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



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



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

.




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)




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




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




= 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 .




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




= 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




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)




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 ] .




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.

radial viscous flow between two parallel annular plates

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