On a second parameter in the solution of the flow near a ... - ispacs

Available online at www.ispacs.com/cna Volume 2012, Year 2012 Article ID cna-00111, 13 Pages doi:10.5899/2012/cna-00111 Research Article

On a second parameter in the solution of the flow near a rotating disk by homotopy analysis method P. Donald Ariel

∗

Department of Mathematical Sciences, Trinity Western University, Langley, BC, Canada.

c P. Donald Ariel. This is an open access article distributed under the Creative ComCopyright 2012 ⃝ mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract It is shown that if the solution of the flow near a rotating disk is derived using the homotopy analysis method (HAM) with the usual single parameter to adjust the convergence, it gives rather unsatisfactory results. The results can be considerably improved by including a second parameter in the form of a scaling factor that can be suitably adjusted. Keywords : Rotating Disk; Homotopy Analysis Method; Convergence; Scaling Parameter.

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Introduction

The steady flow of a viscous, incompressible fluid caused by a rotating disk is a classic problem which is discussed in most of the well-known books on viscous fluid flows such as Schlichting [26], Pai [24], White [30] etc. It is one of those rare problems in fluid dynamics whose governing equations can be reduced to a set of ordinary differential equations. Von Karman [28] was the first to give its solution by using an approximate method involving the integral approach. Cochran [11] gave a numerical solution in which he obtained two solutions, both in the series form, one near the disk and the other far from the disk. He then matched the two solutions in an intermediate domain to find the unknowns occurring in each series. It must have been a love of labor as at the time of analysis of the problem by Cochran, the computing facilities were in a nascent state. Later as the digital computers came in vogue, the solutions were obtained by Rogers and Lance [25], and Benton [10]. Whereas Rogers and Lance employed the straightforward shooting method, marching away from the disk, Benton found that the series solution of Cochran, valid away from the disk, could indeed be extended all the way to the disk. Ackroyd [3] further expanded on the series solution of Benton and suggested that the series solution could be replaced by an initial value problem (IVP) which could be integrated without taking recourse to iterations. ∗

Corresponding author. Email address: [email protected], Tel: +1(604)5045842

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Of late there has been a renewed interest in obtaining analytical solutions of the nonlinear problems in engineering and technology. Two methods of recent vintage have drawn special attention from researchers both drawing their inspiration from the concept of homotopy. The idea is to have a real parameter p which takes the value zero for which a simple solution is assumed to exist. The parameter increases continuously to one, and the original solution successively deforms to the required solution. One of the methods is known as the homotopy analysis method (HAM). It has been developed and refined by Liao [19]. Liao and many other researchers [1, 2, 12, 13, 14, 20, 21, 22, 23, 27, 29] have found an extensive use of the HAM to find the solutions of numerous physical problems from widely varying disciplines of science and technology. The other method, known as the homotopy perturbation method (HPM) has been introduced and polished by He [15]. The HPM has also a large number of adherents [6, 9, 7, 16, 17] who have used it to obtain the solutions of several equally challenging problems. The problem of flow due to a rotating disk has been attempted using both the homotopy methods. Yang and Liao [31] gave the solution using the HAM, while Ariel [8] derived the solution using the extended version of the HPM. The solution given by Yang and Liao required as many as 45 terms to obtain a four-digit accuracy, whereas the solution derived by Ariel required only eight terms to obtain a much superior accuracy. It may be mentioned here that previously there have been attempts to obtain analytical solutions of the problem of flow due to rotating disk using the weighted residual methods. In particular a mention can be made of the extremal collocation method due to Jain [18] and the method of least-square minimization of the residual of the differential equation governing the flow by Ariel [5]. In their work Yang and Liao [31] applied the classical version of the HAM, which includes a single parameter ~ that is supposed to control the convergence of the series occurring in the solution. It is recommended by Liao [31] in his monograph on the HAM that ~-curves should be drawn and a value of ~ should be picked from those segments of the curves which are horizontal for the longest lengths. Yang and Liao [31], however, did not follow this recipe and instead chose a fixed value of ~ namely ~ = −1 and kept on increasing the value of N , the order of approximation. Eventually they got a four-digit accuracy for the solution of the normal velocity at infinity. Clearly this approach leaves something to be desired, as for the most other problems that have been solved using the HAM, ~-curves have been plotted and from those curves the proper value of ~ has been selected to get the “best” solution. In the present work we have reworked the solution given in [31]. We have drawn the ~-curves and found that even for the 40th approximation they are not flat enough to provide a satisfactory choice of ~. The value −1 chosen by Yang and Liao [31] therefore can at best be considered to be a fortuitous and not a rational choice. We surmize that the one-parameter solution as developed by Yang and Liao [31] is not fully satisfactory, and some modifications are mandatory to obtain a better solution in terms of computational efficiency and accuracy. One straightforward modification that suggests itself is to introduce a second parameter, typically a scale factor, into the solution. The idea dates back to Cochran [24] who showed that a scale factor characterizes the flow, and he then went on to calculate the same. Ackroyd [4], in general, showed that the flows caused by moving boundaries in unbounded domains have a scale factor associated by them, and it greatly helps in getting a more efficient solution by taking this fact into account. Accordingly, we have developed a two-parameter solution, the second parameter being the scale factor. We may further point out that our approach of solving the differential equations is not 2

quite the same as that of Yang and Liao [31] in whose treatment the pair of differential equations could be solved in either of the two orders. In our approach the equations have to be solved in a specific order. We have followed the same approach in [8] and came up with a highly efficient solution using the extended HPM.

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Equations of motion

We consider the steady flow of a viscous, incompressible fluid of density ρ induced by a disk rotating about the z-axis with a uniform angular velocity Ω. Working in the cylindrical coordinates (r, θ, z), if the velocity at any point is (u, v, w), then by means of the transformations √ u = rΩF (η), v = rΩG(η), w = νΩH(η), (2.1) where ν is the coefficient of kinemetic viscosity, and √ Ω z, η= ν

(2.2)

the Navier-Stokes equations governing the flow, can be reduced to the following boundary value problem (BVP): F ′′ − HF ′ − F 2 + G2 = 0, (2.3)

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G′′ − HG′ − 2F G = 0,

(2.4)

2F + H ′ = 0

(2.5)

F (0) = 0, G(0) = 1, H(0) = 0,

(2.6)

F (∞) = 0, G(∞) = 0.

(2.7)

Yang and Liao’s Solution

Yang and Liao [31], first eliminated the variable H, and further reduced the system of equations (2.3)-(2.7) to H ′′′ − HH ′′ + 12 H ′2 − 2G2 = 0, (3.8) G′′ − HG′ + H ′ G = 0,

(3.9)

H(0) = 0, H ′ (0) = 0, G(0) = 1,

(3.10)

′

H (∞) = 0, G(∞) = 0.

(3.11)

Then, in keeping with the guidelines enunciated by Liao [19], they reformulated the above system in the HAM format by rewriting it as

and

(1 − p)LH (H − H0 ) = ~p(H ′′′ − HH ′′ + 21 H ′2 − 2G2 ),

(3.12)

(1 − p)LG (G − G0 ) = ~p(G′′ − HG′ + H ′ G),

(3.13)

where LH and LG are the linear operators defined by LH =

d3 d d2 − , and L = − 1. G dη 3 dη dη 2 3

(3.14)

For the initial approximation, they chose H0 (η) = −[1 − (1 + η)e−η ], and G0 (η) = e−η .

(3.15)

H and G are now expanded in a power series of p as H(η) =

∞ ∑

Hn (η)pn , and G(η) =

n=0

∞ ∑

Gn (η)pn .

(3.16)

n=0

The final solutions for H and G are obtained by setting p = 1 in equation (3.16). In practice the series are terminated after a finite number of terms, say N , when it is presumed that the convergence of the series has taken place. The convergence is controlled by the homotopy parameter ~. The value of ~ is chosen from the segment of the so-called ~ curves where the later are sufficiently flat which is taken to be a sign of convergence. In their work Yang and Liao [31] have not produced any ~-curve for any of the important physical parameters. Rather they picked up ~ = −1 and judged the convergence of their solution by monitoring the value of H∞ as N was increased. In the absence of a ~-curve, any other value of ~ could have been chosen that ensures the convergence of the series. Below we tabulate in Table 1 the values of H∞ with N for different values of ~. It is clear than as N is increased a convergence takes place for more than one value of ~. Lest one might get an impression that there is a horizontal segment of the curve in the neighborhood of ~ = −1 for N = 40, we also present these values correct to six-digits in Table 2 at a finer spacing of ~, and it becomes evident that for N = 40 there is no horizontal segment of the curve near ~ = −1 which ensures a four digit accuracy. This fact is further corroborated in Figure 1 in which ~-curves for H∞ are plotted for different values of N . From the figure it can be seen that the oscillations do subside as N is increased, though they are not yet subdued enough to guarantee a four-digit accuracy for N = 40. On further increasing the value of N it is quite possible to get the curves which would become flat enough to give a four-digit or even higher accuracy, but that would stretch the system resources to the maximum and is not worth it, especially when there are other means of getting a sufficiently accurate solution with less of a computational effort. Table 1* Illustrating the variation of H∞ with N for different N \h -1.2 -1.1 -1.0 -0.9 -0.8 5 -0.9130 -0.9174 -0.9173 -0.9124 -0.9026 10 -0.8850 -0.8777 -0.8747 -0.8770 -0.8837 15 -0.8803 -0.8848 -0.8878 -0.8868 -0.8830 20 -0.8867 -0.8853 -0.8833 -0.8837 -0.8855 25 -0.8841 -0.8837 -0.8849 -0.8848 -0.8839 30 -0.8841 -0.8849 -0.8843 -0.8844 -0.8847 35 -0.8848 -0.8843 -0.8845 -0.8845 -0.8844 40 -0.8841 -0.8845 -0.8845 -0.8845 -0.8845

values of ~ -0.7 -0.8884 -0.8928 -0.8803 -0.8855 -0.8845 -0.8843 -0.8846 -0.8844

* There seems to be some discrepancy between the values listed above and those reported by Yang and Liao [31] for some values of N when ~ = −1. This could be owing to our using a different CAS, MAPLE rather than MATHEMATICA. We have used only the rational numbers which ensures that the final results are correct to within infinite precision.

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There is also an aesthetic reason for taking recourse to the ~-curves. Ignoring them and setting up ~ = −1 is against the very spirit of the HAM, as ~ = −1 simply leads to a regular perturbation solution, albeit based on p as a perturbation parameter. The HAM is much more powerful than that. Table 2. Illustrating the behavior of H∞ with ~ in the neighborhood of ~ = −1 for N = 40. Note that there is no consistency of the fourth digit as ~ is varied. ~ H∞ -1.1 -0.884502 -1.075 -0.884549 -1.05 -0.884554 -1.025 -0.884522 -1 -0.884475 -0.975 -0.884436 -0.95 -0.884421 -0.925 -0.884433 -0.9 -0.884463

Fig. 1. Illustrating the variation of H∞ with ~ for various values of N .

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Two Parameter Solution

In this section we present a variation of the classical HAM, as given by Yang and Liao [31], by including an additional parameter in the form of a scale factor. We show that the presence of an adjustable scale factor leaves the researcher with sufficient leeway to produce a solution that can make judicious use of the ~-curves to arrive at an optimum solution. We also digress from the principles laid down by Liao [31] in our formulation of the HAM in the sense that for the H-equation we do not conform to an operator that applies 5

only on to the H-terms in the starting solution. We follow the same idea as introduced by Ariel [8] for computing the flow near a rotating disk using the extended HPM. We formulate the HAM solution by writing equations (2.3)-(2.5) as (1 − p)[(G′′ − λ2 G) − (G′′0 − λ2 G0 )] = ~p(G′′ − HG′ − 2F G),

(4.17)

(1 − p)[(F ′′ − λ2 F + G2 ) − (F0′′ − λ2 F0 + G20 )] = ~p(F ′′ − HF ′ − F 2 + G2 ), 2F + H ′ = 0.

(4.18) (4.19)

Firstly note the presence of a second parameter λ in equations (4.17) and (4.18). This parameter will characterize the scale factor of the flow. Next, note that the presence of the G2 term in the left side of equation (4.18) requires that equation (4.17) be solved first. Further observe that especially for the flow due to a rotating disk the two components of the velocity F and G are closely intertwined and therefore it seems very reasonable not to choose another parameter to control the convergence of the two series for F and G – one expects them to have the same convergence properties. Finally it might be noticed that since F and H are related linearly (Equation (4.19)), there is no need to introduce the homotopy parameter in their relationship. We now seek the power series solution for F , G and H as under: F (η) =

∞ ∑

Fn (η)pn , G(η) =

n=0

∞ ∑

Gn (η)pn and H(η) =

n=0

∞ ∑

Hn (η)pn

(4.20)

n=0

The zeroth order system is G′′0 − λ2 G0 = 0, F0′′ − λ2 F0 + G20 = 0, and 2F0 + H0 , G0 (0) = 1,

(4.21)

G0 (∞) = 0; F0 (0) = 0, F0 (∞) = 0; H0 (0) = 0.

(4.22)

Its solution is rather trivial and is given by G0 (η) = e−λη , F0 (η) =

1 −λη 1 (e − e−2λη ), H0 (η) = − 3 (1 − e−λη )2 . 2 3λ 3λ

(4.23)

Substituting for F , G and H from equation (4.20) into equations (4.17)-(4.19) and equating various powers of p, we obtain the following system of equations: For n = 1: G′′1 − λ2 G1 = ~(G′′0 − H0 G′0 − 2F0 G0 ), (4.24) For n ≥ 2: [ G′′n

− λ Gn = 2

G′′n−1

− λ Gn−1 + ~ 2

G′′n−1

−

n−1 ∑

] (Hm G′n−m−1

+ 2Fm Gn−m−1 ) ;

(4.25)

m=0

For n = 1:

F1′′ − λ2 F1 + 2G0 G1 = ~(F0′′ − H0 F0′ − F02 + G20 ),

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(4.26)

For n ≥ 2: Fn′′

− λ Fn = 2

′′ Fn−1

− λ Fn−1 − G0 Gn − 2

m=0

[ +~

n−1 ∑

′′ Fn−1

−

n−1 ∑

′ (Hm Fn−m−1

Gm [Gn−m − (1 + ~)Gn−m−1 ] + ]

+ Fm Fn−m−1 ) ;

(4.27)

m=0

For n ≥ 1:

2Fn + Hn′ = 0;

(4.28)

Gn (0) = 0, Gn (∞) = 0, Fn (0) = 0, Fn (∞) = 0, Hn (0) = 0.

(4.29)

with the boundary conditions

Since the structure of equations (4.24)-(4.27) is identical, including the boundary conditions, the same procedure can be applied to obtain their solutions for successive values of n, and the HAM solution can be completed by setting p = 1 in equation (4.20). Again in practice, the series in equation (4.20) must be terminated at some finite value of n. Now we have two parameters ~ and λ to adjust the convergence of the series occurring in the solution. Two approaches are possible for choosing the values of these parameters. Either, we can construct the ~−λ surfaces for the critical physical parameters, or we can construct both, the traditional ~-curves, and the λ-curves, keeping the value of the other parameter fixed. In Figures 2 through 4, the ~ − λ-surfaces for the parameters F ′ (0), G′ (0) and H(∞) are plotted. For “ideal” values of ~ and λ, one must look for plateaus on the ~ − λ surfaces. They are indeed there on each of the three surfaces. However, to pinpoint those values it is a better idea to look into the two-way tables for each parameter represented by these surfaces. In Tables 3 through 5 we give the fragments of the tables for the three parameters. It can be seen that the most stationary values of the parameters F ′ (0), G′ (0) and H(∞) are realized for approximately λ = 0.7 and ~ = −0.55.

Fig. 2. The ~ − λ surface representing F ′ (0) for N = 16.

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Fig. 3. The ~ − λ surface representing G′ (0) for N = 16.

Fig. 4. The ~ − λ surface representing H(∞) for N = 16.

Table 3. Illustrating the dependence of F ′ (0) on ~ and λ for N = 16 ~\λ 0.65 0.7 0.75 0.8 0.85 -0.7 0.510448 0.510233 0.510235 0.510208 0.510339 -0.65 0.510249 0.510233 0.510233 0.510237 0.510257 -0.6 0.510233 0.510233 0.510232 0.510245 0.510171 -0.55 0.510233 0.510233 0.510233 0.510228 0.510181 -0.5 0.510233 0.510232 0.510233 0.510221 0.510274 -0.45 0.510232 0.510232 0.510231 0.510240 0.510320 -0.4 0.510230 0.510229 0.510229 0.510247 0.510211

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Table 4. ~\λ -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 -0.4

Illustrating the dependence of G′ (0) on ~ and 0.65 0.7 0.75 0.8 -0.615738 -0.615922 -0.615921 -0.615949 -0.615908 -0.615922 -0.615921 -0.615932 -0.615921 -0.615922 -0.615923 -0.615908 -0.615922 -0.615922 -0.615922 -0.615917 -0.615923 -0.615923 -0.615922 -0.615937 -0.615925 -0.615925 -0.615926 -0.615929 -0.615933 -0.615935 -0.615936 -0.615910

λ for N = 16 0.85 -0.615843 -0.615831 -0.615934 -0.616005 -0.615949 -0.615828 -0.615838

Table 5. ~\λ -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 -0.4

Illustrating 0.65 -0.889373 -0.884839 -0.884491 -0.884474 -0.884474 -0.884475 -0.884476

the dependence of H(∞) on ~ and 0.7 0.75 0.8 -0.884474 -0.884527 -0.884122 -0.884474 -0.884474 -0.884674 -0.884474 -0.884459 -0.884684 -0.884474 -0.884485 -0.884324 -0.884474 -0.884473 -0.884270 -0.884475 -0.884460 -0.884712 -0.884477 -0.884521 -0.884912

λ for N = 16 0.85 -0.886148 -0.884341 -0.883187 -0.883849 -0.885494 -0.886050 -0.884112

On the other hand if the second approach is followed then we need the ~-curves and the λ-curves for suitable values of the other adjustable parameter. In Figure 5, the ~-curves are shown for λ = 0.7 when N = 16, and in Figure 6, the λ-curves are shown for ~ = −0.55 again when N = 16. From both the set of curves it is easy to conclude that one can choose ~ = −0.55 and λ = 0.7 for nearly optimum results. Ackroyd [3], using the infinite series of exponentially decaying functions, has probably given the most accurate solution for the steady flow of a viscous, incompressible fluid near a rotating disk in the literature so far. His results are F ′ (0) = 0.5102326189, G′ (0) = −0.6159220144, and H(∞) = −0.8844741102 correct to ten decimal places. For the indicated values of ~ and λ, our values are in complete agreement with those given by Ackroyd up to six decimal places.

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Fig. 5. Illustrating the dependence of F ′ (0), G′ (0) and H(∞) on ~ for λ = 0.7 when N = 16.

Fig. 6. Illustrating the dependence of F ′ (0), G′ (0) and H(∞) on λ for ~ = −0.55 when N = 16.

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