Radial Viscous Flow between Two Parallel Annular Plates
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Radial Viscous Flow between Two Parallel Annular Plates
Mya San May
Abstract
We seek the approximate solution to the Navier-Stokes equation for the
radial flow between two parallel annular plates since there is no exact
solution which satisfies the boundary conditions for the problem.
1. Introduction
In this paper, we shall deduce the velocity distribution of steady flow of an
incompressible fluid of densit y ρ and viscosity μ between two parallel, coaxialannular plates of inner radius r 1, outer radius r 2 and separation h when pressure
difference ∆p is applied between the inner and outer radii.
As an exact solution of the Navier-Stokes equation appears to be difficult, it
suffices to give an approximate solution assuming that the velocity is purely radial,
r ˆ
z)u(r,q , in a cylindrical coordinate system (r, φ , z) whose z-axis coincides with
that of the two annuli. We will deduce a condition for validity of the approximation.
This problem arises, for example, in considerations of a rotary joint betweentwo sections of a pipe. Here, we ignore the extra complication of the effect of the
rotation of one of the annuli on the fluid flow.
2. Two-Dimensional Flow between Parallel Plates
Figure (1)
The velocity distribution obeys the continuity equation for an incompressible
fluid,
Dr, Tutor, Department of Mathematics, University of Magway
z=0
z=h
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72 Magway University Research Journal 2012, Vol. IV, No. 1
0q. ,
(1)
and if the motion is steady and no body force then the Navier-Stokes equation reduces
toqμ pq).q(ρ 2 .
(2)
It is already known that an analytic solution is readily obtained for the
problem of two-dimensional viscous flow between two parallel plates. For example,
suppose that the plates are at the planes z = 0 and z = h, and that the flow is in the x-
direction, i.e., i
ˆ
z)(x,uq . Then the boundary conditions are as follows:
q (z = 0) = 0 = q (z = h) ,
(3)
which means that the flow velocity vanishes next to the plates.
The equation of continuity (1) then tells us that 0xu
, so that the velocity is a
function of z only,
iˆ
(z)uq .
(4)
Then the Navier-Stokes equation (2) becomes
i
ˆ
zu(z)
iˆ
xu(z)
μk ˆ
z p
iˆ
x p
)iˆ
(u(z)x
u(z)ρ 2
2
2
2
.
The z-component of the Navier-Stokes equation (2) reduces to 0z
p, so that
the pressure is a function of x only.
The x-component of (2) is
2
2
zu(z)
μx
p(x).
(5)
Since the left-hand side is a function of x, and the right-hand side is a function
of z, (5) can be satisfied only if both sides are constant. Supposing that the pressure
decreases with increasing x, we write
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Magway University Research Journal 2012, Vol. IV, No. 1 73
0.ΔxΔp
constantx
p
From (5),it follows that
B,Az2z
x p(x)
μ1
u(z)
2
(6)
where A and B are constants.
Since 0,0)(zq we find B = 0 and similarly, since q (z h) 0, we get
1 p(x) hA .
μ x 2
Then
21 p(x) z 1 p(x) hzu(z) μ x 2 μ x 2
= h)(z2z
x
p(x)
μ1
= z)(h2z
)x
p(x)(
μ1
u(z)=2μ
z)(hz
ΔxΔp
.
(7)
Hence the average velocity u a is given by
ua =h
0dzu(z)
h1
= h
0dz
2μz)(hz
x p
h1
= dz)z(zh 2μ1
ΔxΔp
h1 h
0
2
=h
0
h
0 3z
2hz
2μ1
ΔxΔp
h1 32
=3h
2h
ΔxΔp
h1
2μ1 33
=6h
ΔxΔp
hμ21 3
.
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74 Magway University Research Journal 2012, Vol. IV, No. 1
ua =ΔxΔp
12μh 2
.
(8)
Then 2a
huμ12
ΔxΔp .
From (7) , we obtain2μ
z)(hz
huμ12
u(z) 2a
and hence
u(z) .)hz
(1hz
u6 a
(9)
3. Radial Flow between Parallel Annular Plates
For the problem of radial flow between two annular plates, we seek a solution
in which the velocity is purely radial, i.e., .r ˆ
z)u(r,q The continuity equation (1) for
this hypothesis tells us that
div q = ,0rur r 1
(10)
so that 0r
(ru) .
By integrating both sides, we get
ru =f(z) since u = u (r, z).
Therefore , we have
u= r
f(z),
so that
q r ˆ
r f(z)
.
(11)
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Magway University Research Journal 2012, Vol. IV, No. 1 75
Following the example of two-dimensional flow between parallel plates, we expect a
parabolic profile in z as in (7),
z)(hz(z)f ,
(12)
which satisfies the boundary conditions (3).
Using the trial solution (11) in cylindrical polar coordinates, (2) becomes
).r ˆ
r f(z)
()z
r zφr
1φr
r r
(r 1
μ)z
p k
ˆ
φ p
r φ
ˆ
r p
r ˆ
(
)r ˆ
r f(z)
()z
k ˆ
φ
r φ
ˆ
r
r ˆ
(.)r ˆ
r f(z)
(ρ
).r ˆ
r f(z)
()zφr
1r r
1r
μ()z
p k
ˆ
φ p
r φ
ˆ
r p
r ˆ
(r ˆ
r f(z)
r r f(z)
ρ 2
2
2
2
22
2
ˆ
f(z) f(z) p φ p p 2f(z) f(z)ˆˆ ˆ ˆ ˆ
( )r (r k ) μ r r 2 3 3r r r φ zr r r 2ˆ
1 f(z) r f(z) f(z)ˆ ˆ
( r ) r 2 2φ r φ φ r r r z
r ˆ
r
(z)f ρ 3
2
r ˆ
r
f(z)
z )φ
ˆ
r
f(z)(
φ
r
1r ˆ
r
f(z)μ)
z
p k
ˆ
φ
p
r
φˆ
r
p r
ˆ
( 2
2
23
=
r ˆ
r f(z)
z
φφ
ˆ
r f(z)
r f(z)
φφ
ˆ
r 1
r ˆ
r f(z)
μ)z
p k
ˆ
φ p
r φ
ˆ
r p
r ˆ
( 2
2
23 .
Therefore
r ˆ
r (z)f
ρ 3
2
= r ˆ
r f(z)
zr ˆ
r f(z)
r ˆ
r f(z)
μ)z
p k
ˆ
φ p
r φ
ˆ
r p
r ˆ
( 2
2
33 .
The z-component of the Navier-Stokes equation (2) again tells us that the
pressure must be independent of z: p =p(r).
Thus (2) becomes
r ˆ
r (z)f
ρ 3
2
r ˆ
r f(z)
zμr
ˆ
r p
2
2
.
(13)
The radial component of (13) yields the nonlinear form
dr
dpr f ρ
dz
f d r μ 32
2
22
.
(14) The hoped-for separation of this equation can only be achieved if f(z) =F is
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76 Magway University Research Journal 2012, Vol. IV, No. 1
constant, which requires the pressure profile to be p(r) = .2r ρF
A 2
2
The boundary
conditions (3) cannot be satisfied by this solution because the fluid must be at rest
(i.e., q = 0) to satisfy (3). Further, this solution exists only for the case that the
pressure is increasing with increasing radius. The fluid flow must then be inward, so
the constant F must be negative. The Navier-Stokes equation is not time-reversal
invariant due to the dissipation of energy associated with the viscosity, and so
reversing the velocity of a solution does not, in general lead to another solution.
While we have obtained an analytic solution to the non-linear Navier-Stokes
equation (2), it is not a solution to the problem of radial flow between two annuli. It is
hard to imagine a physical problem involving steady radially inward flow of a long
tube of fluid, to which the solution could apply.
Instead of an exact solution, we are led to seek an approximate solution in
which the non-linear term 2f of (14) can be ignored. In this case, the differential
equation takes the separable form
mconstantdr dp
μr
dz
f d2
2
.
(15)
Integrating both sides, we get
f = BAz2z
dr dp
μr 2
, where A and B are constants.
From these equations and boundary conditions, we obtain A =2h
dr dp
μr
and B =
0.
Substituting, f = 2zh
dr dp
μr
2z
dr dp
μr 2
.
f = h)(z2z
dr dp
μr
.
(16)
The average of f(z) = f a is given by
f a=h
0dzf(z)
h1
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Magway University Research Journal 2012, Vol. IV, No. 1 77
= dzhz)(zdr dp
2μr
h1 2
h
0
=h
0
h
0 2
hz
3
z
dr
dp
μh2
r 23
.
= 0)2h
(0)3h
(dr dp
μh2r 33
=
63h2h
dr dp
μh2r 33
f a = 2hdr dp
μ12
r .
(17)
From (16), we get f = h)(z2z
h
f 122
a
= a
z z6f 1
h h
.
Following (7), we write the solution for f that satisfies the boundary conditions
(3) as
f (z) = a
z z6f 1
h h
(18)
where f a is the average of f(z) over the interval h.z0
From (15), we obtain
r μm
dr dp
.
Integrating both sides, we find p = Ar lnμm .
If r = r 1 and p = p 1 , then p 1= 1r lnμm +A.
Therefore A= .r lnμm p 11
Similarly, if r = r 2 and p = p 2 , then p 2 = 2r lnμm +A.
Therefore A= p 2+ 2r lnμm .
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78 Magway University Research Journal 2012, Vol. IV, No. 1
Thus we get
1
2
21
r r
ln
p pμm
and A=
1
2
1221
r r
ln
r ln pr ln p .
The part of (15) that describes the pressure leads to the solution
p ( r )=
1
2
21
r r
ln
p p ln r +
1
2
1221
r r
ln
r ln pr ln p .
p ( r ) =
1
2
122121
r r
ln
r ln pr ln pr ln pr ln p
=
1
2
12
21
r r ln
r r
ln pr r
ln p
,
(19)
where p i = p (r i). Substituting (18) and (19) back into (15), we find
af = 2h
r mμ
μ12
r
=
μ12
h 2
1
2
21
r r ln
p p
=
μ12h 2
1
2
r r
ln
p,
(20)
where p = 21 p p . Hence, the flow velocity is
r ˆ
r f(z)
zr,q
= a
1 z z ˆ
6 f 1 r r h h
= r ˆ
z)(hhz
r r
lnμ12
Δph 6
r 1
2
1
2
2
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Magway University Research Journal 2012, Vol. IV, No. 1 79
= r ˆ
r r
lnrμ2
Δpz)z(h
1
2
,
(21) whose average with respect to z is u a r f a
. As with all solution to the
linearized Navier-Stokes equation, the velocity is independent of the density. It is also
to be noted that the direction of the flow is from the high pressure region to the low.
For the approximate solution (21) to be valid, the term 2a
2 f f must be small
in (14), which requires
dr dp
r f ρ 32 .
It follows 1
dr dp
r
f ρ
3
2
dr dp
r
f ρ
3
2a << 1
1
dr dp
r
144 μ
)dr dp
(hr ρ
3
2
242
1r μ144
hdr dp
ρ 2
4
1 r μ144
hr mμ
ρ 2
4
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80 Magway University Research Journal 2012, Vol. IV, No. 1
1
r r
lnr μ144
h) pρ(p
1
222
421
1
r r
lnr μ144
h) pρ(p
1
222
421 .
1
r r
lnr μ144
Δphρ
1
222
4
where p = ) p(p 21 .
(22)
When this condition is not satisfied, the solution must include velocity
component in the z- direction that are significant near the inner and outer radii, while
the flow pattern at intermediate radii could be reasonably well described by (21).
In the case of low viscosity μ we make an approximate analysis of (14) by
taking the factor f to be the constant 2af of (18). Then,
,hdr dp
μ12
r f since
r f ρ
r hf μ12
dr dp 2
a3
2a
2a
(23)
which integrates to2
a a2 2
12 μ f ρ f p ln r A
h 2r .
When r = r 1 and p = p 1 , A p1 21
2a
12a
2r f ρ
r lnh
f μ12.
Similarly, if r = r 2 and p = p 2 then A p 2 22
2a
22a
2r
f ρ r lnh
f μ12.
So, we obtained
2a a
2 2
12 μ f ρ f p(r) ln r
h 2r p1
2
1
2a
12a
2r f ρ
r lnh
f μ12,
p (r)
221
2a
12
a1 r
1
r
1
2f ρ
r r
lnh
f μ12 p ,
(24)
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Magway University Research Journal 2012, Vol. IV, No. 1 81
where p 1 is the pressure at radius r 1. Evaluating this at radius r 2 where the pressure is
p2, we can write
p pΔp 21 2
1
2a
12a
2r f ρ
r lnh
f μ122
2
2a
22a
2r f ρ
r lnh
f μ12,
p pΔp 21 ρμ21
22
2a
1
22
a ΔpΔpr
1
r
12f ρ
r r
lnh
f μ12
(25)
where the term
1
2 2a
ρ
a a2a
ρ(u ) ρ(u ) f Δp since u ,
2 2 r
(26)is the familiar change in pressure associated with the change in velocity described by
Bernoulli’s law for fluids with zero viscosity. From (25), we have
2
12
2
2a
r
1
r
12f ρ
+
1
22
a
r r
lnh
f μ120Δp
2 222 2 1 2
2 2 2 21 1 1 2
2 2
1 22 2
1 2
a
r r r r 12 μ 12 μ ln ( ln ) 2 ρ ( ) ( Δp)
h r h r r r f
r r ρ( )r r
.
The constant f a = r r au is determined by equation (25) to be
22 2 2 22 2
1 2 1 2 2 21 1 1 2
a 2 2 2 2 2 2 2 22 1 2 1 2 1
r r 12 μ r r ln 12 μ r r ln
r r 2 r r Δpf
ρ h (r r ) ρ h (r r ) ρ (r r )
.
(27)
The direction of the flow can be either from high pressure to low or vice versa, but
with different velocities in these two cases.
Conclusion
We have obtained an analytic form of solution for the radial flow between
annular plates in the linear approximation to the Navier-Stokes equation . An
experimental study of a case well approximated by the solution is reported in [ 4 ] .
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82 Magway University Research Journal 2012, Vol. IV, No. 1
There are only three examples in which analytic solutions to the equation have
been obtained when the non-linear term (q )q is non-vanishing [ 5 ].These three
cases are
( i ) the flow due to an infinite plane disk rotating uniformly about its axis,( ii ) the steady flow between two plane walls meeting at an angle α and
( iii ) the flow in a jet emerging from the end of a narrow tube into an infinite
space filled with the fluid.
References
Batchelor, G.K., 1997, “An Introduction to Fluid Dynamic”, Cambridge University Press, New York,
Burgess, D and Van Elst, H., 2002 “MAS 209: Fluid Dynamics”, University of
London, London.
Chorlton, F., 1967, “Fluid Dynamics”, D. Van Nostrand Co.Ltd, Londan.
Kirk McDonald, T., 2008, “ Radial Viscous Flow between Two Parallel Annular Plates” , Princeton
University, Princeton, June 25, 2000; updated July 30.
Landau, L. D and Lifshitz, E. M., 1987, “Fluid Mechanic”, 2 nd ed, chap (2) Pergamon Press, Oxford,
Wilson, D. H., 1964, “Hydrodynamics”, Edward Arnold (Publishers) Ltd, London.