Couple stress flow of exponentially stretching sheet with Cattaneo–Christov heat flux model

This paper deals with the effect of three-dimensional magnetohydrodynamic flow for a couple of stress fluids on an exponentially stretching sheet. The magnetic field is implemented normally to the surface. To observe the transfer of heat phenomenon, the Cattaneo–Christov flux model of heat is employed. Using similarity transformation, the substantial differential equations are reformed into ordinary differential equations. Eventually, the effects of different physical parameters are studied graphically. The drawback in Fourier heat flux model is removed by adding a new paramter known as thermal relaxation time by Cattaneo. This perimeter allows heat transportation by way of propagation of waves thermally at a defined speed. After this, the Cattaneo law is further modified by Christov–Christov to replace the ordinary derivative along Oldroyd's upper-convective derivative.     

Cattaneo–Christov double-diffusion model with Stefan blowing effect on copper–water nanofluid flow over a stretching surface

We often encounter many processes where the cooling rate is a key factor in deciding the features of a desired product. Due to increasing demands of controlled cooling systems, an effort is made to theoretically study the effect of volume fraction on mixed convective Cu–water nanofluid flow over a stretching surface with activation energy and thermal radiation. The nonlinear dynamical system is simplified using apt similarity variables and the obtained ordinary differential equations are dealt numerically using Runge–Kutta–Fehlberg method and shooting scheme. The thermal and solutal equations are modeled considering Cattaneo–Christov double-diffusion model. The flow problem is studied considering velocity slip and zero mass flux state at the surface. As a novelty, the present case considers the blowing effect at the surface to study massive species transport during nanofluid flow with Cattaneo–Christov double-diffusion model. The results show that an increase in strength of thermal radiation increases temperature and buoyancy ratio parameter, thereby escalating the skin friction coefficient. When thermal relaxation parameter changes from 0.001 to 0.005, heat transfer coefficient improves by 24.36%. Furthermore, with the change in value of the blowing parameter from 0.1 to 0.1015, the maximum value concentration of nanoparticles that is attained during the flow is increased by 7.15%.     

Cattaneo–Christov flux and entropy in thermofluidics involving shrinking surface

The paper examines radiative Casson boundary layer flow over an exponentially shrinking permeable sheet in a Cattaneo–Christov heat flux environment. The sheet is placed at the bottom of the fluid-saturated porous medium and suction is applied normally to the sheet to contain the vorticity. The radiative heat flux in the energy equation is assumed to follow the Rosseland approximation. Similarity transformation is performed to convert the governing partial differential equations into ordinary differential equations. The resulting boundary value problem is treated numerically employing Runge–Kutta fourth-order integration scheme along with the shooting method. The effects of pertinent parameters on quantities of interest are showcased graphically/in tabular form and are discussed. The dual profiles for velocity and temperature lead to a dual solution regime for entropy. It is found that critical mass suction rate and Nusselt number are substantially responsive to various parameters' values. Critical suction values decrease with a rise in Casson parameter β and permeability parameter K. Skin friction coefficient and Nusselt number show peculiar behavior for distinct branches of solutions.     

Cattaneo–Christov heat flux model for nanofluid flow over a curved stretching sheet: An application of Stefan blowing

The present work investigates the thermophoresis and Brownian motion effects in nanofluid flow over a curved stretching sheet (CSS). Also, the Cattaneo–Christov heat flux and Stefan blowing (SB) conditions are considered for studying heat and mass transport characteristics. The present work's novelty is associated with considerations of convective boundary and SB conditions in nanomaterial flow over a CSS. The coupled partial differential equations are changed to ordinary differential equations by employing suitable similarity variables, and the resultant model is numerically handled using Runge–Kutta–Fehlberg's fourth fifth-order method with the shooting scheme. The stimulation of the involved parameters/numbers on the flow, mass, and heat fields is broadly deliberated using suitable graphs. The present analysis's significant relevant outcomes are that the inclination in thermophoresis and Brownian motion parameters increases the heat transfer. The inclined values of the Brownian motion parameter decay the mass transfer. Furthermore, the increased values of both Schmidt number and SB parameter drop the mass transport. The increased values of the Brownian motion parameter and Schmidt number decays the rate of mass transference.     

Irreversibility analysis for Casson thermo-fluidics inside a cone: Cattaneo–Christov heat flux

In this communication, thermodynamic irreversibility arising in dissipative Casson fluid flow inside a cone is investigated. The boundary–layer flow is considered wherein the motion is caused due to a point sink at the cone's vertex and the movement of the wall of the cone. The wall of the cone is subjected to mass transpiration that alters the flow and thermal regime. The cone having fluid-saturated porous medium experiences Cattaneo–Christov heat flux. The configuration admits a similarity transformation that yields a boundary value problem (BVP) comprising an ordinary differential equation. The BVP is treated by the fourth-order R-K method along with the shooting algorithm. The system yields a dual solution for momentum and energy, which gives rise to a dual regime for entropy distribution. Numerical computations provide quantities of interest viz. velocity and temperature distributions, skin friction coefficient, Nusselt number, and entropy distribution. Phenomena exhibited through profiles/tables for velocity, temperature, entropy, streamlines, and other quantities of interest reveal interesting results.     

Significance of Dufour and Soret effects on the non-Darcy flow of Cross fluid by a tilted plate with radiation and chemical reaction: A Cattaneo–Christov heat flux model

The Darcy–Forchheimer flow model is substantial in the fields where a high flow rate effect is a common phenomenon, for instance, in petroleum engineering. In this paper, we aim to scrutinize the aspects of cross-diffusion effects on the non-Darcy flow of Cross fluid by a tilted plate with thermal radiation and chemical reaction. Metamorphosed equations are resolved with the combination of shooting and Runge–Kutta fourth-order procedures. The correlation coefficient is used to discuss the impact of pertinent parameters on friction factor and transfer rates (heat and mass). The main findings of this study are that the Dufour number escalates fluid temperature and the Soret number ameliorates the fluid concentration. It is observed that the fluid velocity minifies with the elevation in the Forchheimer number. And also, it is perceived that the heat transfer rate has a generous positive relationship with the thermal relaxation parameter. Furthermore, validation of current results with the earlier results under specified conditions is performed and an adequate concord is seen.     

Hybrid nanofluid flow over a stretched cylinder with the impact of homogeneous–heterogeneous reactions and Cattaneo–Christov heat flux: Series solution and numerical simulation

The Catteno–Christov heat flux plays a dynamic role in flow of heat enhancement in various manufacturing, industrial, and engineering applications. This present work focuses on the influence of Catteno–Christov heat flux model on Darcy–Forchheimer flow of a hybrid nanofluid placed in a porous medium. The formulation of the mathematical model is done by considering a fluid with two different nanoparticles Al₂O₃ and Cu dispersed in the water as the base fluid. The set of partial differntial equations is reduced by using similarity variables and boundary conditions to obtain ordinary differntial equations. The coupled nonlinear governing differential equations are solved using Runge–Kutta fourth–fifth order (RKF-45). The impact of numerous dimensionless parameters on the velocity, thermal, and concentration profiles are plotted and studied. Furthermore, the coefficient of skin friction for the relevant parameters are analysed through graphs. Result reveals that, increase in the porosity parameter declines the velocity gradient and shoots up the thermal and concentration gradients. Inclination in magnetic parameter declines velocity and concentration profiles due to the Lorentz force. Enhancement in the thermal relaxation parameter declines the thermal profile. Inclination in homogeneous-heterogeneous reaction parameters declines the mass transfer rate. Also, the well-known differential transform method is used for the validity of RKF-45 method and an impressive agreement is noticed between the results of RKF-45 and DTM.     

Cattaneo–Christov heat flux model and multiple slip effect on carbon nanofluid over a stretching sheet in a Darcy–Forchheimer porous medium

In several biotechnological processes, multiple slips are the most paramount, such as blood pumping from the heart to different body components, endoscopy treatment, pabulum distribution, and the heat transport phenomenon regulation. In the current research, we have studied the multiple slips, Darcy–Forchheimer, and Cattaneo–Christov heat flux model on a stretching surface exposed to magnetic carbon nanotube nanofluid. We have additionally included a heat source or sink, a chemical reaction for manipulating the heat and mass transport phenomena. The resulting governing partial differential equations have been transformed into ordinary differential equations. With the Runge–Kutta–Fehlberg fourth–fifth-order procedure, the transformed governing equations are numerically solved. Numerical solutions for different parameters for velocity, temperature, and concentration profiles (Eckert number, velocity slip, thermal slip, mass slip, etc.) are highlighted. Graphical and numerical results for the various parameters in the modeled problem have been outlined. The present numerical results are compared with the published ones for some limiting cases. The slip has been found to control the flow of the boundary layer.     

Numerical study of activation energy and thermal radiation effects on Oldroyd-B nanofluid flow using the Cattaneo–Christov double diffusion model over a convectively heated stretching sheet

This paper investigates a theoretical model of a mixed convective Oldroyd-B nanofluid with thermal radiation and activation energy effects. A thorough analysis is done by employing the nonhomogeneous Buongiorno model in the presence of velocity slip and suction. The surface is porous in nature, and nanoparticle mass flux is maintained passively at the surface. The thermal and concentration equations are modeled with the Cattaneo–Christov theory of heat and mass flux, respectively. Proper transformations are utilized for the conversion of transport equations and boundary conditions. The similarity solution is obtained through a numerical approach by utilizing the Runge–Kutta–Fehlberg method and shooting technique. The vital outcomes of this study and the influence of controlling parameters on the flow field, temperature, and concentration profiles are discussed graphically and in a tabular manner. Furthermore, a detailed discussion is provided to explain the results physically. The velocity of the nanofluid increases when the porosity parameter is increased, and temperature decreases with increasing thermal relaxation parameter. The outcomes elucidate that the suction parameter, thermal radiation parameter, and thermal relaxation parameter are positively correlated with the heat transfer coefficient. The result of passive control of nanoparticles at the surface is that the Brownian motion parameter has no influence on the temperature of the Oldroyd-B nanofluid flow and rate of heat transfer at the surface.     

Impacts of Cattaneo–Christov heat flux on the radiative MHD nanofluid flow past a stretched cylinder under multiple slip conditions

This investigation explores the features of velocity distribution, mass and heat transmissions of nanoliquid stream over a permeable cylinder accompanied by Cattaneo–Christov heat model and thermal radiation with nonlinear sort. Multiple slip conditions have been also encountered here. A magnetic force is oriented along vertically upward. The existence of thermophoresis together with Brownian motion has been assumed here. The foremost equations and associated boundary conditions have been normalized through the similarity technique. Then we solve the system numerically along with the fourth-order Runge–Kutta shooting scheme by using the software MAPLE-17 and round it with our preassigned accuracy level. The obtained outcomes are epitomized by tables and graphs. All of the impacts have been compared in suction and injection correspondingly and explained with proper reasoning. In charts, the physical consignments (such as Sherwood number, Nusselt number and skin friction) reveal the transference of mass and temperature and amount of friction by nanoparticles in the nanocomposition. For suction, the nanofluidic temperature gradually diminished due to the advanced thermal relaxation, whereas the contrary fact is exhibited in injection. The relaxation parameter of concentration provides a positive influence on mass transmission. The rates of amplification of this transportation are 1.99% and 3.87%, measured in consideration of injection and suction, respectively. Thermal radiation influenced the fluid's temperature in a positive direction. It increases Nusselt number with 41.75% in suction, and 45.21% is recorded for injection.     

Features of the exponential form of internal heat generation,Cattaneo–Christov heat theory on water-based graphene–CNT–titanium ternary hybrid nanofluid flow

To increase energy efficiency, the flow of fluids containing nanoparticles is crucial in industrial applications notably in nuclear reactors and nuclear system cooling. In light of this, this study examines the flow of a water-based ternary hybrid nanofluid (graphene, single-walled carbon nanotubes, and titanium dioxide) across a curved stretching sheet with suction. The non-Fourier heat flux model is also considered in the modeling. The existing partial differential equations are converted into ordinary differential equations through the use of similarity variables. These ordinary differential equations are then numerically solved using the Runge–Kutta–Fehlberg fourth- and fifth-order method along with a shooting approach. The collection of graphical findings for the key variables on the temperature and velocity profiles is investigated. Results reveal that the heat transport in ternary hybrid nanoliquid rises as the heat source/sink parameter rises. The Biot number influences the thermal profile positively, whereas the increasing curvature parameter values reduce heat transport. The curvature parameter has a positive impact on skin friction but the suction parameter has a negative impact on skin friction.     

Computation of a turbulent Rayleigh–Benard convection with the elliptic-blending second-moment closure

This communication reports briefly on the computational results of a turbulent Rayleigh–Benard convection with the elliptic-blending second-moment closure (EBM). The primary emphasis of the study is placed on an investigation of accuracy and numerical stability of the elliptic-blending second-moment closure for the turbulent Rayleigh–Benard convection. The turbulent heat fluxes in this study are treated by the algebraic flux model where the molecular dissipation rate of turbulent heat flux is included. The model is applied to the prediction of the turbulent Rayleigh–Benard convection for Rayleigh numbers ranging from Ra = 2 × 10⁶ to Ra = 10⁹, and the computed results are compared with the previous experimental correlations, T-RANS and LES results. The predicted cell-averaged Nusselt number follows the correlation by Peng et al. [S.H. Peng, K. Hanjalic, L. Davidson, Large-eddy simulation and deduced scaling analysis of Rayleigh–Benard convection up to Ra = 10⁹, J. Turbulence 7 (2006) 1–29.] (Nu = 0.162Ra^0.286) in the ‘soft’ convective turbulence region (2 × 10⁶ ≤ Ra ≤ 4 × 10⁷) and it follows the experimental correlation by Niemela et al. [J.J. Niemela, L. Skrbek, K.R., Sreenivasan, R.J. Donnelly, Turbulent convection at very high Rayleigh numbers, Nature 404 (2000) 837–840.] (Nu = 0.124Ra^0.309) in the ‘hard’ convective turbulence region (10⁸ ≤ Ra ≤ 10⁹) within 5% accuracy. This result shows that the elliptic-blending second-moment closure with an algebraic flux model predicts very accurately the Rayleigh–Benard convection.     

An enhanced thermal conduction model for the prediction of convection dominated solid–liquid phase change

An enhanced thermal conduction model for predicting convection dominated solid–liquid phase change is presented. The main feature of the model is to predict (1) the overall thermal behavior of the system and (2) the phase front position without recurring to the full solution of the Navier–Stokes equations. The model rests entirely on the conduction equation for both the solid and liquid phases. The effect of convection in the melt is mimicked via an enhanced thermal conductivity that depends on the dimensionless numbers and the geometry of the flow. The model is tested and confronted to full CFD solutions for a freezing duct flow problem and for buoyancy driven melting in an enclosure. In both cases, the predictions of the enhanced thermal conduction model show excellent agreement with that of the CFD model. Not only is the enhanced thermal conduction model simpler to implement but its simulations run at least ten times as fast as those of the CFD model. Consequently, the enhanced thermal conduction model is well suited for controlling real-time solid–liquid phase change processes that occur in industrial applications as well as in latent heat thermal energy storage systems.     

Effects of Cattaneo–Christov heat flux analysis on heat and mass transport of Casson nanoliquid past an accelerating penetrable plate with thermal radiation and Soret–Dufour mechanism

In this analysis, the effect of Catteneo–Christov model on heat alongside mass transport magnetohydrodynamics of a Casson nanoliquid with thermal radiation and Soret–Dufour mechanism is considered. The fluid flow is considered through porous media as the thermophysical attributes such as viscosity along with thermal conductivity are considered to be constant. Suitable similarity transformations were employed on the governing coupled flow equation to yield total differential equations (ODE). An accurate and newly developed spectral method called spectral homotopy analysis method (SHAM) was employed to provide solution to the simplified equations. The numerical method of homotopy analysis method (HAM) is SHAM. SHAM portrays the division of nonlinear equations into linear as well as nonlinear parts. The findings in this study show that an increment in the Casson parameter is seen to elevate the velocity plot at the wall and lessen the velocity far away from the plate. An increase in the Brownian motion and thermophoresis term is observed to speed up the local skin friction coefficient.     

Natural convection in a shallow porous cavity—the Brinkman model

The natural convection in a shallow porous rectangular cavity with differentially heated sidewalls is examined using the Brinkman model. The heat transfer rate through the cavity is determined in terms of a Nusselt number, in the limit of vanishingly small aspect ratio. Two types of boundary conditions are considered. Case I deals with a cavity with all rigid boundaries so that the no-slip boundary conditions can be imposed. In case II, the cavity has a free upper surface. The present analysis shows that the Brinkman model and Darcy's law give virtually the same result for the heat transfer rate when the Darcy number, based on the depth of the cavity, is less than the order of 10⁻⁴. We also find that the presence of a free surface can significantly increase the heat transfer rate through the cavity, especially when the permeability of the medium is high.     

Thermal stochastic collision model in turbulent gas–solid pipe flows

New thermal stochastic particle collision model in gas–solid flow in a riser is developed. The simulation is based on four-way coupling of phases considering inter-particle collision and heat transfer. It is shown that the limitation of excessive computational time in Eulerian–Lagrangian simulation of gas–solid flows for the high loading ratios is eliminated by using the stochastic particle collision model. The simulation results demonstrate that the predictions of the developed thermal stochastic particle collision modem are in good agreement with those obtained by the direct particle collision model and the available experimental data. The new stochastic modeling is used and nearly dense gas–solid flow is simulated for high loading ratios up to eight and the results are presented and discussed.     

Thermal conductivity variation on natural convection flow of water–alumina nanofluid in an annulus

This work is focused on the numerical modeling of steady laminar natural convection flow in an annulus filled with water–alumina nanofluid. The inner surface of the annulus is heated uniformly by a uniform heat flux q and the outer boundary is kept at a constant temperature T_c. Two thermal conductivity models namely, the Chon et al. model and the Maxwell Garnett model, are used to evaluate the heat transfer enhancement in the annulus. The governing equations are solved numerically subject to appropriate boundary conditions by a penalty finite-element method. A parametric study is conducted and a selective set of graphical results is presented and discussed to illustrate the effects of the presence of nanoparticles, the Prandtl number and the Grashof number on the flow and heat transfer characteristics for both nanofluid models. It is found that significant heat transfer enhancement can be obtained due to the presence of nanoparticles and that this is accentuated by increasing the nanoparticles volume fraction and Prandtl number at moderate and large Grashof number using both models. However, for the Chon et al. model the greatest heat transfer rate is obtained.     

Using bubble functions in the multi-scale finite element modeling of the convection–diffusion–reaction equation

The Convection–diffusion–reaction (CDR) equation shows multi-scale behaviour in cases where it represents convection or reaction dominated transport processes. Bubble function enriched finite elements are used to generate stable and accurate solutions for this equation. To validate the approach, the numerical results obtained for a benchmark problem are compared with their corresponding analytical solution for both exponential and propagation regimes.     

Natural convection in a cavity with undulated walls filled with water-based non-Newtonian power-law CuO–water nanofluid under the influence of the external magnetic field

Natural convection heat transfer in a square cavity (with wavy or plane wall) filled with non-Newtonian power-law nanofluid has been elucidated for several input parameters like Ra spanning from 10⁵ to 10⁶, power-law index (n) from 0.6 to 1.4, and volume fraction of CuO nanoparticles (?) from to 0 to 0.12. Effect of external magnetic field on heat transfer has been illustrated by varying the Ha from 0 to 90. In the present study, our main objective is to explore the effect of nanoparticles on heat transfer enhancement in non-Newtonian power-law fluid. It is found that the addition of nanoparticles (?) to shear thinning fluid enhances the heat transfer approximately 15% when ? increases from 0 to 0.12 for Ha less than 60 at all Ra. For a shear thickening fluid, the same thing happens for all Ha at any Ra. The average surface Nusselt number for a cavity with wavy wall is less than that of a plane wall for all cases which is not true for the case of local Nusselt number.     

Transient and non-Darcian effects on natural convection flow in a vertical channel partially filled with porous medium: Analysis with Forchheimer–Brinkman extended Darcy model

The present study addresses the transient as well as non-Darcian effects on laminar natural convection flow in a vertical channel partially filled with porous medium. Forchheimer–Brinkman extended Darcy model is assumed to simulate momentum transfer within the porous medium. Two regions are coupled by equating the velocity and shear stress in the case of momentum equation while matching of the temperature and heat flux is taken for thermal energy equation. Approximate solutions are obtained using perturbation technique. Variations in velocity field with Darcy number, Grashof number, kinematic viscosity ratio, distance of interface and variations in temperature distribution with thermal conductivity ratio, distance of interface are obtained and depicted graphically. The skin-friction and rate of heat transfer at the channel walls are also derived and the numerical values for various physical parameters are tabulated.