Explicit Finite Difference Approach Research Articles

Dynamics of chemically reactive Carreau nanomaterial flow along a stretching Riga plate with active bio-mixers and Arrhenius catalysts

Nanomaterial flow has fascinated the concern of scientists across the globe due to its innovative applications in various manufacturing, industrial, and engineering domains. Bearing aforementioned uses in mind, the focal point of this study is to examine the Carreau nanofluid flow configured by the Riga surface with Arrhenius catalysts. Microorganisms are also suspended in nanofluid to strengthen the density of the regular fluid. Time-dependent coupled partial differential equations that represent the flow dynamics are modified into dimensionless patterns via appropriate non-dimensional variables, and handled through an explicit finite difference approach with stability appraisal. The performances of multiple flow variables are examined graphically and numerically. Representation of 3D surface and contour plots for heat transportation and entropy generation are also epitomized. The findings express that the modified Hartmann number strengthens the motion of nanomaterial. Reverse outcomes for heat transport rate and entropy are seen for the radiation variable. Concentration diminishes for chemical reaction variable. Activation energy enhances the concentration of nanomaterial, whereas reduction happens in the movement of microbes for bio-Lewis number. Greater Brinkman variable heightens the entropy.

Numerical treatment and global error estimation for thermal electro-osmosis effect on non-Newtonian nanofluid flow with time periodic variations

The essential purpose of this study is to discuss the impact of time-periodic variations on mixed convection heat transfer for MHD Eyring-Powell nanofluid. The fluid flows through a non-Darcy porous medium over an infinite vertical plate. The effects of viscous dissipation, Ohmic dissipation, electro-osmosis force, heat source, thermal radiation, Dufour feature, and chemical reaction are presumed. The system of partial differential equations which governs the problem is transformed into a system of non-linear algebraic equations and then an explicit finite difference approach is espoused to solve these nonlinear algebraic equations. The numerical results for the velocity, temperature, and nanoparticles concentration distributions are computed and displayed through a set of graphs. Also, the skin friction coefficient, reduced Nusselt number, and Sherwood number are computed numerically for various values of the physical parameters. It is found that the velocity becomes greater with an elevation in the value of the Helmholtz–Smoluchowski velocity. Meanwhile, it enlarges with rising in the value of the electro-osmotic parameter. The rise in the value of the thermal radiation parameter causes a dwindling influence on both temperature and nanoparticles concentration. Investigations of these effects together are very useful due to their important vital applications in various scientific fields, especially in medicine and medical industries, such as endoscopes, respirators, and diverse medical implementations, as nanoparticles can be utilized in the remedy of cancer tumors. Additionally, electroosmotic flow is important due to its ability to control fluid movement and enhance mass transport, making it valuable in various application such as sample separation, drug delivery, and DNA analysis, offering enhanced efficiency and sensitivity.

A Numerical Assessment of Time-Dependent Magneto-Convective Thermal-Material Transfer over a Vertical Permeable Plate

The objective of this work is to investigate the influences of thermal radiation, heat generation, and buoyancy force on the time-dependent boundary layer (BL) flow across a vertical permeable plate. The fluid is unsteady, incompressible, viscous, and electrically insulating. The heat transfer mechanism happens due to free convection. The nondimensional partial differential equations of continuity, momentum, energy, and concentration are discussed using appropriate transformations. The impressions of thermal radiation and buoyancy forces are exposed in the energy and momentum equation, respectively. For numerical model, a set of nonlinear dimensionless partial differential equations can be solved using an explicit finite difference approach. The stability and convergence analyses are also established to complete the formulation of the model. The thermophysical effects of entering physical parameters on the flow, thermal, and material fields are analyzed. The variations in local and average skin friction, material, and heat transfer rates are also discussed for the physical interest. The analysis of the obtained findings is shown graphically, and relevant parameters pointedly prejudice the flow field. Studio Developer FORTRAN 6.2 and Tecplot 10.0 are applied to simulate the schematic model equations and graphical presentation numerically. The intensifying values of the magnetic field are affected decreasingly in the flow field. The temperature profiles decrease within the BL to increase the value of radiation parameters. The present study is on the consequences for petroleum engineering, agriculture engineering, extraction, purification processes, nuclear power plants, gas turbines, etc. To see the rationality of the present research, we compare these results and the results available in the literature with outstanding compatibility.

Numerical Simulation of Heat and Moisture Transfer in Corrugated Walls Dryer

The primary goal of the current work is to model heat and mass transfer during mango drying in a wavy airflow dryer. By modifying the dryer walls, we were able to produce the undulating airflow (with V-shaped obstacles). With convective boundary conditions applied to all product surfaces, the explicit finite difference approach was used to study heat and mass exchanges in two dimensions during the drying of mango slices. During drying, the transfer coefficients are thought to fluctuate. Using EasyCFD software, the external flow, temperature, velocity, and pressure fields were then analyzed. This provided the profile of the heat transfer coefficient. These profiles were then utilized to calculate the mass transfer coefficient using the Shilton-Colburn analogy. Moreover, the code created to determine the heat and mass transfer coefficients in the product was used to derive the evolution of temperature and moisture content over time. The results allowed for the discovery of a new air flow in dryers called an undular flow and demonstrated how modifying the drying air stream enhanced heat transfer efficiency. By changing the air flow in the dryer, it was possible to achieve heat transfer coefficients ranging from 47.55 W/m²K to 357.38 W/m²K and mass transfer coefficients of 3.21 x 10-5 to 3.21 x 10-4 m²/s. When the outcomes of this investigation were compared to experimental results from the literature (under identical drying circumstances), a reasonable level of adequacy was discovered.

Heat and mass diffusion to williamson fluid streaming through a tube with multiple stenoses while subjected to periodic body acceleration

This article examines a mathematical framework that describes the versatile behavior of heat and mass exchange in blood flowing through a narrowed vessel having multiple stenoses. The geometry of a channel having multiple stenoses with an asymmetrical axial axis and a symmetrical radial axis can be visualized by applying a suitable mathematical expression. The geometry of the chosen model considers the height and shape of stenoses. The modification in shape parameter is used to capture variations in the shape of the stenoses in the artery. The blood is supposed to be isochoric (incompressible), while its rheological behavior is characterized by Williamson’s fluid model. The transfer of momentum is analyzed using the equation of motion in cooperation with the continuity equation. In addition, the equations of heat conduction and diffusion are utilized, respectively, to illustrate the distributions of heat and mass. Simplified forms of momentum, mass, and heat transport equations are achieved by incorporating dimensionless quantities and moderate stenosis conditions. A well-known explicit finite difference approach is utilized to solve the emergent non-linear system of governing equations numerically. The influence of different evolving parameters on the flow field along with mass and heat distributions is illustrated through various plots.

Derivative pricing as a transport problem: MPDATA solutions to Black–Scholes-type equations

We discuss applications of the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) to numerically solve partial differential equations arising from stochastic models in quantitative finance. In particular, we develop a framework for solving Black–Scholes-type equations by transforming them into advection–diffusion problems. The equations are then numerically integrated backward in time using an iterative explicit finite-difference approach, in which the Fickian term is represented as an additional advective term. We discuss the correspondence between transport phenomena and financial models, uncovering the possibility of expressing the no-arbitrage principle as a conservation law. We show second-order accuracy in time and space of the embraced numerical scheme. This is done via a convergence analysis comparing MPDATA numerical solutions with classic Black–Scholes analytical formulæ for the valuation of European options. We demonstrate in addition a way of applying MPDATA to solve the free boundary problem (leading to a linear complementarity problem) for the valuation of American options. We finally comment on the potential of MPDATA methods with respect to more complex models typically used in quantitative finance.

The Study of Flow Resulting from Wave on Lhonga Beach, Aceh Besar

Dynamic interaction between waves and topographic generate flow dynamics that affect Lhonga coast, Aceh Besar. To analyze the dynamics, the numerical model of currents induced by wave was developed with the explicit finite difference approach method. The model in this study was tested by verifying the result directly with the field data obtained previously. The model is applied in Lhoknga coast of Aceh Besar district with the aim to study how the dynamics of hydro-oceanography as one of the elements to support the preservation of the marine resources of the area. Based on the simulation, the results obtained that the wave occurred in Lhoknga beach relatively small with the dominant direction to the northwest and the maximum of wave height occurs during the high tide condition. The result also informed that the distribution of longshore currents is generally moving from west to east along the coast. The direction of dominant current occurs in the area of 10-15 meters from the shore with an average depth of 6 meters. Keywords: wave flow, flow dynamics, beach, hydro-oceanography

Coupled rate-dependent superelastic behavior of shape memory alloy bars induced by combined axial-torsional loading: a semi-analytic modeling

In this article, the coupled thermomechanical response of superelastic shape memory alloy bars and tubes in combined tension and torsion is studied both analytically and experimentally. Using the Gibbs free energy as the thermodynamic potential and choosing appropriate internal state variables, a three-dimensional phenomenological macroscopic constitutive model for shape memory alloys is derived. Taking into account the effect of latent heat during the forward and reverse martensitic phase transformation, the appropriate form of the energy balance relation is obtained. The three-dimensional coupled relations for the energy balance in the presence of the internal heat flux and the constitutive equations are reduced to a two-dimensional form for the tension-torsion case. An explicit finite difference approach is utilized to discretize the governing boundary conditions of bars. An empirical expression for the free heat convection from the surface of a shape memory alloy bar was proposed and experimentally va...

Numerical Analysis of Heat Transfer Characteristics in Microwave Heating of Magnetic Dielectrics

A numerical simulation of heat transfer during the microwave heating process of magnetite, which is a two-dimensional (2-D) magnetic dielectric, subjected to heat conduction, convection, and radiation was performed. The heat transfer process was modeled using an explicit finite-difference approach, and the temperature profiles for different heating parameters were generated through developing a code in Mathematica 7.0 (Wolfram Research, Inc., Champaign, IL). The temperature in the sample increases rapidly in 1 minute and nonuniform temperature distribution inside the object is observed. An obvious temperature hot spot is formed in the corner of the predicted temperature profile initially, which shifts to the center of the object as heating power increases. Microwave heating at 915 MHz exhibits better heating uniformity than 2450 MHz mainly because of the larger microwave penetration depth. It is also observed that the heating homogeneity in the object can be improved by reducing the dimension of object. The effects of heating time, microwave power, microwave frequency, and object dimension need to be considered to obtain high heating performance and avoid/minimize thermal runaway resulting from temperature nonuniformity in large-scale microwave heating.

Numerical Simulation of Heat Transfer during Microwave Heating of Magnetite

Numerical simulation of heat transfer during the microwave heating process of a one-dimensional (1-D) magnetite slab subjected to convective, radiative boundary conditions was performed. The governing equations representing the heating process in the slab were discretized using an explicit finite-difference approach, and a computer code was developed to predict the temperature distributions inside the slab. The heat generation from microwave irradiation dominates the initial temperature rise in the heating and the heat radiation heavily affects the temperature distribution, giving rise to a temperature peak in the predicted temperature profile. As heating continues, the temperature peak migrates inward. The microwave power level is crucial to obtain a high temperature increase rate in the initial heating period (i.e. < 60 s for magnetite). Microwave heating at 915 MHz exhibits better heating homogeneity than 2450 MHz due to larger microwave penetration depth. To minimize/avoid temperature non-uniformity during the microwave heating the optimization of the object dimension should be considered.

Numerical solution of the sequential investment model: a note on Dixit and Pindyck's (1994) analysis

This paper discusses the model and solution approach adopted by Majd and Pindyck (1987. Time to build, option value, and investment decisions. Journal of Financial Economics 18, March: 7–27) and Dixit and Pindyck (1994. Investment under uncertainty. Princeton, NJ: Princeton University Press), when considering the sequential investment decision. It is shown that specific results presented in these two sources are based on invalid solutions to the relevant partial differential equation. The problem stems from the possibility that economically feasible parameter values and apparently acceptable step sizes for the explicit finite difference approach used can combine to generate non-convergent, invalid solutions.

An Explicit Finite Difference Approach to the Pricing Problems of Perpetual Bermudan Options

The pricing problem of options with an early exercise feature, such as American options, is one of the important topics in mathematical finance. Pricing formulas for options with the early exercise feature, however, are not easy to obtain and the numerical methods are thus frequently required to derive the price of these options. The value function of perpetual Bermudan options is characterized with the partial differential equation and this is solved by the finite difference method in this article.

A Diffusion-based Model for Transient High Temperature Treatment of Wood

A three-dimensional (3D) mathematical model describing simultaneous unsteady heat and moisture transfer between a gas phase and a solid phase during high temperature treatment of wood has been developed. The model is based on a diffusion equation with variable diffusion coefficients. The governing equations representing the heating process in a 3D rectangular object are discretized using an explicit finite-difference approach, and a computer code is developed to predict the temperature and moisture distributions inside the wood sample. The sample was subjected to high temperature treatment under different operating conditions. The model predictions are compared with experimental results obtained for temperature and average moisture content during high temperature treatment of birch wood. Satisfactory agreement is obtained over a range of heating conditions. A parametric study was also carried out to determine the effects of several parameters such as initial moisture content, heat and mass transfer coefficients, and the sample thickness on the temperature and moisture content distributions within the samples during heat treatment.

A Range-Based Finite Difference Approach to Valuing Options in Markov Switching Regimes*

Recognizing the inflexibility inherent in standard capital budgeting analysis, recent research has provided new insights using a real options framework. This paper uses the explicit finite difference approach to value real options. However, instead of assuming a constant return and volatility term, we assume that these variables are sensitive to changes in the economic environment. Accordingly, we adapt our approach to incorporate a Markov switching feature. Further, we recognize that some of the modeling assumptions can be violated in a practical application. Therefore, we recommend using range based estimates of the real option value, as opposed to a point estimate. *This research was initiated when Andrea Vidozzi and Luca Vidozzi were graduate students at the College of Business and Management, Northeastern Illinois University, Chicago.

NUMERICAL SIMULATION OF TWO-DIMENSIONAL HEAT AND MOISTURE TRANSFER DURING DRYING OF A RECTANGULAR OBJECT

The present article deals with the numerical modeling of heat and moisture transfer during the drying process of a two-dimensional (2-D) rectangular object subjected to convective boundary conditions. As is common in solids drying, it is assumed that drying takes place as a simultaneous heat and moisture transfer whereby moisture is vaporized by means of a drying fluid (e.g., air), which passes over a moist object. The governing equations representing the drying process in a 2-D rectangular object are discretized using an explicit finite-difference approach, and a computer code is developed to predict the temperature and moisture distributions inside the object. Moreover, the results obtained from the present model are compared with the experimental data available in the literature, and considerably high agreement is found.

Two-dimensional heat and moisture transfer analysis of a cylindrical moist object subjected to drying: A finite-difference approach

In this paper a two-dimensional numerical analysis of heat and moisture transfer during drying of a cylindrical object is presented. Drying is a process of simultaneous heat and moisture transfer whereby moisture is vaporized by means of a drying fluid (e.g., air), as it passes over a moist object. The two-dimensional analysis of heat and moisture transfer during drying of a cylindrical object is carried out using an explicit finite-difference approach. Temperature and moisture distributions inside the moist objects are obtained for different time periods and the results predicted from the present analysis are compared with two sets of experimental data available in the literature. A considerably high agreement is found between the predicted and measured values.

Analysis of two-dimensional heat and moisture transfer during drying of spherical objects

This paper deals with the numerical and analytical modelling of two-dimensional heat and moisture transfer during drying of a spherical object. Drying is considered to be a process of simultaneous heat and moisture transfer whereby moisture is vapourized by means of a drying fluid (e.g. air), as it passes over a moist object. Numerical modelling of two-dimensional heat and moisture transfer during drying of a spherical object is carried out using an explicit finite-difference approach. Temperature and moisture distributions inside the object are determined by using the developed computer code. Moreover, the results predicted from the present model are compared with the experimental data available in the literature and a considerably high agreement is found. Copyright © 2003 John Wiley & Sons, Ltd.

An explicit finite difference approach to the pricing of barrier options

A modified explicit finite difference approach to the pricing of barrier options is developed. To obtain accurate prices, the grid is constructed such that the barrier is located in a suitable position relative to horizontal layers of nodes on the grid. This means that the barrier passes through a horizontal layer of nodes for continuous-time barrier options and is located halfway between two horizontal layers of nodes for discrete-time barrier options. Both single and double barrier cases can be accommodated. The option price at each node on the grid may be obtained by implementing a standard trinomial tree procedure. As the initial asset price will generally not lie exactly on the grid, the current value of the option is obtained using a quadratic interpolation of the option prices at the three adjacent nodes. The approach is shown to be robust and to provide accurate option prices and hedge ratios (such as delta, gamma, and theta) regardless of whether or not the barrier is close to the initial asset price, and it works effectively for both continuous-time and discrete-time barrier options. This device of adjusting the grid so that the barrier and the asset price lie on the grid is well known in the numerical analysis area.

Melting of a cylindrical polymeric medium subjected to cyclic torsional shear stress

The problem of phase change in a cylindrical polymeric material subjected to a constant amplitude cyclic torsional shear stress is analyzed. The steady state as well as the transient behavior of a long solid cylinder and a cylindrical shell are examined when the hysteresis heat generation G, in the medium is spacially variable and exponentially dependent on the local temperature in the form G = y/(r)exp(A/T). f(r) accounts for the stress distribution within the solid, y and Ar are related in part to the amplitude and frequency of the cyclic torque, and T is the sample local temperature. The analysis is carried through for heat generation parameters high enough to induce melting within the medium. The onset of melting and the propagation of the solid-liquid interface (two interfaces can arise), are examined for various heat generation parameters, external and internal boundary conditions for the case of a cylindrical shell. Analytical solutions for the steady-state temperature profiles are developed, and an explicit finite difference approach is used for the study of the full transient problem.

An explicit finite difference approach for the Mindlin plate analysis

An explicit finite difference method for the analysis of plates subjected to dynamic loads is presented. The theoretical approach is based on the Mindlin plate theory which includes the effects of shear deformation and rotatory inertia. Such a tool will enable the analyst to study plate vibration when higher response modes are to be considered. A stability criterion for the numerical approach is given, the method is then illustrated by an example, and a comparison to the classical plate theory is presented.