In this work, a mathematical model of generalized porothermoelasticity with one relaxation time for poroelastic half-space saturated with fluid will be constructed in the context of Youssef model (2007). We will obtain the general solution in the Laplace transform domain and apply it in a certain asphalt material which is thermally shocked on its bounding plane. The inversion of the Laplace transform will be obtained numerically and the numerical values of the temperature, stresses, strains and displacements will be illustrated graphically for the solid and the liquid.
ASPHALTIC MATERIAL IN THE CONTEXT OF GENERALIZED POROTHERMOELASTICITYijsc
This document summarizes a mathematical model of generalized porothermoelasticity for a porous asphalt material. The model considers a poroelastic half-space saturated with fluid and constructs governing equations for the stresses, strains, displacements, and temperatures of the solid and fluid phases. The equations are derived using the theory of generalized porothermoelasticity with one relaxation time. Numerical solutions to the equations are obtained by applying the model to an asphalt material subjected to thermal shock on its surface. Graphs of the temperature, stresses, strains, and displacements within the material are presented.
Chemical reaction and radiation effect on mhd flow past an exponentially acce...Alexander Decker
This document describes a mathematical analysis of MHD fluid flow past an exponentially accelerated vertical plate embedded in a porous medium. The analysis considers the effects of variable temperature, mass diffusion, radiation, and a heat source on the flow characteristics. The governing equations for this problem are derived and non-dimensionalized. The non-dimensional equations are then solved using the Laplace transform technique. The effects of various physical parameters like the magnetic field, radiation, heat generation, and chemical reaction on the velocity, temperature, and concentration profiles are determined.
Homotopy Analysis to Soret and Dufour Effects on Heat and Mass Transfer of a ...iosrjce
The objective of this paper is to study the Soret and Dufour effects on the free convection boundary
layer flow of an incompressible, viscous and chemically reacting fluid over a vertical plate in the presence of
viscous dissipation. The governing partial differential equations are converted to a set of ordinary differential
equations using suitable similarity transformations. The resulting equations are solved analytically using
homotopy analysis method (HAM). The convergence of obtained analytical solutions is explicitly discussed. The
effects of various parameters on dimensionless velocity, temperature and concentration profiles are discussed
with the help of graphs. The numerical values of skin friction, Nusselt number and Sherwood number for
different parameters are presented in tabular form. Our results are compared with the previously published
results and are found to be in good agreement.
This document summarizes a study that examines heat and mass transfer over a vertical plate in a porous medium with Soret and Dufour effects, a convective surface boundary condition, chemical reaction, and magnetic field. The governing equations for the fluid flow, heat transfer, and mass transfer are presented. Similarity solutions are used to transform the governing partial differential equations into ordinary differential equations, which are then solved numerically. The results are presented graphically to show the influence of various parameters on velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number.
EFFECT OF SLIP PARAMETER OF A BOUNDARY-LAYER FLOW FOR NANOFLUID OVER A VERTIC...IAEME Publication
In this paper we analyze the effect of momentum slip, thermal slip and solutal slip on stagnation point flow of MHD nanofluid towards stretching sheet .The governing partial differential equation of flow, heat and mass transfer on considered flow are converted into the ordinary differential equations by means of similarity trans formations .The resulting equations are solved by the Runge-Kutta fourth order method with efficient shooting technique. Effects of various governing parameters on flow, heat and mass transfer are studied through the plots. The various numerical tables which are calculated and tabulated. A comparison of our present results with a previous study has been done and we found that an excellent agreement is there with the earlier results and of ours.
Analysis of mhd non darcian boundary layer flow and heat transfer over an exp...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
In the present paper we carried out several experiments in oxygen or dry air, at low
temperature of some metallic samples. In order to be able to extend or estimate the corrosion
phenomenon we made use of the modelling of oxygen diffusion through rust layers (oxides) and of
solving the parabolic equations of diffusion, respectively. The diffusion equation is important for
modelling the oxygen diffusion within biological systems and for modelling the neutron flux from
nuclear reactors.
ASPHALTIC MATERIAL IN THE CONTEXT OF GENERALIZED POROTHERMOELASTICITYijsc
This document summarizes a mathematical model of generalized porothermoelasticity for a porous asphalt material. The model considers a poroelastic half-space saturated with fluid and constructs governing equations for the stresses, strains, displacements, and temperatures of the solid and fluid phases. The equations are derived using the theory of generalized porothermoelasticity with one relaxation time. Numerical solutions to the equations are obtained by applying the model to an asphalt material subjected to thermal shock on its surface. Graphs of the temperature, stresses, strains, and displacements within the material are presented.
Chemical reaction and radiation effect on mhd flow past an exponentially acce...Alexander Decker
This document describes a mathematical analysis of MHD fluid flow past an exponentially accelerated vertical plate embedded in a porous medium. The analysis considers the effects of variable temperature, mass diffusion, radiation, and a heat source on the flow characteristics. The governing equations for this problem are derived and non-dimensionalized. The non-dimensional equations are then solved using the Laplace transform technique. The effects of various physical parameters like the magnetic field, radiation, heat generation, and chemical reaction on the velocity, temperature, and concentration profiles are determined.
Homotopy Analysis to Soret and Dufour Effects on Heat and Mass Transfer of a ...iosrjce
The objective of this paper is to study the Soret and Dufour effects on the free convection boundary
layer flow of an incompressible, viscous and chemically reacting fluid over a vertical plate in the presence of
viscous dissipation. The governing partial differential equations are converted to a set of ordinary differential
equations using suitable similarity transformations. The resulting equations are solved analytically using
homotopy analysis method (HAM). The convergence of obtained analytical solutions is explicitly discussed. The
effects of various parameters on dimensionless velocity, temperature and concentration profiles are discussed
with the help of graphs. The numerical values of skin friction, Nusselt number and Sherwood number for
different parameters are presented in tabular form. Our results are compared with the previously published
results and are found to be in good agreement.
This document summarizes a study that examines heat and mass transfer over a vertical plate in a porous medium with Soret and Dufour effects, a convective surface boundary condition, chemical reaction, and magnetic field. The governing equations for the fluid flow, heat transfer, and mass transfer are presented. Similarity solutions are used to transform the governing partial differential equations into ordinary differential equations, which are then solved numerically. The results are presented graphically to show the influence of various parameters on velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number.
EFFECT OF SLIP PARAMETER OF A BOUNDARY-LAYER FLOW FOR NANOFLUID OVER A VERTIC...IAEME Publication
In this paper we analyze the effect of momentum slip, thermal slip and solutal slip on stagnation point flow of MHD nanofluid towards stretching sheet .The governing partial differential equation of flow, heat and mass transfer on considered flow are converted into the ordinary differential equations by means of similarity trans formations .The resulting equations are solved by the Runge-Kutta fourth order method with efficient shooting technique. Effects of various governing parameters on flow, heat and mass transfer are studied through the plots. The various numerical tables which are calculated and tabulated. A comparison of our present results with a previous study has been done and we found that an excellent agreement is there with the earlier results and of ours.
Analysis of mhd non darcian boundary layer flow and heat transfer over an exp...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
In the present paper we carried out several experiments in oxygen or dry air, at low
temperature of some metallic samples. In order to be able to extend or estimate the corrosion
phenomenon we made use of the modelling of oxygen diffusion through rust layers (oxides) and of
solving the parabolic equations of diffusion, respectively. The diffusion equation is important for
modelling the oxygen diffusion within biological systems and for modelling the neutron flux from
nuclear reactors.
Using resonant ultrasound spectroscopy (RUS), the author will determine the complete elastic constant matrices of two thermoelectric single crystal samples, Ce.75Fe3CoSb12 and CeFe4Sb12. RUS involves measuring the resonant frequencies of a sample's vibrations, which depend on the sample's elastic constants, shape, orientation, and density. The author aims to obtain the elastic moduli from a single RUS spectrum for each sample. Understanding the elastic properties may help identify better thermoelectric materials by correlating low elastic stiffness with low thermal conductivity and higher thermoelectric efficiency. The author will compute the resonant frequencies using the samples' properties and compare to measurements.
This document summarizes a research paper that analyzes boundary layer flow of an incompressible fluid past a moving vertical plate in a porous medium. The analysis considers variable permeability, thermal conductivity, and viscous dissipation effects. Governing equations are non-dimensionalized and solved numerically. Results show that the permeability parameter and Grashof number significantly impact velocity and temperature profiles. Increasing the Eckert number decreases velocity and temperature. Accounting for variable thermal conductivity and permeability produces new patterns in velocity and temperature variation.
Convective Heat And Mass Transfer Flow Of A Micropolar Fluid In A Rectangular...IJERA Editor
In this chapter we make an investigation of the convective heat transfer through a porous medium in a Rectangular enclosure with Darcy model. The transport equations of liner momentum, angular momentum and energy are solved by employing Galerkine finite element analysis with linear triangular elements. The computation is carried out for different values of Rayleigh number – Ra micropolar parameter – R, spin gradient parameter, Eckert number Ec and heat source parameter. The rate of heat transfer and couple stress on the side wall is evaluated for different variation of the governing parameters.
ANALYSIS OF POSSIBILITY OF GROWTH OF SEVERAL EPITAXIAL LAYERS SIMULTANEOUSLY ...ijoejournal
We analyzed nonlinear model with varying in space and time coefficients of growth of epitaxial layers
from gas phase in a vertical reactor with account native convection. We formulate several conditions to
increase homogeneity of epitaxial layers with varying of technological process parameters.
Heat Transfer on Steady MHD rotating flow through porous medium in a parallel...IJERA Editor
We discussed the combined effects of radiative heat transfer and a transverse magnetic field on steady rotating flow of an electrically conducting optically thin fluid through a porous medium in a parallel plate channel and non-uniform temperatures at the walls. The analytical solutions are obtained from coupled nonlinear partial differential equations for the problem. The computational results are discussed quantitatively with the aid of the dimensionless parameters entering in the solution.
Measuring the Thermal Conductivities of Low Heat Conducting Disk Samples by M...IJERA Editor
This article aims to establish an experimental procedure to measure heat transmission coefficients in low heat conductive materials. The newly developed model takes as starting point the application of Fourier’s law to a disk sample when a temperature gradient is established between its faces. The power of a heating element is determined as the heat transfer coefficient of the problem disk. Initially, a glass vessel containing water is placed in direct contact with the heating element; then, a problem plastic disk is placed between this element and the glass vessel, treating the set as a composite wall. Prior to the above the water equivalent of a calorimetric set (vessel + water + accessories) and the thermal conductivity of the vessel must be determined. The thermal conductivity of the problem plastic disk sample is obtained for temperatures ranging from 30 to 70° C. The results reveal the existence of some type of structural transition for the problem material.
Effects Of Heat Source And Thermal Diffusion On An Unsteady Free Convection F...IOSR Journals
This document presents a mathematical analysis of unsteady free convection flow along a porous vertical plate with constant heat and mass flux in a rotating system. The analysis considers the effects of heat source and thermal diffusion. Perturbation technique is used to obtain expressions for velocity, temperature, concentration profiles, skin friction, and Nusselt number. Key findings include that increasing Schmidt number decreases skin friction, increasing Grashoff number increases skin friction, and increasing Prandtl number decreases Nusselt number. Graphs and tables are used to discuss the effects of various parameters on flow variables.
This document contains 19 multiple choice questions regarding mechanical properties of fluids. The questions cover topics such as pressure, density, buoyancy, and their relationships. Key details assessed include the definitions of fluid, gauge pressure, factors that influence pressure in liquids, and applications of fluid properties such as hydraulic jacks.
The document describes experiments conducted to determine the relative density of glucose and ethylene glycol using a pycnometer, and the dynamic and kinematic viscosity of glycerin using a falling sphere viscometer. The specific gravity of glucose and ethylene glycol were calculated from measurements of the pycnometer weight with and without the samples. Data from the viscometer test including steel ball mass, elapsed time, and diameter were used to calculate the dynamic and kinematic viscosity of glycerin. A graph of observed velocity versus the ratio of steel ball diameter to viscometer diameter showed consistency in the viscometer data.
Mathematical modelling and analysis of three dimensional darcyIAEME Publication
This document presents a mathematical model and numerical analysis of three-dimensional natural convection in an inclined porous box using the Darcy-Brinkman flow model. Governing equations for mass, momentum and energy are derived in dimensionless form using parameters like Rayleigh number, Darcy number, and aspect ratios. Numerical solutions are obtained for varying parameters like Rayleigh number, Darcy number, and angle of inclination. Results show that average Nusselt number first increases and then decreases with inclination angle, due to a transition from multicellular to unicellular flow patterns. Three-dimensional effects are more pronounced at lower aspect ratios.
Natural convection in a two sided lid-driven inclined porous enclosure with s...IAEME Publication
This document summarizes a study of natural convection in a two-sided lid-driven inclined porous enclosure with a sinusoidal thermal boundary condition on one wall. The governing equations for the fluid flow and heat transfer were solved numerically using a finite difference method. Results are presented for streamlines, isotherms, local Nusselt number, and average Nusselt number for various values of thermal radiation and heat generation parameters at different inclination angles of the enclosure. The key findings are that the flow pattern and temperature field depend significantly on the thermal parameters and inclination angle. Grid independence studies were performed to validate the numerical results.
Effect of rotation on the onset of Rayleigh-Bénard convection in a layer of F...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
This document discusses the hydrodynamic equations that describe neutral gas and plasma, and how they are modified to become the magnetohydrodynamic (MHD) equations when a conducting fluid is in a magnetic field. It introduces the continuity, momentum, and entropy equations for neutral gas hydrodynamics. It then explains how these are updated to the MHD equations by adding magnetic forces and Ohm's law relating current and fields. The key MHD equations derived include equations for momentum, entropy, and the magnetic field evolving due to motion and diffusion.
This document presents a simple phenomenological approach to modeling nanoindentation creep using conventional spring and dashpot elements. It describes how creep, which is the time-dependent increase in depth under a held load, can be modeled for a variety of materials using Maxwell and Voigt models. Equations are presented that relate the depth increase over time to the elastic modulus and viscosity of the material being tested. A method is described for fitting experimental nanoindentation data, including holding periods, to these equations to determine material properties while accounting for creep. The approach aims to provide an accessible way to analyze creep in nanoindentation that can be incorporated into computer programs.
B. Dragovich: On Modified Gravity and CosmologySEENET-MTP
This document discusses modified gravity and cosmological solutions. It introduces Einstein's theory of gravity and some of its problems, including lack of renormalizability and prediction of dark energy and dark matter. It then presents a nonlocal modified gravity model to address these issues. The model yields nonsingular bouncing cosmological solutions for the scale factor in the form a(t) = a0(σeλt + τe−λt), which exist for different values of spatial curvature. These solutions depend on the cosmological constant and satisfy the equations of motion for certain conditions on the parameters of the nonlocal gravity action.
Effect of radiation and chemical reaction on transient mhd free convective fl...Alexander Decker
This document analyzes the effects of radiation, chemical reaction, and transient magnetohydrodynamic (MHD) free convective flow over a vertical porous plate. The governing equations for mass, momentum, energy and concentration are presented and solved using perturbation methods. Parameters like chemical reaction rate, thermal and mass Grashof numbers, rarefaction, magnetic field, radiation, and suction are studied. Graphical results are discussed for practical parameter ranges regarding heat and mass transfer over the plate embedded in a porous medium under the influence of thermal radiation and chemical reaction.
Effect of radiation and chemical reaction on transient mhd free convective fl...Alexander Decker
This document analyzes the effects of radiation, chemical reaction, and transient magnetohydrodynamic (MHD) free convective flow over a vertical plate through porous media. The governing equations for mass, momentum, energy and concentration are presented and solved using a perturbation method. Parameters like chemical reaction rate, thermal and mass Grashof numbers, rarefaction, magnetic field, radiation, and suction are studied. Graphical results are discussed for practical parameter ranges regarding heat and mass transfer characteristics of the transient flow.
11.effect of radiation and chemical reaction on transient mhd free convective...Alexander Decker
This document analyzes the effects of radiation, chemical reaction, and transient magnetohydrodynamic (MHD) free convective flow over a vertical plate through porous media. The governing equations for this unsteady, viscous flow are presented and non-dimensionalized. The equations are then linearized using a perturbation method about a small parameter. Solutions are obtained by reducing the coupled, nonlinear partial differential equations to a set of ordinary differential equations that can be solved analytically. Parameters such as the chemical reaction parameter, thermal Grashof number, and radiation parameter are discussed in relation to their effects on the convective heat transfer along the plate.
This document presents a numerical solution for unsteady heat and mass transfer flow past an infinite vertical plate with variable thermal conductivity, taking into account Dufour number and heat source effects. The governing equations are non-linear and coupled, and were solved numerically using an implicit finite difference scheme. Various parameters, including Dufour number and heat source, were found to influence the velocity, temperature, and concentration profiles. Skin friction, Nusselt number, and Sherwood number were also calculated.
Effects of Variable Viscosity and Thermal Conductivity on MHD free Convection...theijes
This document summarizes a study that numerically investigates the effects of variable viscosity and thermal conductivity on magnetohydrodynamic (MHD) free convection and mass transfer flow over an inclined vertical surface in a porous medium with heat generation. The governing equations are reduced to ordinary differential equations using similarity transformations and then solved numerically using a shooting method. The results show that increasing the viscosity variation parameter, thermal conductivity parameter, magnetic parameter, permeability parameter, or Schmidt number decreases the fluid velocity, while increasing the heat generation parameter, local Grashof number, or mass Grashof number increases the fluid velocity. Skin friction, Nusselt number, and Sherwood number are also computed and presented in tabular form.
Thermal Effects in Stokes’ Second Problem for Unsteady Second Grade Fluid Flo...IOSR Journals
In this paper, we investigated the effects of magnetic field and thermal in Stokes’ second problem for unsteady second grade fluid flow through a porous medium. The expressions for the velocity field and the temperature field are obtained analytically. The effects of various pertinent parameters on the velocity field and temperature field are studied through graphs in detail.
Using resonant ultrasound spectroscopy (RUS), the author will determine the complete elastic constant matrices of two thermoelectric single crystal samples, Ce.75Fe3CoSb12 and CeFe4Sb12. RUS involves measuring the resonant frequencies of a sample's vibrations, which depend on the sample's elastic constants, shape, orientation, and density. The author aims to obtain the elastic moduli from a single RUS spectrum for each sample. Understanding the elastic properties may help identify better thermoelectric materials by correlating low elastic stiffness with low thermal conductivity and higher thermoelectric efficiency. The author will compute the resonant frequencies using the samples' properties and compare to measurements.
This document summarizes a research paper that analyzes boundary layer flow of an incompressible fluid past a moving vertical plate in a porous medium. The analysis considers variable permeability, thermal conductivity, and viscous dissipation effects. Governing equations are non-dimensionalized and solved numerically. Results show that the permeability parameter and Grashof number significantly impact velocity and temperature profiles. Increasing the Eckert number decreases velocity and temperature. Accounting for variable thermal conductivity and permeability produces new patterns in velocity and temperature variation.
Convective Heat And Mass Transfer Flow Of A Micropolar Fluid In A Rectangular...IJERA Editor
In this chapter we make an investigation of the convective heat transfer through a porous medium in a Rectangular enclosure with Darcy model. The transport equations of liner momentum, angular momentum and energy are solved by employing Galerkine finite element analysis with linear triangular elements. The computation is carried out for different values of Rayleigh number – Ra micropolar parameter – R, spin gradient parameter, Eckert number Ec and heat source parameter. The rate of heat transfer and couple stress on the side wall is evaluated for different variation of the governing parameters.
ANALYSIS OF POSSIBILITY OF GROWTH OF SEVERAL EPITAXIAL LAYERS SIMULTANEOUSLY ...ijoejournal
We analyzed nonlinear model with varying in space and time coefficients of growth of epitaxial layers
from gas phase in a vertical reactor with account native convection. We formulate several conditions to
increase homogeneity of epitaxial layers with varying of technological process parameters.
Heat Transfer on Steady MHD rotating flow through porous medium in a parallel...IJERA Editor
We discussed the combined effects of radiative heat transfer and a transverse magnetic field on steady rotating flow of an electrically conducting optically thin fluid through a porous medium in a parallel plate channel and non-uniform temperatures at the walls. The analytical solutions are obtained from coupled nonlinear partial differential equations for the problem. The computational results are discussed quantitatively with the aid of the dimensionless parameters entering in the solution.
Measuring the Thermal Conductivities of Low Heat Conducting Disk Samples by M...IJERA Editor
This article aims to establish an experimental procedure to measure heat transmission coefficients in low heat conductive materials. The newly developed model takes as starting point the application of Fourier’s law to a disk sample when a temperature gradient is established between its faces. The power of a heating element is determined as the heat transfer coefficient of the problem disk. Initially, a glass vessel containing water is placed in direct contact with the heating element; then, a problem plastic disk is placed between this element and the glass vessel, treating the set as a composite wall. Prior to the above the water equivalent of a calorimetric set (vessel + water + accessories) and the thermal conductivity of the vessel must be determined. The thermal conductivity of the problem plastic disk sample is obtained for temperatures ranging from 30 to 70° C. The results reveal the existence of some type of structural transition for the problem material.
Effects Of Heat Source And Thermal Diffusion On An Unsteady Free Convection F...IOSR Journals
This document presents a mathematical analysis of unsteady free convection flow along a porous vertical plate with constant heat and mass flux in a rotating system. The analysis considers the effects of heat source and thermal diffusion. Perturbation technique is used to obtain expressions for velocity, temperature, concentration profiles, skin friction, and Nusselt number. Key findings include that increasing Schmidt number decreases skin friction, increasing Grashoff number increases skin friction, and increasing Prandtl number decreases Nusselt number. Graphs and tables are used to discuss the effects of various parameters on flow variables.
This document contains 19 multiple choice questions regarding mechanical properties of fluids. The questions cover topics such as pressure, density, buoyancy, and their relationships. Key details assessed include the definitions of fluid, gauge pressure, factors that influence pressure in liquids, and applications of fluid properties such as hydraulic jacks.
The document describes experiments conducted to determine the relative density of glucose and ethylene glycol using a pycnometer, and the dynamic and kinematic viscosity of glycerin using a falling sphere viscometer. The specific gravity of glucose and ethylene glycol were calculated from measurements of the pycnometer weight with and without the samples. Data from the viscometer test including steel ball mass, elapsed time, and diameter were used to calculate the dynamic and kinematic viscosity of glycerin. A graph of observed velocity versus the ratio of steel ball diameter to viscometer diameter showed consistency in the viscometer data.
Mathematical modelling and analysis of three dimensional darcyIAEME Publication
This document presents a mathematical model and numerical analysis of three-dimensional natural convection in an inclined porous box using the Darcy-Brinkman flow model. Governing equations for mass, momentum and energy are derived in dimensionless form using parameters like Rayleigh number, Darcy number, and aspect ratios. Numerical solutions are obtained for varying parameters like Rayleigh number, Darcy number, and angle of inclination. Results show that average Nusselt number first increases and then decreases with inclination angle, due to a transition from multicellular to unicellular flow patterns. Three-dimensional effects are more pronounced at lower aspect ratios.
Natural convection in a two sided lid-driven inclined porous enclosure with s...IAEME Publication
This document summarizes a study of natural convection in a two-sided lid-driven inclined porous enclosure with a sinusoidal thermal boundary condition on one wall. The governing equations for the fluid flow and heat transfer were solved numerically using a finite difference method. Results are presented for streamlines, isotherms, local Nusselt number, and average Nusselt number for various values of thermal radiation and heat generation parameters at different inclination angles of the enclosure. The key findings are that the flow pattern and temperature field depend significantly on the thermal parameters and inclination angle. Grid independence studies were performed to validate the numerical results.
Effect of rotation on the onset of Rayleigh-Bénard convection in a layer of F...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
This document discusses the hydrodynamic equations that describe neutral gas and plasma, and how they are modified to become the magnetohydrodynamic (MHD) equations when a conducting fluid is in a magnetic field. It introduces the continuity, momentum, and entropy equations for neutral gas hydrodynamics. It then explains how these are updated to the MHD equations by adding magnetic forces and Ohm's law relating current and fields. The key MHD equations derived include equations for momentum, entropy, and the magnetic field evolving due to motion and diffusion.
This document presents a simple phenomenological approach to modeling nanoindentation creep using conventional spring and dashpot elements. It describes how creep, which is the time-dependent increase in depth under a held load, can be modeled for a variety of materials using Maxwell and Voigt models. Equations are presented that relate the depth increase over time to the elastic modulus and viscosity of the material being tested. A method is described for fitting experimental nanoindentation data, including holding periods, to these equations to determine material properties while accounting for creep. The approach aims to provide an accessible way to analyze creep in nanoindentation that can be incorporated into computer programs.
B. Dragovich: On Modified Gravity and CosmologySEENET-MTP
This document discusses modified gravity and cosmological solutions. It introduces Einstein's theory of gravity and some of its problems, including lack of renormalizability and prediction of dark energy and dark matter. It then presents a nonlocal modified gravity model to address these issues. The model yields nonsingular bouncing cosmological solutions for the scale factor in the form a(t) = a0(σeλt + τe−λt), which exist for different values of spatial curvature. These solutions depend on the cosmological constant and satisfy the equations of motion for certain conditions on the parameters of the nonlocal gravity action.
Effect of radiation and chemical reaction on transient mhd free convective fl...Alexander Decker
This document analyzes the effects of radiation, chemical reaction, and transient magnetohydrodynamic (MHD) free convective flow over a vertical porous plate. The governing equations for mass, momentum, energy and concentration are presented and solved using perturbation methods. Parameters like chemical reaction rate, thermal and mass Grashof numbers, rarefaction, magnetic field, radiation, and suction are studied. Graphical results are discussed for practical parameter ranges regarding heat and mass transfer over the plate embedded in a porous medium under the influence of thermal radiation and chemical reaction.
Effect of radiation and chemical reaction on transient mhd free convective fl...Alexander Decker
This document analyzes the effects of radiation, chemical reaction, and transient magnetohydrodynamic (MHD) free convective flow over a vertical plate through porous media. The governing equations for mass, momentum, energy and concentration are presented and solved using a perturbation method. Parameters like chemical reaction rate, thermal and mass Grashof numbers, rarefaction, magnetic field, radiation, and suction are studied. Graphical results are discussed for practical parameter ranges regarding heat and mass transfer characteristics of the transient flow.
11.effect of radiation and chemical reaction on transient mhd free convective...Alexander Decker
This document analyzes the effects of radiation, chemical reaction, and transient magnetohydrodynamic (MHD) free convective flow over a vertical plate through porous media. The governing equations for this unsteady, viscous flow are presented and non-dimensionalized. The equations are then linearized using a perturbation method about a small parameter. Solutions are obtained by reducing the coupled, nonlinear partial differential equations to a set of ordinary differential equations that can be solved analytically. Parameters such as the chemical reaction parameter, thermal Grashof number, and radiation parameter are discussed in relation to their effects on the convective heat transfer along the plate.
This document presents a numerical solution for unsteady heat and mass transfer flow past an infinite vertical plate with variable thermal conductivity, taking into account Dufour number and heat source effects. The governing equations are non-linear and coupled, and were solved numerically using an implicit finite difference scheme. Various parameters, including Dufour number and heat source, were found to influence the velocity, temperature, and concentration profiles. Skin friction, Nusselt number, and Sherwood number were also calculated.
Effects of Variable Viscosity and Thermal Conductivity on MHD free Convection...theijes
This document summarizes a study that numerically investigates the effects of variable viscosity and thermal conductivity on magnetohydrodynamic (MHD) free convection and mass transfer flow over an inclined vertical surface in a porous medium with heat generation. The governing equations are reduced to ordinary differential equations using similarity transformations and then solved numerically using a shooting method. The results show that increasing the viscosity variation parameter, thermal conductivity parameter, magnetic parameter, permeability parameter, or Schmidt number decreases the fluid velocity, while increasing the heat generation parameter, local Grashof number, or mass Grashof number increases the fluid velocity. Skin friction, Nusselt number, and Sherwood number are also computed and presented in tabular form.
Thermal Effects in Stokes’ Second Problem for Unsteady Second Grade Fluid Flo...IOSR Journals
In this paper, we investigated the effects of magnetic field and thermal in Stokes’ second problem for unsteady second grade fluid flow through a porous medium. The expressions for the velocity field and the temperature field are obtained analytically. The effects of various pertinent parameters on the velocity field and temperature field are studied through graphs in detail.
MATHEMATICAL MODEL FOR HYDROTHERMAL CONVECTION AROUND A RADIOACTIVE WASTE DEP...Jean Belline
A mathematical model of thermally induced water movement in the vicinity of a hard rock
depository for radioactive waste is presented and discussed. For the low permeability rocks envisaged for
geological disposal the equations describing heat and mass transfer become uncoupled and linear.
Analytic solutions to these linearized equations are derived for an idealized spherical model of a
depository in a uniformly permeable rock mass. As the hydrogeological conditions to be expected at a
disposal site are uncertain, examples of flow paths are presented for a range of different permeabilities,
porosities, boundary conditions and regional cross-flows.
Effects of conduction on magneto hydrodynamics mixed convection flow in trian...Alexander Decker
This document summarizes research on magnetohydrodynamic (MHD) mixed convection flow in triangular enclosures. Key points:
1) The study investigates the effects of conduction on MHD mixed convection flow in triangular enclosures using a finite element method.
2) Parameters like the Hartmann number, Prandtl number, Reynolds number, and Rayleigh number are found to strongly influence the flow and thermal fields.
3) Validation of the numerical code is done by comparing average Nusselt numbers to previous research on natural convection in triangular enclosures.
Radiation and magneticfield effects on unsteady naturalAlexander Decker
This document discusses research on the effects of thermal radiation and magnetic fields on unsteady natural convective flow of nanofluids past an infinite vertical plate with a heat source. The following key points are discussed:
- Governing equations for the unsteady, two-dimensional flow are derived taking into account radiation, magnetic fields, and thermophysical properties of nanofluids.
- The equations are solved numerically using Laplace transform techniques. Parameters like radiation, magnetic field, heat source, and nanoparticle volume fraction are examined.
- It is found that increasing the magnetic field decreases fluid velocity, while radiation, heat source, and nanoparticle volume fraction more strongly influence velocity and temperature profiles. Nanoparticle shape
Radiation and magneticfield effects on unsteady naturalAlexander Decker
This document discusses research on the effects of thermal radiation and magnetic fields on unsteady natural convective flow of nanofluids past an infinite vertical plate with a heat source. The following key points are discussed:
- Governing equations for the unsteady, two-dimensional flow are derived taking into account radiation, magnetic fields, and thermophysical properties of nanofluids.
- The equations are solved numerically using Laplace transform techniques. Parameters like radiation, magnetic field, heat source, and nanoparticle volume fraction are examined.
- It is found that increasing the magnetic field decreases fluid velocity, while radiation, heat source, and nanoparticle volume fraction have a greater influence on fluid velocity and temperature profiles. Nan
Two-temperature elasto-thermo diffusive response inside a spherical shell wit...IJERA Editor
The present work deals with the investigation of elasto-thermo diffusion interaction of a homogeneous isotropic
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Asphaltic Material in the Context of Generalized Porothermoelasticity
1. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
DOI:10.5121/ijsc.2017.8103 27
ASPHALTIC MATERIAL IN THE CONTEXT
OF GENERALIZED
POROTHERMOELASTICITY
Mohammad H. Alawi
College of engineering and Islamic architecture, p.o.box:7398 Makkah, Saudi Arabia
ABSTRACT
In this work, a mathematical model of generalized porothermoelasticity with one relaxation time for
poroelastic half-space saturated with fluid will be constructed in the context of Youssef model (2007). We
will obtain the general solution in the Laplace transform domain and apply it in a certain asphalt material
which is thermally shocked on its bounding plane. The inversion of the Laplace transform will be obtained
numerically and the numerical values of the temperature, stresses, strains and displacements will be
illustrated graphically for the solid and the liquid.
KEY-WORDS:
Porothermoelasticity; asphaltic Material; Thermal shock.
NOMENCLATURE
i i
u , U The displacements of the skeleton and fluid phases
, ,R, Q
λ µ The poroelastic coefficients
11 12 21 22
R , R , R , R The mixed and thermal coefficients
s s
0
T T
θ = − The temperature increment of the solid where s
T is the solid
f f
0
T T
θ = − The temperature increment of the fluid where f
T is the fluid
0
T The reference temperature
β The porosity of the material
s* f *
,
ρ ρ The density of the solid and the liquid phases respectively
( )
s s*
1
ρ = −β ρ The density of the solid phase per unit volume of bulk
f f *
ρ = βρ The density of the solid phase per unit volume of bulk
s
11 12
ρ = ρ −ρ The mass coefficient of solid phase
f
22 12
ρ = ρ − ρ The mass coefficient of fluid phase
12
ρ The dynamics coupling coefficient
s* f *
k , k The thermal conductivity of the solid and the fluid phases
( )
s s*
k 1 k
= −β The thermal conductivity of the solid phase
f f *
k k
= β The thermal conductivity of the fluid phase
k The interface thermal conductivity
2. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
28
s f
o o
,
τ τ The relaxation time of the solid and the fluid phases
ij
σ The stress components apply to the solid surface
σ The normal stress apply to the fluid surface
ij
e The strain component of the solid phase
ε The strain component of the fluid phase
s f
,
α α The coefficients of the thermal expansion of the phases
sf fs
,
α α The thermoelastic couplings between the phases
s f
E E
C ,C The specific heat of the solid and the fluid phases
sf
E
C The specific heat couplings between the phases
s s
s E
s
C
k
ρ
η = The thermal viscosity of the solid
f f
f E
f
C
k
ρ
η = The thermal viscosity of the fluid
sf
12 E
C
k
ρ
η = The thermal viscosity couplings between the phases
P 3 2
= λ + µ
11
R s fs
p Q
= α + α
22
R f sf
R 3 Q
= α + α
12
R f sf
Q P
= α + α
11
F s s
E
C
= ρ
22
F f f
E
C
= ρ
12
F ( )
s fs
12 22 o
3 R R T
= − α + α
21
F ( )
sf f
11 21 o
3 R R T
= − α + α
1. INTRODUCTION
Due to many applications in the fields of geophysics, plasma physics and related topics, an
increasing attention is being devoted to the interaction between fluid such as water and thermo
elastic solid, which is the domain of the theory of porothermoelasticity. The field of
porothermoelasticity has a wide range of applications especially in studying the effect of using
the waste materials on disintegration of asphalt concrete mixture.
Porous materials make their appearance in a wide variety of settings, natural and artificial and in
diverse technological applications. As a consequence, a variety of problems arise while dealing
with static and strength, fluid flow, heat conduction and the dynamics of such materials. In
connection with the later, we note that problems of this kind are encountered in the prediction of
behavior of sound-absorbing materials and in the area of exploration geophysics, the steadily
growing literature bearing witness to the importance of the subject [1]. The problem of a fluid-
saturated porous material has been studied for many years. A short list of papers pertinent to the
present study includes Biot [2-3], Gassmann [4], Biot and Willis [5], Biot [6], Deresiewicz and
Skalak [7], Mandl [8], Nur and Byerlee [9], Brown and Korringa [10], Rice and Cleary [11],
Burridge and Keller [12], Zimmerman et al. [13-14], Berryman and Milton [15], Thompson and
Willis [16], Pride et al. [17], Berryman and Wang [18], Tuncay and Corapcioglu [19], Alexander
3. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
29
and Cheng [20], Charlez, P. A., and Heugas, O. [21], Abousleiman et al. [22], Ghassemi, A.[23]
and Diek, A S. Tod [24].
The thermo-mechanical coupling in the poroelastic medium turns out to be of much greater
complexity than that in the classical case of impermeable elastic solid. In addition to thermal and
mechanical interaction within each phase, thermal and mechanical coupling occurs between the
phases, thus, a mechanical or thermal change in one phase results in mechanical and thermal
changes throughout the aggregate of asphaltic concrete mixtures. Following Biot, it takes one
physical model to consist a homogeneous, isotropic, elastic matrix whose interstices are filled
with a compressible ideal liquid both solid and liquid form continuous (and interacting) regions.
While viscous stresses in the liquid are neglected, the liquid is assumed capable of exerting a
velocity-dependent friction force on the skeleton. The mathematical model consists of two
superposed continuous phases each separately filling the entire space occupied by the aggregate.
Thus, there are two distinct elements at every point of space, each one characterized by its own
displacement, stress, and temperature. During a thermo-mechanical process they may interact
with a consequent exchange of momentum and energy.
Our development Proceeds by obtaining, the stress-strain-temperature relationships using the
theory of the generalized thermo elasticity with one relaxation time “Lord-Shulman” [25].
Moreover, to the usual isobaric coefficients of thermal expansion of the single-phase materials,
two coefficients appear which represent measures of each phase caused by temperature changes
in the other phase.
As a result of the presence of these "coupling" coefficients, it follows that coefficient of thermal
expansion of the dry material which differs than that of the saturated ones and the expansion of
the liquid in the bulk is not the same as of the liquid phase. Putting into consideration the
applications of geophysical interest, it takes the coefficient of proportionality in the dissipation
term to be independent of frequency, that is, we confine ourselves to low-frequency motions. The
last constituent of the theory is the equations of energy flux. Because the two phases in general,
will be at different temperatures in each point of the material, there is a rise of a heat-source term
in the energy equations representing the heat flux between the phases. It has been taken this
"interphase heat transfer" to be proportional to the temperature difference between the phases.
Finally, by using the uniqueness theorem the proof has been done.
Recently, Youssef has constructed a new version of theory of porothermoelasticity, using the
modified Fourier law of heat conduction. The most important advantage for this theory, is
predicting the finite speed of the wave propagation of the stress and the displacement as well as
the heat [26].
In this paper, a mathematical model of generalized porothermoelasticity with one relaxation time
for poroelastic half-space saturated with fluid will be constructed in the context of Youssef
model. We will obtain the general solution in the Laplace transform domain and apply it in a
certain asphalt material which is thermally shocked on its bounding plane. The inversion of the
Laplace transform will be obtained numerically and the numerical values of the temperature,
displacement and stress will be illustrated graphically.
BASIC FORMULATIONS
Starting by Youssef model of generalized porothermoelasticity [26], the linear governing
equations of isotropic, generalized porothermoelasticity in absence of body forces and heat
sources, are
4. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
30
(i)- Equations of motion
( ) s f
i,jj j,ij i,ii 11 ,i 12 ,i 11 i 12 i
u u QU R R u U
µ + λ +µ + − θ − θ = ρ +ρ &&
&& , (1)
s f
i,ii j,ij 21 ,i 22 ,i 12 i 22 i
RU Qu R R u U
+ − θ − θ = ρ +ρ &&
&& . (2)
(ii)- Heat equations
( )
2
s s s s f
,ii o 11 12 o 11 ii o 21
2
k F F T R e T R
t t
∂ ∂
θ = + τ θ + θ + + ε
∂ ∂
(3)
( )
2
f f f s f
,ii o 21 22 o 12 ii o 22
2
k F F T R e T R
t t
∂ ∂
θ = + τ θ + θ + + ε
∂ ∂
(4)
(iii)- Constitutive equations
( )
s f
ij ij kk ij 11 12 ij
2 e e Q R R
σ = µ + λ δ + ε − θ − θ δ , (5)
f s
kk 22 21
R Qe R R
σ = ε + − θ − θ . (6)
( ) i
,
i
ii
i
,
j
j
,
i
ij u
e
e
,
u
u
2
1
e =
=
+
= (7)
i,i
U
ε = . (8)
FORMULATION THE PROBLEM
We will consider one dimensional half-space 0 x
≤ < ∞ is filled with porous, isotropic and
elastic material which is considered to be at rest initially. The displacement will be considered to
be in one dimensional as follows:
( ) ( ) ( )
1 2 3
u u x,t , u x,t u x,t 0
= = = , (9)
( ) ( ) ( )
1 2 3
U U x,t , U x,t U x,t 0
= = = . (10)
Then the governing equations (1)-(8) will take the forms:
(a) Equations of motion
( ) ( ) ( ) ( ) ( )
s f
2 2
11 12 11 12
2 2
R R
u Q U
u U
x 2 x 2 x 2 x 2 2
∂θ ∂ θ ρ ρ
∂ ∂
+ − − = +
∂ λ + µ ∂ λ + µ ∂ λ + µ ∂ λ + µ λ + µ
&&
&& , (11)
s f
2 2
21 22 21 22
2 2
R R
U Q u
u U
x R x R x R x R R
∂ θ ∂θ ρ ρ
∂ ∂
+ − − = +
∂ ∂ ∂ ∂
&&
&& . (12)
(b) Equation of heat
2 s 2
s s f o 11 o 21
11 12
o
2 2 s s s s
T R T R
F F u U
x t t k k k x k x
∂ θ ∂ ∂ ∂ ∂
= + τ θ + θ + +
∂ ∂ ∂ ∂ ∂
, (13)
5. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
31
2 f 2
f s f o 12 o 22
21 22
o
2 2 f f f f
T R T R
F F u U
x t t k k k x k x
∂ θ ∂ ∂ ∂ ∂
= + τ θ + θ + +
∂ ∂ ∂ ∂ ∂
. (14)
(c) The constitutive relations
( ) ( ) ( ) ( )
s f
xx 11 12
R R
u Q U
2 x 2 x 2 2
σ ∂ ∂
= + − θ − θ
λ + µ ∂ λ + µ ∂ λ + µ λ + µ
, (15)
f s
22 21
R R
U Q u
R x R x R R
σ ∂ ∂
= + − θ − θ
∂ ∂
. (16)
u
e
x
∂
=
∂
(17)
U
x
∂
ε =
∂
. (18)
Using the non-dimensional variables as follows:
( ) ( ) ( ) ( ) ( ) ( ) ij
2 s f s f
o 0 o 0 0 ij
u ,U ,x c u,U,x , t , c t, , , T , , ,
2 R
σ σ
′ ′ ′ ′ ′ ′ ′ ′ ′
= η τ = η τ θ θ = θ θ σ = σ =
λ + µ
where
sf
2 12 E
o
12
C
2
c ,
k
ρ
λ + µ
= η =
ρ
.
Then, we get
( ) ( ) ( )
2 2 s f
0 11 0 12 11
2 2
12
T R T R
u Q U
u U
x 2 x 2 x 2 x
ρ
∂ ∂ ∂ θ ∂ θ
+ − − = +
∂ λ + µ ∂ λ + µ ∂ λ + µ ∂ ρ
&&
&& , (19)
( ) ( )
2 2 s f
0 21 0 22 22
2 2
12
2 2
T R T R
U Q u
u U
x R x R x R x R R
λ + µ λ + µ
ρ
∂ ∂ ∂ θ ∂ θ
+ − − = +
∂ ∂ ∂ ∂ ρ
&&
&& . (20)
2 s 2 s
s s f
12 11 21
o
2 2 s s s
F R R
u U
x t t k k x k x
∂ θ ∂ ∂ η ∂ ∂
= + τ θ + θ + +
∂ ∂ ∂ η η η ∂ η ∂
, (21)
2 f 2 f
f s f
21 12 22
o
2 2 f f f
F R R
u U
x t t k k x k x
∂ θ ∂ ∂ η ∂ ∂
= + τ θ + θ + +
∂ ∂ ∂ η η η ∂ η ∂
. (22)
( ) ( ) ( )
s f
0 11 0 12
xx
T R T R
u Q U
x 2 x 2 2
∂ ∂
σ = + − θ − θ
∂ λ + µ ∂ λ + µ λ + µ
, (23)
f s
0 22 0 21
T R T R
U Q u
x R x R R
∂ ∂
σ = + − θ − θ
∂ ∂
. (24)
In the above equation, we dropped the prime for convenient.
6. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
32
FORMULATION THE PROBLEM IN LAPLACE TRANSFORM DOMAIN
Applying the Laplace transform for the both sides of the equations (19)-(24) which is defined as
follows:
( ) ( ) st
0
f s f t e dt
∞
−
= ∫ ,
then, we get
2 s f
11 12 13 14
2
d u d d
L u L U L L
d x dx d x
θ θ
= + + + , (25)
2 s f
21 22 23 24
2
d U d d
L u L U L L
d x dx d x
θ θ
= + + + , (26)
2 s
s f
31 32 33 34
2
d d u d U
L L L L
d x d x d x
θ
= θ + θ + + , (27)
2 f
s f
41 42 43 44
2
d d u d U
L L L L
d x d x d x
θ
= θ + θ + + , (28)
s f
xx 11 12
d u d U
A A A
d x d x
σ = + − θ − θ , (29)
s f
21 22
d U du
B A A
d x dx
σ = + − θ − θ , (30)
d u
e
d x
= (31)
d U
d x
ε = . (32)
where
11 21 12 21 11 21 12 22
11 12 13 14
C AC C AC A AA A AA
L , L , L , L
1 AB 1 AB 1 AB 1 AB
− − − −
= = = =
− − − −
,
21 11 22 12 21 11 22 12
21 22 23 24
C BC C BC A BA A BA
L , L , L , L
1 AB 1 AB 1 AB 1 AB
− − − −
= = = =
− − − −
,
( ) ( ) ( ) ( )
s 2 s s 2 s 2 s 2
o o 12 o 11 o 21
31 32 33 34
s s s
s s s s F s s R s s R
L , L ,L , L
k k k
+ τ η + τ + τ + τ
= = = =
η η η η
,
( ) ( ) ( ) ( )
f 2 f 2 f f 2 f 2
o 21 o o 12 o 22
41 42 43 44
f f f
s s F s s s s R s s R
L , L , L , L
k k k
+ τ + τ η + τ + τ
= = = =
η η η η
.
( ) ( ) ( )
2 2
0 11 0 12 11
11 12 11 12
12
T R T R
Q
A ,A , A , C s , C s
2 2 2
ρ
= = = = =
λ + µ λ + µ λ + µ ρ
( ) ( )
2 2
0 21 0 22 22
21 22 21 22
12
2 2
T R T R
Q
B , A , A , C s , C s ,
R R R R R
λ + µ λ + µ
ρ
= = = = =
ρ
By using equations (25)-(28), we get
7. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
33
8 6 4 2
x x x x
D aD bD cD d u 0
− + − + =
, (33)
8 6 4 2
x x x x
D aD bD cD d U 0
− + − + =
, (34)
8 6 4 2 s
x x x x
D aD bD cD d 0
− + − + θ =
, (35)
8 6 4 2 f
x x x x
D aD bD cD d 0
− + − + θ =
, (36)
where
11 13 33 14 43 22 23 34 24 44 31 42
a = L + L L + L L + L + L L + L L + L + L
( ) ( )
( )
( )
( )
( )
11 22 23 34 24 44 31 42 12 21 23 33 24 43
13 21 34 22 33 24 34 43 33 44 32 43 33 42
14 21 44 22 43 23 33 44 34 43 31 43 33 41
22 31 42 23 34 42
b= L L +L L +L L +L +L - L L + L L + L L -
L L L - L L + L (L L - L L ) + L L - L L -
L L L - L L + L L L - L L - L L + L L
+ L L + L + L (L L ( )
32 44 24 31 44 34 41 31 42 32 41
- L L )+ L L L - L L + L L - L L
11 22 31 42 23 34 42 32 44 24 31 44 34 41 31 42 32 41
12 21 31 42 23 33 42 32 43 24 31 43 33 41
13 21 32 44 34 42 22 33 42 32 43
c= L (L (L + L ) + L (L L - L L ) +L (L L - L L ) +L L - L L )
- L (L (L + L ) + L (L L - L L ) + L (L L - L L ))
+ L (L (L L - L L ) +L (L L - L L ) 14 21 31 44 34 41
22 33 41 31 43 22 31 42 32 41
) - L (L (L L - L L )
+ L (L L - L L )) + L (l L - L L )
11 22 31 42 32 41 12 21 32 41 31 42
d =L L (L L - L L ) + L L (L L - L L ) ,
and
n
n
x n
d
D
d x
= .
According to equations (33)-(36) and to bounded state of functions at infinity, we can consider
the following forms
( ) i
4
x
i
i 1
u x,s e−λ
=
= α
∑ , (37)
( ) i
4
x
i
i 1
U x,s e−λ
=
= β
∑ , (38)
( ) i
4
x
s
i
i 1
x,s e−λ
=
θ = γ
∑ , (39)
( ) i
4
x
f
i
i 1
x,s e−λ
=
θ = ω
∑ , (40)
where i , i 1,2,3,4
±λ = are the roots of the characteristic equation of the system (33)-(36) which
takes the form
8 6 4 2
a b c d 0
λ − λ + λ − λ + = , (41)
To get the relations between the parameters i i i
, ,
β γ ω and i
α , we will use equations (26)-(28) in
the following forms
8. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
34
2 s f
x 22 23 x 24 x 21
D L U L D L D L u
− − θ − θ =
, (42)
2 s f
34 x 31 32 33 x
L U D L L L D u
− + − θ − θ =
, (43)
s 2 f
44 x 41 x 42 43 x
L D U L D L L D u
− − θ + − θ =
, (44)
Inserting the formulas in (37)-(40) into equations (42)-(44), we get
( )
2
i 22 i 23 i i 24 i i 21 i
L L L L , i 1,2,3,4
λ − β + λ γ + λ ω = α = , (45)
( )
2
34 i i 31 i 32 i 33 i i
L L L L , i 1,2,3,4
− β + λ − γ − ω = − λ α = , (46)
( )
2
44 i i 41 i i 42 i 43 i i
L L L L , i 1,2,3,4
λ β − γ + λ − ω = − λ α = , (47)
By solving the system in (45)-(47), we obtain
i
i i
i
H
, i 1,2,3,4
W
β = α = ,
i
i i
i
G
, i 1,2,3,4
W
γ = α = ,
i
i i
i
F
, i 1,2,3,4
W
γ = α = ,
Where
4 2
i i 21 23 33 24 43 i 21 31 42 23 33 42 32 43
24 31 43 33 41 21 31 42 32 41
H ( (L + L L + L L ) - (L (L + L ) + L (L L - L L ) +
L (L L - L L )) + L (L L - L L )) , i 1,2,3,4
= − λ λ
=
5 3 2
i 33 i i 22 33 24 33 44 32 43 33 42 34 i 21 24 43
21 32 44 22 33 42 32 43 21 34 42
G = L - (L L + L L L - L L + L L ) - L (L + L L )
+ (L L L + L (L L - L L )) + L L L , i 1,2,3,4
λ λ λ
λ =
5 3 2
i 43 i i 21 44 22 43 23 33 44 31 43 33 41 23 43 34 i
i 21 31 44 22 33 41 31 43 21 34 41
F = L + (L L - L L + L L L - L L +L L ) + L L L
- (L L L + L (L L - L L )) - L L L , i 1,2,3,4
λ λ λ
λ =
6 4 3 2
i i i 22 24 44 31 42 23 34 i i 22 31 42 23 32 44
24 31 44 31 42 32 41 34 i 23 42 24 41
22 31 42 32 41
W = - + (L + L L + L + L ) - L L - (L (L + L ) - L L L
+L L L + L L - L L ) + L (L L - L L )
+ L (L L - L L ) , i 1,2,3,4
λ λ λ λ
λ
=
Hence, we have
( ) i
4
x
i
i
i 1 i
H
U x,s e
W
−λ
=
= α
∑ , (48)
( ) i
4
x
s i
i
i 1 i
G
x,s e
W
−λ
=
θ = α
∑ , (49)
( ) i
4
x
f i
i
i 1 i
F
x,s e
W
−λ
=
θ = α
∑ , (50)
To get the values of the parameters i
α , we have to apply the boundary conditions as follows;
9. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
35
(1) The thermal conditions
We will consider the bounding plane surface of the medium at x = 0 has been thermally loaded by
thermal shock as follows:
( ) ( ) ( )
s
0
0,t 1 H t
θ = −β θ , (51)
and
( ) ( )
f
0
0,t H t
θ = βθ , (52)
where H(t) is the Heaviside unite step function and 0
θ is constant which gives after using the
Laplace transform the following conditions
( )
( ) 0
s 1
0,s
s
−β θ
θ = , (53)
and
( )
f 0
0,s
s
βθ
θ = , (54)
(2) The mechanical conditions
We will consider the bounding plane surface of the medium at x = 0 has been connected to a rigid
surface which prevents any displacement to accrue on that surface, i.e.
( )
u 0,t 0
= , (55)
and
( )
U 0,t 0
= , (56)
which gives after using the Laplace transform the following conditions
( )
u 0,s 0
= , (57)
and
( )
U 0,s 0
= . (58)
After using the boundary conditions in (53), (54), (57) and (58), we get the following system
4
i
i 1
0
=
α =
∑ , (59)
4
i
i
i 1 i
H
0
W
=
α =
∑ , (60)
( )
4
0
i
i
i 1 i
1
G
W s
=
−β θ
α =
∑ , (61)
4
0
i
i
i 1 i
F
W s
=
βθ
α =
∑ , (62)
Then we get
10. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
36
( )
( )
( )
2 3 4 4 3 3 4 4 3 3 4 4 3
0 1
1 3 2 4 4 2 2 4 4 2 2 4 4 2
4 2 3 3 2 2 3 3 2 2 3 3 2
W (F H - F H +G H G H )-F H F H -
W
= W (F H - F H +G H G H )-F H F H
s
W (F H - F H +G H G H )-F H F H
β − +
θ
α β − + +
∆
β − +
,
( )
( )
( )
1 3 4 4 3 3 4 4 3 3 4 4 3
0 2
2 3 1 4 4 1 1 4 4 1 1 4 4 1
4 1 3 3 1 1 3 3 1 1 3 3 1
W (F H - F H +G H G H )-F H F H -
W
= W (FH - F H +G H G H )-FH F H
s
W (FH - F H +G H G H )-FH F H
β − +
θ
α − β − + +
∆
β − +
,
( )
( )
( )
1 2 4 4 2 2 4 4 2 2 4 4 2
0 3
3 2 1 4 4 1 1 4 4 1 1 4 4 1
4 1 2 2 1 1 2 2 1 1 2 2 1
W (F H - F H +G H G H )-F H F H -
W
= W (FH - F H +G H G H )-FH F H
s
W (FH - F H +G H G H )-FH F H
β − +
θ
α β − + +
∆
β − +
,
( )
( )
( )
1 2 3 3 2 2 3 3 2 2 3 3 2
0 4
4 2 1 3 3 1 1 3 3 1 1 3 3 1
3 1 2 2 1 1 2 2 1 1 2 2 1
W (F H - F H +G H G H )-F H F H -
W
= W (FH - F H +G H G H )-FH F H
s
W (FH - F H +G H G H )-FH F H
β − +
θ
α − β − + +
∆
β − +
,
where
1 2 3 4 4 3 3 4 2 2 4 4 2 3 3 2
2 1 3 4 4 3 3 4 1 1 4 4 1 3 3 1
3 1 2 4 4 2 2 4 1 1 4 4 1 2 2 1
4 1 2 3 3 2
= - W (F (G H - G H ) + F (G H - G H ) + F (G H - G H )) +
W (F (W H - G H ) + F (G H - G H ) + F (G H - G H )) -
W (F (G H - G H ) + F (G H - G H ) + F (G H - G H )) +
W (F (G H - G H ) +
∆
2 3 1 1 3 3 1 2 2 1
F (G H - G H ) + F (G H - G H ))
,
Those complete the solution in the Laplace transform domain.
NUMERICAL INVERSION OF THE LAPLACE TRANSFORMS
In order to invert the Laplace transforms, we adopt a numerical inversion method based on a
Fourier series expansion [27].
By this method the inverse )
t
(
f of the Laplace transform ( )
s
f is approximated by
( ) ( )
ct N
1
k 1
1 1 1
e 1 i k i k t
f t f c R1 f c exp , 0 t 2t,
t 2 t t
=
π π
= + + < <
∑
Where N is a sufficiently large integer representing the number of terms in the truncated Fourier
series, chosen such that
( ) 1
1 1
i N i N t
exp c t R1 f c exp
t t
π π
+ ≤ ε
,
where ε1 is a prescribed small positive number that corresponds to the degree of accuracy
required. The parameter c is a positive free parameter that must be greater than the real part of all
the singularities of ( )
s
f . The optimal choice of c was obtained according to the criteria described
11. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
37
in [27].
NUMERICAL RESULTS AND DISCUSSION
The Ferrari's method has been constructed by using the FORTRAN program to solve equation
(41). The material properties of asphaltic material saturated by water have been taken as follow
[28], [29]:
*
o 11 2 11 2 11 2
0
11 2 s -5 o 1 s* 3 3
11
s 1 o 1 s 1 o 1 s
E o
T 27 C, Q 0.4853 10 dyne.cm , R 0.0362 10 dyne.cm , 0.2160 10 dyne.cm
0.0926 10 dyne.cm , 2.16 10 C , 2.35 gm.cm , 0.002 gm.cm
k 0.8W m k , C 800 J.kg . C , 0.02s, k 0.001
− − −
− − − −
− − − −
= = × = × λ = ×
µ = × α = × ρ = ρ =
= = τ = = 1 o 1
W m k
− −
*
f sf fs o 1 f * 3 f 1 o 1 f 1 o 1
E
f
o
0.0001 C , 0.82 gm.cm , k 0.3W m k , C 1.9cal.gm . C ,
0.00001s,
− − − − − −
α = α = α = ρ = = =
τ =
We will take the non-dimensional x variable to be in interval 0 x 1
≤ ≤ and all the results will be
calculated at the same instance t 0.1
= for two different values of the porosity β of the material
when 0.25
β = and 0.35
β = .
The temperature, the stress, the strain and the displacement for the solid and the liquid have been
shown in figures 1-8 respectively. We can see that, the value of the porosity has a significant
effect on all the studied fields.
Figure 1 shows the temperature increment distribution of the solid with two different values of
the porosity; 0.25
β = and 0.35
β = . It shows that the porosity parameter has a significant effect.
Figure 2 shows the temperature increment distribution of the liquid with two different values of
the porosity; 0.25
β = and 0.35
β = . We can see that the porosity parameter has a significant
effect where the liquid temperature increases when the porosity increases.
Figure 3 shows the stress distribution of the solid with two different values of the porosity;
0.25
β = and 0.35
β = . We can see that the porosity parameter has a significant effect where the
absolute value of the stress acts on the solid increases when the porosity increases.
Figure 4 shows the stress distribution of the liquid with two different values of the porosity;
0.25
β = and 0.35
β = . We can see that the porosity parameter has a significant effect where the
absolute value of the stress acts on the liquid increases when the porosity increases for wide range
of x.
Figures 5-8, show that the porosity parameter has significant effects on the deformation and the
displacement for both medium solid and liquid. The absolute value of the peak points (sharp
points) increase when the value of the porosity parameter increases for the both medium solid and
water.
12. International Journal on Soft Computing (IJSC) Vol.8, No. 1, February 2017
38
CONCLUSION
In studying a mathematical model of generalized porothermoelasticity with one relaxation time
for poroelastic half-space saturated with fluid in the context of Youssef model we found that:
1- The porosity parameter of the poroelastic material has significant effects on the
temperature, the stress, the deformation and the displacement distributions for the both
medium the solid and the liquid.
2- Youssef model of porothermoelasticity with one relaxation time introduce finite speed of
thermal wave propagation which agree with realistic physical behavior for the solid and
the liquid.
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