M is MA
M =
8aEa3 (Oe.
s2 Oe 2s .
Oe .
) M = Oe (Oe.
s 2 2 ( s Oe 2s .
Oe .
2 Oe 2s .
) For the case of the pervious sphere rotating an infinite expanse of fluid Ie 0, so, the non dimensional moment of the required couple M O in infinite expanse of fluid 2 (.
Oe .
eOes ) s MO = 2 3s Oe 6 .
eOes Oe .
Also, the result above reduces to the result obtained by Datta & Srivastava .
Here s Ie 0 in above equation describe the case for rotation of solid sphere in an unbounded fluid.
Figure 2.
Variation azimuthal velocity w with r at = A/2 for various values of s .
Ie 0(Blu.
, s = 1(Gree.
,s = 10(Brow.
, s = 100(Blac.
, s Ie O(Re.
) when the pervious sphere rotating in an infinite expanse of fluid .
Ie .
Now we present few graphs in figures 2 to 5 to show the effects of parameters on the velocity w and moment of the couple M .
Figure 2 shows variations in the azimuthal velocity w with r at = A/2 for various values of source parameter s in an infinite expanse of fluid.
In this graph w increases as s increases but in each graph rate of variation of the azimuthal velocity w with r diminishes as r increases.
Figure 3 shows variations in the azimuthal velocity w with r for various values of source parameter s when the pervious sphere rotating in a concentric cavity whose radius is 2 times of the pervious sphere.
In this graph w increases as s increases but in each graph rate of variation of w with r diminishes as s increases.
Figure 4 shows variations in the couple M with source parameter s for various values of separation parameter .
In this graph M increases as increases but in each graph rate of variation of the couple with s diminishes as s increases.
Figure 5 shows variations in the couple M with separation parameter for various values of source parameter s.
In this graph M increases as s decreases but in each graph rate of variation of the couple increases as increases and increases sharply as Ie 1.
Figure 3.
Variation azimuthal velocity w with r at = A/2 for various values of s.
Ie 0(Blu.
,s = 1(Gree.
,s = 10(Brow.
,s = 100(Blac.
,s = 107 (Re.
) when the pervious sphere rotating in a concentric cavity whose radius is 2 times of the pervious sphere .
= 0.
Figure 4.
Variation of torque M with s for various values of = a/b ( =.
001(Blu.
, =.
4(Gree.
, =.
6(Brow.
, =.
8(Blac.
, =.
9(Re.
) when the pervious sphere rotating in a concentric Figure 5.
Variation of torque M with = a/b for various values of source parameter s .
Ie 0(Blu.
,s = 1(Gree.
,s = 10(Brow.
,s = 100(Blac.
,s = 1000(Re.
)when the pervious sphere rotating in a concentric cavity.
Secondary Flow Secondary flow is determined by the equations .
, .
with w replaced by f .
thus, we have the equations ON2 u Oe OC u OCp1 f 2 .
sin2 , r2 sin OC OCr r2 OCr s OC 1 OCp1 f 2 .
2 OCu Oe .
Oe sin cos , .
r OCr r OC r2 sin r OC where f .
is given by equation .
These equations have to be solved under boundary conditions:
At the common interface r = 1 ON2 v Oe u = 0, v = 0, .
u = 0, v = 0.
At cavity wall r = 1/ It will be convenient using Stokes stream function O to express velocity components 1 OCO r sin OCr Next, eliminating p1 in between equations .
and using .
, we get )2 A E4O Oe s .
sin2 cos , .
OCr r2
OCO
r2 sin OC
,v = Oe
OC2
sin OC
E = 2 2
OCr
r OC
1 OC
sin OC where using the value of f .
as given by equation .
, we have f .
f A .
= .
s2 Oe 2s .
2 e2s(Oe.
) .
r3 Oe 8r2 s 6rs2 Oe 2s3 ) Oe r.
Oe 2.
2 2 s2 Oe 2s 2 es(Oe.
) ))/.
3 (.
2 Oe 2s .
Oe e(Oe.
s2 Oe 2s .
)2 ).
Now, keeping in view the form of equation .
, we may assume the solution as O = G.
sin2 cos.
In terms of G.
written as 2 dr r2 dr2 = 2f .
f A .
on collecting terms of the same order in s r4 Ga .
Oe 12r2 GAA .
24rGA .
Oe s.
2 Ga .
Oe 2rGAA .
Oe 6GA .
A G.
) = 2r f .
where the right hand term is evaluated above in .
Since finding the analytical solution of the equation .
is very tedious, we use the following perturbation expansion for small s G.
= G0 .
sG1 .
s2 G2 .
Also, we need the expansion up to O.
) term of the right hand side of equation .
A .
Oe .
)/.
where f .
given by equation .
thus we have 2f .
A .
Oe .
)/.
r3 Oe .
/(.
Oe .
2 r5 )) s((.
Oe 3 ).
r3 9r Oe .
3 3 6
Oe 4 .
r( r Oe .
)/.
Oe ) r )) s (.
( Oe .
((.
r3 Oe .
Oe 10( Oe .
Oe .
Oe .
)/40( Oe .
4 r7 ) O.
3 ).
Substituting G.
and f .
and collecting zero order terms in s, we get the equation for G0 .
r4 G0 .
Oe 12r2 GAA0 .
24rGA0 .
= 6(.
r3 Oe .
/(.
Oe .
), .
with boundary conditions :
G0 .
= GA0 .
= G0 .
/) = GA0 .
/) = 0,
we get the solution .
G0 .
= (.
Oe .
Oe .
2 ((((.
Oe .
Oe .
((.
( .
Oe .
Oe .
r3 2( .
((.
( .
Oe .
Oe .
Oe .
r Oe 4.
) Oe 4.
) Oe .
) /.
Oe .
2 (((.
(( .
r2 ).
A similar process for G1 .
yields differential equation:
2 AA
2 a AA r4 Ga 1 .
Oe 12r G1 .
24rG1 .
Oe .
G0 .
Oe 2rG0 .
Oe 6G0 .
G0 .
) = (.
Oe 3 ).
r3 9r Oe .
Oe 43 .
r3 Oe .
)/.
Oe 3 )3 r6 ).
Use G0 .
, we get r4 G1 .
Oe 12r2 GAA1 .
24rGA1 .
= r4 ((.
Oe .
165 404 553 402 16 .
Oe 3 ).
r3 43 r3 9r Oe 20 .
Oe 43 .
r3 Oe .
) .
Oe 3 )3 .
165 404 553 402 16 .
83 52 Oe 2 Oe .
r6 43 .
165 404 553 402 16 .
r3 Oe 12.
256 405 574 483 202 8 .
)/.
Oe 3 )3 .
Oe .
165 404 553 402 16 .
r6 )).
Boundary conditions G1 .
= GA1 .
= G1 .
/) = GA1 .
/) = 0.
Solution for G2 .
are form differential equation r4 G2 .
Oe 12r2 GAA2 .
24rGA2 .
Oe .
2 G1 .
Oe 2rGAA1 .
Oe 6GA1 .
G1 .
) = .
( Oe .
2 ((.
r3 Oe .
Oe .
3 Oe 10( Oe .
Oe .
2 )/.
( Oe .
4 r7 ).
Use G1 .
, we get r4 G2 .
Oe 12r2 GAA2 .
24rGA2 .
= .
( Oe .
( Oe .
2 ((.
r3 Oe .
Oe .
3 Oe 10( Oe .
Oe .
2 ) .
(((.
(( .
5 r2 log.
/)((.
( Oe .
(Oe1530( Oe .
(((.
(( .
2 75 (((( (((((.
(((.
( .
Oe 2.
Oe 1.
r7 Oe 210( Oe .
2 3 ((.
Oe .
Oe .
Oe .
((( .
(( .
2 r4 Oe 2520( Oe .
2 3 .
(((.
(( .
2 r3 Oe 6((((((((((((.
(((( 2.
Oe .
Oe 19.
Oe 139.
Oe 259.
Oe 247.
Oe 19.
Oe 571.
Oe 2383.
Oe 2813.
Oe 1782.
Oe 553.
r2 1050( Oe .
2 ( .
(((.
( ( .
(((((.
))/((((.
(( .
4 )2 ))/.
( Oe .
4 r7 ) .
Boundary conditions G2 .
= 0.
GA2 .
= 0.
G2 .
/) = 0.
GA2 .
/) = 0.
Solution G2 .
is given in appendix of this chapter.
Substituting G0 .
G1 .
and G2 .
, we get G.
which is given in appendix.
We discuss only two cases for separation parameter for various flow/physical One for rotation of pervious sphere in infinite expanse of fluid .
Ie .
and other for rotation of pervious sphere in concentric spherical cavity which is at rest and of radius two times of sphere .
= 0.
Case.
: = 0.
= Oe(.
Oe .
s2 2592s 5.
Oe r.
9s2 7488s 5.
Oe 1384s2 ) 144rs.
)/.
20r3 ), .
At s = 0 G.
= Oe .
Oe .
which conforms to the solution of .
Haberman .
Collins .
and Chakraborty and Roy .
Case.
: = 0.
= (Oe3r8 .
63904 log.
Oe 14740.
9520 log.
Oe 5870.
3577r7 s.
9520 log.
Oe 10985.
Oe 3679.
928580192 log.
Oe 85925650.
05024 log.
Oe 37619 .
Oe 57530178.
s2 2408s 3.
Oe 9149679840r4 s.
Oe 6r3 .
8027328 log.
Oe 6273170.
7337 83395 20 log.
) 357464204.
8405 3335808 log.
) 1822430 4s 16711.
7r5 14308r2 Oe 167.
s Oe 208.
Oe 8r.
64924147 18552562224 log.
) Oe 36792s.
1432 log.
Oe 905 8.
Oe 422105229952s2 )/.
5020555360r3 ).
It may be seen that above result at s = 0 conforms to the corresponding solution of the flow due to the rotation of a sphere in a Newtonian fluid .
s in .
, .
Discussion of the secondary flow solution Having obtained G.
and hence the stream function associated with the secondary flow, we present below certain quantities viz.
pressure, streamlines, vortex lines that give more information about of the flow.
Graphs have been drawn to depict pressure, streamlines, vortex lines for twelve values .
, 1, 10, 20, 30, .
of the source parameter s and two values .
, 0.
of the separation parameter .
Here it may be noted that = 0 corresponds to an infinite expanse of fluid media outside of the pervious sphere and decreasing values of = 0.
5 imply shrinking of the space in-between the pervious sphere and the cavity.
Also that a increase in the value of s results in an increase in the effect of source.
thus, the value s=0 approximates the Source modified Stokes equation to Stokes equation.
Pressure.
In order to determine the pressure p1 in x-z plane we need to use the differential equation OCp1 OCp1 d.
OCr On inserting the values of u and v from equations .
in the equations .
, .
with use of .
, we get r4 f .
2 Oe 3r3 GAA .
3rsGA .
Oe 12sG.
OCp1 sin2 OCr 2.
3 GAA .
Oe rsGA .
s Oe 6.
) , .
OCp1 = 3 .
3 f .
2 Oe r3 G.
rsGAA .
6rGA .
Oe 12G.
) sin cos .
Above equations indicate that p1 is to be written as p1 .
, ) = p11 .
sin2 p12 .
Integrating .
with respect to gives .
2 Oe r3 G.
rsGAA .
6rGA .
Oe 12G.
) sin2 g.
Differentiating above equation with respect to r and equating with right hand side of .
and using the differential equation .
, we get 2 r3 GAA .
Oe rsGA .
s Oe 6.
= A .
Since r Ou 1 here, so equation .
provide Z r g.
= 2 r3 GAA .
Oe rsGA .
s Oe 6.
Due to very much larger expression of G.
in terms of .
Also, the needed f .
in form as f .
= .
r3 Oe .
/(.
Oe .
r2 ) s.
2 r4 Oe 43 .
Oe 2 Oe Oe .
( Oe .
2 r3 ) Oe .
2 Oe 3 .
1013 572 17 .
r5 203 .
r4 Oe 403 .
2 r3 ( Oe .
243 102 4 .
24 3 Oe 32 Oe 2 Oe .
) /.
(( Oe .
3 r4 )).
Using g.
and f .
, we get p1 completely known from.
Since expression of p1 in terms of is very huge, therefore, we discuss for two values 0 and 0.
5 of .
Expression for pressure for = 0 p1 = (.
r Oe .
r4 ) s(.
r3 Oe 350r2 340r Oe .
r6 )) s2 (.
r3 Oe 3180r2 7320r Oe 2448 log.
Oe 4.
0r6 ))) sin2 Oe (.
Oe .
r4 )) Oe s(.
r3 Oe 105r2 104r Oe .
r6 )) s2 ((Oe485r3 3033r2 Oe 6984r 2448 log.
80r6 )).
Expression for pressure for = 0.
p1 = (.
7r6 Oe 8176r3 68448r Oe 65.
78r4 ) s(.
9840 log.
Oe 1672.
Oe 8809640r7 Oe 73543120r5 208896800r4 24r3 .
7337 8339520 log.
) Oe 962151680r2 811713280r Oe 204072.
797160r6 )) s2 (.
19556576 log.
Oe 8142486.
Oe 3066r7 .
55680 log.
Oe 40893.
16377509120r6 73584r5 .
8752 log.
Oe 5091.
983590582800r4 4r3 ( 292867557696 log.
Oe 442967754.
Oe 29278975488.
Oe 98112r2 .
9681 12509280 log.
) 4712998295040r 65408.
6444 log.
Oe 4891 5.
)/.
8340185120r6 ))) sin2 Oe 2(.
r6 765r4 Oe 4088r3 11408r Oe 8.
/ .
39r4 )) Oe s((.
2 Oe 3r .
9520 log.
Oe 5870.
9520 log( .
Oe 5294.
18560 log.
Oe 10549.
968 log.
Oe 94092 .
Oe 68163312r2 219378432r Oe 102036.
)/.
695740r6 )) Oe s2 (.
99r8 ( 20063904 log.
Oe 14740.
Oe 9198r7 .
18560 log.
Oe 29547.
Oe r6 .
64 2900960 log.
Oe 390848648.
52512 log.
Oe 54021.
1475 385874200r4 12r3 .
22519232 log.
Oe 170428454.
Oe 1876392r2 .
249 654080 log.
) Oe 3659871936.
r6 56r3 .
4551963929856r 32 704.
32888 log.
Oe 94436.
)/.
7510277680r6 )).
Now we discuss variation of pressure flow pattern.
showing in figures 6 to 17 are discussed here.
This is done for two values of separation parameter = a/b with various values of source parameter s as depicted on the diagrams.
Equi-pressure surfaces are obtained by setting surfaces are given by setting the pressure function p1 = c for different values of the constant c.
Since the flow is axial symmetric about the z-axis, their intersection with the x-z plane may be termed as streamlines.
Thus they are given by equation .
using above expressions.
we present below the graphs clearly depicting the variation in the pressure in Figure 6-17.
Figure 6.
Ie 0 and s = 0 Figure 7.
Ie 0 and s = 1 Figure 8.
Ie 0 and s = 10.
Figure 9.
Ie 0 and s = 20.
Figure 10.
Ie 0 and s = 30.
Figure 11.
Ie 0 and s = 40.
Figure 12.
= 0.
5 and s = 0 Figure 13.
= 0.
5 and s = 1 Figure 14.
= 0.
5 and s = 10 Figure 15.
= 0.
5 and s = 20.
Figure 16.
= 0.
5 and s = 30.
Figure 17.
= 0.
5 and s = 40.
Streamlines.
Stream surfaces are given by setting the Stokes stream function O = c for different values of the constant c.
Since the flow is axial symmetric, about the zaxis, their intersection the x-z may be termed as streamlines.
Thus they are given by equation .
x2 z O = G.
sin2 cos = .
This shows that the flow pattern is symmetric about z-axis and anti symmetric about x-axis.
Therefore, it suffices to draw the streamline pattern graphically only in the first quadrant of x-z plane.
This is done for two values of separation parameter (= a/.
= 0 for the pervious sphere rotating in an infinite expanse of fluid (Figs.
and = 0.
5 for the case when the fluid is confined to a cavity has radius 2 (Figs.
In both cases the streamlines are depicted for 6 values of the source parameter viz.
s = 0, 1, 10, 20, 30, 40.
First we observe and discuss the disturbance induced by the presence of source in the two cases.
The stream lines only refer to the exterior flow and hence do not appear in the interior of the sphere.
Case .
: The fluid extends to infinity, = 0.
It may be noted that s = 0 corresponds to the case when source is absent.
It is observed that in Figs.
18, 19, 20
: 0, 1, .
, the stream lines are in the anticlockwise direction appearing to diverge to infinity with hardly any difference in the flow pattern.
It is also noteworthy that these secondary vorticities are in clockwise direction as depicted in Figs.
20, 21,
22, 23.
Figure 18.
Ie 0 and s = 0.
Figure 19.
Ie 0 and s = 1.
Figure 20.
Ie 0 and s = 10.
Figure 21.
Ie 0 and s = 20.
Figure 22.
Ie 0 and s = 30.
Figure 23.
Ie 0 and s = 40.
Case .
: The fluid is confined in the region 1 < r < 2, = 0.
Since fluid remains confined to a closed region, the flow is markedly different from the = 0 For s from the value 0 to the value 10 (Figs.
24, 25, .
there is only a single eddy rotating anticlockwise but streamlines became more compact as s increases.
The base flow here is the primary flow when s = 0, i.
when the source is absent and a solid sphere rotating steadily.
The flow lines consist of circles round the axis of rotation that we have taken as x-axis.
Then centrifugal force as in Fig.
18 throws the fluid outwards in the radial direction and on account of continuity equation the fluid returns down ward in axial direction as shown in Fig.
18 for the infinite region case and Fig.
24 in the confined region case.
It is seen that in the latter case the streamlines form closed curves, while in the former case they are closed at infinity.
It may also be inferred from the expression .
of velocity that as we move away from the equatorial plane towards the pole above, it decreases and so does the circular streamline there.
Thus, velocity is maximum at the equators and tends to zero at the pole.
Then the secondary flow pattern emerges and is modified by the source flow which is in the spheres radial r-direction, and wanes as it moves away from the surface of the sphere and it remains the same for all the emerging It may be seen that at points near the pole outgoing source flow meets the incoming secondary flow and the former gets, for sufficient larger values of s, stronger than the latter appoints and reverse happens at points near the equator.
this process goes to generate eddies .
ee Figs.
21, 22, 23 and 27, 28, 29 ) near the inner sphere.
Figure 24.
= 0.
5 and s = 0 Figure 25.
= 0.
5 and s = 1 Figure 26.
= 0.
5 and s = 10 Figure 27.
= 0.
5 and s = 20.
Figure 28.
= 0.
5 and s = 30.
Figure 29.
= 0.
5 and s = 40.
Vorticity.
The vorticity is in the azimuthal direction and is given by 1 OC
OCu
r OCr
E2O
r sin G .
sin cos .
Now we present graphically .
for two values of one for Ie 0 and other for = 0.
As for streamlines.
In both cases the streamlines are depicted for 6 values of the source parameter viz.
s = 0, 1, 10, 20, 30, 40.
First we observe and discuss the disturbance induced by the presence of source in the two cases.
The graphical results are depicted below.
In both the cases inside the sphere there are no vortices as source flow itself is Case .
The fluid extends to infinity, = 0.
Fig.
30 provides the base state when s = 0.
it is seen that there are two sections of vortices, a small section, in clockwise direction, near the surface of the inner sphere and a large section, in anticlockwise direction, away from the sphere and extending to infinity.
With increase in the value of s upto 10 (Fig.
, the same pattern persists except that the inner section becomes more and more prominent and the centers of the two sections shifting away from the surface of the sphere.
Then with further increase of s, a third section, again in anticlockwise direction, starts appearing and becoming prominent (Figs.
The primary circulatory motion around the sphere inhibited by its presence of the sphere generate the vortices in the secondary flow and to preserve the circulation the next vortex appears circulating in opposite direction.
Interaction with the source flow enhances this phenomena.
this causes the appearance of the third section.
Case .
The fluid is confined in the region 1 < r < 2, = 0.
In this case the 3 sections are present from the very beginning and circulating in the same sense as in case .
noticeable difference being that the central portions get elongated as s Vortices are generated by the presence of a solid boundary.
There are two in this case at r = 1 and at r = 1/ and that explains the presence of third region from the very start.
Figure 30.
Ie 0 and s = 0.
Figure 31.
Ie 0 and s = 1.
Figure 32.
Ie 0 and s = 10.
Figure 33.
Ie 0 and s = 20.
Figure 34.
Ie 0 and s = 30.
Figure 35.
Ie 0 and s = 40.
Figure 36.
= 0.
5 and s = 0.
Figure 37.
= 0.
5 and s = 1.
Figure 38.
= 0.
5 and s = 10.
Figure 39.
= 0.
5 and s = 20.
Figure 40.
= 0.
5 and s = 30.
Figure 41.
= 0.
5 and s = 40.
REFERENCES