Thermal Relaxation Parameter Research Articles

Unsteady Hydromagnetic Boundary Layer Flow of a Nanofluid over a Stretching Sheet: Using Cattaneo-Christov Heat Flux Model

The present investigation has put a focus on the hydromagnetic boundary layer unsteady flow of a nanofluid over a stretching sheet. A new heat flux model named Cattaneo-Christov is applied as the substitution of classical Fourier’s law. Buongiorno’s model is incorporated. The coupled non-linear transformed equations are solved numerically by using shooting technique with MATLAB bvp4c package. The obtained results are presented and discussed through graphs and tables in detail. Our results reveal that the unsteady parameter reduces all the three boundary layer thickness. The thermal relaxation parameter exhibits a non-conducting nature that makes the decline in fluid temperature.

Numerical computations of fractional nonlinear Hartmann flow with revised heat flux model

In this communication we have examined the magneto-hydrodynamic flow of viscoelastic liquid with revise version of thermal flux. Thermal conductivity of the fluid is variable. Effects of body forces such as gravity are also examined in terms of convection. Non-integer time derivatives are used for the better understanding of viscoelastic flow characteristics. In literature, these derivatives have been proved suitable for the predictions of hereditary systems such as viscoelastic flows. Heat transfer is analyzed in the flow field via modified fractional thermal gradient. Temperature gradient exists between the flow boundaries and increases with the passage of time. Flow is generated by the variable movement of flow boundary. All the segments of numerical modeling are completely addressed in this paper i.e. physical system, mathematical model, simulation, prediction, validation and verification. Unsteady motion of in-compressible fluid is directed by differential equations with fractional partial derivatives, for the solutions of these equations numerical techniques are necessary to use. Governing equations with appropriate flow conditions have been discretized by finite difference along with finite element algorithms. Theoretical error appraisals are hypothesized and created numerical plan has been approved by doing error investigations numerically regarding spatial directions. We have shown the estimate error and plotted the error curves in logarithmic scales of L2(Ω) and H1(Ω) norms by taking absolute approximations of the space discretization. Local Nusselt number and coefficients of skin friction are determined for non-integer viscoelastic model. Flow field is simulated for various values of physical parameters. The obtained results showed that fractional exponent α has opposite effects on the velocity, temperature profiles for time t=2. It is also noted that temperature gradient increases by the increase of fractional number α and thermal relaxation parameter γ. Viscoelastic flows in polymer and plastics industries can be addressed by similar technique.

MHD Flow of a Carreau Fluid Past a Stretching Cylinder with Cattaneo-Christov Heat Flux Using Spectral Relaxation Method

A theoretical investigation of a hydromagnetic boundary layer flow of Carreau fluid over a stretching cylinder with surface slippage and temperature jump is presented in this paper. It is assumed that heat transfer characteristics of the flow follows Cattaneo-Christov heat flux model base on conventional Fourier’s law with thermal relaxation time. The spectral relaxation method (SRM) is being utilized to provide the solution of highly nonlinear system of coupled partial differential equations converted into dimensionless governing equations. The behaviour of flow parameters on velocity, temperature distributions are sketched as well as analyzed physically. The result indicates that the temperature distribution decay for higher temperature jump and thermal relaxation parameters respectively.

Thermal stratification effects on mixed convective Maxwell fluid flow with variable thermal conductivity and homogeneous/heterogeneous reactions

In this article, the magnetohydrodynamic mixed convection Maxwell fluid flow over a vertical stretching sheet with variable thickness close to a stagnation point is explored numerically. The variable thermal conductivity has been assumed for the analysis of heat transfer characteristics. Furthermore, homogeneous–heterogeneous chemical reactions and thermal stratification have also been considered in the present study. Both the assisting and the opposing flows are discussed through tables and graphs. The ordinary differential equations are obtained via suitable similarity transformations. The numerical solutions of the non-dimensional velocity, temperature and concentration profiles are obtained by employing the shooting method and examined for different physical parameters such as the thermal stratification, thermal relaxation, mixed convection, thermal conductivity, stretching ratio, homogeneous reaction and heterogeneous reaction parameters. It is found that the temperature upturns for the higher values of the Maxwell and thermal conductivity parameters. Moreover, the ascending values of the thermal stratification parameter decay the temperature profile. The local Nusselt number or the heat transfer rate at the wall is higher in the assisting flow regime as compared with the opposing flow regime.

MHD flow of Carreau fluid over a variable thickness melting surface subject to Cattaneo-Christov heat flux

PurposeThe purpose of this paper is to focus on MHD unsteady flow of Carreau fluid over a variable thickness melting surface in the presence of chemical reaction and non-uniform heat sink/source.Design/methodology/approachThe flow governing partial differential equations are transformed into ordinary ones with the help of similarity transformations. The set of ODEs are solved by a shooting technique together with the R.K.–Fehlberg method. Further, the graphs are depicted to scrutinize the velocity, concentration and temperature fields of the Carreau fluid flow. The numerical values of friction factor, heat and mass transfer rates are tabulated.FindingsThe results are presented for both Newtonian and non-Newtonian fluid flow cases. The authors conclude that the nature of three typical fields and the physical quantities are alike in both cases. An increase in melting parameter slows down the velocity field and enhances the temperature and concentration fields. But an opposite outcome is noticed with thermal relaxation parameter. Also the elevating values of thermal relaxation parameter/ wall thickness parameter/Prandtl number inflate the mass and heat transfer rates.Originality/valueThis is a new research article in the field of heat and mass transfer in fluid flows. Cattaneo–Christov heat flux model is used. The surface of the flow is assumed to be melting.

A three-dimensional generalized shock plate problem with four thermoviscoelastic relaxations

The three-dimensional generalized thermoviscoelastic shock plate problem is presented. Four thermoviscoelastic relaxations are used during this study, three of them are due to the generalized thermoelasticity models and the fourth is due to the viscosity parameter. The plate is thermally isolated at its bottom surface while its upper one is under a thermal shock. The transient thermal shock plate problem is presented according to a unified theory of generalized thermoelasticity. The classical coupled thermoelasticity model is used and two of its generalizations, namely Green–Lindsay and Lord–Shulman models, are also used. Normal mode analysis is adopted to get an analytical general solution of the present plate problem. The distributions of all variables are investigated along the plate directions. Different thermoelasticity theories are compared to present suitable conclusions. Some special findings are pointed out to show the effects of viscosity and other thermal relaxation parameters on thermoelastic interactions. Numerical results are plotted and tabulated to serve as benchmark results for future comparisons.

Analysis of Eyring–Powell liquid flow in curved channel with Cattaneo–Christov heat flux model

The article concentrates on the analysis of heat transfer phenomena incorporated with Cattaneo–Christov model of heat flux in magnetohydrodynamic Eyring–Powell fluid flow in a semipermeable curved channel. The flow equations are modeled by introducing the curvilinear coordinates system. The governing mathematical equations are reformed into ordinary differential equations by utilizing nonlinear type of similarity variables. The obtained mathematical model is solved numerically by utilizing shooting method. The obtained results are also validated with the well-known finite difference algorithm known as Keller-box method. The impact of diverse parameters on the flow and physical quantities like rate of heat transport and shear stress is investigated and discussed in detail via graphs and table. It is noticed that an increase in the fluid parameter increases the velocity of the fluid, whilst an increase in the thermal relaxation parameter decreases the temperature of the fluid.

Radiative nanofluid flow and heat transfer between parallel disks with penetrable and stretchable walls considering Cattaneo–Christov heat flux model

AbstractIn this work, we explore the unsteady squeezing flow and heat transfer of nanofluid between two parallel disks in which one of the disks is penetrable and the other is stretchable/shrinkable, in the presence of thermal radiation and heat source impacts, and considering the Cattaneo–Christov heat flux model instead of the more conventional Fourier's law of heat conduction. A similarity transformation is utilized to transmute the governing momentum and energy equations into nonlinear ordinary differential equations with the proper boundary conditions. The achieved nonlinear ordinary differential equations are solved by the Duan–Rach Approach (DRA). This method modifies the standard Adomian Decomposition Method by evaluating the inverse operators at the boundary conditions directly. The impacts of diverse active parameters, such as the suction/injection parameter, the solid volume fraction, the heat source parameter, the thermal relaxation parameter, and the radiation parameter on flow and heat transfer traits are examined. In addition, the value of the Nusselt number is calculated and portrayed through figures.

Investigation of second grade fluid through temperature dependent thermal conductivity and non-Fourier heat flux

Abstract Here we investigated stagnation point flow of second grade fluid over a stretchable cylinder. Heat transfer is characterized by non-Fourier law of heat flux and thermal stratification. Temperature dependent thermal conductivity and activation energy are also accounted. Transformations procedure is applying to transform the governing PDE’s into ODE’s. Obtained system of ODE’s are solved analytically by HAM. Influence of flow variables on velocity, temperature, concentration, skin friction and Sherwood number are analyzed. Obtained outcome shows that velocity enhanced through curvature parameter, viscoelastic parameter and velocities ratio variable. Temperature decays for larger Prandtl number, thermal stratification, thermal relaxation and curvature parameter. Sherwood number and concentration field show opposite behavior for higher estimation of activation energy, reaction rate, curvature parameter and Schmidt number.

Three-dimensional flow of Prandtl fluid with Cattaneo-Christov double diffusion

Abstract This research paper intends to investigate the 3D flow of Prandtl liquid in the existence of improved heat conduction and mass diffusion models. Flow is created by considering linearly bidirectional stretchable sheet. Thermal and concentration diffusions are considered by employing Cattaneo-Christov double diffusion models. Boundary layer approach has been used to simplify the governing PDEs. Suitable nondimensional similarity variables correspond to strong nonlinear ODEs. Optimal homotopy analysis method (OHAM) is employed for solutions development. The role of various pertinent variables on temperature and concentration are analyzed through graphs. The physical quantities such as surface drag coefficients and heat and mass transfer rates at the wall are also plotted and discussed. Our results indicate that the temperature and concentration are decreasing functions of thermal and concentration relaxation parameters respectively.

Analysis of Arrhenius activation energy in magnetohydrodynamic Carreau fluid flow through improved theory of heat diffusion and binary chemical reaction

In this work, we addressed the characterization of stagnation-point Carreau fluid flow induced due to stretching of chemically reactive surface. The energy expression is incorporated with a new theory of heat diffusion named Cattaneo-Christov, which is an advanced form of Fourier’s heat flux formula. The additional term of thermal relaxation time is arisen in Cattaneo-Christov model of heat diffusion. We considered modified Arrhenius energy function with chemical reaction effect. The theory of boundary layer is employed to govern the mathematical phenomenon. The ordinary differential system is obtained by the implementation of suitable similarity variables. The governing system of mathematical expressions is solved by using Runge–Kutta based MATLAB bvp4c package. The derived solutions are sketched for various values of physical constraints on quantities of interest. A comparative study is presented for validating the results. It is found that the larger value of the thermal relaxation parameter act as non-conductor. It also noted that the destructive binary reaction is favorable to enhance the species concentration.

Impacts of variable thermal conductivity on stagnation point boundary layer flow past a Riga plate with variable thickness using generalized Fourier’s law

Abstract This article explores the problem of two-dimensional, laminar, steady and boundary layer stagnation point slip flow over a Riga plate. The incompressible upper-convected Maxwell fluid has been considered as a rheological fluid model. The heat transfer characteristics are investigated with generalized Fourier’s law. The fluid thermal conductivity is assumed to be temperature dependent in this study. A system of partial differential equations governing the flow of an upper-convected Maxwell fluid, heat and mass transfer using generalized Fourier’s law is developed. The main objective of the article is to inspect the impacts of pertinent physical parameters such as the stretching ratio parameter ( 0 ⩽ A ⩽ 0.3 ) , Deborah number ( 0 ⩽ β ⩽ 0.6 ) , thermal relaxation parameter ( 0 ⩽ γ ⩽ 0.5 ) , wall thickness parameter ( 0.1 ⩽ α ⩽ 3.5 ) , slip parameter ( 0 ⩽ R ⩽ 1.5 ) , thermal conductivity parameter ( 0.1 ⩽ δ ⩽ 1.0 ) and modified Hartmann number ( 0 ⩽ Q ⩽ 3 ) on the velocity and temperature profiles. Suitable local similarity transformations have been used to get a system of non-linear ODEs from the governing PDEs. The numerical solutions for the dimensionless velocity and temperature distributions have been achieved by employing an effective numerical method called the shooting method. It is seen that the velocity profile shows the reduction in the velocity for the higher values of viscoelastic parameter and the thermal relaxation parameter. In addition, to enhance the reliability at the maximum level of the obtained numerical results by shooting method, a MATLAB built-in solver bvp4c has also been utilized.

Chemically reactive species in squeezed flow through modified Fourier’s and Fick’s laws

The squeezing flow of a Newtonian fluid with variable viscosity over a stretchable sheet embedded in Darcy porous medium is addressed. Cattaneo-Christov double diffusion models are adopted to disclose the salient features of heat and mass transport via variable thermal conductivity and variable mass diffusivity instead of conventional Fourier’s and Fick’s laws. Further, the concept of heat generation/absorption coefficient and first-order chemical reaction are also imposed to illustrate the characteristics of heat and mass transfer. Highly nonlinear computations are developed in dimensionless form and analyzed via the homotopic technique. The variation of flow parameters on velocity, concentration, and temperature distributions are sketched and disclosed physically. The results found that both concentration and temperature distributions decay for higher solutal and thermal relaxation parameters, respectively. Moreover, a higher chemical reaction parameter results in the reduction of the concentration field whereas the temperature profile enhances for a higher heat generation/absorption parameter.

Analytical investigation of third grade nanofluidic flow over a riga plate using Cattaneo-Christov model

Abstract This article addresses third grade nanofluidic flow instigated by riga plate and Cattaneo-Christov theory is adopted to investigate thermal and mass diffusions with the incorporation of newly eminent zero nanoparticles mass flux condition. The governing system of equations is nondimensionalized through relevant similarity transformations and significatory findings are attained by using optimal homotopy analysis method. The behaviors of affecting parameters for velocity, temperature and concentration profiles are depicted graphically and also verified through three dimensional patterns for some parameters. Values of skin friction coefficient and Nusselt number with the apposite discussion have been recorded. The current results reveal that temperature and concentration profiles experience decline when thermal and concentration relaxation parameters are augmented respectively.

Heat transfer analysis of Blasius and Sakiadis flow of MHD Radiated Carreau fluid with Cattaneo-Christov heat flux

A theoretical investigation is performed for studying the flow of heat transfer features of Sakiadis and Blasius flow of magnetohydrodynamic Carreau fluid with thermal radiation. We also considered the Cattaneo-Christov heat flux model to control the heat transfer phenomena. Numerical solutions are carried out Runge-Kutta based Shooting technique. The effects of various governing parameters on the flow quantities are demonstrated graphically. Also, calculated the friction factor coefficient and local Nusselt numbers for the various non-dimensional governing parameters such as radiation parameter, magnetic field parameter, power index parameter and thermal relaxation parameter. We found that the thermal relaxation and thermal radiation parameters are help to improve the rate of heat transfer.

Cattaneo-Christov heat flux effect on hydromagnetic radiative Oldroyd-B liquid flow across a cone/wedge in the presence of cross-diffusion

The present article scrutinizes the prominent characteristics of the Cattaneo-Christov heat flux on magnetohydrodynamic Oldroyd-B radiative liquid flow over two different geometries. The effects of cross-diffusion are considered in the modeling of species and energy equations. Similarity transformations are employed to transmute the governing flow, species and energy equations into a set of nonlinear ordinary differential equations (ODEs) with the appropriate boundary conditions. The final system of dimensionless equations is resolved numerically by utilizing the R-K-Fehlberg numerical approach. The behaviors of all physical pertinent flow controlling variables on the three flow distributions are analyzed through plots. The obtained numerical results have been compared with earlier published work and reveal good agreement. The Deborah numbers $ \gamma_{1}$ and $ \gamma_{2}$ have quite opposite effects on velocity and energy fields. The increase in thermal relaxation parameter $ \beta$ corresponds to a decrease in the fluid temperature. This study has salient applications in heat and mass transfer manufacturing system processing for energy conversion.

Three dimensional rotating flow of Powell-Eyring nanofluid with non-Fourier’s heat flux and non-Fick’s mass flux theory

This article numerically examines three dimensional boundary layer flow of a rotating Powell-Eyring nanofluid. In modeling heat transfer processes, non-Fourier heat flux theory and for mass transfer non-Fick’s mass flux theory are employed. This theory is recently re-initiated and it becomes the active research area to resolves some drawback associated with the famous Fourier heat flux and mass flux theory. The mathematical model of the flow problem is a system of non-linear partial differential equations which are obtained using the boundary layer analysis. The non-linear partial differential equations have been transformed into non-linear high order ordinary differential equations using similarity transformation. Employing bvp4c algorithm from matlab software routine, the numerical solution of the transformed ordinary differential equations is obtained. The governing equations are constrained by parameters such as rotation parameter λ, the non-Newtonian parameter N, dimensionless thermal relaxation and concentration relaxation parameters δt and δc. The impacts of these parameters have been discussed thoroughly and illustrated using graphs and tables. The findings show that thermal relaxation time δt reduces the thermal and concentration boundary layer thickness. Further, the results reveal that the rotational parameter λ has the effect of decreasing the velocity boundary layer thickness in both x and y directions. Further examination pinpoints that the skin friction coefficient along x-axis is an increasing and skin friction coefficient along y-axis is a decreasing function of rotation parameter λ. Furthermore, the non-Newtonian fluid parameter N has the characteristic of reducing the amount of local Nusselt numbers -f″(0) and -g″(0) both in x and y -directions.

Effect of Cattaneo–Christov heat flux on buoyancy MHD nanofluid flow and heat transfer over a stretching sheet in the presence of Joule heating and thermal radiation impacts

In this study, buoyancy MHD nanofluid flow and heat transfer over a stretching sheet in the presence of Joule heating and thermal radiation impacts, are studied. Cattaneo–Christov heat flux model instead of conventional Fourier’s law of heat conduction is applied to investigate the heat transfer characteristics. A similarity transformation is used to transmute the governing momentum and energy equations into non-linear ordinary differential equations with the appropriate boundary conditions. The obtained non-linear ordinary differential equations are solved numerically. The impacts of diverse active parameters such as the magnetic parameter, the radiation parameter, the buoyancy parameter, the heat source parameter, the volume fraction of nanofluid and the thermal relaxation parameter are examined on the velocity and temperature profiles. In addition, the value of the Nusselt number is calculated and presented through figures. The results demonstrate that the temperature profile is lower in the case of Cattaneo–Christov heat flux model as compared to Fourier’s law. Moreover, the Nusselt number raises with the raising volume fraction of nanofluid and it abates with the ascending the radiation parameter.

Thermal and concentration diffusion in Jeffery nanofluid flow over an inclined stretching sheet: A generalized Fourier's and Fick's perspective

Recent study illustrates the systematic survey of boundary layer heat and mass diffusion (Cattaneo-Christov model) of Jeffery fluid passed by an inclined stretching surface in the occurrence of magnetic field. Prevailing non-linear PDEs are converted into non-linear ODEs and then the problem is solved via RK-4 technique (using coefficients of Cash and Craps). The related important physical parameters such as M magnetic parameter, λT thermal buoyancy parameter, λC concentration buoyancy parameter, γ inclined stretching sheet parameter, PrPrandtl number, Nb Brownian motion parameter, Nt thermophoresis parameter and Le Lewis number are plotted graphically for velocity, temperature and concentration distributions. In order to check the double diffusive phenomenon, the impact of fluid relaxation parameter δm, thermal relaxation parameter δe and nanoparticle concentration relaxation parameter δc are reflected through tables and graphs. The main theme of present article is to explore its unique attempt towards the generalized version of conventional Fourier's law and Fick's law at nanostructure level. Assumption is drawn on the source of entire analysis and it is clear that the velocity and temperature profiles reduces for increasing values of fluid relaxation parameter, inclined sheet, magnetic field and Prandtl number.

Exploring the Cattaneo-Christov heat flux phenomenon on a Maxwell-type nanofluid coexisting with homogeneous/heterogeneous reactions

This specific article unfolds the efficacy of Cattaneo-Christov heat flux on the heat and mass transport of Maxwell nanofluid flow over a stretched sheet with changeable thickness. Homogeneous/heterogeneous reactions in the fluid are additionally considered. The Cattaneo-Christov heat flux model is initiated in the energy equation. Appropriate similarity transformations are taken up to form a system of nonlinear ODEs. The impact of related parameters on the nanoparticle concentration and temperature is inspected through tables and diagrams. It is renowned that temperature distribution increases for lower values of the thermal relaxation parameter. The rate of mass transfer is enhanced for increasing in the heterogeneous reaction parameter but the reverse tendency is ensued for the homogeneous reaction parameter. On the other side, the rate of heat transfer is getting enhanced for the Cattaneo-Christov model compared to the classical Fourier’s model for some flow factors. Thus the implication of the current study is to delve its unique effort towards the generalized version of traditional Fourier’s law at nano level.