403
Internat. J. Math. & Math. Sci.
VOL. ii NO. 2 (1988) 403-404
ON THE DISSIPATION LAYER OF RADIAL BEARINGS
R.O. AYENI, S.S. OKOYA and C.J. AGORZIE
Department of Mathematics
University of Ife
lle-lfe, Nigeria
(Received May 14, 1986)
The dissipation boundary layer of certain radial bearings is identified.
It is shown that, under certain conditions, the temperature outside this layer is
ABSTRACT.
constant.
KEY WORDS AND PHRASES. Dissipation boundary layer, variable viscosity, coupled momentum and energy equations, thin lubricating oils.
1980 AMS SUBJECT CLASSIFICATION CODE. 80A20.
I.
INTRODUCTION.
-
In paper [I], we discussed the solution to the coupled momentum and energy equations
30
I’ 2
where
and
A
Pr
y(eXp(8-2
320
30 2
(.i)
+ A exp(8,
(y)
3Y 2
),
i
3
2
02) (y)
(1.2)
given, #(,t)= 0
(I.3)
are constants.
The boundary and initial conditions are
-(0
3y
B
t)
(y,O)
(0,t)
tt
0
e(o,t)
e(-,t)
e(y,O)
0
(z.5)
0
(1.6)
The above equations arose from investigations on the flow of thin lubricating oils.
Earlier in [2], we discussed the thermal runaway of variable viscosity flows between concentric cylinders
there is a wide gap between the cylinders and we show
that, under certain conditions which affect the Peclet number, the reduced Reynolds
number and the Nahme-Griffith number, the width of the thermal boundary layer is 0(R)
R
where
is the radius of the inner cylinder.
exists a dissipation boundary layer of width
Under the same assumptions, there
0(R/G 2)
where
G
is the Nahme-Griffith
number.
if
is small and thus
In the case of radial bearings, the width, h, of the lubricant
are the radii of the shaft and the bearing then
R
RI,
2
R
i
>> h.
i
I, 2
(1.7)
404
R.O. AYENI, S.S. OKOYA and C.J. AGORZIE
(1.7) implies that as a first approximation, we may regard the shaft and the bearing
as flat surfaces, that is R
when h is the standard measure. We therefore need
i
to
investigate the boundary layer of the lubrication problem although
quate for [2].
2.
-
0(R/G =) is ade-
DISSIPATION BOUNDARY LAYER.
Shampine [3] investigated the problem
c
c
-x(D(c)-x),
I,
c(0,t)
He showed that if
this, see
n
c(0
x
t)
c
-D(1)
then there is an
(2. I)
+
(0,t)
<
max D(c),
c(n I)
such that
(2.2)
given
0, where
n
x//
For the proof of
[3].
In this paper as in [i], we have
Pr
D(c)
in the plane of
exp(--2)
above, but a careful study of the proof advanced by Shampine
showed that his result is true for (I.I).
That is, there exists an
I
<
such that
if
I
+
B
then there exists an
e
<
I
k2
(2.3)
such that
@(nl
0
()
and
0
for
ql"
This
Then there exists an
2I
B
leads to
Let (I.I), (1.2) (1.3)
THEOREM.
8(n 2)
such that
PROOF.
0
(i
for
implies
y
0
for
n
That is,
8
y
0
for
n
8 E 0
for
Hence
8( )E 0
and
E 0
(1.6), and (2.3) hold.
and
2"
I
>
n I.
n2
+
I
n2
3. PHYSICAL INTERPRETATION.
The above theorem shows that at time
to
y
n2/
t
the heat generated has only penetrated
This identifies the dissipation layer.
The viscosity of the lubricant
outside this layer is not affected by heat under the conditions assumed in this pro-
blem.
REFERENCES
I.
2.
3.
4.
5.
AKINRELERE, E. A., and AMAO, J. O., On the Temperature of Radial
I, Matematlka 32 (1985), 317-324.
Bearings
AYENI, R. O. On the Thermal Runaway of Variable Viscosity Flows Between Concentric
Cylinders, Z. Angew. Math. Phys. 33(1982), 408-413.
SHAMPINE, L. F. Concentratlon-dependent Diffusion II. Singular Problems, Quart..
Appl. Math. 31(1973), 287-293.
SHAMPINE, L. F. Concentratlon-dependent Diffusion, Quart. Appl. Math. 30(1973),
441-452.
AYENI, R. O.
WAGNER, C. Diffusion of Lead Chloride in Solid Silver Chloride, J. Chem. Phys. 18
(1950), 1227-1230.