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Numerical Investigation of the Effect of Unsteadiness on Three-Dimensional Flow of an Oldroyb-B Fluid.

Motsa SS, Makukula ZG, Shateyi S - PLoS ONE (2015)

Bottom Line: The problem is governed by a set of three highly coupled nonlinear partial differential equations.The modified approach involves seeking solutions that are expressed as bivariate Lagrange interpolating polynomials and applying pseudo-spectral collocation in both independent variables of the governing PDEs.Numerical simulations were carried out to generate results for some of the important flow properties such as the local skin friction and the heat transfer rate.

View Article: PubMed Central - PubMed

Affiliation: School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa.

ABSTRACT
A spectral relaxation method used with bivariate Lagrange interpolation is used to find numerical solutions for the unsteady three-dimensional flow problem of an Oldroyd-B fluid with variable thermal conductivity and heat generation. The problem is governed by a set of three highly coupled nonlinear partial differential equations. The method, originally used for solutions of systems of ordinary differential equations is extended to solutions of systems of nonlinear partial differential equations. The modified approach involves seeking solutions that are expressed as bivariate Lagrange interpolating polynomials and applying pseudo-spectral collocation in both independent variables of the governing PDEs. Numerical simulations were carried out to generate results for some of the important flow properties such as the local skin friction and the heat transfer rate. Numerical analysis of the error and convergence properties of the method are also discussed. One of the benefits of the proposed method is that it is computationally fast and gives very accurate results after only a few iterations using very few grid points in the numerical discretization process.

No MeSH data available.


Related in: MedlinePlus

Variation of the residual error, Res(θ) with ξ.β = 1, β1 = 0.1, β2 = 0.1, ϵ = 0.1, Pr = 0.8, S = 0.3.
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pone.0133507.g004: Variation of the residual error, Res(θ) with ξ.β = 1, β1 = 0.1, β2 = 0.1, ϵ = 0.1, Pr = 0.8, S = 0.3.

Mentions: The accuracy of the proposed BSRM can be estimated by considering the residual error which measures the extent to which the numerical solutions approximate the true solution of the governing differential Eqs (7)–(9). Accordingly, we define the following residual error functionsRes(f)=//Δf[Fi,Gi,Θi]//∞,Res(g)=Δg[Fi,Gi,Θi]//∞Res(θ)=Δf[Fi,Gi,Θi]//∞(40)where Δf, Δg and Δf represent the governing nonlinear PDEs Eqs (7), (8) and (9), respectively, and Fi, Gi, Θi are the BSRM approximate solutions at the time collocation points ξi. Figs 2–4 show the variation of the residual errors over the time scale ξ for different number of iterations. The decrease in the residual errors with an increase in the number of iterations is an indication of the convergence of the method. It can also be noted that the residual error is nearly uniform across ξ. This is an advantage of the proposed method over some other methods whose accuracy deteriorates when ξ becomes large.


Numerical Investigation of the Effect of Unsteadiness on Three-Dimensional Flow of an Oldroyb-B Fluid.

Motsa SS, Makukula ZG, Shateyi S - PLoS ONE (2015)

Variation of the residual error, Res(θ) with ξ.β = 1, β1 = 0.1, β2 = 0.1, ϵ = 0.1, Pr = 0.8, S = 0.3.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4510369&req=5

pone.0133507.g004: Variation of the residual error, Res(θ) with ξ.β = 1, β1 = 0.1, β2 = 0.1, ϵ = 0.1, Pr = 0.8, S = 0.3.
Mentions: The accuracy of the proposed BSRM can be estimated by considering the residual error which measures the extent to which the numerical solutions approximate the true solution of the governing differential Eqs (7)–(9). Accordingly, we define the following residual error functionsRes(f)=//Δf[Fi,Gi,Θi]//∞,Res(g)=Δg[Fi,Gi,Θi]//∞Res(θ)=Δf[Fi,Gi,Θi]//∞(40)where Δf, Δg and Δf represent the governing nonlinear PDEs Eqs (7), (8) and (9), respectively, and Fi, Gi, Θi are the BSRM approximate solutions at the time collocation points ξi. Figs 2–4 show the variation of the residual errors over the time scale ξ for different number of iterations. The decrease in the residual errors with an increase in the number of iterations is an indication of the convergence of the method. It can also be noted that the residual error is nearly uniform across ξ. This is an advantage of the proposed method over some other methods whose accuracy deteriorates when ξ becomes large.

Bottom Line: The problem is governed by a set of three highly coupled nonlinear partial differential equations.The modified approach involves seeking solutions that are expressed as bivariate Lagrange interpolating polynomials and applying pseudo-spectral collocation in both independent variables of the governing PDEs.Numerical simulations were carried out to generate results for some of the important flow properties such as the local skin friction and the heat transfer rate.

View Article: PubMed Central - PubMed

Affiliation: School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa.

ABSTRACT
A spectral relaxation method used with bivariate Lagrange interpolation is used to find numerical solutions for the unsteady three-dimensional flow problem of an Oldroyd-B fluid with variable thermal conductivity and heat generation. The problem is governed by a set of three highly coupled nonlinear partial differential equations. The method, originally used for solutions of systems of ordinary differential equations is extended to solutions of systems of nonlinear partial differential equations. The modified approach involves seeking solutions that are expressed as bivariate Lagrange interpolating polynomials and applying pseudo-spectral collocation in both independent variables of the governing PDEs. Numerical simulations were carried out to generate results for some of the important flow properties such as the local skin friction and the heat transfer rate. Numerical analysis of the error and convergence properties of the method are also discussed. One of the benefits of the proposed method is that it is computationally fast and gives very accurate results after only a few iterations using very few grid points in the numerical discretization process.

No MeSH data available.


Related in: MedlinePlus