Cmjv03i04p0446 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

Computer Extended Series Solution of Free Surface Boundary Layer Flow on a Curved Bed VISHWANATH B. AWATI1, N. M. BUJURKE2 and RAMESH B. KUDENATTI3 1

Department of Mathematics, Govt. First Grade College,K. R. Puram,Bangalore-560 036, INDIA. 2 Department of Mathematics, Karnatak University, Dharwad-580 003, INDIA. 3 Department of Mathematics, Bangalore University, Bangalore-560 001, INDIA. (Received on: June 30, 2012) ABSTRACT Analysis of large-reynolds number flows on a curved bed with free surface at constant pressure is considered. The dimensionless film thickness

δ and reynolds number r based on film thickness Rδ = λ = o(1). For moderately large values of the

are related by reynolds number r, boundary-layer equations with appropriate boundary conditions hold. It is assumed that the film-thickness

parameter δ is small but that λ is not. The leading term of the series expansion is the one given by lubrication theory for the expansion of stream function in powers of λ . The boundary layer and series expansion results (for streamwise velocity, scaled velocity and varying film thickness) agree with earlier findings. Pade’ approximants have been used to accelerate the convergence of the series. Useful results valid up to as earlier findings were only up to

λ = 7 are obtained where

λ = 4.

Keywords: Large-reynolds number, free surface, boundary layer flow and curved bed.

1. INTRODUCTION The problem of high-reynolds number flows on a curved bed with a free

surface at constant pressure is considered. The major difficulty of the problems of fluid flows with a free surface is that the location of the free surface is not known in priori and

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Vishwanath B. Awati, et al., J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

a considerable body of literature on thin-film flows down an inclined bed on their instability and on consequent wavy disturbances is available. Lin11 lists many of the relevant references and mentions applications to rates of transport of mass 13 (Stainthorpe and Wild ) and momentum Duckler6. Fulford8 reports applications in chemical engineering processes and states that the velocity profiles under various flow conditions are of great value and it is possible to calculate the rates of convective heat and mass transfer processes in flowing films. Wang16 has considered film flows over an inclined bed with wavy striations and reports many important applications in chemical engineering process viz. Film cooling, dip painting, swept film evaporators and vapour condensers. Gajjar9 has considered high-reynolds number interactive flows caused by curved sections of nearly horizontal beds. Merkin12 has considered flow over some fairly general curved beds with oncoming parabolic velocity profiles. Acheson1 has considered the problem for two-dimensional layer of viscous fluid spread down a slope under gravity in thin film flows. Benjamin7 and Yih17 predicted long-wave instability to flow down a straight bed at an angle β 0 to the horizontal and the reynolds number greater than ( 5 4 ) cot β 0 .

Here the reynolds number is based on film thickness and fluid velocity at free surface. De Bruin5 has also confirmed this result. Bertschy et al.3 observed complete laminar flows for reynolds number as high as 5000, for β 0 up to 50 . The use of series solution represents an attractive alternative approach

provided convergence properties of the series are guaranteed. In the present study, we investigate a problem for small and moderately large reynolds number (r) and present interesting results based on series analysis. Van Dyke14,15 and his associates have shown the potential applications of these methods in fluid dynamics. Bujurke et al.4 (1996) have successfully used series analysis in their study associated with comparison of series expansion and finite-difference computations of internal laminar flow separation in a nonuniform channel flow. The physical problem considered in this chapter is of great importance in lubrication theory. So, calculation of streamwise velocity, scaled velocity and varying film thickness is of interest in all these cases. The present analysis is primarily concerned with possible extension of Eagles et al.7 high-reynolds number perturbation series by computer and its analysis. They considered only two terms of the series for small λ . However, as the problem has to be analysed for sufficiently large λ , it requires large number of terms of the series. As complex expressions involving elementary functions appearing in successive terms of the series, it is possible to calculate only eight effective terms for the shape of the stream-wise velocity profile and varying film-thickness. For the calculation of scaled velocity profile and the film thickness, it is possible to calculate only 11 effective terms. Using these coefficients we obtain series solution and calculate various physical parameters of interest for moderately large reynolds number. The series expected to be limited in convergence by the presence of singularity may be extended to moderately high-reynolds

Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

Vishwanath B. Awati, et al., J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

number by extracting the singularity and using analytic continuation. The proposed semi-analytical and semi-numerical scheme is convenient in obtaining series solution for this problem.

448

are u and v respectively. Consider the scaled co-ordinates ξ=

( )

, u = U (S , ξ ) + O δ 2 , v = δ V ( S , ξ ) + O δ 3 , p = P(S ,

ξ = , u = U ( S , ξ ) + O δ 2 , v = δ V ( S , ξ ) + O (δ 3 ) , p = P( S , ξ ) + O (δ 2 ) 2. MATHEMATICAL FORMULATION (2.3a) Let l be the reference length, d a typical film-thickness, and U 0 a typical fluid Where p is the dimensionless pressure. Also, δ → 0 with where Rδ = λ = O(1) U d velocity parallel to the bed and R = 0 , U d υ R = 0 and ν is the kinematic viscosity. ν the reynolds number and the dimensionless Substitute (2.3) into the incompressible gap width navier-stokes equations to get

δ=

d L

(2.1)

Consider the parameter λ = δ R and limit as + VU ξ = - P + 2 +U , δ → 0 with λ = o(1). Let the fixed rigid bed be (2.2) β = fˆ ( S )

Where β is the acute angle made by the bed with the horizontal and s is the arc length so that it increases as the vertical height decreases, and l the dimensionless reference length and suppose β → β 0 as S → ∞ so that the bed tends to a straight bed resulting into an exact semi-poiseuille flow. The schematic diagram of the geometry is illustrated in fig. 1 Eagles et al.7. Let d be the film thickness of the exact oncoming flow in (2.1), s the co-ordinate along the bed and q the dimensionless parameter, ql the perpendicular distance from the bed. The dimensionless fluid velocity U 0 and the edge velocity of the exact oncoming flows

λ (UU S + VU ξ ) = - λ PS + 2

sin β + U qq , sin β 0

(2.4)

an d Pξ = 0

In the gravitational term containing sin β ,

the relation arises from solution at S = −∞ . Let the free surface at constant pressure p0

ξ = G(S ) . By expanding the full be boundary conditions for continuity of stress at the free surface and neglecting surface tension, we get p = p0 , U ξ = 0 on

ξ = G (S )

(2.5)

From equation (2.4), p is a function of s only, and from equation (2.5) it is equal to constant p 0 . Therefore, the term p S in equation (2.4) is zero. Let the velocity components be represented in terms of a stream function ψ , viz.

∂ψ , ∂ξ

V =−

∂ψ . ∂S

Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

(2.6)

and Pξ = 0

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Vishwanath B. Awati, et al., J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

By using equations (2.4-2.6) and no slip boundary conditions on the rigid bed, we get +sinGβ S

ψ ξξξ = −2

sin β 0

+λ

+ G S ψ ξψ ξ S

(

−

−ψ Sψ ξξ

)

(2.7)

ψ ξξ = 0 ,

ψ =0 ,

2 ψ= 3 ψξ = 0

on ξ = G ( S ) , and

on ξ = 0

(2.8a~b)

Where g(s) is unknown. The boundary

2 in (2.8) arises from the 3 exact oncoming flow and that ξ = G( S ) is a streamline. In this problem β is a function

condition ψ =

of s only.

+ 2G3 ( S )

sin β =0 sin β0

(3.1)

With boundary conditions

2 3

ψ ηη = 0 ,

ψ =

on η = 1

ψ =0,

ψ η = 0 on η = 0 (3.2b)

(3.2a)

The solution of (3.1) in power series of λ are taken in the forms ∞

ψ = ψ 0 (η ) + ∑ λ nψ n ( S , η ) and n =0

∞

G = ∑ λ nGn ( S )

(3.3a~b)

n =0

METHOD OF SOLUTION

Equation (2.7) is to be solved as a power series in λ . Introducing the cross stream variable η =

G (S )

, where g is

unknown film-thickness and ψ is a function

of η and

ψ ηηη + λ G ′ ( S ) ψη2 + λ G ( S ) (ψ Sψ ηη − ψ ηψ η S ) + 2 G 3 S

ξ , the equation (2.7) becomes

ψ 0ηη = ψ nηη = 0 , ψ0 =ψn = 0 ,

Substituting (3.3) into equation (3.1) and equating like powers of λ on both sides, we get

1 O (λ 0 ) : ψ 0 = η 2 − η 3 , 3 1

 sin β 0  3 G0 =    sin β 

(3.4)

The relevant boundary conditions are

ψ 0 = , ψ n = 0 on η = 1, 3 ψ 0η = ψ nη = 0 on η = 0, n =1,2,3.....

(3.5a~b)

( )

Similarly, the solutions of the above equations up to the O λ3 are

O (λ ) :

ψ 1 = G0′ ( S ) L0 (η ) Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

(3.6a)

Vishwanath B. Awati, et al., J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

L0 (η ) = −

O (λ 2 ) :

8 2 12 3 1 5 1 6 1 7 η + η − η + η − η 105 105 15 30 210 12 G1 ( S ) = − G0 ( S ) G0′ ( S ) 105

450 (3.6b) (3.6c)

ψ 2 = G0 ( S ) G0′′ ( S ) L1 (η ) + ( G0′ ( S ) ) L2 (η ) 2

Where

L1 (η ) = 0.0104666 η 2 − 0.0147763 η3 + 0.00507937 η5 − 0.0015873 η8

+ 0.0010582 η 9 − 0.00026455 η10 + 0.00002405 η11 L2 (η ) = 0.00523329 η 2 − 0.0107744 η 3 + 0.0177778 η 5 − 0.0177778 η 6 + 0.00380952 η 7 + 0.00396825 η8 − 0.00291005 η 9 + 0.000740741 η10 − 0.0000673401 η11

G2 ( S ) = −0.00228681 G0 ( S ) G0′2 ( S ) + 0.0147763G02 ( S ) G0′′ ( S )}

(3.7 a~c)

O ( λ 3 ) : ψ 3 = G02 ( S ) G0′′′ ( S ) L3 (η ) + G0 ( S ) G0′ ( S ) G0′′ ( S ) L4 (η ) +

G0′3 ( S ) L5 (η )

(3.8)

Where

L3 (η ) = −0.00138382η 2 + 0.0019377η 3 − 0.000636203η 5 + 0.000120937η 8 − 0.0000302343η 9 − 0.0000224467η 11 + 0.0000200417η 12 − 7.09167 ×10−6η 13 + 1.18929 ×10−6η 14 − 7.92858 × 10−8η 15 L4 (η ) = −0.00155702η 2 + 0.00277304η 3 − 0.00304029η 5 + 0.00231906η 6 − 0.000492544η 7 + 0.000362812η 8 − 0.000655077η 9 + 0.000262031η 10 + 0.000133306η 11 − 0.000151515η 12 + 0.0000548217η 13 − 9.25001 × 10−6 η 14 + 6.16667 × 10−7 η 15 L5 (η ) = 0.000703635η 2 − 0.000583656η 3 − 0.00209331η 5 + 0.00364278η 6 − 0.0012299η 7 − 0.00181406η 8 + 0.00210632η 9 − 0.000733686η 10 − 0.000155844η 11 + 0.000228475η 12 − 0.0000840518η 13 + 0.000014245η 14 − 9.49668 × 10−7 η 15

G3 ( S ) = 0.000558526G0 ( S ) G0′3 ( S ) + 0.000604404G02 ( S ) G0′ ( S ) G0′′ ( S ) − 0.0019377G03 ( S ) G0′′′ ( S )

4. ANALYSIS OF SERIES SOLUTION The calculation of higher order coefficients becomes too tedious beyond ψ 4 and G4 because of the complexity of the

algebra involved. We need large number of coefficients for the credible calculation of the shape of streamwise velocity, scaled velocity profiles and varying film-thickness. Higher order terms in ψ involve more

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Vishwanath B. Awati, et al., J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

functions of s and increasing number of powers of η . Enumerating of functions of s is possible, since for each n, the different functions involve all the possible combinations of n functions G 0 and its derivatives multiplied together (ψ 3 involves

G02G0′′′, G0G0′G ′′ and G0′3 ; ψ 4 involves G03G0iv , G02G0′G0′′′, G02G0′′2 , G0 G0′2G0′′ and G0′4 ). Therefore, the functions of s in

ψn

an arbitrary reference length horizontal with the x-axis. If the equation of the bed is y = f(x) where f ′( x ) < 0 and f ′ ( x ) → negative constant as x → −∞ , then an arc length is given by x

S=∫

(4.1)

Where all am are non-negative integers satisfying the diophantine equations. n

∑a

m=0

= ∑ ma m = n

(4.2)

m =1

Thus, the number of such combinations satisfying (4.1), is p(n) and the integer j in (4.2) runs from 1 to p(n) where p(n) is the number of partitions of n. For any n, partitions can be calculated systematically in a variety of ways and the algorithm for this algebra is given by Gupta10. Bujurke et al.4 have also used the similar analysis in the exhaustive study of internal laminar flow separation associated with flows in nonuniform channel. The dimensionless streamwise fluid velocity is G −1 ∂ψ ∂η ,

(

)

where the term G −1 is a scaling factor and ( ∂ψ ∂η ) gives the shape of the velocity profile. Let x, y be the dimensionless rectangular cartesian co-ordinates, and l be

sin β = −

span all members of the set n a Gn j ( S ) = G0a0 G0′a1 G0′′a2 .........G0( ) n

1 + [ f ′(u )] du also, f ′ ( x) 1 +  f ′ ( x ) 

(4.3~4.4)

Where β is the acute angle made by the bed with the horizontal. Equations (4.3) and (4.4) are parametric representations of the curve β = fˆ ( S ) . For illustration of the results and theory chosen, and for comparison with earlier predictions (Eagles et al. 1986) we consider the bed specified by the equation

y = Ax + B ln(cosh x ) With A = −

(4.5)

2 1 and B = − gives a gradient 5 5

1 3 at x = −∞ and − at x = ∞ , so 5 5 that β is steadily increasing with s and expect the semi-poiseuille flow at S = ±∞ . of −

Table 1 shows the values of the thickness coefficients G 0 , G 1 and G2 . Eagles et al.7 observed that thickness coefficients have decreasing trend ( G1 is compared with G 0 and G2 is compared with G 1 ). Eagles et al.7 have obtained three terms of the series expansion of varying film-thickness and

Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)

Vishwanath B. Awati, et al., J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

streamwise velocity profile in terms of λ . It is not sufficient to analyse the problem with just three approximations. It is essential to get higher approximations in the series if it is to reveal the analytic structure of the function represented by the series. As we proceed for higher approximations, the algebra becomes cumbersome and it is difficult to calculate higher order terms manually beyond eight effective terms. Total number of the coefficients in the expansion ψ calculated in this case is of 7

∑m P m =1

and

Gˆ

(where Gˆ =

= 952 . In fig. 2 the profiles of for λ = 4, 5 and 6 are given 7

∑λ G 7

n =0

n ).

452

streamwise velocity approximations

 1  ∂ψ 0   and U 0 =    G0  ∂η  1 ∂  7 n  U7 =    ∑ λ ψ n  for λ =1, 4, 5, 6  Gˆ  ∂η  n =0  and 7 for s = 0.548 are given. In the analysis pertaining to varying film-thickness and streamwise velocity profiles converging pade’ sums are taken for values of λ up to 7. If only the direct sum (in series representation) is taken then it is not found to be accurate and reliable. The effect of the fluid velocity is to decrease the filmthickness G0 with increasing s.

In table 3,

Fig.1 Schematic diagram of the flow over a curved bed (The dashed curve represents the free surface of the bed)

Another example considered is the one whose geometry is given by Eagles et al.7.

sin β =

sin β1 , 1 < S < S0 S3

(4.6)

Which joins on to section of the bed for s<1

and is such that β → β 0 as S → −∞ and

β = β 1 at s = 1. For s > 1,  sin β 0 G0 =   sin β

 3  sin β 0  =    sin β1

 S . If 

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453

Vishwanath B. Awati, et al., J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

β1 =

1 11 π and β 0 = π at s =1.5, then 12 120 10

G0 and Gˆ = ∑ λn G n are good n=0

approximations to the film thickness for λ = 4, 5 and 6. The scaled velocity profile

Table 1: Thickness coefficients

 ∂ψ   ∂η

  ∂  10 n   =   ∑ λ ψ n  is also given in    ∂η  n =0

fig. 7. In both the calculations of filmthickness and scaled velocity profile, it is of interest to note that pade’ approximants give converging sum up to λ = 7.

G0 , G1 , G 2 and Gˆ 2 (For λ = 4 ) and At X = - 2.5 (0.5) 2.0 for example (4.5)

G1 × 103

G2 × 103

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-2.58350 -2.07314 -1.56200 -1.04878 -0.53022 0.0 0.54803 1.11291 1.68811 2.26810

0.995989 0.988990 0.971711 0.934589 0.873978 0.808473 0.763035 0.740430 0.731022 0.727394

0.933544 2.385840 5.534190 10.32020 12.99370 9.965160 5.115470 2.130380 0.818559 0.305816

-0.232275 -0.574462 -1.219680 -1.792540 -1.144810 0.178595 0.497114 0.289894 0.124090 0.048139

Table 2: The film thickness coefficients X

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-2.58350 -2.07314 -1.56200 -1.04878 -0.53022 0.0 0.54803 1.11291 1.68811 2.26810

G0 0.995989 0.988990 0.971711 0.934589 0.873978 0.808473 0.763035 0.740430 0.731022 0.727394

Gˆ 2 at

λ=4

0.996007 0.989342 0.974333 0.947189 0.907636 0.851191 0.791451 0.753590 0.736281 0.729388

G0 and Gˆ 7 at x = -2.5 (0.5) 1.5 for example (4.5)

Gˆ 7 at

0.9967370 0.9909104 0.9762240 0.9432770 0.8857054 0.8185745 0.7686412 0.7428840 0.7319851 0.7277570

0.9978589 0.9937803 0.9828718 0.9553180 0.9101385 0.8504658 0.7911472 0.7569580 0.7389198 0.7306595

0.9980666 0.9942924 0.9839236 0.9562360 0.9166419 0.8616558 0.8005010 0.7663720 0.7459334 0.734357

0.9982324 0.9946903 0.9846551 0.9561090 0.9227281 0.8731440 0.8107809 0.7815212 0.7669360 0.7555797

0.9983679 0.9950043 0.9851455 0.9551359 0.92850127 0.8849432 0.82199901 ----------

λ =1

λ=4

λ =5

λ =6

λ =7

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Vishwanath B. Awati, et al., J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

Table 3: Streamwise velocity components in example (4.5) at S = 0.584 using Pade’ sum

η 0.2 0.4 0.6 0.8 1.0

U0 0.471776 0.838713 1.100810 1.258070 1.310490

U 2 at

U 7 at

λ=4

λ =1

λ=4

λ =5

λ =6

λ =7

0.462223 0.814121 1.060130 1.205220 1.253120

0.469796 0.833677 1.092550 1.247380 1.298900

0.462379 0.814396 1.060530 1.205730 1.253680

0.459491 0.806753 1.047750 1.189050 1.235550

0.456425 0.798573 1.03404 1.171150 1.216100

0.453193 0.789892 1.019460 1.152120 1.195420

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Vishwanath B. Awati, et al., J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

complex expressions involving elementary functions appear in successive terms of the series and as such it is possible to calculate only seven terms (in case of the example (4.5) considered above). To this order there are 952 non-zero coefficients. These coefficients in turn give universal polynomials Ln (η ) which determine

ψ n (S ,η ) and Gn (S ) (n = 0, 1, 2, ….,7).

The series enables to predict streamwise velocity profile and film-thickness. The desired analytic continuation can be achieved by taking various pade’ approximants. The region of validity of the series is enhanced from λ = 4 found earlier by eagles et al (1986) to λ = 6 (for filmthickness) and λ = 4 to 7 (for streamwise velocity). In fig. 2 for λ = 4, 5 and 6, profiles pertaining to the first approximation G0 , third approximation Gˆ 2 and eighth approximation Gˆ 7 with λ (using pade’ approximants) (to the film-thickness) as a function of s are given. In fig. 3, the eighth approximation Gˆ 7 (direct sum) to the film-

5. RESULTS AND DISCUSSION A series solution is presented for studying the problem of moderately large reynolds number flow on a curved bed with a free surface at constant pressure. In the low-reynolds perturbation expansion, large number of coefficients is generated. The

thickness as a function of s for various λ is shown. The discrepancy in summation increases as singularity is approached in s and λ . Also, it suggests that a wave-like structure may be generated upstream and downstream of the position of the singularity, although it is difficult to be ascertaining about this phenomenon. In table 1 and 2 the thickness

coefficients G0 , G1 , G2 , Gˆ 2 and Gˆ 7 (using pade’ approximants) for different values of λ are given. The thickness coefficients are found to be decreasing in

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Vishwanath B. Awati, et al., J. Comp. & Math. Sci. Vol.3 (4), 446-457 (2012)

magnitude and also it is found that as the values of s increase the film-thickness decreases. For the example considered above the streamwise velocity profiles using pade’ approximants and direct sum respectively are obtained. In fig. 4 and 5, the eighth

approximation Uˆ 7 as a function of η are

shown (for λ =1, 2,…,7). For profiles in fig. 4 it is observed that as λ increases the streamwise velocity decreases, where as in fig. 5 streamwise velocity profiles are found to be fluctuating for different values of λ . Table 3 shows the streamwise velocity components in respect of example (4.5) and it is found to be decreasing as λ increases. Thinning of film thickness with increase in s results in decrease of streamwise velocity. In the case of second example (4.6) the series representing film-thickness and scaled velocity profiles (series of ten terms and 4424 non-zero coefficients) are obtained. Now, the region of validity of the series is enhanced by the use of pade’ approximants from λ = 4 (earlier finding eagles et al. 1986) to λ = 6 for film-thickness and from λ = 4 to λ = 7 for scaled velocity profiles which are shown in fig. 6 and fig. 7 respectively.

REFERENCES 1. 2.

Acheson, T. J. Elementary fluid dynamics, Oxford Univ. Press (1990).. Benzamin, T. B. Wave formation in laminar flow down an inclined plane, J. Fluid Mech., 2, 554 (1957). Bertschy, J. R., chin, r.w., abernathy, F. H. The development of an oncoming uniform flow down a straight bed, J. Fluid Mech.,126, 443 (1983).

10. 11. 12.

13.

14.

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Bujurke, N. M, Pedley, T.J., Tutty, O.R., Comparison series expansion and finite difference computations of internal laminar flow separation, Phil. Trans. R. Soc. Lond. A, 354, 1751 (1996). Debruin G. J., Stability of a layer of liquid flowing down an inclined plane, J. Eng. Maths., 8(3), 259 (1974). Duckler, A. E., Progress in heat and mass transfer, (eds hetrosoni G., Sideman. S., Hartnet J. P.), Pergamon press (1972). Eagles, P. M., Daniels, P. G., Freesurface boundary-layer flow on a curved bed, Ima Jl. of Appl. Math., 36, 101 (1986). Fulford, G. D., The flow of liquid in thin films, Adv. Chem. Eng., 5, 151 (1964). Gajjar, J. High-reynolds number interactive flows caused by curved section of nearly horizontal beds, Ph.D thesis, University College, London (1983). Gupta, H., Selected topics in number theory, abacus press (1982). Lin, S. P., Waves on fluid interfaces, academic press (1983). Merkin, J. H., Flow over some fairly general curved beds with oncoming parabolic velocity profiles, J. Eng. Sci., 7, 319 (1973). Stainthorpe, F.P, Wild, G.J., Application of rate of transport of mass, Chem. Eng. Sci., 22, 701 (1967). Van Dyke M., Analysis and improvement of perturbation series, Q. Jl. Mech., 27, 423 (1974).

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Van Dyke M., Fluid mechanics off real axis, Eng. Sc. Fluid Dynamics, 1, 356 (1990). Wang, C. Y., Liquid film flowing slowly down a wavy incline, Aichej,

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27(2), 207 (1981). Yih, C. S., Stability of liquid flow down an inclined plane, Phys. Fluids, 6, 321 (1963).

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