On-line version ISSN 1807-0302
Comput. Appl. Math. vol.31 no.1 São Carlos 2012
Zodwa G. Makukula; Precious Sibanda*; Sandile S. Motsa
School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa E-mails: firstname.lastname@example.org / email@example.com / firstname.lastname@example.org
The steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipationis studied. We present a novel technique for accelerating the convergence of the spectral-homotopy analysis method. Solutions of the nonlinear momentum and energy equations are obtained using the improved spectral homotopy analysis method. Solutions were also generated using the spectral-homotopy analysis method and benchmarked against results in the literature.
Mathematical subject classification: Primary: 76A05, 76N05; Secondary: 76M25.
Key words: Reiner-Rivlin fluid, Chebyshev spectral method, spectral-homotopy analysis method, improved spectral-homotopy analysis method.
The boundary layer induced by a rotating disk arises in many engineering applications, for example, in computer storage devices, viscometry, turbo-machinery and in crystal growth processes (Attia ). Since the pioneering study by von Kármán , research on swirling flows has been carried out by, among others, Cochran  who proposed an improved solution to the von Kármán formulation based on a mixture of analytical and numerical techniques. Benton  studied the impulsive rotation from rest of a disk in an infinite viscous fluid. He improved Cochran's solutions by first expanding the variables in a power series and solving for the first two orders analytically, and then numerically computing the next two orders.
The shooting method was used to solve the von Kármán equations and to investigate heat transfer in porous medium in [19, 29, 30, 33, 35]. Numericalschemes involving Runge-Kutta methods, finite element and finite difference approximations were used in [14, 18, 24] to study the effects of a rough disk surface on the flow. The Crank-Nicholson implicit scheme was used in studies involving non-Newtonian characteristic of the fluid [4, 5, 6, 8, 10, 33] and in Newtonian fluids [7, 9, 11]. Perturbation techniques including the differential transform method and Padé approximations (DTM-Padé), the variational iteration method (VIM) and the homotopy perturbation method (HPM) were used in [1, 2, 31, 32, 34]. Analytical methods such as the DTM and the homotopy analysis method (HAM) were applied in, among other studies, [40, 3, 17, 37, 38]. These methods may result in secular terms in the solutions, and may converge very slowly or may even fail to converge for problems with strong non-linearity and/or with very large parameter values. Some approaches may not be applicable at all for certain problems, for example, the DTM for unbounded domainproblems. Motsa et al. [22, 23, 25, 26, 27] proposed and applied a modification of the homotopy analysis method that improves the performance of this method and removes some restrictions associated with it. In general, the convergence of many numerical methods depends on how good the initial approximation is to the true solution. In this paper we present an algorithm that first seeks to improve the initial "guess" and then uses the spectral-homotopy analysis method to find solutions to systems of nonlinear equations that governthe Reiner-Rivlin swirling flow. This procedure considerably accelerates the convergence rate of the spectral homotopy analysis method. Here we apply the improved spectral-homotopy analysis method (ISHAM) to solve the nonlinear equations that govern the flow of an electrically conducting Reiner-Rivlin fluid in the presence of Joule heating and viscous dissipation. The governing equations were solved earlier by Sahoo  using a numerical scheme that blends the finite difference scheme and the shooting method. Solutions obtained are compared with those of Sahoo  and against the 'standard' spectral homotopyanalysis method.
We consider an infinite rotating disk coinciding with the plane z = 0 with the space z > 0 occupied by a viscous, incompressible Reiner-Rivlin fluid. Thefluid motion and heat transfer are governed by the equations (see [4, 5, 33]);
with the following no-slip boundary conditions
where the disk is rotating with a constant angular velocity Ω about the line r = 0 and an external uniform magnetic field is applied perpendicular to theplane of the disk with a constant magnetic flux density B0. The velocity components in the directions of increasing r, ϕ, z are u, v, w respectively. ρ is the density of the fluid, σ is the electrical conductivity of the fluid, µ is the coefficient of viscosity, κ is the thermal conductivity, cp is the specific heat at constant pressure of the fluid. The temperature of the fluid T, equals Tw at the surface of the disk. At large distances from the disk, T tends to T∞ where T∞ is the temperature of the ambient fluid. The second term on the right hand side of equation (5) represents the viscous dissipation while the last term represents the Joule heating. The constitutive equation for the Reiner-Rivlin fluidis given by
where p represents the pressure, is the stress tensor, is the rate of straintensor and µc is the coefficient of cross viscosity. The Reiner-Rivlin model is a simple model which can provide some insight into predicting the flow characteristics and heat transfer performance for viscoelastic fluid above a rotating disk . The first term on the right hand side of (8) represents the viscous property of the fluid and the third term, the elastic property of the fluid. We introduce the non-dimensional distance η = z measured along the axis of rotation and the von Kármán transformations ;
where F, G, H, P and Θ are non-dimensional functions of η, ν = µ/ρ is the kinematic viscosity. With these transformations equations (1)-(5) take the form
where K = µc Ω/µ is the parameter that describes the non-Newtonian characteristic of the fluid, M = σ/ρΩ is the magnetic interaction number, Pr is the Prandtl number and Ec is the Eckert number. The system (10)-(12) with the prescribed boundary conditions (15) are sufficient to solve for the three velocity components. Equation (13) can be used to find the pressure distribution at any point if required. Simplifying the equation system by substituting equation (10) into (11), (12) and (14) yields
subject to the boundary conditions
In the following section we solve the nonlinear coupled system (17)-(19) with boundary conditions (20) by the ISHAM.
3 Method of solution
The main thrust of the method of solution [21, 28], is the improvement of the initial approximation used in the higher order deformation equations of the spectral homotopy analysis method. A systematic approach is used to find optimal initial "guesses" which are then used in the SHAM algorithm to accelerate convergence. In the first instance we assume that solutions for H(η), G(η) and Θ(η) in equations (17)-(19) can be found in the form
where hi, gi and θi are unknown functions whose solutions are obtained usingthe SHAM approach at the ith iteration and hm, gm and θm (m > 1) are known from previous iterations. For m = 0, suitable initial guesses satisfying the boundary conditions (20) are
The initial guesses (22) are improved upon as follows. Substituting (21) intothe governing equations (17)-(19) gives
subject to the boundary conditions
The coefficient parameters ak, i - 1, bk, i - 1, ck, i - 1 (k = 0, ..., 6), r1, i - 1, r2,i - 1 and r3,i - 1 are defined as
Starting from the initial guesses (22), the subsequent solutions hi, gi and θi (i > 1) are obtained by recursively solving equations (23)-(25). To solve equations (23)-(25), we start by defining the following linear operators
where q ∈ [0, 1] is the embedding parameter, and
are unknown functions. The zeroth order deformation equations are given by
where ħ is the non-zero convergence controlling auxiliary parameter and h, g and θ are nonlinear operators given by
Differentiating (43)-(45) m times with respect to q and then setting q = 0 and finally dividing the resulting equations by m! yields the mth order deformation equations
subject to the boundary conditions
The initial approximations hi,0, gi,0 and θi,0 that are used in the higher order equations (46)-(48) are obtained by solving the linear part of equations (23)-(25) given by
with the boundary conditions
It is worthwhile to note at this stage that the initial approximate solutions are no longer just h0, g0 and θ0 but hi,0, gi,0 and θi,0 at the ith iteration. Essentially, this procedure allows for the improvement of the initial guesses at each iteration. Since the right hand side of equations (54)-(56) for i = 1, 2, 3, ..., areknown from previous iterations, the equations may be solved using any numerical method. In this work, we apply the Chebyshev spectral collocation method to integrate equations (54)-(56). The method is based on the Chebyshev polynomials defined on the interval [-1, 1] by
We first transform the physical region [0, ∞) into the region [-1, 1] using the domain truncation technique. The problem is solved in the interval [0, L] instead of [0, ∞). This leads to the following algebraic mapping
where L is the scaling parameter used to invoke the boundary condition at infinity. The Chebyshev nodes in [-1, 1] are defined by the Gauss-Lobatto collocation points [13, 36] given by
where N is the number of collocation points. The variables hi,0(η), gi,0(η) and θi,0(η) are approximated as truncated series of Chebyshev polynomials of the form
where T1,k, T2,k and T3,k are the kth Chebyshev polynomials. Derivatives of the variables at the collocation points are represented as
where r is the order of differentiation, D = D and D is the Chebyshev spectral differentiation matrix. Substituting equations (61)-(64) in (53)-(56) yields
subject to the boundary conditions
In the above definitions T stands for transpose, I is an (N + 1) × (N + 1) identity matrix and ak,i - 1, bk,i - 1 and cs,i - 1 (k = 0,...,5, s = 0,...,6) are diagonal matrices of size (N + 1) × (N + 1). After modifying the matrix system (65) to incorporate the boundary conditions, the solution is obtained as
Similarly, applying the Chebyshev spectral transformation on the higher order deformation equations (46)-(48) gives
subject to the boundary conditions
where Bi - 1 and Qi - 1, are as defined in (69) and
The boundary conditions (72)-(74) are implemented in matrix Bi - 1 on the left hand side of equation (71) in rows 1, N, N + 1, N + 2, 2(N + 1) 2N + 3 and 3(N + 1) respectively as before with the initial solution above. The corresponding rows, all columns, of Bi - 1 on the right hand side of (71), Qi - 1and Pm - 1 are all set to be zero. This results in the following recursive formula for m > 1.
where is the modified matrix Bi - 1 on the right hand side of (71) afterincorporating the boundary conditions (72)-(74). Thus starting from the initial approximation, which is obtained from (70), higher order approximations Xi,m(ξ) for m > 1, can be obtained through the recursive formula (77). The solutions for hi, gi and θi are then generated using the solutions for hi,m, gi,m and θi,m as follows
The [i, m] approximate solutions for h(η), g(η) and θ(η) are then obtained by substituting hi, gi and θi which are obtained from (78), (79) and (80) into equation (21), where i represents the ith iteration of the initial approximation and m represents the mth iteration of the spectral homotopy analysis method.
4 Convergence theorem
The approximate solutions of the nonlinear equations are generated using the higher order deformation equations (46)-(48). The right hand sides of theseequations are governed by the unknown functions i(η; q), i(η; q) and Θ(η; q). As the embedding parameter q gradually increases from 0 to 1, the solutions vary from the initial approximations to the exact solutions, i.e.
Expanding i(η; q), i(η; q) and Θi(η; q) using the Taylor series expansion about q yields
We note that at q = 1 the series becomes the exact solutions
For validity of the solutions generated by these equations, it is important to show that these series converge at q = 1. As stated earlier, the SHAM is a hybrid method founded on the HAM. We kindly refer readers to Liao's proof [20, ch. 3] since the higher order deformation equations are similar in the two methods.
5 Results and discussion
In this section we present and discuss results computed using the improvedspectral homotopy analysis method, the original spectral homotopy analysis method and the numerical bvp4c routine which is based on Runge-Kuttaschemes. Comparison is also made between the current results and those in the literature. For our simulations we used ħ = -1, L = 30 and N = 150. The CPU run times (RT) in seconds are shown for the ISHAM and SHAM for comparison of computational efficiency.
Tables 1 and 2 present approximate solutions of the shear stresses in the radial F'(0) and tangential -G'(0) directions respectively. The results are computed for a Newtonian fluid (K = 0) and for different values of the magnetic parameter M. We note that the ISHAM approximate solutions for both F'(0) and -G'(0) converge to the numerical solutions at 2nd order approximations for up to 8 decimal places. Comparison with Sahoo  shows a good agreement. The effect of the magnetic parameter on the Newtonian fluid shows that F'(0) decreases while -G'(0) increases as M is increased.
The radial shear stress when M = 2 and for different values of K is presented in Table 3. For validation of the current method, and to determine the effect of improving the initial guesses, we compare the results against the numerical solution. For convergence of the method the results are compared with the 'standard' spectral homotopy analysis method for the same values of N, L and ħ. For 0 < K < 2 the ISHAM converges at 2nd order while the SHAM would not have converged even at the 8th order for some values of K. Comparatively therefore the ISHAM converges much faster than the SHAM. This is clearly seen in Table 4 where the absolute errors in the solution are given.
In Table 5, the tangential stress results obtained using the ISHAM and the SHAM are compared with the numerical results when M = 2 and for different values of K. Convergence of the ISHAM to the numerical solutions was achieved at the 2nd order of approximation. When using the SHAM convergence to the numerical solution was achieved at the 8th order for values of K up to 2. It is also clear that the ISHAM is computationally much more efficient compared to the SHAM. A comparison of the absolute errors is made in Table 6 where the fast convergence of the ISHAM when compared with the SHAM is confirmed. It is worth noting that the SHAM converges faster for -G'(0) compared to F'(0). This is due to the differences in the level of nonlinearity of the equations of F(η) and G(η). For the ISHAM however, convergence has not been affected by this difference in the nonlinearity of functions. This there appears to be an added advantage of the ISHAM over the SHAM.
Table 7 gives a comparison of the convergence rate of the ISHAM and the SHAM versus the numerical solutions for F'(0) for different values M when K = 1. Convergence to the numerical solution is achieved at 2nd order of the ISHAM and at the 8th order of the SHAM. The absolute errors are shown in Table 8. In Table 9 the ISHAM and the SHAM solutions are compared with the numerical solutions for different values M when K = 1. An increase in the tangential shear stress is observed with an increase in M. The absolute errors are shown in Table 10. The nonlinearity of the equations has no effect on the convergence of the ISHAM.
Table 11 shows the heat transfer coefficient -Θ'(0) at different orders of the ISHAM compared against numerical results for different values of M, Pr, and Ec when K is fixed. For all cases convergence of the ISHAM approximate solutions to the numerical solution is achieved at 2nd orders of approximations. For a fixed value of K, increasing M, Pr, and Ec decreases -Θ'(0). For all results with CPU run times, the ISHAM has shown great computer efficiency as well. In all simulations the run times are significantly lower with the ISHAM than with the SHAM.
Figures 1-2 show the effect of M and K on the radial and axial velocity profiles respectively. Increasing M reduces the radial component of the velocity while increasing K enhances F. The axial velocity H(η) increases with the magnetic parameter but decreases when K is increased. There is excellent agreement between the second order ISHAM solutions for F(η) and H(η) and thenumerical result.
The tangential velocity component and the temperature profiles are show in Figures 3-4 for different values of M and K respectively. An increase in M reduces G(η) while enhancing the Θ(η). When K values are increased both G(η) and Θ(η) decrease.
In Figure 5 temperature profiles are presented for varying values of Pr and Ec. The temperature decreases with increasing Prandtl numbers while an increase in the Eckert number enhances the temperature.
A novel approach for accelerating the convergence of the spectral homotopyanalysis method that is used to solve nonlinear equations in science and engineering has been proposed and applied successfully to the nonlinear system of equations governing the Reiner-Rivlin fluid in with Joule heating and viscousdissipation. The primary objective of the algorithm is to improve the initial approximate solution. The improved approximations are then used in the algorithm of the spectral-homotopy analysis method to reduce the number of iterations required to achieve convergence and better accuracy. The shear stresses in the radial and azimuthal directions were computed and the corresponding absolute errors determined. Convergence to the numerical solutions of the ISHAM approximate solutions was achieved at the 2nd orders for all flow parameters while the SHAM converged at the 8th order for some of the flow parameters.
The effects of flow parameters was also investigated for the radial and tangential shear stresses for both the Newtonian (K = 0) and non-Newtonian (K ≠ 0) cases. For the Newtonian case, increasing M reduces F'(0) and enhances -G'(0) while in the non-Newtonian case increasing M enhances both F'(0) and -G'(0).
The effect of K and M was determined and it was observed that an increase in K results in an increase in F(η), and a decrease in H(η), G(η) and Θ(η) while increasing M increases H(η) and Θ(η) while both F(η) and G(η) decreases. Θ(η) decreased with an increase in the Ec and decreased with an increase in Pr. The success of the ISHAM in solving the non-linear equations governing the von Kármán flow of an electrically conducting non-Newtonian Reiner-Rivlin fluid in the presence of viscous dissipation, Joule heating and heat transfer proves that the ISHAM fits as a newly improved method of solution that can be used to solve non-linear problems arising in science and engineering.
Acknowledgements. The authors wish to acknowledge financial support from the University of KwaZulu-Natal and the National Research Foundation (NRF).
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