It is usually desirable to approximate the solution of mathemati- cal problems with high-order of... more It is usually desirable to approximate the solution of mathemati- cal problems with high-order of accuracy and preferably using com- pact stencils. This work presents an approach for deriving high-order compact discretization of heat equation with source term. The key contribution of this work is the use of Hermite polynomials to reduce second order spatial derivatives to lower order derivatives. This does not involve the use of the given equation, so it is universal. Then, Tay- lor expansion is used to obtain a compact scheme for first derivatives. This leads to a fourth-order approximation in space. Crank-Nicholson scheme is then applied to derive a fully discrete scheme. The result- ing scheme coincides with the fourth-order compact scheme, but our derivation follows a different philosophy which can be adapted for other equations and higher order accuracy. Two numerical experiments are provided to verify the fourth-order accuracy of the approach.
St. Petersburg Polytechnical University Journal: Physics and Mathematics, 2017
Abstract In this article the influence of thermal radiation on magnetohydrodynamic (MHD) flow of ... more Abstract In this article the influence of thermal radiation on magnetohydrodynamic (MHD) flow of Cu-water nanofluid past a wedge in the occurrence of viscous-Ohmic dissipation and chemical reaction has been analyzed. The non-linearity numerical approach called RKF of the 4–5th order have been used with a shooting technique to find the results of velocity, temperature and concentration field for several points of employing parameters. The skin friction coefficient, Nusselt number and Sherwood number are examined in detail and the results are illustrated by figures and tables. The outcomes declared that the concentration boundary layer width diminishes with an increase in the values of the chemical reaction parameter and velocity profiles increase with increase in the magnetic field parameter.
Many fixed point iterative methods have been proposed. Their authors provide, theoretically, the ... more Many fixed point iterative methods have been proposed. Their authors provide, theoretically, the rate of convergence which indicates how much the error changes after one iteration step. Of practical importance are the CPU time required by each method and their accuracy. This has never been investigated. In this work we investigate nine iterative processes and compare their CPU time and accuracy. The schemes are the Picard, Mann, Ishikawa, Noor, Argawal, Abbas-Nazir, Thakur, Ullah, and S* methods. By applying a recently proposed fourth order quadrature rule to the integral equation and replacing the analytical operator with the discretized one, the iterative schemes are derived. They are tested on five Hammerstein equations. The results show that (i) all the methods converge to the exact solution at the correct discretization order of convergence, (ii) the Picard scheme is the fastest and also has good accuracy, and (iii) the S* and Abbas-Nazir schemes are the least efficient. It is ...
This paper investigates the nonlinear dispersion of a pollutant in a non-isothermal incompressibl... more This paper investigates the nonlinear dispersion of a pollutant in a non-isothermal incompressible flow of a temperature-dependent viscosity fluid in a rectangular channel filled with porous materials. The Brinkman-Forch-heimer effects are incorporated and the fluid is assumed to be variably permeable through the porous channel. External pollutant injection, heat sources and nonlinear radiative heat flux of the Rossland approximation are accounted for. The nonlinear system of partial differential equations governing the velocity, temperature and pollutant concentration is presented in non-dimensional form. A convergent numerical algorithm is formulated using an upwind scheme for the convective part and a conservative-type central scheme for the diffusion parts. The convergence of the scheme is discussed and verified by numerical experiments both in the presence and absence of suction. The scheme is then used to investigate the flow and transport in the channel. The results show that...
Journal of Advances in Mathematics and Computer Science
This study examines the effect of viscosity and magnetic field on a non-isothermal cylindrical ch... more This study examines the effect of viscosity and magnetic field on a non-isothermal cylindrical channel flow. This work considered a model of convective-thermal-diffusion with constant viscosity and magnetic field. The governing model equations are nondimensionalized using the dimensionless quantities and then solved analytically using power series method of Frobenius type so as to tackle the singularity in the model equations. Furthermore, the analytical solutions are displayed via graphs to show the effects of the flow parameters on the flow velocity, temperature and concentration profiles. The graphical results show that increase in viscosity, magnetic field and solutal Grashof number parameters retard the fluid flow. While increase in thermal Grashof number enhances the flow velocity. Thermal conductivity and solute injection parameters increase the fluid temperature and concentration respectively while the cooling and diffusive parameters decrease the fluid temperature and con...
Applied Mathematics-a Journal of Chinese Universities Series B, 2019
We investigate the effects of moving channel wall, thermal radiation and variable-thermal conduct... more We investigate the effects of moving channel wall, thermal radiation and variable-thermal conductivity on the flow of a non-Newtonian fluid in a porous channel. The effects on fluid temperature variations are also studied. By assuming that both the fluid viscosity and thermal conductivity are temperature-dependent, and incorporating viscous dissipation, uniform magnetic field and constant pressure gradient, the governing equations are presented. An implicit-explicit finite difference scheme is formulated and the numerical results are presented and discussed. The results show that near the moving wall, the transport of momentum, from the moving wall into the channel, is retarded by decrease in the fluid viscosity. While the opposite happens -velocity increases with decreasing viscosity- away from the moving wall where the pressure forces dominate the wall shearing forces.
This study investigates the fluid flow and transport in a vertical channel with an exponentially ... more This study investigates the fluid flow and transport in a vertical channel with an exponentially decaying suction and mobile wall. The governing equations are derived based on the assumptions of incompressible flow with buoyancy forces and viscous dissipation. A finite-difference scheme is formulated and implemented. The numerical results are presented graphically. The results show that the increase in Brinkman number, Suction Parameter and Prandtl number increase the temperature distribution, while the increase in Thermal Grashof number, Mass Grashof number and Suction parameter increase the flow velocity.
This study investigates the fluid flow and transport in a vertical channel with an exponentially ... more This study investigates the fluid flow and transport in a vertical channel with an exponentially decaying suction and mobile wall. The governing equations are derived based on the assumptions of incompressible flow with buoyancy forces and viscous dissipation. A finite-difference scheme is formulated and implemented. The numerical results are presented graphically. The results show that the increase in Brinkman number, Suction Parameter and Prandtl number increase the temperature distribution, while the increase in Thermal Grashof number, Mass Grashof number and Suction parameter increase the flow velocity.
It is usually desirable to approximate the solution of mathemati- cal problems with high-order of... more It is usually desirable to approximate the solution of mathemati- cal problems with high-order of accuracy and preferably using com- pact stencils. This work presents an approach for deriving high-order compact discretization of heat equation with source term. The key contribution of this work is the use of Hermite polynomials to reduce second order spatial derivatives to lower order derivatives. This does not involve the use of the given equation, so it is universal. Then, Tay- lor expansion is used to obtain a compact scheme for first derivatives. This leads to a fourth-order approximation in space. Crank-Nicholson scheme is then applied to derive a fully discrete scheme. The result- ing scheme coincides with the fourth-order compact scheme, but our derivation follows a different philosophy which can be adapted for other equations and higher order accuracy. Two numerical experiments are provided to verify the fourth-order accuracy of the approach.
St. Petersburg Polytechnical University Journal: Physics and Mathematics, 2017
Abstract In this article the influence of thermal radiation on magnetohydrodynamic (MHD) flow of ... more Abstract In this article the influence of thermal radiation on magnetohydrodynamic (MHD) flow of Cu-water nanofluid past a wedge in the occurrence of viscous-Ohmic dissipation and chemical reaction has been analyzed. The non-linearity numerical approach called RKF of the 4–5th order have been used with a shooting technique to find the results of velocity, temperature and concentration field for several points of employing parameters. The skin friction coefficient, Nusselt number and Sherwood number are examined in detail and the results are illustrated by figures and tables. The outcomes declared that the concentration boundary layer width diminishes with an increase in the values of the chemical reaction parameter and velocity profiles increase with increase in the magnetic field parameter.
Many fixed point iterative methods have been proposed. Their authors provide, theoretically, the ... more Many fixed point iterative methods have been proposed. Their authors provide, theoretically, the rate of convergence which indicates how much the error changes after one iteration step. Of practical importance are the CPU time required by each method and their accuracy. This has never been investigated. In this work we investigate nine iterative processes and compare their CPU time and accuracy. The schemes are the Picard, Mann, Ishikawa, Noor, Argawal, Abbas-Nazir, Thakur, Ullah, and S* methods. By applying a recently proposed fourth order quadrature rule to the integral equation and replacing the analytical operator with the discretized one, the iterative schemes are derived. They are tested on five Hammerstein equations. The results show that (i) all the methods converge to the exact solution at the correct discretization order of convergence, (ii) the Picard scheme is the fastest and also has good accuracy, and (iii) the S* and Abbas-Nazir schemes are the least efficient. It is ...
This paper investigates the nonlinear dispersion of a pollutant in a non-isothermal incompressibl... more This paper investigates the nonlinear dispersion of a pollutant in a non-isothermal incompressible flow of a temperature-dependent viscosity fluid in a rectangular channel filled with porous materials. The Brinkman-Forch-heimer effects are incorporated and the fluid is assumed to be variably permeable through the porous channel. External pollutant injection, heat sources and nonlinear radiative heat flux of the Rossland approximation are accounted for. The nonlinear system of partial differential equations governing the velocity, temperature and pollutant concentration is presented in non-dimensional form. A convergent numerical algorithm is formulated using an upwind scheme for the convective part and a conservative-type central scheme for the diffusion parts. The convergence of the scheme is discussed and verified by numerical experiments both in the presence and absence of suction. The scheme is then used to investigate the flow and transport in the channel. The results show that...
Journal of Advances in Mathematics and Computer Science
This study examines the effect of viscosity and magnetic field on a non-isothermal cylindrical ch... more This study examines the effect of viscosity and magnetic field on a non-isothermal cylindrical channel flow. This work considered a model of convective-thermal-diffusion with constant viscosity and magnetic field. The governing model equations are nondimensionalized using the dimensionless quantities and then solved analytically using power series method of Frobenius type so as to tackle the singularity in the model equations. Furthermore, the analytical solutions are displayed via graphs to show the effects of the flow parameters on the flow velocity, temperature and concentration profiles. The graphical results show that increase in viscosity, magnetic field and solutal Grashof number parameters retard the fluid flow. While increase in thermal Grashof number enhances the flow velocity. Thermal conductivity and solute injection parameters increase the fluid temperature and concentration respectively while the cooling and diffusive parameters decrease the fluid temperature and con...
Applied Mathematics-a Journal of Chinese Universities Series B, 2019
We investigate the effects of moving channel wall, thermal radiation and variable-thermal conduct... more We investigate the effects of moving channel wall, thermal radiation and variable-thermal conductivity on the flow of a non-Newtonian fluid in a porous channel. The effects on fluid temperature variations are also studied. By assuming that both the fluid viscosity and thermal conductivity are temperature-dependent, and incorporating viscous dissipation, uniform magnetic field and constant pressure gradient, the governing equations are presented. An implicit-explicit finite difference scheme is formulated and the numerical results are presented and discussed. The results show that near the moving wall, the transport of momentum, from the moving wall into the channel, is retarded by decrease in the fluid viscosity. While the opposite happens -velocity increases with decreasing viscosity- away from the moving wall where the pressure forces dominate the wall shearing forces.
This study investigates the fluid flow and transport in a vertical channel with an exponentially ... more This study investigates the fluid flow and transport in a vertical channel with an exponentially decaying suction and mobile wall. The governing equations are derived based on the assumptions of incompressible flow with buoyancy forces and viscous dissipation. A finite-difference scheme is formulated and implemented. The numerical results are presented graphically. The results show that the increase in Brinkman number, Suction Parameter and Prandtl number increase the temperature distribution, while the increase in Thermal Grashof number, Mass Grashof number and Suction parameter increase the flow velocity.
This study investigates the fluid flow and transport in a vertical channel with an exponentially ... more This study investigates the fluid flow and transport in a vertical channel with an exponentially decaying suction and mobile wall. The governing equations are derived based on the assumptions of incompressible flow with buoyancy forces and viscous dissipation. A finite-difference scheme is formulated and implemented. The numerical results are presented graphically. The results show that the increase in Brinkman number, Suction Parameter and Prandtl number increase the temperature distribution, while the increase in Thermal Grashof number, Mass Grashof number and Suction parameter increase the flow velocity.
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