7 research outputs found
An approximate solution of the MHD Falkner-Skan flow by Hermite functions pseudospectral method
Based on a new approximation method, namely pseudospectral method, a solution
for the three order nonlinear ordinary differential laminar boundary layer
Falkner-Skan equation has been obtained on the semi-infinite domain. The
proposed approach is equipped by the orthogonal Hermite functions that have
perfect properties to achieve this goal. This method solves the problem on the
semi-infinite domain without truncating it to a finite domain and transforming
domain of the problem to a finite domain. In addition, this method reduces
solution of the problem to solution of a system of algebraic equations. We also
present the comparison of this work with numerical results and show that the
present method is applicable.Comment: 15 pages, 4 figures; Published online in the journal of
"Communications in Nonlinear Science and Numerical Simulation
Heat transfer analysis for falkner-skan boundary layer flow past a stationary wedge with slips boundary conditions considering temperature-dependent thermal conductivity
We studied the problem of heat transfer for Falkner-Skan boundary layer flow past a stationary wedge with momentum and thermal slip boundary conditions and the temperature dependent thermal conductivity. The governing partial differential equations for the physical situation are converted into a set of ordinary differential equations using scaling group of transformations. These are then numerically solved using the Runge-Kutta-Fehlberg fourth-fifth order numerical method. The momentum slip parameter Ī“ leads to increase in the dimensionless velocity and the rate of heat transfer whilst it decreases the dimensionless temperature and the friction factor. The thermal slip parameter leads to the decrease rate of heat transfer as well as the dimensionless temperature. The dimensionless velocity, rate of heat transfer and the friction factor increase with the Falkner-Skan power law parameter m but the dimensionless fluid temperature decreases with m. The dimensionless fluid temperature and the heat transfer rate decrease as the thermal conductivity parameter A increases. Good agreements are found between the numerical results of the present paper with published results
Computational and numerical analysis of differential equations using spectral based collocation method.
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.In this thesis, we develop accurate and computationally eļ¬cient spectral collocation-based methods,
both modiļ¬ed and new, and apply them to solve diļ¬erential equations. Spectral collocation-based
methods are the most commonly used methods for approximating smooth solutions of diļ¬erential
equations deļ¬ned over simple geometries. Procedurally, these methods entail transforming the gov
erning diļ¬erential equation(s) into a system of linear algebraic equations that can be solved directly.
Owing to the complexity of expanding the numerical algorithms to higher dimensions, as reported
in the literature, researchers often transform their models to reduce the number of variables or
narrow them down to problems with fewer dimensions. Such a process is accomplished by making
a series of assumptions that limit the scope of the study. To address this deļ¬ciency, the present
study explores the development of numerical algorithms for solving ordinary and partial diļ¬erential
equations deļ¬ned over simple geometries. The solutions of the diļ¬erential equations considered are
approximated using interpolating polynomials that satisfy the given diļ¬erential equation at se
lected distinct collocation points preferably the Chebyshev-Gauss-Lobatto points. The size of the
computational domain is particularly emphasized as it plays a key role in determining the number
of grid points that are used; a feature that dictates the accuracy and the computational expense of
the spectral method. To solve diļ¬erential equations deļ¬ned on large computational domains much
eļ¬ort is devoted to the development and application of new multidomain approaches, based on
decomposing large spatial domain(s) into a sequence of overlapping subintervals and a large time
interval into equal non-overlapping subintervals. The rigorous analysis of the numerical results con
ļ¬rms the superiority of these multiple domain techniques in terms of accuracy and computational
eļ¬ciency over the single domain approach when applied to problems deļ¬ned over large domains.
The structure of the thesis indicates a smooth sequence of constructing spectral collocation method
algorithms for problems across diļ¬erent dimensions. The process of switching between dimensions
is explained by presenting the work in chronological order from a simple one-dimensional problem
to more complex higher-dimensional problems. The preliminary chapter explores solutions of or
dinary diļ¬erential equations. Subsequent chapters then build on solutions to partial diļ¬erential
i
equations in order of increasing computational complexity. The transition between intermediate
dimensions is demonstrated and reinforced while highlighting the computational complexities in
volved. Discussions of the numerical methods terminate with development and application of a
new method namely; the trivariate spectral collocation method for solving two-dimensional initial
boundary value problems. Finally, the new error bound theorems on polynomial interpolation are
presented with rigorous proofs in each chapter to benchmark the adoption of the diļ¬erent numerical
algorithms. The numerical results of the study conļ¬rm that incorporating domain decomposition
techniques in spectral collocation methods work eļ¬ectively for all dimensions, as we report highly
accurate results obtained in a computationally eļ¬cient manner for problems deļ¬ned on large do
mains. The ļ¬ndings of this study thus lay a solid foundation to overcome major challenges that
numerical analysts might encounter
On the numerical solution of the Lane-Emden, Bratu and Troesch equations.
Masters Degree. University of KwaZulu-Natal, Pietermaritzburg.Many engineering and physics problems are modelled using differential equations, which
may be highly nonlinear and difficult to solve analytically. Numerical techniques are often
used to obtain approximate solutions. In this study, we consider the solution of three nonlinear
ordinary differential equations; namely, the initial value Lane-Emden equation, the
boundary value Bratu equation, and the boundary value Troesch problem. For the Lane-
Emden equation, a comparison is made between the accuracy of solutions using the finite
difference method and the multi-domain spectral quasilinearization method along with
the exact solution. We found that the multi-domain spectral quasilinearization method
gave a better solution. For the Bratu problem, a comparison is made between the spectral
quasilinearization method and the higher-order spectral quasilinearization method. The
higher-order spectral quasilinearization method gave more accurate results. The Troesch
problem is solved using the higher-order spectral quasilinearization method and the finite
difference method. The solutions obtained are compared in terms of accuracy. Overall,
the higher-order spectral quasilinearization method and multi-domain spectral quasilinearization
method gave the accurate solutions, making these two methods to be the most
reliable for these three problems
A numerical study of entropy generation in nanofluid flow in different flow geometries.
This thesis is concerned with the mathematical modelling and numerical solution of equations
for boundary layer flows in different geometries with convective and slip boundary conditions.
We investigate entropy generation, heat and mass transport mechanisms in non-Newtonian
fluids by determining the influence of important physical and chemical parameters on
nanofluid flows in various flow geometries, namely, an Oldroyd-B nanofluid flow past a Riga
plate; the combined thermal radiation and magnetic field effects on entropy generation in
unsteady fluid flow in an inclined cylinder; the impact of irreversibility ratio and entropy
generation on a three-dimensional Oldroyd-B fluid flow along a bidirectional stretching
surface; entropy generation in a double-diffusive convective nanofluid flow in the stagnation
region of a spinning sphere with viscous dissipation and a study of the fluid velocity, heat and
mass transfer in an unsteady nanofluid flow past parallel porous plates. We assumed that the
nanofluids are electrically conducting and that the velocity slip and shear stress at the
boundary have a linear relationship. We also consider different boundary conditions for all the
flow models. The study further analyzes and quantifies the influence of each source of
irreversibility on the overall entropy generation.
The transport equations are solved using two recent numerical methods, the overlapping grid
spectral collocation method and the bivariate spectral quasilinearization method, first to
determine which of these methods is the most accurate, and secondly to authenticate the
numerical accuracy of the results. Further, we determine the skin friction coefficient and the
changes in the heat and mass transfer coefficients with various system parameters. The results
show, inter alia that reducing the heat transfer coefficient, the particle Brownian motion
parameter, chemical reaction parameter, Brinkman number, thermophoresis parameter and the
Hartman number all lead individually to a reduction in entropy generation. The overlapping
grid spectral collocation method gives better computational accuracy and converge faster than
the bivariate spectral quasilinearization method. The fluid flow problems have engineering and
industrial applications, particularly in the design of cooling systems and in aerodynamics
Chebyshev spectral pertutrbation based method for solving nonlinear fluid flow problems.
M. Sc. University of KwaZulu-Natal, Pietermaritzburg 2014.In this dissertation, a modi cation of the classical perturbation techniques for solving nonlinear ordinary
di erential equation (ODEs) and nonlinear partial di erential equations (PDEs) is presented.
The method, called the Spectral perturbation method (SPM) is a series expansion based technique
which extends the use of the standard perturbation scheme when combined with the Chebyshev
spectral method. The SPM solves a sequence of equations generated by the perturbation series
approximation using the Chebyshev spectral methods. This dissertation aims to demonstrate that,
in contrast to the conclusions earlier drawn by researchers about perturbation techniques, a perturbation
approach can be e ectively used to generate accurate solutions which are de ned under
the Williams and Rhyne (1980) transformation. A quasi-linearisation technique, called the spectral
quasilinearisation method (SQLM) is used for validation purpose. The SQLM employs the quasilinearisation
approach to linearise nonlinear di erential equations and the resulting equations are
solved using the spectral methods. Furthermore, a spectral relaxation method (SRM) which is a
Chebyshev spectral collocation based method that decouples and rearrange a system of equations
in a Gauss - Seidel manner is also presented. In the SRM, the di erential equations are decoupled,
rearranged and the resulting sequence of equations are numerically integrated using the Chebyshev
spectral collocation method. The techniques were used to solve mathematical models in
uid dynamics.
This study consists of an introductory chapter which gives the description of the methods
and a brief overview of the techniques used in developing the SPM, SQLM and the SRM. In Chapter
2, the SPM is used to solve the equations that model magnetohydrodynamics (MHD) stagnation
point
ow and heat transfer problem from a stretching sheet in the presence of heat source/sink
and suction/injection in porous media. Using similarity transformations, the governing partial differential
equations are transformed into ordinary di erential equations. Series solutions for small
velocity ratio and asymptotic solutions for large velocity ratio were generated and the results were
also validated against those obtained using the SQLM. In Chapter 3, the SPM was used to solve
the momentum, heat and mass transfer equations describing the unsteady MHD mixed convection
ow over an impulsively stretched vertical surface in the presence of chemical reaction e ect. The
governing partial di erential equations are reduced into a set of coupled non similar equations and
then solved numerically using the SPM. In order to demonstrate the accuracy and e ciency of the
SPM, the SPM numerical results are compared with numerical results generated using the SRM
and a good agreement between the two methods was observed up to eight decimal digits which is
a reasonable level of accuracy. Several simulation are conducted to ascertain the accuracy of the
SPM and the SRM. The computational speed of the SPM is demonstrated by comparing the SPM
computational time with the SRM computational time. A residual error analysis is also conducted
for the SPM and the SRM, in order to further assess the accuracy of the SPM. In Chapter 4, the
SPM was used to solve the equations modelling the unsteady three-dimensional MHD
ow and mass
transfer in a porous space previously reported in literature. E ciency and accuracy of the SPM is
shown by validating the SPM results against the results obtained using the SRM and the results were
found to be in good agreement. The computational speed of the SPM is demonstrated by comparing
the SPM and the SRM computational time. In order to further assess the accuracy of the SPM, a
residual error analysis is conducted for the SPM and the SRM. In Chapter 2, we show that the SPM
can be used as an alternative to the standard perturbation methods to get numerical solutions for
strongly nonlinear boundary value problems. Also, it is demonstrated in Chapter 2 that the SPM
is e cient even in the case where the perturbation parameter is large, as the convergence rate is
seen to improve with increase in the large parameter value. In Chapters 3 and 4, the study shows
that SPM is more e cient in terms of computational speed when compared with the SRM. The
study also highlighted that the SPM can be used as an e cient and reliable tool for solving strongly
nonlinear partial di erential equations de ned under the Williams and Rhyne (1980) transformation.
In addition, the study shows that accurate results can be obtained using the perturbation method
and thus, the conclusions earlier drawn by researchers regarding the accuracy of the perturbation
method is corrected