This study investigates chemical reaction and thermal radiation effects on hydromagnetic nanoflui... more This study investigates chemical reaction and thermal radiation effects on hydromagnetic nanofluid flow over an exponentially stretching sheet. The governing partial differential equations were transformed to ordinary differential equations by using similarity transformation and the resulting equations were solved using asymptotic series method. Graphical results showing the influence of the governing parameters on the velocity, temperature and concentration are displayed. Our results indicated that an increase in stretching sheet, thermal Grashof parameters leads to the increase in the rate of fluid flow while it decreases when magnetic field factor is increased. Also, increasing the thermophoresis number brings about increase in temperature and concentration while the reverse is the case as Prandtl number, Schmidt factor and chemical reaction rate increases. Increase in radiation leads to increase in the temperature.
In this paper, we propose a novel approach for coupling 2D/1D shallow water flows based on a two ... more In this paper, we propose a novel approach for coupling 2D/1D shallow water flows based on a two layer model in the channel, a 1D lower layer model and a 2D upper laayer model. The upper layer is only used in regions where flooding occurs otherwise the model reduces to a standard 1D channel model. To switch between the one layer and the two layer models the user prescribes an elevation above which the channel is considered to be full, i.e., a flooding event may be taking place. In the case of flooding the 2D upper layer model make it strightforward to couple the channel flow solver to a 2D shallow water solver used in the floodplain. We show that the resulting method (i) is well-balanced (ii) preserves a no-numerical flooding property (iii) preserves conservation properties of the underlying 1D and 2D finite volume schemes used for the flow in the channel and the floodplain. Numerical tests show that the method performs well compared to two horizontal coupling methods found in the literature. The results show that the method recovers the 2D flow structure in the channel in flooding regions, retains 1D flow structure in non-flooding regions while maintains good efficiency.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
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 ...
Journal of advances in mathematics and computer science, Feb 10, 2023
A fixed point method is developed on a mesh for the solution of nonlinear Fredholm equation. Firs... more A fixed point method is developed on a mesh for the solution of nonlinear Fredholm equation. First, the problem is collocated at mesh points and a second order quadrature rule is used to approximate the nonlinear integral. Under the assumption of nonexpansivity of self-map, we construct an Ishikawa iteration to linearize the resulting system and approximate the solution at the mesh points. Four numerical examples are given to verify the accuracy and practicability of the method. The results show that indeed the method converges with second order of accuracy. One important lesson from this study is that the results support the claim, in previous studies, that fixed point iterations can provide reliable means of solving several nonlinear problems. It is recommended to extend this work to functional integral equations using higher order quadrature rules.
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...
International Journal of Mathematics Trends and Technology, 2022
This study analyses the effects of chemical reaction, slip effect and heat source on the MHD flow... more This study analyses the effects of chemical reaction, slip effect and heat source on the MHD flow of blood through an inclined permeable artery with stenosis under body acceleration present. The blood is treated as a non-Newtonian electrically conduction fluid with accumulated substances of fatty substance in the blood cells creating porosity at the artery walls. The mathematical model for the blood flow is developed with inclusion of buoyancy force for both energy and diffusion with variations in heat and mass transfer having an effect on the blood flow. The partial differential equation of the governing model is transformed to ordinary differential equation using the boundary conditions. Variations in parameters all had effects on the blood flow, temperature and diffusion. Results showed that chemical reaction, magnetic field and slip reduces the blood flow while the body acceleration, heat source and pressure gradient increases the blood flow.
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 chemical reaction and thermal radiation effects on hydromagnetic nanoflui... more This study investigates chemical reaction and thermal radiation effects on hydromagnetic nanofluid flow over an exponentially stretching sheet. The governing partial differential equations were transformed to ordinary differential equations by using similarity transformation and the resulting equations were solved using asymptotic series method. Graphical results showing the influence of the governing parameters on the velocity, temperature and concentration are displayed. Our results indicated that an increase in stretching sheet, thermal Grashof parameters leads to the increase in the rate of fluid flow while it decreases when magnetic field factor is increased. Also, increasing the thermophoresis number brings about increase in temperature and concentration while the reverse is the case as Prandtl number, Schmidt factor and chemical reaction rate increases. Increase in radiation leads to increase in the temperature.
In this paper, we propose a novel approach for coupling 2D/1D shallow water flows based on a two ... more In this paper, we propose a novel approach for coupling 2D/1D shallow water flows based on a two layer model in the channel, a 1D lower layer model and a 2D upper laayer model. The upper layer is only used in regions where flooding occurs otherwise the model reduces to a standard 1D channel model. To switch between the one layer and the two layer models the user prescribes an elevation above which the channel is considered to be full, i.e., a flooding event may be taking place. In the case of flooding the 2D upper layer model make it strightforward to couple the channel flow solver to a 2D shallow water solver used in the floodplain. We show that the resulting method (i) is well-balanced (ii) preserves a no-numerical flooding property (iii) preserves conservation properties of the underlying 1D and 2D finite volume schemes used for the flow in the channel and the floodplain. Numerical tests show that the method performs well compared to two horizontal coupling methods found in the literature. The results show that the method recovers the 2D flow structure in the channel in flooding regions, retains 1D flow structure in non-flooding regions while maintains good efficiency.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
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 ...
Journal of advances in mathematics and computer science, Feb 10, 2023
A fixed point method is developed on a mesh for the solution of nonlinear Fredholm equation. Firs... more A fixed point method is developed on a mesh for the solution of nonlinear Fredholm equation. First, the problem is collocated at mesh points and a second order quadrature rule is used to approximate the nonlinear integral. Under the assumption of nonexpansivity of self-map, we construct an Ishikawa iteration to linearize the resulting system and approximate the solution at the mesh points. Four numerical examples are given to verify the accuracy and practicability of the method. The results show that indeed the method converges with second order of accuracy. One important lesson from this study is that the results support the claim, in previous studies, that fixed point iterations can provide reliable means of solving several nonlinear problems. It is recommended to extend this work to functional integral equations using higher order quadrature rules.
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...
International Journal of Mathematics Trends and Technology, 2022
This study analyses the effects of chemical reaction, slip effect and heat source on the MHD flow... more This study analyses the effects of chemical reaction, slip effect and heat source on the MHD flow of blood through an inclined permeable artery with stenosis under body acceleration present. The blood is treated as a non-Newtonian electrically conduction fluid with accumulated substances of fatty substance in the blood cells creating porosity at the artery walls. The mathematical model for the blood flow is developed with inclusion of buoyancy force for both energy and diffusion with variations in heat and mass transfer having an effect on the blood flow. The partial differential equation of the governing model is transformed to ordinary differential equation using the boundary conditions. Variations in parameters all had effects on the blood flow, temperature and diffusion. Results showed that chemical reaction, magnetic field and slip reduces the blood flow while the body acceleration, heat source and pressure gradient increases the blood flow.
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.
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