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On the New Bivariate Local Linearisation Method for Solving Coupled Partial Differential Equations in Some Applications of Unsteady Fluid Flows with Heat and Mass Transfer

By Sandile Motsa

Submitted: October 28th 2014Reviewed: February 23rd 2015Published: October 22nd 2015

DOI: 10.5772/60444

Downloaded: 1051

Abstract

This work presents a new numerical approach for solving unsteady two-dimensional boundary layer flow with heat and mass transfer. The flow model is described in terms of a highly coupled and nonlinear system of partial differential equations that models the problem of unsteady mixed convection flow over a vertical cone due to impulsive motion. The proposed method of solution uses a local linearisation approach to decouple the original system of PDEs to form a sequence of equations that can be solved in a computationally efficient manner. Approximate functions defined in terms of bivariate Lagrange interpolation polynomials are used with spectral collocation to approximate the solutions of the decoupled linearised equations. To test the accuracy and to validate the results of the proposed method, numerical error analysis and convergence tests are conducted. The present results are also compared with results from published literature for some special cases. The proposed algorithm is shown to be very accurate, convergent and very effective in generating numerical results in a computationally efficient manner.

Keywords

  • Unsteady boundary layer flow
  • heat and mass transfer
  • bivariate spectral collocation
  • impulsive flow

1. Introduction

In this investigation we revisit the problem of unsteady mixed convection flow over a vertical cone due to impulsive motion that was previously discussed in Singh and Roy [7]. In the flow model, the external stream is impulsively set into motion and the surface temperature is suddenly changed from its ambient temperature. The resulting constitutive equations describing the flow properties are a system of highly coupled non-linear partial differential equations (PDEs) which cannot be solved exactly. In [7], the solution of the PDEs system was approximated numerically using implicit finite differences after linearising using the quasi-linearisation (QLM) technique of Bellman and Kalaba [1]. The QLM simplifies the solution process by simply reducing the original PDEs to a linearised form which, nevertheless, is coupled. In view of the coupled nature of the QLM iteration scheme, matrices of very large dimensions are obtained when using any numerical method is used for discretization. For systems of coupled equations, the method proposed by [3] was used in [7] to decouple the governing equations. There is generally a large overhead in computing resources when performing computational tasks on large matrix systems. To avoid dealing with large matrix systems, relaxation methods can be used. However, the disadvantage of relaxation methods is that they convergence much slower than quasi-linearisation schemes which are known to converge quadratically.

A linearisation method, termed the local linearisation method (LLM), was recently introduced in [4] as an efficient method for solving coupled systems of non-linear ordinary differential equations that model boundary layer equations. This method was extended to a PDE system in [5] where the Chebyshev spectral collocation method [2, 8] was used for discretization in one independent variable and finite differences was used for discretization in the second independent variable of the PDE. In this study, we present a new approach that seeks to improve on the original LLM approach of [5] by eliminating the need for using finite differences for discretizing in one of the independent variables and, instead, apply spectral collocation independently in all independent variables of the PDE system. Finite differences require very fine grids (with a large number of grid points) to give more accurate solutions. In contrast, spectral methods are computationally less expensive, converge faster and are more accurate than finite difference methods particularly for problems with smooth solutions. The collocation method applied in this work uses bivariate Lagrange interpolation polynomials as basis functions. The proposed method converges fast and gives very accurate results which are obtained in a computationally efficient manner. The accuracy is established through residual and solution error analysis. Further validation of present results in established by comparing with existing results from literature.

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2. Governing equations

The model under consideration is that of an unsteady mixed convection flow over a vertical cone that is impulsively set into motion to cause unsteadiness in the flow (see [6, 7]). It is assumed that buoyancy forces are present due to temperature and concentration variation of the fluid flow. The governing boundary layer momentum, energy and concentration equations were reduced to dimensionless form in [7] using transformations that were initially proposed in Williams and Rhyne [9] to give,

3fη3+η2(1ξ)2fη2+f2fη2[ξ(m+36)(1ξ)(m36)log(1ξ)]+m3ξ[1(fη)2]λξ(θ+Nϕ)=ξ(1ξ)2fηξ+ξ3(m3)(1ξ)log(1ξ)[2fη2fξfη2fηξ],E1
2θη2+η2Pr(1ξ)θη+Prfθη[ξ(m+36)(1ξ)(m36)log(1ξ)]Pr(2m33)ξθfη=ξ(1ξ)Prθξ+Pr(1ξ)log(1ξ)ξ(m33)[θηfξfηθξ],E2
2ϕη2+η2Sc(1ξ)ϕη+Scfϕη[ξ(m+36)(1ξ)(m36)log(1ξ)]Sc(2m33)ξϕfη=ξ(1ξ)Scϕξ+Sc(1ξ)log(1ξ)ξ(m33)[ϕηfξfηϕξ],E3

where Pris the Prandtl number, Scis the Schmidt number, fis the dimensionless stream function, θand ϕare the dimensionless temperature and concentration, N=λ/λ*is the ratio of the thermal (λ)and concentration λ*buoyancy parameter.

The boundary conditions are

f(0,ξ)=0,f(0,ξ)η=0,θ(0,ξ)=1,ϕ(0,ξ)=1,f(,ξ)η=1,θ(,ξ)=0,ϕ(,ξ)=0.E4

Other quantities of physical interest include the local skin friction coefficient, Nusselt number and Sherwood numbers which are given, in dimensionless forms as,

ReL1/2Cf=2ξ1/2f(0,ξ),E5
ReL1/2Nu=ξ1/2θ(0,ξ),E6
ReL1/2Sh=ξ1/2ϕ(0,ξ).E7

3. Derivation of solution method

The derivation of the solution method is described for a general system of non-linear PDEs in this section. Consider a system of 3coupled PDEs in f(η,ξ), θ(η,ξ)and ϕ(η,ξ)

Ωk[F,T,H]=0,fork=1,2,3,E8

where Ω1, Ω2and Ω3are non-linear operators that represent the non-linear PDEs (1), (2) and (3), respectively and F,T,Hare defined as

F={f,fη,2fη2,3fη3,2fξη},T={θ,θη,2θη2,θξ},H={ϕ,ϕη,2ϕη2,ϕξ}.

Equation (8) can be simplified and decoupled by using the following algorithm called local linearisation method (LLM):

LLM Algorithm

  1. Solve for Fin 1st equation assuming that Tand Hare known from previous iteration. Fr+1.

  2. With Fr+1now known, solve for Tin 2nd equation, assuming that His known from previous iteration. Tr+1.

  3. Solve for Hin 3rd equation Hr+1.

The LLM algorithm was recently reported in [4] as an efficient method for solving coupled system of non-linear ordinary differential equations that model boundary layer equations. The method was later extended to partial differential equations in [5]. Applying the algorithm in equations (8) gives

FfΩ1[Fr,Tr,Hr]=FrfΩ1[Fr,Tr,Hr]Ω1[Fr,Tr,Hr],E9
TθΩ2[Fr+1,Tr,Hr]=TrθΩ2[Fr+1,Tr,Hr]Ω2[Fr+1,Tr,Hr],E10
HϕΩ3[Fr+1,Tr+1,Hr]=HrϕΩ3[Fr+1,Tr+1,Hr]Ω3[Fr+1,Tr+1,Hr].E11

The above equation forms a system of three decoupled linear PDEs that are to be solved iteratively for f(η,ξ), θ(η,ξ)and ϕ(η,ξ), where is a vector of partial derivatives defined as

={f,θ,ϕ},f={f,f,f,f,fξ},θ={θ,θ,θ,θξ},ϕ={ϕ,ϕ,ϕ,ϕξ}.

Applying the LLM iteration scheme (9) - (11) on the governing non-linear PDE system (1) - (3) gives

fr+1'+a1,r(η,ξ)fr+1'+a2,r(η,ξ)fr+1'+a3,r(η,ξ)fr+1+a4,r(η,ξ)fr+1'ξ=R1,r(η,ξ),E12
θr+1'+b1,r(η,ξ)θr+1'+b2,r(η,ξ)θr+1+b3,r(η,ξ)θr+1ξ=R2,r(η,ξ),E13
ϕr+1'+c1,r(η,ξ)ϕr+1'+c2,r(η,ξ)ϕr+1+c3,r(η,ξ)ϕr+1ξ=R3,r(η,ξ),E14

where the primes denote partial differentiation with respect to ηand the coefficients ai,r,bi,rand ci,r(i=1,2,3,4) are known coefficients whose quantities are known from the previous iteration level r. The definition of the coefficients is given in the Appendix.

The boundary conditions are given by

fr+1(0,ξ)=0,fr+1(0,ξ)=0,θr+1(0,ξ)=1,ϕr+1(0,ξ)=1,E15
fr+1(,ξ)=1,θr+1(,ξ)=0,ϕr+1(,ξ)=0.E16

Starting from given initial approximations, denoted by f0(η,ξ),θ0(η,ξ)and ϕ0(η,ξ), equations (12) – (14) are solved iteratively for r=1,2,,until approximate solutions that are consistent to within a certain tolerance level are obtained.

To solve the linearised equations (12) – (16), an approximate solution defined in terms of bivariate Lagrange interpolation polynomials in sought. For example, the approximate solution for f(η,ξ)takes the form

f(η,ξ)m=0Nxj=0Ntf(τm,ζj)Lm(τ)Lj(ζ),E17

where the functions Lm(τ)and Lj(ζ)are the well-known characteristic Lagrange cardinal polynomials. The function (17) interpolates f(η,ξ)at the collocation points (known as Chebyshev-Gauss-Lobatto) defined by

τi=cos(πiNx),ζj=cos(πjNt),i=0,1,,Nx;j=0,1,,Nt.E18

The choice of collocation points (18) makes it possible for the Chebyshev spectral collocation method to be used as a solution procedure. The Chebyshev collocation method requires that the domain of the problem be transformed to [1,1]×[1,1]. Accordingly, linear transformations have been used to transform η[0,η]and ξ[0,Lt]to τ[1,1]and ζ[1,1], respectively. Here ηis a finite value that is introduced to facilitate the application of the numerical method at infinity and Ltis the largest value of ξ.

Substituting equation (17) in equation (12)-(14) and making use of the derivatives formulas for Lagrange functions at Gauss-Lobatto points given in [2, 8], results in

AiFr+1,i+a4,ij=0Ntdi,jDFr+1,j+a5,ij=0Ntdi,jFr+1,j=R1,i,E19
BiQr+1,i+b3,ij=0Ntdi,jQr+1,j=R2,i,E20
CiFr+1,i+c3,ij=0Ntdi,jFr+1,j=R3,i,E21

where di,j(i,j=0,1,,Nt)are entries of the standard Chebyshev differentiation matrix d=[di,j]of size (Nt+1)×(Nt+1)(see, for example [2, 8]), D=(2/ηe)[Dr,s](r,s=0,1,2,,Nx) with [Dr,s]being an (Nx+1)×(Nx+1)Chebyshev derivative matrix. The vectors and matrices are defined as

Fr+1,i=[fr+1,i(τ0),fr+1,i(τ1),,fr+1,i(τNx)]T,Qr+1,i=[θr+1,i(τ0),θr+1,i(τ1),,θr+1,i(τNx)]T,Fr+1,i=[ϕr+1,i(τ0),ϕr+1,i(τ1),,ϕr+1,i(τNx)]T,Ai=D3+a1,iD2+a2,iD+a3,i,Bi=D2+b1,iD+b2,i,Ci=D2+c1,iD+c2,i,n2

am,r(ξi)(m=1,2,3,4,5)is the diagonal matrix of the vector [am,r(τ0),am,r(τ1),,am,r(τNx)]T. The matrices b1,i,b2,i,c1,i,c2,iare also diagonal matrices corresponding to b1,r,b2,r,c1,r,c2,revaluated at collocation points. Additionally, R1,i, R2,iand R3,iare (Nx+1)×1vectors corresponding to R1,r, R2,rand R3,r, respectively, evaluated at the collocation points.

After imposing the boundary conditions for i=0,1,,Nt, equation (20) can be written as the following matrix equation:

[A0,0A0,1A0,NtA1,0A1,1A1,NtANt,0ANt,1ANt,Nt][Fr+1,0Fr+1,1Fr+1,Nt]=[R1,0R1,1R1,Nt],E22

where

Ai,j=Ai+a4,idi,iD+a5,idi,iI,Ai,j=a4,idi,iD+a5,idi,iI,whenij,n1

where Iis an (Nx+1)×(Nx+1)identity matrix.

Applying the same approach to (20) and (21) gives

[B0,0B0,1B0,NtB1,0B1,1B1,NtBNt,0BNt,1BNt,Nt][Qr+1,0Qr+1,1Qr+1,Nt]=[R2,0R2,1R2,Nt],E23
[C0,0C0,1C0,NtC1,0C1,1C1,NtCNt,0CNt,1CNt,Nt][Fr+1,0Fr+1,1Fr+1,Nt1]=[R3,0R3,1R3,Nt],E24

where

Bi,j=Bi+b3,idi,iI,Bi,j=b3,idi,jI,whenij,Ci,j=Ci+c3,idi,iI,Ci,j=c3,idi,jI,whenij,

Starting from a given initial guess, the approximate solutions for f(η,ξ), θ(η,ξ)and ϕ(η,ξ)are obtained by iteratively solving the matrix equations (22), (23) and (24), in turn, for r=0,1,2,.Convergence to the expected solution can be affirmed by considering the norm of the difference between successive iterations. If this quantity is less than a given tolerance level, the algorithm is assumed to have converged. The following error norms are defined for the difference between values of successive iterations,

Ef=max0iNtFr+1,iFr,i,Eθ=max0iNtQr+1,iQr,i,Eϕ=max0iNtFr+1,iFr,i.E25

4. Results and discussion

The local linearisation method (LLM) described by equations (12) - (14 was used to generate approximate numerical solutions for the governing systems of equations (1) - (3). The linearised equations were subsequently solved using the bivariate spectral collocation method as described in the previous section. The whole solution process is termed bivariate spectral local linearisation method (BSLLM). In this section, we present the results of the numerical computations for the various flow profiles. The BSLLM results are compared with published results from literature. In computing the numerical results presented in this paper, Nx=60and Nt=15collocation points in the ηand ξdomain, respectively, were found to be sufficient to give accurate and consistent results. The minimum number of iterations required to give consistent results to within a tolerance level of 107were determined using equation (25).

Figure 1.

Effect ofλandPron the velocity profilef(η,ξ)

Figure 2.

Effect ofλandPron the temperature profileθ(η,ξ)

The effects of buoyancy parameter (λ)and Prandtl number (Pr)on velocity and temperature profiles are shown in Figs. 1 and 2, respectively for when N=0.5, Sc=0.94at ξ=0.5. It can be seen from the figures that the results from the present work match those reported in [7] exactly. We remark that in [7], the quasi-linearisation approach was used with implicit finite differences as a solution method. Details of the analysis of the effects of various governing parameters on the governing equations (1) – (3) can be found in [7] and have not been repeated here.

Figure 3.

Effect ofNvelocity profilef(η,ξ)

Figure 4.

Effect ofNon the temperature profileθ(η,ξ)

Figure 5.

Effect ofNconcentration profileϕ(η,ξ)

Figs. 3, 4 and 5 show the effect of buoyancy ratio Non the velocity, temperature and concentration profiles, respectively, when Pr=0.7, Sc=0.94and λ=5at ξ=0.5. The graphs are qualitatively similar to those reported in [7]. These figures are a qualitative validation of the numerical results generated using the proposed BSLLM.

Figure 6.

Effect ofλon the skin friction coefficientCfReL1/2

The effect of λon the skin friction coefficient is shown in Fig. 6 for Pr=0.7, Sc=0.22and N=0.5. Again, it is observed that the results match those of [7]. In particular, the oscillating trend in the skin friction coefficient for near the stagnation region can be observed.

5. Conclusion

The purpose of the current study was to develop a bivariate Lagrange polynomial based spectral collocation method for solving system of coupled non-linear partial differential equations. The applicability of the proposed method, termed bivariate spectral local linearisation method (BSLLM) was tested on the well-known problem of unsteady mixed convection flow over a vertical cone due to impulsive motion. The validity of the approximate numerical results was confirmed against known results from literature. The results of this study were found to be qualitatively congruent with those from published literature. The study revealed that the proposed method can be used as a viable approach for solving coupled partial differential equations with fluid mechanics applications.

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Appendix

Definition of coefficients

a1,r=η2(1ξ)+β1(ξ)frβ2(ξ)frξ,E26
a2,r=2m3ξfr+β2(ξ)frξ,E27
a3,r=β1(ξ)fr,E28
a4,r=ξ(1ξ)+β2(ξ)fr,E29
a5,r=β2(ξ)fr,E30
R1,r=β1(ξ)frfrm3ξ[1+(fr)2]λξ(θr+Nϕr)β2(ξ)[frfrξfrfrξ],E31
β1(ξ)=ξ(m+36)(1ξ)(m36)log(1ξ),E32
β2(ξ)=ξ(1ξ)log(1ξ)(m3)3,E33
b1,r=η2Pr(1ξ)+Prβ1(ξ)frPrβ2(ξ)frξ,E34
b2,r=Pr(2m3)3ξfr,E35
b3,r=ξ(1ξ)Pr+Prβ2(ξ)fr,E36
c1,r=η2Sc(1ξ)+Scβ1(ξ)frScβ2(ξ)frξ,E37
c2,r=Sc(2m3)3ξfr,E38
c3,r=ξ(1ξ)Sc+Scβ2(ξ)fr,E39

© 2015 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Sandile Motsa (October 22nd 2015). On the New Bivariate Local Linearisation Method for Solving Coupled Partial Differential Equations in Some Applications of Unsteady Fluid Flows with Heat and Mass Transfer, Mass Transfer - Advancement in Process Modelling, Marek Solecki, IntechOpen, DOI: 10.5772/60444. Available from:

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