Proe. Indian Acad. Sci., Vol. 88 A, Part III, Number 2, May 1979, pp. 163-167.
9 printed in India.
Effect of porous lining on the flow between two concentric rotating cylinders
M N CHANNABASAPPA, K G U M A P A T H Y and I V N A Y A K * Department of Mathematics, Karnataka Regional Engineering College, Surathkal P,O, Srinivasnagar 574 157
*Department of Applied Mechanics and Hydraulics MS received 17 February 1978
Abstract. The effect of non-credible porous lining on the flow between two con- centrie rotating cylinders is investigated using Beavers and Joseph slip boundary condition. It is shown that the shearing stress at the walls increases with the porous lining thickness parameter ~.
Keywords. Porous lining; rotating cylinders; shearing stress.
1. Introduction
The control of shearing stress is important in the design of rotating machinery (like totally ertclosed --fart-cooled motors and lubrication industry) in which the centri- fugal force plays a major role. To achieve this, following the rectangular geo- metry model of Chartrtabasappa et a/[3], we consider the flow between two cortcert- tric rotatirtg cylinders with rtort-erodible porous lining on the irmer wall of the outer cylinder. The velocity field, using Beavers and Joseph [1] slip boundary cortditiort (hereafter called the BJ condition), is determined and ;t is shown that the velocity irtcreases with the porous lining thickness parameter e. Shearing stress at the walls is calculated for different values o f e and it is shown that it increases with e.
2. Formulation of the woblem
The physical model illustratittg the problem under consideratiort is shown in figure 1.
We consider art axi-symmetric steady incompressible viscous flow in an annulus between two concerttric rotating cylirtders. The inrter impermeable cylinder o f radius a rotates with art artgular velocity o9, while the outer cylinder o f radius b ( > a) rotating with art artgular velocity 12 has a nort-erodible porous lining o f thickness h on its inner wall. The flow in the artnulus (called zone 1) is governed by the usual 163
164
M N Channabasappa, K G Umapathy and I V Nayak
R dO dR 1. Physical model.
Navier-Stokes equations and that in the porous medium (called zone 2) by the Darcy law. The basic equations of the flow in zone 1, assuming that the flow is caused by the rotation of the cylinders, in cylindrical coordinates are [2]
p ( u d U R~ ) d P , Qd'-U , 1 dU
a u ~ + = / ' t , a - ~ : ' ~ R dzr ~ "
The equation of continuity is
( 2 )
dU
+ ~ = 0,U
(3)where U and V are the radial and azimuthal components of the velocity and R the radial distance. The boundary conditions are
U = 0 at
R = a
andR = b - - h
(4)V = a o ) at R = a (5)
V = Va at R = b -- h (6)
whore
Va
is obtained by using the BJ condition~-~-~ ~ - ~ - ~ v B - Q ~-~ , (71
where Q is the Darcy velocity irt zone 2 given by
Q = Rt? + E. (8)
Here RO is the velocity in the porous medium due to the rotation of the porous medium itself with an angular velocity 12 and E is given by
~ r r b
E = K_ ~ ~-hI pRO"- (R dO dR)
2'11" b
P .f I RdOdR
= ~ - 5 p ~ , . . . . . ( 9 )
P o r o u s lining on f l o w b e t w e e n cylinders 165 The quantities K and a in (7) are respectively the permeability arid the slip parameter o f the porous material. For a given porous material K can be determined experi- mentally by using the usual permeameter apparatus, while a, since it is independent o f the geometry of flow, cart be determined by using the experimental set up o f Beavers and Joseph [1].
We note that the equation of continuity (3) implies
U = C / R , (10)
where C is a constant. Since there is no suction or injection normal to the walls, this C has to be zero to satisfy the no slip boundary condition (4).
Therefore U ~- 0 everywhere. (11)
Irt view of (11), the solution of (2) satisfying the boundary conditions (5) and (6) is v ~--- r {).~ -- (1 -- eV} [22 fl t r -- (1 -- g)o] .5 vB (1 -- e) ().~ -- r~-)] (12) 1 f "
where va =
(1 - - e ) { 2 2 - - (1 - - e)2} {3aa"~ (e2 _ 3 e + 2 ) + 2a R e ( e 2 - - 3 e + 3 ) } + 6a2 ~- f i ( & -- 3e + 2)
3aa='(e2--3e + 2 ) {2 z - ( l - e)~} + 3a(2-- e) {2 2 + ( I - E ) ~}
(13)
artd [v, r, 2 , a , fl, e, R e , p, -r]
r V R a b
= ~ ' U 6 ' ~ ' -~ ' ,, / K '
Using (11) in (1), we get
c ~ P u ~ ]
' b ' v ' p b 2 g22 " "
(14)
v ~. dp ( 1 5 )
which gives the radial pressure.
The shearing stress cart be computed from (12), using
-r = r (d/dr) ( v / r ) (! 6)
and is of the form
2,~ 2 (1 - e) [/~ (1 - ~) - vB}
= - r : ' { ; . : - (J - ~)")
( 1 7 )
Titus the shearing stresses at the inner artd the outer walls are respectively given by
2 ( 1 - ~) (/~ ( 1 - ~) - v . } (18)
2 2 2 { f l ( l - - e ) - - v s }
T.. = ( l - - e ) { Z ' - - (1 - - ~f-} (19}
P. (A)--6
166 M N Channabasappa, K G Umapathy and I V Nayak
These in the limit e -* 0 and a --* oo reduce to the results of impermeable case [4]
2 ( f l - - 1)
"r,~-- 2 2 - 1 (20)
222 (/~ - l)
r,a = 22 - - 1 ( 2 1 )
3 . C o n c l u s i o n s
The velocity field given by (12) and the shearing stresses given by (18) and (19) are numerically computed for certain combinations o f the parameters and the results are presented graphically in figures 2 and 3.
From figure 2 it is clear that the velocity increases with the porous lining thick- ness e. Figure 3 represents the effect of ~ on shearing stress, and from this we conclude that (i) the shearing stresses (at the walls) with porous lining are always greater than the corresponding ones for the impermeable case, (ii) the shearing stresses increase with the thickness of the porous lining.
Typical behaviours are observed for other parameters.
1.5
1.1
0.9
0.7
0-5
0.3
0.5
~=0.01~,
0.3 0 . 4 , . / /
0.1~ I I I [
0.2 0.4 0.6 0.8 1.0
r
Figure 2. Variation o f v with r f o r ) , = 0 . 2 ; f l = 0 " 5 ; a = 100; Re = 1000.
Porous lining on flow between cylinders 167
2"6 I It
2.4 I
9
/
? 2 - /////%Ou~er cyhnder
2.0 - ,/ Inner cylinder /
1.8
1.6
1,4 .._____.i__ I [ i I
o o.~ 0.2 o.3 0.4 0.5
Figure 3. Variation of z with ~ for ). = 0 " 2 ; /~=0"5; ~r= 100; Re : 1000.
Acknowledgements
T h e authors are highly tkartkfal to Prof. N RudraiaJa, D e p a r t m e n t of Mathe- matics, Central College, Bartgalore, for his kio.d help alxd ortcottragemertt irt the p r e p a r a t i o n o f this paper. The fmartcial s u p p o r t for the a b o v e project f r o m the U G C , New Delhi, is gratefully acknowledged.
References
[1] Beavers G S and Joseph D D 1967 3". Fluid Mech. 30 197
[2] Chandrasekhar S 1961 Hydrodynamic and hydromagnetic stability (Oxford University Press) p. 292
[3] Channabasappa M N, Umapathy K G and Nayak I V 1976 AppL Sci. Res. 32 607 [4] Yuan S W 1969 Foundations of Fluid Mechanics (New Delhi : Prentice-Hail of India) pp. 273
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