A semi-analytic approach to the viscous flow between a rotating and a stationary disk

Fluid Dynamics North-Holland

Research

10 (1992) 91-99

A semi-analytic approach to the viscous flow between and a stationary disk N.M. Bujurke Department Received

and P.K. Achar

of Mathematics, 22 October

a rotating

Karnatak

Unil,ersity, Dharwad-580003,

India

1991

Abstract. The steady axisymmetric flow of a viscous incompressible fluid between two coaxial disks, one rotating and the other stationary, is discussed. Similarity solutions of Navier-Stokes equations are obtained for small and moderately large Reynolds numbers. We propose a double series expansion for the perturbation series. The recurrence relation derived allows the generation of large numbers of universal coefficients, in small Reynolds number perturbation series of the solution function by computer. The convergence of this computer-extended series is found to be limited by a simple pole at R2 = -593.8091 for the case of non-dimensional pressure gradient. Using Euler transformation, the series is recast into a new form whose region of validity is enlarged, resulting in analytic continuation of the series solution. In addition, Pad6 approximants, via continued fraction representation, provide results that are in excellent agreement with the pure numerical studies of Mellor et al.

1. Introduction

The steady flow of a viscous incompressible fluid between two infinite coaxial rotating disks has been discussed by several authors. Batchelor (19511, while giving an extension of von K&mBn’s solution, briefly explains the general nature of such flows. His predictions are based (after reducing Navier-Stokes equations to a set of ordinary differential equations using similarity transformation) on the general properties of the ordinary differential equations and the physics of the problem. Later, Stewartson (1953) investigated the nature of a solution based on small-perturbation expansion. Both Batchelor (1951) and Stewartson (1953), illustrating the nature of a solution to the general problem, were primarily interested in knowing the way two-disk flow approaches the one-disk flow of von KBrmhn. The alternative view of these authors sparked a healthy interest by many others in the field. The analytical studies on the problem are due to Stewartson (1953), Hoffman (19741, Phan-Thien and Bush (1984) and others. Some careful numerical studies on this problem are those by Lance and Rogers (19621, Pearson (1965) and Mellor et al. (1968). A clear qualitative picture of the solution of the problem emerged only after the numerical works of Lance and Rogers (1962) and Mellor et al. (1968). Mellor et al. (1968), in their investigation of the steady problem for the case of one fixed disk, predicted the non-uniqueness of the solution at high Reynolds number. They demonstrated that the Batchelor solution evolves from the zero-Reynolds number solution; the Stewartson solution does not appear until R = 217. Hoffman (1974) extended the low Reynolds number perturbation series to a moderately high Reynolds number using a computer. He preferred employing ad hoc recurrence relations in generating coefficients of the Correspondence

to: N.M. Bujurke,

0169-5983/92/$03.25

Department

0 1992 - The Japan

of Mathematics, Society

Karnatak

of Fluid Mechanics.

University,

Dharwad-580003,

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India.

N.M. Bujurke, P. K. Achar / Viscous flow betweenrofating disks

92

series solution. Phan-Thien and Bush (1984) solved the problem using power series in conjugation with unconstrained optimization techniques. They could identify pressure gradients only up to R = 18. The recent reviews by Parter (1982) and Zandbergen et al. (1987) provide the history of the problem and the current status of investigations into the problem of the von Karman flow between two impermeable disks. Brady and Durlofsky (1987) extensively analyze the case of flow between finite disks of arbitrarily large aspect ratio (non-self similar flows) with different end conditions and find close agreement with similar solutions near the axis of rotation. Most numerical methods are comparatively tedious and difficult to implement on a computer, which is partly due to the non-linearity of the equation and partly due to the need to satisfy the boundary conditions. For simple geometries, the semi-analytical numerical techniques have advantages over the purely numerical methods. These methods reveal the analytical structure of the solution that is hidden in numerical methods. Van Dyke (1974, 1990) and his associates have successfully exploited the power of these methods in unveiling important features of various types of flows in fluid mechanics. We calculate a sufficient number of universal coefficients of low Reynolds number perturbation series using a computer. The convergence of such a series is limited by the appearance of a non-physical singularity. We given an accurate analysis of locating the position and predicting the nature of this singularity. The extraction of this singularity yields analytic continuation of the series to high Reynolds number.

2. Formulation

of the problem

We consider The lower disk with an angular consist of the Assuming axial

the flow of a viscous fluid between two rigid circular disks of infinite extent. is stationary and the upper disk, which is separated by a distance d, is rotating velocity R (fig. 1). The governing equations for an incompressible viscous flow continuity equation and the three equations of motion (Batchelor, 1967). symmetry, they are

1 a(rU)

aw (2.1)

Fir --+?G=O. r

(2.2) u d( n:)

--+wG=u r ar aw u-+w-_=_-&

a2r

at% i

aw

1 ap

a2

P az

Fig.

a(c/r)

s+-

ar

I a2r. az2

1

’

+”

I. Schematic

(2.3)

(2.4)

diagram

of the problem.

NM

The velocity

Bujurke, P.K. Achar / Viscous flow between rotating disks

components t=z/d,

satisfy no-slip

For seeking

similarity

solutions,

we define

u =rRh’([),

c = rRg(.$), The no-slip

conditions.

93

w = -2Rdh(t).

boundary

conditions

h(0)

= h’(0)

g(0)

= 0,

(2.5)

are

= h( 1) = h’( 1) = 0, g(1)

= 1.

With the transformation (2.9, ordinary differential equations

(2.6)

eqs. (2.2) and (2.3) reduce

to the pair of coupled

non-linear

g” - 2R( h’g - hg’) = 0, h’” + 2R( g’g +/z/r”‘) = 0, where

R = fid*/v

is the Reynolds

(2.7)

number

and derivatives

are with respect

to 5.

3. Method and solution For small Reynolds form g=g,+ Substituting

numbers

we seek a solution

e R%,(t), tl=l

h=h,+

series in R of the

R’%(5).

(3.1)

n=l

(3.1) into (2.7) and equating g:: = 2 i

f

of eq. (2.7) as a power

(h:-,g,-,

like powers

of R on both sides, we get

-h.-18,:~,)>

r=l

-2

hk”=

k

n=o,

(g:-,g,_,+h,_,h:_,),

1,2,

...

(3.2)

n=l

The relevant

boundary

conditions

are

h”(0)

=hb(O)

=o,

h”(1)

=hb(l)

=o,

h,(O)

= h;(O)

= 0,

h,(l)

= h;(l)

= 0,

g,(O)

= 0,

g,(I)

g,(l)

=o,

n = 1, 2, 3, ..,

The solutions

= I,

of the above equations

g0=5,

h,=O,

g,=O,

h,=&(-2[*+3[‘+),

g, = - &,(8[

g,(O)

= 0, (3.3)

up to OCR*) are

+ 35t4 - 63t5 + 205’),

h, = 0.

(3.4)

It is essential to get higher approximations (large numbers of coefficients) in the series if it has to reveal the true nature of the function represented by it. As we proceed to higher approximations, the algebra involved becomes cumbersome and it is difficult to calculate the terms manually. The general trend of the function (3.4) allows us to propose a systematic

04

N.M. Bujurkr,

P.K. Achar

/

Viscous flow between rotating disks

double series expansion scheme, which is quite useful and efficient in the calculation of higher approximation terms of the series. Towards this goal, we consider g,,, h,, to be of the forms 311-I 3n g, = c

4,&l

- GJ?+‘>

h,, =

k=l n =

c

4&l

- t)‘t”+‘>

k-1

1, 2, 3,

.

(3.5)

in (3.1). These expressions exactly yield the earlier calculated terms of g,, h, (i = 0, 1, 2). It also enables us to find g,, h, for i 2 2 using a computer. We substitute expressions (3.5) into (3.2) and equate various powers of 5 on both sides to obtain a recurrence relation for the unknowns AnCkj, BnCkj in the forms A n(k)

=A,(k+

I) -

(k:

,)k i

where nk=3n-2-k,

m=n-r,

t=3m-j,

k=

1=3r+j-1,

1,2 ,...,

3n,

and B ,l(k)=2B,l(k+l)-B,*(k+2)-

X

i

i i=l n-2

+

AC,-,,w,Q,(k -i> 4

nk’ir

cc c r=l i-0

n-2

+

(k+q(&(k+qk

2

Br(,~nk,~r~B(n*-,)(r-3)ehti(t - 3,

j=l

nk’i2ii

cc c r=l i=o

Ar(,-nk,-,)A~m-,)(r-Z)e3ti(l-

nk’- * - 4

]=I

where nk’=3n-4-k, t=3m-j, P,(k)=2k, P4( k, k,)

m=n-r,

k=l,2 Z’,(k)=

1=3r+j-1, 3n-1,

,...,

-4(k+l),

= 2( k - k, + l),

P,(k,k,)=2(3k-3k,+5),

P3(k)=2(k+2),

Ps(k,

k,)

=3(3k,-3k-4),

P,(k,k,)=2(k,-k-2),

Q,(k) Q3( k) Q,(k) Q,(k,) Qx(k,)

=2(k+ I), Q,(k) = -2(k+2), = 2k, Q,(k) = -2(k + I), = 2(k + I)> Q,(k,) = 2(k, + l)k,(k, - I)> = -4(k, + l)(k,)(k, - 1) -4(k, +2)(k, + l)k,, = 2(k, + l)(k,)(k, - 1) + 8(k, + 2)(k, + l)k, + 2(k, + 3)(k, + 2)(k, + I)> Q,(k,) = -J(k, +2)(k, + l)(k,) -4(k, +3)(k, +2)(k, Qdk,) = 2(k, +3)(k, +2)(k, + I)>

A,,=O,

A,,=O,

A,,=O,

B,,=

-&,

B,,=

+.

+ I)?

(3.7)

N.M. Bujurke, P. K. Achar / Viscous flow between rotating disks

The expressions for g(t), h(t) and I?‘([), representing respectively,

components

c’, w and CL,are,

given by co

3n

c R” c 4&k(l

g(S) = 5 + h(e)

R"3nf'B,,,k,c$k+1(1 Y$)~,

= f

k=l

n=l

3n ~ 1

00

h’(5)

= c

R”

c &,,[( k=l

pressure

gradient

n=l

The dimensionless

- t>>

k-1

n=l

p = h”‘(O) - R[ ht2(0)

4. Analysis

velocity

95

and improvement

k + 1)s” - 2( k + 2)5“+’ per unit radius

-g”(O)

+ (k + 3)5k+2].

(3.8)

is

- 2h(O)h”(O)]

= /z”‘(O).

(3.9)

of the series

We have, from (3.9): p = h”‘(O) = R 5

C2n_,R2”-2

n=l

=

R

f

b,R2”-2

=

(R;E)“2 f

fl=l

R;“-2b,E”-‘,

(4.1)

n-l

where Czn-l = 6Bpn-1p - 12+,-~,~, C,,_,

= b,, R, = 24.36819857, E = (R/R”)=. Using recurrence relations (3.6) and (3.71, we generate the universal coefficients [((A.,,,), k = 1, 2,. . . ,3n), n = 1, 2,. . . ,50, and ((B,,,,), k = 1, 2,. . . , 3n - l), n = 1, 2,. . . ,501 of the series (3.5). The generation of these universal coefficients is an elegant interactive process. Using one of the arithmetic programming languages makes this calculation possible. We run the program in double precision and calculate coefficients up to n = 50 (3825 universal coeffiso generated are used in cients AnCkj and 3775 universal coefficients BnCkj). The coefficients obtaining pressure gradient and velocity profiles (series (3.8) and (3.9), respectively). The coefficients b, of pressure gradient (obtained using universal coefficients AnCkj and BnCkj are

Table 1 Coefficients of the series (4.1).

No.

bn

No.

bn

1 2 3 4 5 6 7 8 9 10 11 12 13

3.OOOOOOOOOOOOOOE-001 - 7.65873042638125E-004 2,26347068721636E-006 - 3,97368486113262E-009 6,81506801629719E-012 - l.l5815167828240E-014 1,94151455182364E-017 -3.27777922142951E-020 5,51494966337760E-023 - 0,29040698123237E-026 1,56440556823279E-028 - 2.63464118134716E-031 4,43684616091149E-034

14 15 16 17 18 19 20 21 22 23 24 25

-7,47194924921144E-037 1.25832004198441E-039 - 2.11908870926058E-042 3,56867397305111E-045 - 6,00986490628219E-048 l.O1209787021707E-050 - 1.70443453199755E-053 2.87037166115505E-056 - 4.83388088380563E-059 8.14055011984519E-062 - 1.37091827198596E-064 2,3087(1995266191E-067

N.M. Bujurke, P.K. Achar / viscous flow between rotating disks

96

Fig. 2. Domb-Sykes

plot for finding

singularity

of the series representing

the pressure

gradient,

h”‘(0).

listed in table 1. These coefficients decrease in magnitude and alternate regularly in sign. This indicates that the nearest singularity restricting the convergence of the series (4.1) lies on the negative real axis of R*. Figure 2 shows the Domb-Sykes plot for the series (4.1). The intercept and slope of the curve indicate the presence of a simple pole, which corresponds to the conjugate pair of square root singularities lying on the imaginary axis of the similarity variable R. For extrapolation (l/n --) 0) we use a rational approximation (Press et al., 1986). The radius of convergence of series (4.1) representing h”‘(O) is found to be R’ = 24.36819857.

5. Euler transformation

The range of validity of the series is restricted by the appearance of a singularity on the negative real axis of R’, which has no physical significance. To overcome this artificial restriction we use the bilinear Euler transformation w = c/(1 + E) and then, E = w/(1 - 01; these will recast the series (4.1) into the new form n-l

h"'(0)

=

(R;E)“’

e

R;“-Zb,,j

e)

n=l

(5.1) where

D, =b,,

D, = Rib,,

D3=b2R;+b,R&

D,=b,R;+2b,R;+b,R;, Aej=ej+l-ej,

0,=(-l)“-’

An-2e2,

e,=(-Ri)‘-lb,.

For the summation of above series we use PadC approximants, which further increase the validity of the transformed series (5.1). A PadC approximant is the ratio of two polynomials that, when expanded, agrees with a power series to as many terms as possible. Such rational fractions are known to have remarkable properties of analytic continuation (Baker, 1975). A Padt approximant can simulate a singularity only by the poles that are zeros of its denominators.

N.M. Bujurke,

6. Results

P.K. Achar

/

Viscous flow between rotating disks

97

and discussion

The flow of a viscous incompressible fluid between a stationary and a rotating disk is governed by the non-linear system of ordinary differential equations (2.7) together with the boundary conditions (2.6). The proposed series expansion scheme (3.11, (3.5) enables vs to obtain the recurrence relations (3.6) and (3.7). Using these interactive relations, we generate a large number (n = 50) of universal coefficients [((A,(,,), k = 1, 2,. . . ,3n), n = 1, 2,. . . ,50, ((BnCk)), k = 1, 2,... ,3n - l), n = 1, 2 ,._., 501. A carefully written FORTRAN program consisting of a number of DO loops makes it possible to perform the complex algebra involved. The series representing the velocity profiles (3.8) is analyzed using PadC approximants. These are shown in figs. 3-5. The velocity profiles are identical with those obtained by Phan-Thien and Bush (1984) (figs. 3 and 4). In fig. 5 the variation of the radial velocity component is shown. We find that these calculations agree favorably with that of Brady and Durlofsky (1987) near the axis of rotation. Using purely numerical techniques, Lance and Rogers (1962) and Mellor et al. (1968) in their studies have calculated torque and pressure gradient, respectively. Phan-Thien and Bush (1984) calculated pressure gradient using a power series in conjugation with unconstrained optimization but they could not proceed

1

0.8

0.2

0

0.4

0.6

1

0.8

TFig. 3. The transverse

velocity

profiles,

g(t),

for different

Reynolds

numbers.

0.05

h 0.01 0

d 0

0.2

0.G

0.L

0.8

1

f-

Fig. 4. The longitudinal

velocity profiles,

h(t),

for different

Reynolds

numbers.

9x

N.M. Bujurkcs. P. K. Achur

Vr.wm~ flow betweenmuting

/

disk.\

0.1 I

0.2

0

04

0.6 .-

0.8

1

5-

Fig. 5. The radial velocity

profiles.

h’(t).

for different

Reynolds

numbers.

beyond R = 18 in their analysis. Using the universal coefficients of the series (3.5) we obtain a series expression for the non-dimensional pressure gradient (3.9). The non-zero coefficients b, of these series are listed in table 1. The decrease in magnitude and alternate in sign. This indicates the presence of the nearest singularity being on the negative real axis which is restricting the convergence of the series. A Domb-Sykes plot (fig. 2) provides the location and nature of the nearest singularity. The rational extrapolation fixes the (location) radius of convergence of the series (4.1) with an error of the order of 10ph to be R2 = -593.8091. The slope of the Domb-Sykes plot shows that the singularity is a simple pole on the negative real axis of R2, which in turn corresponds to a conjugate pair of square root singularities lying on the imaginary axis of similarity variable R itself. This singularity has no physical significance, and it limits the range of applicability of the series. Applying modified Euler transformation and using PadC approximants, the series (5.1) is summed for different values of R. The variation of p with R is shown in fig. 6. The results on the non-dimensional pressure gradient agree most favorably with the numerical results of Mellor et al. (1968) up to R = 100. Also, it is of interest to note that [4/4] and [5/5] Padt approximants for non-dimensional pressure gradient are found to bracket the numerical values of Mellor et al. (1968). Beyond R = 100 our results deviate slightly from numerical findings (Mellor et al., 1968). Our results for non-dimensional pressure gradient agree with the numerical results of Brady and Durlofsky (1987) near the axes of rotation which are shown in fig. 6. Phan-Thien and Bush (1984) calculated pressure gradient only up to R = 18, and our results agree most accurately with theirs. Moreover, the method proposed here is quite flexible and efficiently implemented on a computer, compared with that of Phan-Thien and Bush (1984) and other purely numerical

0

20

LO

60

80

100

Rd

Fig. 6. Pressure

gradient

versus

R. Numerical

results (Mellor et al., 1968; Brady et al., 1987) and Pad6 approximants / ,from the series (5.1).

NM

Bujurke, P.K. Achar / Viscous flow hetwern rotating disks

99

methods. Once the universal coefficients are generated, the rest of the analysis can be done at a single stretch, requiring hardly any computer time and storage, while other analytical and numerical methods require huge storage and long computing time.

Acknowledgement The authors

thank

the referees

for their useful

suggestions.

References Baker, G.A. (1975) Essentials qf Pad& Approximants (Academic Press, New York). Batchelor, G.K. (1951) Note on a class of solutions of the Navier-Stokes equations representing steady rotationallysymmetric flow, Quart. J. Mech. Appl. Math. 4, 29-41. Batchelor, G.K. (1967) An Introduction to Fluid Dynamics (Cambridge Univ. Press, Cambridge). Brady, J.F. and L. Durlofsky (1987) On rotating disk flow, J. Fluid Mech. 175, 363-394. Hoffman, G.H. (1974) Extension of perturbation series by computer: Viscous flow between two infinite rotating disks, J. Comput. Phys. 16, 240-258. Lance, G.N. and M.H. Rogers (1962) The axially symmetric flow of a viscous fluid between two infinite rotating disks, Proc. R. Sot. A 266, 109-121. Mellor, G.L. et al. (1968) On the flow between a rotating and stationary disk, J. Fluid Mech. 31, 95-112. Parter, S.V. (1982) On swirling flow between rotating coaxial disks: a survey, MRC Technical Summary Rej No. 2332. Pearson, C.E. (1965) Numerical solutions for the time-dependent viscous flow between two rotating disks, J. Fluid Mech. 21, 623-633. Phan-Thien, N. and M.B. Bush (1984) On the steady flow of a Newtonian fluid between two parallel disks, ZAMP 35, 912-919. Press, W.H. Et al. (1986) Numerical Recipes (Cambridge, Univ. Press, Cambridge). Stewartson, K. (1953) On the flow between two rotating coaxial disks, Proc. Camhridge, Phil. Sot. 49, 333-341. Van Dyke, M. (1974) Analysis and improvement of perturbation series, Quart. J. Mech. Appl. Math. 27, 423-450. Van Dyke, M. (1990) Fluid mechanics off real axis, engineering science, Fluid Dynamics, 355-366. Zandbergen, P.J. and D. Dijkstra (1987) Von K&man swirling flows, Ann. Rer. Fluid Mech. 19, 465-491.