Semi-discrete Finite Difference Method Research Articles

Magnetohydrodynamics Buoyancy-Driven Unsteady Couette Flow with Fluctuating Viscosity

Magnetohydrodynamics plays a significant role in new development and future perspectives of clean energy power generation systems and flow control technology in the aeronautics, astronautics, plasma, and electrical conducting flow in the electric machinery. This paper examines the inherent irreversibility in an unsteady hydromagnetic Couette flow of a conducting variable viscosity fluid between two vertical parallel plates with thermal buoyancy and heat transfer characteristics. Necessary regulating model equations are obtained and then converted using dimensionless variables to form a set of nonlinear initial boundary valued problem. Employing a semi-discretization finite difference method known as the method of lines, the model is addressed numerically. The velocity and the temperature results obtained are used further to compute the entropy generation rate, the Bejan number, skin friction and Nusselt number. Some effects of the embedded parameters on the above-mentioned results are investigated and displayed by graphs. Our results revealed that entropy generation rate is enhanced with an upsurge in thermal buoyancy but a rise in magnetic field lessened it.

Modeling Heat Transfer Enhancement of Ferrofluid (Fe3O4–H2O) Flow in a Microchannel Filled with a Porous Medium

Heat transfer characteristics and hydrodynamical properties of ferrofluid through microchannels with non-uniform permeable walls temperature and filled with porous media plays an important role in modern microfluidic applications, such as solar collectors, nuclear reactors, micro-electro-chemical cell transport, micro heat exchanging, microchip cooling, and electronic equipment. Therefore, this paper presents the investigation of ferrofluid (Fe3O4-H2O) heat transfer characteristics as well as hydrodynamical properties in a permeable microchannel with non-uniform permeable walls. The semi-discretization finite difference method is utilized to solve the highly non-linear partial differential equations that govern the momentum and energy equations. Accordingly, the numerical outcomes reveal that the ferrofluid velocity and temperature profiles indicate a rising trend as the pressure gradient parameter, the variable viscosity parameter, the Darcy number, the Eckert number, and Prandtl number increase. The Reynolds number, which is a suction/injection parameter, shows a contrary influence on the ferrofluid velocity and temperature whereas nanoparticles volume fraction and the Forchheimer constant show a decreasing effect on the ferrofluid velocity and temperature. The outcomes also depict that the coefficient of skin friction at the cold wall of the microchannel is larger for higher values of the nanoparticles volume fraction, the variable viscosity parameter, the Darcy number, and the Eckert number. Besides, the coefficient of skin friction at the hot wall rises with the Darcy number, and the Prandtl number. Furthermore, the heat transfer rate at both cold and hot walls of the microchannel increases as the variable viscosity parameter, the Darcy number, the Eckert number, and the Prandtl number increase. The nanoparticles volume fraction and Darcy number show a retarding effect on the heat transfer rate at both walls of the microchannel.

A semi-discrete finite difference method to uniform stabilization of wave equation with local viscosity

In this paper, we propose a completely new and pure semi-finite difference scheme for uniform exponential convergence of approximation of 1-D wave equation with local viscosity damping by a semi-discrete finite difference scheme. It is known that for this partial differential equation system, the continuous system is exponentially stable yet the classical semi-discrete finite difference scheme is not uniformly exponentially stable. This paper adopts an order reduction finite difference scheme to semi-discretize the continuous system. There are at least three apparent advantages for this approach: a) it keeps the simplicity of the finite difference method; b) the uniformly exponentially convergence is preserved; c) the proof of the uniform exponentially stability is analogous to the continuous part. Finally, it is shown that the solution of the discrete system is convergent to the solution of the continuous counterpart and the energy of the discrete system is uniformly convergent to the continuous energy. The method is systematic to be applicable to any order of PDEs.

Stabilization Results for Well-Posed Potential Formulations of a Current-Controlled Piezoelectric Beam and Their Approximations

Hysteresis is highly undesired for the vibration control of piezoelectric beams especially in high-precision applications. Current-controlled piezoelectric beams cope with hysteresis substantially in comparison to the voltage-controlled counterparts. However, the existing low fidelity current-controlled beam models are finite dimensional, and they are either heuristic or mathematically over-simplified differential equations. In this paper, novel infinite-dimensional models, by a thorough variational approach, are introduced to describe vibrations on a piezoelectric beam. Electro-magnetic effects due to Maxwell’s equations factor in the models via the electric and magnetic potentials. Both models are written in the standard state-space formulation (A, B, C), and are shown to be well-posed in the energy space by fixing the so-called Coulomb and Lorenz gauges. Different from the voltage-actuated counterparts, the control operator B is compact in the energy space, i.e. the exponential stabilizability is not possible. Considering the compact $$C=B^*-$$type state feedback controller (induced voltage), both models fail to be asymptotically stable if the material parameters satisfy certain conditions. To achieve at least asymptotic stability, we propose an additional controller. Finally, the stabilizability of infinite-dimensional electrostatic/quasi-static model (no magnetic effects) is analyzed for comparison. The biggest contrast is that the asymptotic stability is achieved by an electro-mechanical state feedback controller for all material parameters. Our findings are simulated by the filtered semi-discrete Finite Difference Method.

Modified auxiliary boundary conditions method for an ill‐posed problem for the homogeneous biharmonic equation

In this paper, we are interested in the inverse problem for the biharmonic equation posed on a rectangle, which is of great importance in many areas of industry and engineering. We show that the problem under consideration is ill‐posed; therefore, to solve it, we opted for a regularization method via modified auxiliary boundary conditions. The numerical implementation is based on the application of the semidiscrete finite difference method for a sequence of well‐posed direct problems depending on a small parameter of regularization. Numerical results are performed for a rectangle domain showing the effectiveness of the proposed method.

Exponential stabilization of a smart piezoelectric composite beam with only one boundary controller

Layered smart composite beams involving a piezoelectric layer are traditionally actuated by a voltage source by the extension mechanism. In this paper, we consider only the bending and shear of a cantilevered piezoelectric smart composite beam modeled by the Mead-Marcus sandwich beam assumptions. Uniform exponential stabilitization with only one boundary state feedback controller, simultaneously controlling both bending moment and shear, is proved by using a spectral multiplier approach. The state feedback controller slightly differs from the classical counterparts by a non-trivial compact and nonnegative integral operator. This is due to the strong coupling of the charge equation with the stretching and bending equations. For simulations, the so-called filtered semi-discrete finite difference method is adopted.

Unsteady Mixed Convection Flow of a Reactive Casson Fluid in a Permeable Wall Channel Filled with a Porous Medium

In the current paper, we investigate the thermal decomposition in an unsteady mixed convection flow of a reactive Casson fluid in a vertical channel filled with a saturated porous medium. The channel walls are assumed to be permeable with fluid injection through the left wall and suction out of the right wall. There is heat dissipation caused by exothermic chemical reaction within the flow system. The dimensionless form of the momentum and energy equations will be solved numerically using a semi-discretization finite difference method and a fourth order Runge-Kutta-Fehlberg integration scheme. The influence of the Casson fluid parameter, the buoyancy parameter, the porous medium shape parameter, the Eckert number, the suction/injection Reynolds number, Frank-Kamenetskii parameter and the Prandtl number on velocity and temperature profiles, skin friction and Nusselt number as well as the thermal stability criteria are presented graphically and discussed quantitatively. It is revealed that increasing the Casson fluid parameter enhances the flow velocity, the fluid temperature and the skin friction but has a diminishing effect on the wall heat transfer rate. The suction/injection Reynolds number, the porous medium shape parameter and the buoyancy parameter enhance the rate of heat transfer at the channel walls.

Modelling the Effect of Variable Viscosity on Unsteady Couette Flow of Nanofluids with Convective Cooling

This paper investigates numerically the effects of variable viscosity on unsteady generalized Couette flow of a water base nanofluid with convective cooling at the moving surface. The Buongiorno model utilized for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The nonlinear governing equations of continuity, momentum, energy and nanoparticles concentration are tackled numerically using a semi discretization finite difference method together with Runge-Kutta Fehlberg integration scheme. Numerical results for velocity, temperature, and nanoparticles concentration profiles together with skin friction and Nusselt number are obtained graphically and discussed quantitatively.

Numerical investigation into entropy generation in a transient generalized Couette flow of nanofluids with convective cooling

This work investigates the effects of convective cooling on entropy generation in a transient generalized Couette flow of water-based nanofluids containing Copper (Cu) and Alumina (Al 2 O 3) as nanoparticles. Both First and Second Laws of thermodynamics are utilised to analyse the problem. The model partial differential equations for momentum and energy balance are tackled numerically using a semi-discretization finite difference method together with Runge–Kutta Fehlberg integration scheme. Graphical results on the effects of parameter variation on velocity, temperature, skin friction, Nusselt number, entropy generation rate, irreversibility ratio and Bejan number are presented and discussed.

Unsteady flow of variable viscosity Cu-water and Al2O3-water nanofluids in a porous pipe with buoyancy force

Purpose – The purpose of this paper is to investigate the combined effects of buoyancy force and variable viscosity on unsteady flow and heat transfer of water-based nanofluid containing copper and alumina as nanoparticles through a porous pipe. Design/methodology/approach – Using the Boussinesq and boundary-layer approximations with Buongiorno nanofluid model. The governing nonlinear partial differential equations for the continuity, momentum and energy balance are formulated. The equations obtained are solved numerically using a semi-discretization finite difference method (know) as method of line coupled with Runge-Kutta-Fehlberg integration scheme. Findings – Numerical results for the skin-friction, heat transfer and for the velocity and temperature profiles are obtained. The results show that with suction, Cu-water produces higher skin friction and heat transfer rate than Al2O3-water. Both nanofluids velocity and temperature increase with a decrease in viscosity and an increase in buoyancy force intensity. Practical implications – Buoyancy-driven flow and heat transfer in porous geometries has many significant applications in industrial and engineering such as, electrical and microelectronic equipments, solar-collectors, geothermal engineering, petroleum reservoirs, thermal buildings insulation. This work provides very important information for researchers on this subject. Originality/value – This paper illustrates the effects of buoyancy force and temperature dependent on heat transfer and fluid flow problem using Cu-water and Al2O3-water nanofluids in a porous pipe.

Irreversibility analysis of unsteady couette flow with variable viscosity

This paper investigates numerically the inherent irreversibility in unsteady generalized Couette flow between two parallel plates with variable viscosity. The nonlinear governing equations are derived from the Navier-Stokes equations and solved numerically using a semi-discretization finite difference method together with the Runge-Kutta-Fehlberg integration scheme. The profiles of velocity and the temperature obtained are used to compute the entropy generation number, Bejan number, skin friction and Nusselt number. The effects of embedded parameters on entire flow structure are presented graphically and discussed quantitatively.

Second Law Analysis of Buoyancy Driven Unsteady Channel Flow of Nanofluids with Convective Cooling

We investigate the combined effects of buoyancy force and convective cooling on entropy generation in unsteady channel flow of water based nanofluids containing Copper (Cu) and Alumina (Al2O3) as nanoparticles. Both first and second laws of thermodynamics are utilised to analyze the model problem. Using a semi discretization finite difference method together with Runge-Kutta Fehlberg integration scheme, the governing partial differential equations are solved numerically. Graphical results on the effects of parameter variation on velocity, temperature, skin friction, Nusselt number, entropy generation rate, irreversibility ratio and Bejan number are presented and discussed.

Modelling the Effects of Variable Viscosity in Unsteady Flow of Nanofluids in a Pipe with Permeable Wall and Convective Cooling

In this paper, the combined effects of variable viscosity, Brownian motion, thermophoresis and convective cooling on unsteady flow of nanofluids in a pipe with permeable wall are investigated. It is assumed that the pipe surface exchange heat with the ambient following the Newton’s law of cooling. Using a semi discretization finite difference method coupled with Runge-Kutta Fehlberg integration scheme, the nonlinear governing equations of momentum and energy balance, and the equation for nanoparticles concentration are tackled numerically. Useful results for the velocity, temperature, nanoparticles concentration profiles, skin friction and Nusselt number are obtained graphically and discussed quantitatively.

Unsteady Hydromagnetic Flow of Radiating Fluid Past a Convectively Heated Vertical Plate with the Navier Slip

This paper investigates the unsteady hydromagnetic-free convection of an incompressible electrical conducting Boussinesq’s radiating fluid past a moving vertical plate in an optically thin environment with the Navier slip, viscous dissipation, and Ohmic and Newtonian heating. The nonlinear partial differential equations governing the transient problem are obtained and tackled numerically using a semidiscretization finite difference method coupled with Runge-Kutta Fehlberg integration technique. Numerical data for the local skin friction coefficient and the Nusselt number have been tabulated for various values of parametric conditions. Graphical results for the fluid velocity, temperature, skin friction, and the Nusselt number are presented and discussed. The results indicate that the skin friction coefficient decreases while the heat transfer rate at the plate surface increases as the slip parameter and Newtonian heating increase.

Convergence of a finite difference method for the KdV and modified KdV equations with $L^2$ data

We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in $L^2$), without size restrictions. Our approach uses a fourth order (in space) stabilization term and a special conservative discretization of the nonlinear term. Convergence follows from a smoothing effect and energy estimates. We illustrate our results with numerical experiments, including a numerical investigation of an open problem related to uniqueness posed by Y. Tsutsumi.

Accuracy of Bhatnagaar-Gross-Krook Scheme in Solving Laminar Viscous Flow Problems

This paper describes the development of a gas-kinetic solver to compute laminar viscous flows in two-space dimensions via a finite difference approach. The convection flux terms of the Navier-Stokes equations are discretized by a semidiscrete finite difference method. The resulting inviscid flux function is then determined by a numerical scheme that is based on the Bhatnagaar-Gross-Krook model of the approximate collisional Boltzmann equation. The scheme is based on the direct splitting of the inviscid flux function with inclusion of particle collisions in the transport process. As for the diffusion flux terms, they are discretized by a second-order central difference scheme. The cell interface values required by the gas-kinetic scheme are reconstructed to higher-order spatial accuracy via the monotone upstream-centered schemes for conservation laws variable interpolation method. An explicit-type time integration method known as the modified fourth-order Runge-Kutta is employed for computing steady-state solutions. In the numerical case studies, the results obtained from the flux vector splitting Bhatnagaar-Gross-Krook scheme are compared with available experimental data, analytical solutions, the results from upwind schemes, and the results from central difference scheme to verify the accuracy and robustness of the gas-kinetic solver. The tests have shown that the Bhatnagaar-Gross-Krook scheme is able to resolve the shear layer, the shock structure, and the flow accurately as the results compare favorably with the available experimental and analytical data.

Computer simulation for local temperature control during microwave-induced hyperthermia.

A numerical procedure to calculate and control transient temperature profiles in plane multilayered biological tissue is described. To express the temperature increase in the normal and neoplastic tissue as a function of time-varying incident microwave heating and artificial surface cooling, a semi-discrete finite difference method is used. Two optimization algorithms are applied in order to achieve and maintain a prescribed temperature pattern. They suggest the extent to which the tissue heating profiles can be experimentally controlled. Several examples are included to illustrate the performance of the algorithms.

Computer Simulation Predicting Temperature Distributions Generated by Microwave Absorption in Multilayered iedia

AbstractA numerical procedure to calculate the time dependence of temperature profiles in multi-layered plane media is described. Discretization of the continuous problem is accomplished by dividing each layer into a number of sublayers. A semi-discrete finite difference method is used to express the temperatures of the sublayers in closed form as a function of the microwave input. Applying the method in an iterative way allows treatment of the non-linear case, which arises when material permittivity is temperature dependent. Numerical results are given to illustrate the versatility of the developed computer program. The simulations are verified experimentally by use of a data acquisition system and a good agreement was found.