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This study investigates the dynamics of heat and mass transfer, focusing on the impact of nonlinear convection and thermophoresis in the flow of viscous nanofluid towards a shrinking sheet embedded with a porous matrix. To facilitate understanding of the flow behavior, the study employs similarity variables to simplify the resulting nonlinear partial differential equations. Numerical analysis utilizing the collocation method with Legendre polynomials as basis functions reveals distinct distributions of velocity, temperature, and concentration. The results demonstrate that the dimensionless suction coefficient amplifies nanoparticle volume fraction and momentum along the stretching surface. In addition, an increase in internal frictional force leads to heightened injection of heat energy, accelerates the temperature field, and impedes nanoparticle diffusion due to heat absorption. Elevation of convection terms augments momentum, reduces energy dispersion, and enhances nanoparticle diffusion. Moreover, an increase in the dimensionless Eckert number (Ec) correlates with intensified energy and diminished diffusion concentration of the nanoparticle volume fraction [ABSTRACT FROM AUTHOR]

The amalgamation of motile microbes in nanofluid (NF) is important in upsurging the thermal conductivity of various systems, including micro-fluid devices, chip-shaped micro-devices, and enzyme biosensors. The current scrutiny focuses on the bioconvective flow of magneto-Williamson NFs containing motile microbes through a horizontal circular cylinder placed in a porous medium with nonlinear mixed convection and thermal radiation, heat sink/source, variable fluid properties, activation energy with chemical and microbial reactions, and Brownian motion for both nanoparticles and microbes. The flow analysis has also been considered subject to velocity slips, suction/injection, and heat convective and zero mass flux constraints at the boundary. The governing equations have been converted to a non-dimensional form using similarity variables, and the overlapping grid-based spectral collocation technique has been executed to procure solutions numerically. The graphical interpretation of various pertinent variables in the flow profiles and physical quantities of engineering attentiveness is provided and discussed. The results reveal that NF flow is accelerated by nonlinear thermal convection, velocity slip, magnetic fields, and variable viscosity parameters but decelerated by the Williamson fluid and suction parameters. The inclusion of nonlinear thermal radiation and variable thermal conductivity helps to enhance the fluid temperature and heat transfer rate. The concentration of both nanoparticles and motile microbes is promoted by the incorporation of activation energy in the flow system. The contribution of microbial Brownian motion along with microbial reactions on flow quantities justifies the importance of these features in the dynamics of motile microbes. [ABSTRACT FROM AUTHOR]

This study investigates the dynamics of heat and mass transfer, focusing on the impact of nonlinear convection and thermophoresis in the flow of viscous nanofluid towards a shrinking sheet embedded with a porous matrix. To facilitate understanding of the flow behavior, the study employs similarity variables to simplify the resulting nonlinear partial differential equations. Numerical analysis utilizing the collocation method with Legendre polynomials as basis functions reveals distinct distributions of velocity, temperature, and concentration. The results demonstrate that the dimensionless suction coefficient amplifies nanoparticle volume fraction and momentum along the stretching surface. In addition, an increase in internal frictional force leads to heightened injection of heat energy, accelerates the temperature field, and impedes nanoparticle diffusion due to heat absorption. Elevation of convection terms augments momentum, reduces energy dispersion, and enhances nanoparticle diffusion. Moreover, an increase in the dimensionless Eckert number (Ec) correlates with intensified energy and diminished diffusion concentration of the nanoparticle volume fraction.

In this work, the flow of ternary hybrid nanomaterials filling porous spaces is investigated in the presence of magnetohydrodynamics (MHD) and nonlinear convection. Three dissimilar nanoparticles copper (Cu), alumina (Al2O3), and magnesium oxide (MgO) are employed. Various features including Ohmic heating, heat generation, and variable thermal conductivity are studied, along with nonlinear radiation and entropy analysis. Solutions are developed using the ND Solve (shooting) method in Mathematica software. Analysis of parameters of interest is conducted, with conclusions highlighting key outcomes. Results show an increase in the thermal field for Eckert number and radiation, while a decrease in liquid flow is observed with increasing the magnetic parameter. The heat transport rate exhibits an opposite trend between heat generation and radiation. Temperature ratio parameter and Prandtl number show opposite trends in the thermal field, while the temperature gradient decreases against variable thermal conductivity. The drag force follows a similar trend against nonlinear convective and porosity variables. The entropy rate increases with the Brinkman number, and both porosity and radiation have an increasing impact on entropy rate. An increase in liquid flow is noted with the nonlinear convection variable. A comparative study of heat transport rate through the Prandtl number shows good agreement.

The investigation of the physical processes determining the melting of the lithospheric rocks is of crucial importance for understanding the volcanic dynamics and its related consequences. Rock melting begins when a sufficiently high temperature is experienced by the rock solidus. The heat transfer from the asthenosphere to the lithosphere can be assumed as the main mechanism accountable for the partial melting of rocks, and initiating magma generation. The heat transfer to the lithosphere is considered to be governed mainly by the convective motion inside the asthenosphere. In order to mathematically describe this process, a generalization of a nonlinear convective 1D model, possibly representing a useful though simplified model for the birth of a volcano, and already analyzed from an analytical viewpoint (Godano et al., 2022), is investigated; here, we solve numerically some physically meaningful initial and boundary value problems, and discuss the results.

The time-dependent nonlinear convective Williamson nanofluid flow over a rough conical surface in the presence of numerous diffusions and Arrhenius activation energy effects are the main focus of the investigation in this paper. The unsteady flow is essentially due to the impulsive mainstream flow. Two diffusive species, namely, liquid Hydrogen and liquid Nitrogen, occur in the flow under investigation. A sinusoidal waveform mathematically models the rough conical surface with small amplitude and high frequency. Thus, the surface gradients, namely skin friction, exhibit wavy effects in the boundary layer regime. In the current flow problem, the governing equations for the fluid flow are highly coupled nonlinear PDEs that depend on the proper initial and boundary conditions. Mangler's transformations are used to convert them into non-dimensional forms, and implicit finite difference approximation and quasilinearization are employed to produce numerical semi-similar solutions. The numerical results representing the boundary layer effects are displayed graphically and analyzed for parametric values. The friction experienced by the rough surface is strongly fluctuating for higher values of Williamson parameter W. The heat transfer rate Rex-1/2Nu increases approximately by 61% and 63% at ξ=0.8 when Nt decreases from Nt=0.5 to Nt=0.1, for the cases of W = 0 (Newtonian fluid) and W = 1(Williamson fluid), respectively. Additionally, there is a strong correlation between the current findings and the related results previously published in the literature.

The amalgamation of motile microbes in nanofluid (NF) is important in upsurging the thermal conductivity of various systems, including micro-fluid devices, chip-shaped micro-devices, and enzyme biosensors. The current scrutiny focuses on the bioconvective flow of magneto-Williamson NFs containing motile microbes through a horizontal circular cylinder placed in a porous medium with nonlinear mixed convection and thermal radiation, heat sink/source, variable fluid properties, activation energy with chemical and microbial reactions, and Brownian motion for both nanoparticles and microbes. The flow analysis has also been considered subject to velocity slips, suction/injection, and heat convective and zero mass flux constraints at the boundary. The governing equations have been converted to a non-dimensional form using similarity variables, and the overlapping grid-based spectral collocation technique has been executed to procure solutions numerically. The graphical interpretation of various pertinent variables in the flow profiles and physical quantities of engineering attentiveness is provided and discussed. The results reveal that NF flow is accelerated by nonlinear thermal convection, velocity slip, magnetic fields, and variable viscosity parameters but decelerated by the Williamson fluid and suction parameters. The inclusion of nonlinear thermal radiation and variable thermal conductivity helps to enhance the fluid temperature and heat transfer rate. The concentration of both nanoparticles and motile microbes is promoted by the incorporation of activation energy in the flow system. The contribution of microbial Brownian motion along with microbial reactions on flow quantities justifies the importance of these features in the dynamics of motile microbes.

The time-dependent nonlinear convective Williamson nanofluid flow over a rough conical surface in the presence of numerous diffusions and Arrhenius activation energy effects are the main focus of the investigation in this paper. The unsteady flow is essentially due to the impulsive mainstream flow. Two diffusive species, namely, liquid Hydrogen and liquid Nitrogen, occur in the flow under investigation. A sinusoidal waveform mathematically models the rough conical surface with small amplitude and high frequency. Thus, the surface gradients, namely skin friction, exhibit wavy effects in the boundary layer regime. In the current flow problem, the governing equations for the fluid flow are highly coupled nonlinear PDEs that depend on the proper initial and boundary conditions. Mangler's transformations are used to convert them into non-dimensional forms, and implicit finite difference approximation and quasilinearization are employed to produce numerical semi-similar solutions. The numerical results representing the boundary layer effects are displayed graphically and analyzed for parametric values. The friction experienced by the rough surface is strongly fluctuating for higher values of Williamson parameter W. The heat transfer rate Re x - 1 / 2 N u increases approximately by 61% and 63% at ξ = 0.8 when Nt decreases from Nt = 0.5 to Nt = 0.1 , for the cases of W = 0 (Newtonian fluid) and W = 1(Williamson fluid), respectively. Additionally, there is a strong correlation between the current findings and the related results previously published in the literature. [ABSTRACT FROM AUTHOR]

The proposed planer layer dynamo physical model has real-world applications, especially in the Earth's liquid core. Thus, in this paper, an attempt is made to understand the finite amplitude convection when there exists a coupling between the Lorentz force and the Coriolis force. In particular, the effect of a horizontally applied magnetic field is studied on the Rayleigh–Bénard convection (RBC) that contains the electrically conducting fluid and rotates about its vertical axis. Free–free boundary conditions are assumed on the geometry. Attention is focused on the nonlinear convective flow behavior during the occurrence of cross rolls which are perpendicular to the applied magnetic field and parallel to the rotation axis. The visualization of cross rolls is achieved using the Fourier analysis of perturbations up to the O( ε 8 ). The relationship of the Nusselt number ( N u ) with respect to the Rayleigh number (R), the Ekman number (E), and the Elsasser number (Λ) is investigated. It is observed that E generates a strong damping effect on the flow velocity and on the heat transfer at high rotation rates. Using the heatline concept, it is observed that the temperature within the central regime is enhanced as the Λ increases. The results show that either E decreases or Λ increases, then the heat transfer rate increases. [ABSTRACT FROM AUTHOR]

The current study focuses on how a magnetic field and oxytactic microorganisms influence the flow of a micropolar nanofluid around a wedge in a bioconvective field. It has practical implications in numerous industries and fields, including oil and gas, aircraft, biological waste processing, bio-microsystems etc. We begin the analysis by transforming the dimensionality of the governing equations via non-similar transformations. The obtained equations are linearized using the Quasilinearization method before being discretized using the implicit finite difference approach. The literature study reveals that the earlier studies on wedge are restricted to only similarity solutions rather than non-similarity without the impact of oxytactic microorganisms. This study concentrates on the effect of oxytactic microorganisms in the nonlinear mixed convective micropolar nanofluid flow around a wedge. The Micropolar fluid reduces the velocity while increasing friction between the surrounding fluid and the wedge's wall. The greater Peclet and microbial density difference ratio values blunted the effects of the rise in microorganisms on the density profile changes. The growth in the oxytactic microbial parameter Δ1 improves the microorganism's mass transfer rate(Rex−1/2NSh). Specifically, a rise of approximately 21% is seen in Rex−1/2NSh in the presence of oxytactic microbes as compared to their absence at ξ=2 and Le=10. The microrotation parameter boosts the entropy for enhancing values of the Brinkman number. And the entropy is almost the same for the Newtonian and micropolar fluid cases for Br=0.1.

A nanofluid flow (TiO2/Ethylene Glycol) over a stretched surface caused by the considerable nonlinear convection is investigated with the nanoparticle aggregation effect in the presence of quadratic thermal radiation. The modified Maxwell model and Bruggeman model are utilized, which depict the nanoparticle aggregation effect. In addition, the system employs two separate variable heat sources i.e., linear heat source and exponential heat source. The similarity transformations are utilized to convert the basic governing PDEs to ODEs. The “bvp4c function” in MATLAB is used to numerically handle the resultant nonlinear system. The impact of the involved parameters is explored on the non-dimensional fields of velocity boundary layer patterns, velocity, streamlines, temperature, and heat transfer rate. Graphs compare and display the control of pertinent parameters on the distinct flow properties. A comparison of the behavior of the solution profiles with and without the influence of nanoparticle aggregation is illustrated. The nanoparticle aggregates significantly enhance the velocity and heat transmission mechanism. Furthermore, the influence of varied heat sources (linear and exponential) aids in growing the thermal boundary layer. Nanofluid flow with nanoparticle aggregation has a 5.36% higher Nusselt number value in comparison to flow without aggregation effect when the volume fraction of nanoparticles is varied from 0 to 0.1. The results of the study will be helpful in the fields where applications of a continuously stretching/shrinking surface are used like in many engineering processes and industries, glass-fiber production, aerodynamic extrusion of plastic sheets, and manufacturing of paper and rubber sheets.

Nonlinear free convection in an inclined rectangular porous cavity heated from below has been studied using a two-dimensional spectral decomposition. The code uses pseudo-arclength continuation to follow solution curves around fold bifurcations. The evolution with inclination of the pattern of convection is complicated and it relies strongly on both the Darcy–Rayleigh number and the aspect ratio of the cavity. When the inclination is large it is generally true that only one cell appears, and that it has a circulation that is consistent with the direction of the buoyancy forces along the heated and cooled boundaries. However, as the inclination decreases back towards the horizontal, this unicellular pattern evolves, sometimes initially via fold bifurcations, into patterns with different numbers of cells. Such evolutions always conserve the parity of the number of cells (such as one cell becoming three and then five, or two cells becoming four), but bifurcations also arise between patterns with different parities. These phenomena are illustrated using a suitable selection of solution curves that show the dependence of the Nusselt number on the inclination. [ABSTRACT FROM AUTHOR]

In this work, the flow of ternary hybrid nanomaterials filling porous spaces is investigated in the presence of magnetohydrodynamics (MHD) and nonlinear convection. Three dissimilar nanoparticles copper (C u) , alumina (Al 2 O 3) , and magnesium oxide (M g O) are employed. Various features including Ohmic heating, heat generation, and variable thermal conductivity are studied, along with nonlinear radiation and entropy analysis. Solutions are developed using the ND Solve (shooting) method in Mathematica software. Analysis of parameters of interest is conducted, with conclusions highlighting key outcomes. Results show an increase in the thermal field for Eckert number and radiation, while a decrease in liquid flow is observed with increasing the magnetic parameter. The heat transport rate exhibits an opposite trend between heat generation and radiation. Temperature ratio parameter and Prandtl number show opposite trends in the thermal field, while the temperature gradient decreases against variable thermal conductivity. The drag force follows a similar trend against nonlinear convective and porosity variables. The entropy rate increases with the Brinkman number, and both porosity and radiation have an increasing impact on entropy rate. An increase in liquid flow is noted with the nonlinear convection variable. A comparative study of heat transport rate through the Prandtl number shows good agreement. [ABSTRACT FROM AUTHOR]

This paper is concerned with the multidimensional stability of V-shaped traveling fronts for a reaction-diffusion equation with nonlinear convection term in $\mathbb{R}^n$ ($n\geq3$). We consider two cases for initial perturbations: one is that the initial perturbations decay at space infinity and another one is that the initial perturbations do not necessarily decay at space infinity. In the first case, we show that the V-shaped traveling fronts are asymptotically stable. In the second case, we first show that the V-shaped traveling fronts are also asymptotically stable under some further assumptions. At the same time, we also show that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which means that the traveling fronts are not asymptotically stable under general bounded perturbations.

The study of fluid dynamics due to the stretching surface is one of the most eminent topics due to its potential industrial applications viz. drawing wire and plastic films, metal and polymer extrusion, fiber and glass production. In the present article, the author is going to study the effects of hybrid nanofluids flow on an inclined plate including CuO (Copper Oxide), and Cu (Copper). The Casson fluid with a couple-stress term has been used in the flow analysis. The surface of the plate is considered slippery. The convection has been taken nonlinear with thermal radiation. The governing equation of the flow of hybrid nanofluids with energy equation has been transformed into highly nonlinear ODEs through similarity transformation. The proposed model has been solved through a numerical RK-4 method. Significant variables of the physical process such as solar radiation, nonlinear convection parameters, heat transfer rates, and their effect on the solar power plant have been noticed.

In this study, a Keller box calculation of magnetic nanopolymer coating flow over a circular cylinder in the presence of nonlinear convection is performed. A two-phase nanofluid model (Buongiorno model) is used to model the equations for nanofluid flow and heat and mass transfer. The problem at hand is first formulated in the dimensional form of nonlinear partial differential equations (PDEs) and then transformed into the dimensionless PDEs form by manipulating non-similar variables. The Keller box approach (implicit finite differences) is used for the calculations of the transformed PDEs. The characteristics of critical thermophysical physical parameters on the flow field, viz thermal convection, Brownian motion, Weissenberg number, thermophoresis, magnetic field, heat source/sink, Biot number, buoyancy ration parameter, Eckert number, and mixed convection, are graphically manifested. In addition, the flow control parameters estimate the sheath friction, heat transfer rate, and mass transfer rate in the flow coordinate. A significant increase in the velocity of the Williamson nanofluid is accompanied by an increase in the Biot number, magnetic field, and mixed convection. The isotherms show a higher fluid temperature near the wall in the absence of convection. The Weissenberg number, Biot number, and thermophoresis significantly increase the temperature of the Williamson nanofluid. The density of the streamlines is lowered with a larger buoyancy ratio parameter. The total entropy generation of the Williamson nanofluid is improved by increasing the Brinkman number and the Weissenberg number. [ABSTRACT FROM AUTHOR]

The present study addresses the heat transfer efficiency and entropy production of electrically conducting kerosene-based liquid led by the combined impact of electroosmosis and peristalsis mechanisms. Effects of nonlinear mixed convection heat transfer, temperature-dependent viscosity, radiative heat flux, electric and magnetic fields, porous medium, heat sink/source, viscous dissipation, and Joule heating are presented. The Debye–Huckel linearization approximation is employed in the electrohydrodynamic problem. Mathematical modeling is conducted within the limitations of δ << 1 and Re → 0. Coupled differential equations after implementing a lubrication approach are numerically solved. The essential characteristics of the production of entropy, the factors influencing it, and the characteristics of heat and fluid in relation to various physical parameters are graphically evaluated by assigning them a growing list of numeric values. This analysis reveals that heat transfer enhances by enhancing nonlinear convection and Joule heating parameters. The irreversibility analysis ensures that the minimization of entropy generation is observed when the parameters of viscosity and radiation are held under control. Fluid velocity can be regulated by adjusting the Helmholtz–Smoluchowski velocity and magnetic field strength. [ABSTRACT FROM AUTHOR]

The studies conducted in the recent past reveals that the non-Newtonian are extensively applied in several engineering applications. The atypical Oldroyd-B fluid, a specific sub-category of non-Newtonian fluids, with fractional derivative constitutive equations, is very expedient to determine the approximation of many dilute polymer liquids. Since very few studies regarding the numerical solution of the aforesaid model are conducted, so the literature that can be referred to in this regard is scarce. The current article investigates the unsteady magneto-hydrodynamic flow of irregular Oldroyd-B fluid with a modified thermal flux restricted between two infinite parallel plates. Here, it is assumed that the thermal conductivity of the fluid is variable, and the impact of body forces like gravity is accordingly analyzed in terms of non-linear convection. The linear acceleration of the lower permeable plate in its plane prompts the flow. Further, we have investigated the heat transfer in the flow field through a modified fractional thermal gradient and internal heat source. The temperature gradient accelerates with time and exists among the flow boundaries. In order to determine the approximate solution unified with finite difference approximation for Caputo fractional time derivatives, we have utilized the standard Galerkin finite element method. Moreover, we have conducted a convergence analysis of the suggested numerical scheme and offered some valid error approximations. In the non-integer viscoelastic model, the local number and coefficient of skin friction are also deliberated. We have also performed some upright numerical simulations to highlight the ascendency of characteristic flow parameters of velocity and temperature fields in the proposed scheme. Here, we have analyzed that the resistive force increases by 40%, corresponding to an increment in α ranging from 0 to 0.4. Further, the heat transfer rate is diminish by 10.94% against an increase in variable thermal conductivity while it is enhanced by 37.78% corresponding to increased in Pr.

This study aims to investigate heat transfer and entropy production in the non-similar flow of an incompressible fluid, with nonlinear convection and viscous dissipation. To obtain precise solutions, the Sparrow-Quack-Boerner local non-similarity method is implemented. The non-similarity arises from the nonlinear convection term present in the momentum equation. The non-similarity arises from the nonlinear convection term present in the momentum equation. Consequently, the inclusion of non-similarity terms into the energy equation is achieved through the interconnection of momentum and energy equations. Entropy generation is explored by employing the second law of thermodynamics. Numerical results from several truncation levels are presented in tabular form, and equations for both first and second-level truncations are generated. The one- and two-equation models are solved by implementing the Generalized Differential Quadrature Method (GDQM). Comparison with a second level of truncation reveals significantly greater inaccuracies in numerical results obtained from the first level. This discrepancy arose due to the omission of non-similar terms in the primary governing equations at the first truncation stage. The validity and accuracy of the derived numerical solutions are further demonstrated by the application of the GDQM and the midpoint method with Richardson extrapolation. Additionally, a plot of the numerical data produced from the second level of truncations is presented together with a discussion of the various physical factors. Increasing the mixed convection parameter accelerates fluid velocity, also when the viscous dissipation parameter increases, temperature and the Bejan number increase. The Prandtl number and the thickness of the thermal boundary layer are observed to be inversely related. Moreover, the quantity of entropy generated at the stretching surface and inside the boundary layer rises in proportion to the increases in the Prandtl and Eckert numbers. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with traveling curved fronts of bistable reaction–diffusion equations with nonlinear convection in a two-dimensional space. By constructing super- and subsolutions, we establish the existence of traveling curved fronts. Furthermore, we show that the traveling curved front is globally asymptotically stable.

Nonlinear free convection in an inclined rectangular porous cavity heated from below has been studied using a two-dimensional spectral decomposition. The code uses pseudo-arclength continuation to follow solution curves around fold bifurcations. The evolution with inclination of the pattern of convection is complicated and it relies strongly on both the Darcy–Rayleigh number and the aspect ratio of the cavity. When the inclination is large it is generally true that only one cell appears, and that it has a circulation that is consistent with the direction of the buoyancy forces along the heated and cooled boundaries. However, as the inclination decreases back towards the horizontal, this unicellular pattern evolves, sometimes initially via fold bifurcations, into patterns with different numbers of cells. Such evolutions always conserve the parity of the number of cells (such as one cell becoming three and then five, or two cells becoming four), but bifurcations also arise between patterns with different parities. These phenomena are illustrated using a suitable selection of solution curves that show the dependence of the Nusselt number on the inclination.

The present study addresses the heat transfer efficiency and entropy production of electrically conducting kerosene-based liquid led by the combined impact of electroosmosis and peristalsis mechanisms. Effects of nonlinear mixed convection heat transfer, temperature-dependent viscosity, radiative heat flux, electric and magnetic fields, porous medium, heat sink/source, viscous dissipation, and Joule heating are presented. The Debye–Huckel linearization approximation is employed in the electrohydrodynamic problem. Mathematical modeling is conducted within the limitations of δ << 1 and Re → 0. Coupled differential equations after implementing a lubrication approach are numerically solved. The essential characteristics of the production of entropy, the factors influencing it, and the characteristics of heat and fluid in relation to various physical parameters are graphically evaluated by assigning them a growing list of numeric values. This analysis reveals that heat transfer enhances by enhancing nonlinear convection and Joule heating parameters. The irreversibility analysis ensures that the minimization of entropy generation is observed when the parameters of viscosity and radiation are held under control. Fluid velocity can be regulated by adjusting the Helmholtz–Smoluchowski velocity and magnetic field strength.

In the existence of an aligned magnetic field over the inclined shrinking/stretching stratified sheet in a non-Darcy porous medium, the two-dimensional boundary layer flow of an upper-convected Maxwell fluid is analyzed. The heat transfer effects are acknowledged by using the nonlinear convection. The system of partial differential equations, which administrates the distinctive properties of flow and heat transfer, is depleted into ordinary differential equations with the use of similarity variables. The governing equations are determined numerically by utilizing the shooting technique. The response of varied implicated parameters on velocity, skin friction, and temperature accounts is inspected graphically and displayed in the table. It is noted that local inertia coefficient is accountable for the reduction in the velocity profile and the aligned magnetic field has the opposite relation for the shrinking and stretching sheet. [ABSTRACT FROM AUTHOR]

The study deals with nonlinear convection in a three-layered porous-air-porous enclosure with zero temperature difference at the external impermeable thermally conductive boundaries. The upper and lower porous matrices are internally heated with a uniform energy source. To describe redistribution of heat energy between the top and bottom surfaces of the enclosure, we introduce a relative heat transfer coefficient q r that has the conduction and convection parts. The former part tends to unity in the fully filled porous domain. The symmetry of heat transfer from the inner area towards the outer surfaces breaks down if one adds an intermediate air layer with low thermal conductivity. Two sandwiched systems which have equal depth ratio d but distinct air layer location are considered. In system 1, the air layer is located in the upper unstably stratified half of the enclosure. In system 2, this layer is in the lower stably stratified half. The total value of q r always increases due to penetrative convection. At d = 0.1, local convection in system 1 is easily initiated in contrast to the hard-to-generate large-scale convection in system 2 due to a strong destabilization. The latter most effectively enhances heat transport though the top surface with increasing the supercriticality. • Nonlinear convection in a porous-air-porous enclosure, induced by energy source. • Air layer effect on redistribution of heat energy between top and bottom surfaces. • Comparing a relative heat transfer rate at different air layer locations and depths. • Local convection is easily initiated due to strong destabilization by depth ratio. • Hard-to-generate large-scale convection is more effective in heat transferring. [ABSTRACT FROM AUTHOR]

The fluid flow over a yawed cylinder is useful in understanding practical significance for undersea applications, for example, managing transference and/or separation of the boundary layer above submerged blocks and in suppressing recirculating bubbles. The present analysis examines nonlinear mixed convection flow past a moving yawed cylinder with diffusion of liquid hydrogen. The coupled nonlinear control relations and the border restrictions pertinent to the present flow problem are nondimensionalized by using nonsimilar reduction. Further, implicit finite difference schemes and Quasilinearization methods are employed to solve the nondimensional governing equations. Impact of several nondimensional parameters of the analysis on the dimensionless velocity, temperature and species concentration patterns and also on Nusselt number, Sherwood number and friction parameter defined at the cylinder shell is analyzed through numerical results presented in various graphs. Velocity profiles can be enhanced, and the coefficients of friction at the surface can be reduced, for increasing values of velocity ratio parameters along chordwise as well as spanwise directions. Species concentration profile is reduced, while the Sherwood number is enhanced, for growth of the Schmidt number and yaw angles. Furthermore, for an increasing value of yaw angle, skin-friction coefficient in chordwise direction diminishes in opposing buoyancy flow case, whereas the results exhibit the opposite trend in assisting buoyancy flow case. Moreover, very importantly, for increasing magnitude of nonlinear convection characteristic, the liquid velocity and surface friction enhance in spanwise direction. Further, for increasing magnitude of combined convection characteristics, velocity profiles and coefficient of friction at the surface enhance in both spanwise and chordwise directions. Moreover, we have observed that there is no deviation for zero yaw angle in Nusselt number and Sherwood number.

The objective of the present study is to investigate the effect of nonlinear variation of density with temperature and concentration on the mixed convective flow of a micropolar fluid over an inclined flat plate in a non-Darcy porous medium in the presence of the convective boundary condition. In order to analyze all the essential features, the governing non-dimensional partial differential equations are transformed into a system of ordinary differential equations using a local non-similarity procedure and then the resulting boundary value problem is solved using a successive linearisation method (SLM). By insisting the comparison between vertical, horizontal and inclined plates, the physical quantities of the flow and its characteristics are exhibited graphically and quantitatively with various parameters. An increase in the micropolar parameter and non-Darcy parameter tend to increase the skin friction and the reverse change is observed in wall couple stress, mass and heat transfer rates. The influence of the nonlinear concentration parameter is more prominent on all the physical characteristics of the present model, compared with that of nonlinear temperature parameter.

The effects of the exponential space based heat source on quadratic convective flow of Casson fluid in a microchannel with an induced magnetic field is studied through a statistical approach. The flow is considered in vertical microchannel formed by two vertical plates. The solution for the governing equations has been obtained for the velocity, induced magnetic field and temperature field using Homotopy Perturbation Method (HPM). The current density, skin friction co-efficient and Nusselt number expressions are also estimated. The impact of various physical parameters on the velocity, temperature, induced magnetic field, current density, skin friction co-efficient and Nusselt number distributions have been discussed with the help of graphs. The results obtained by using HPM, are compared to those obtained by using the Runge-Kutta-Fehlberg 4-5th order method and an excellent agreement is found. The impact of Casson fluid parameter and the exponential heat source is qualitatively agreed for all flow fields. [ABSTRACT FROM AUTHOR]

A nonlinear flow of Jeffrey liquid with Cattaneo-Christov heat flux is investigated in the presence of nanoparticles. The features of thermophoretic and Brownian movement are retained. The effects of nonlinear radiation, magnetohydrodynamic (MHD), and convective conditions are accounted. The conversion of governing equations into ordinary differential equations is prepared via stretching transformations. The consequent equations are solved using the Runge-Kutta-Fehlberg (RKF) method. Impacts of physical constraints on the liquid velocity, the temperature, and the nanoparticle volume fraction are analyzed through graphical illustrations. It is established that the velocity of the liquid and its associated boundary layer width increase with the mixed convection parameter and the Deborah number. [ABSTRACT FROM AUTHOR]

Highlights • MHD Nonlinear convective flow of Carreau nanofluid is modeled. • Effects of thermal radiation and chemical reaction are present. • Newtonian conditions for heat and mass transfer are imposed. • Brownian motion and thermophoresis features are analyzed. • Skin friction, Nusselt and Sherwood numbers are analyzed through graphs. Abstract This article addresses magnetohydrodynamic (MHD) nonlinear convective stagnation point flow of Carreau nanofluid towards a stretching sheet with chemical reaction. Analysis is carried out in the presence of Newtonian conditions of heat and mass transfer. Brownian motion and thermophoresis phenomena have been utilized in energy and concentration expressions. Appropriate variables lead to development of nonlinear ordinary differential systems. Results for velocity, thermal field, nanoparticles concentration, skin friction and local Nusselt and Sherwood numbers are presented and investigated. Here higher values of material parameter corresponds to high velocity. Moreover the variations of temperature and nanoparticles concentration are opposite in frame of Brownian motion. [ABSTRACT FROM AUTHOR]

This is a theoretical study of unsteady nonlinear convection on magnetohydrodynamic fluid in a suspension of dust and graphene nanoparticles. For boosting the heat transport phenomena we consider the Cattaneo-Christov heat flux and thermal radiation. Dispersal of graphene nanoparticles in dusty fluids finds applications in biocompatibility, bio-imaging, biosensors, detection and cancer treatment, in monitoring stem cells differentiation etc. Initially the simulation is performed by amalgamation of dust (micron size) and nanoparticles into base fluid. Primarily existing partial differential system (PDEs) is changed to ordinary differential system (ODEs) with the support of usual similarity transformations. Consequently, the highly nonlinear ODEs are solved numerically through Runge-Kutta and Shooting method. The computational results for Non-dimensional temperature and velocity profiles are offered through graphs ( ϕ = 0 and ϕ = 0.05 ) cases. Additionally, the numerical values of friction factor and heat transfer rate are tabulated numerically for various physical parameters obtained. We also validated the current outcomes with previously available study and found to be extremely acceptable. From this study we conclude that in the presence of nanofluid heat transfer rate and temperature distribution is higher compared to micro fluid. [ABSTRACT FROM AUTHOR]

The present study aims to investigate the thermal radiation heat transfer effect on unsteady magnetohydrodynamic flow of micropolar fluid over a uniformly heated vertical hollow cylinder using Bejan's heat function concept. The normalized conservation equations emerge as a system of time-dependent non-linear coupled partial differential equations. Under appropriate wall and free stream conditions these equations are solved with an efficient unconditionally stable implicit scheme of Crank-Nicolson type. Important thermo-physical parameters featured include the magnetic body force parameter ( M ), Grashof (free convection) parameter ( Gr ), Eringen micropolar material parameter ( K ), Prandtl number ( Pr ), conjugate heat transfer parameter ( P ) and radiative-conductive Rosseland parameter ( N ), are analyzed on the flow-field with ranges 0–3, 10 5 –10 6 , 0–1.2, 0.7–7.0, 0–0.5 and 0–15, respectively. The time-histories of average values of momentum and heat transport coefficients, as well as the steady-state flow variables are presented for selected values of these non-dimensional parameters. With elevation in magnetic parameter or radiation parameter, the time taken for the flow-field variables to attain the time-independent state increases. The dimensionless thermal radiative heat function values are closely correlated with the overall rate of heat transfer on the outer hot cylindrical wall. Bejan's heat flow visualization implies that the thermal radiative heat function contours are compact in the neighbourhood of leading edge of the boundary layer on the outer hot cylindrical wall. Increasing radiation or magnetic parameter values result in an increase in the deviation of heat lines from the hot wall. Also, the heat lines are observed to depart slightly away from the hot wall with greater values of vortex viscosity. Furthermore, the deviations of flow variables from the hot wall for a micropolar fluid are significant compared to the Newtonian fluid (vanishing micropolar vortex viscosity). [ABSTRACT FROM AUTHOR]

In this communication we utilized a modified Fourier approach featuring thermal relaxation effect in nonlinear convective flow by a vertical exponentially stretchable surface. Temperature-dependent thermal conductivity describes the heat transfer process. Thixotropic liquid is modeled. Convergent local similar solutions by homotopic approach are obtained. Graphical results for emerging parameters of interest are analyzed. Skin friction is calculated and interpreted. Consideration of larger local buoyancy and nonlinear convection parameters yields an enhancement in velocity distribution. Temperature and thermal layer thickness are reduced for larger thermal relaxation factor. [ABSTRACT FROM AUTHOR]

In this paper we study the following Cauchy problem: u t = u x x + ( u n ) x , ( x , t ) ∈ R × ( 0 , ∞ ) , u ( x , 0 ) = 0 , x ≠ 0 , where parameter n ≥ 0 . Its nonnegative solution is called singular solution when u ( x , t ) satisfies the equation in the sense of distribution, initial conditions in the classical sense and also u ( x , t ) exhibits a singularity at the origin ( 0 , 0 ) . As we know, the singular solution is called source-type solution if the initial is M δ ( x ) , where δ ( x ) is Dirac measure and constant M > 0 . The solution is called very singular solution if it possesses more singularity than that of source-type solution at the origin. Here we focus on what happens in the interactive effect between the diffusion and convection in a whole physical process. We find critical values n 2 < n 1 < n 0 such that there exists unique source-type solution in the exponent range of 0 < n < n 0 , while there exists no nonnegative singular solution if n ≥ n 0 . Only in the case of n 2 < n < n 1 there exists a very singular solution, but in the case of n ≥ n 1 or n ≤ n 2 there is no solution that exhibits more singular than source-type solution at the origin. Furthermore we describe the short time asymptotic behavior of the singular solutions when such Cauchy problem is solvability for source-type solution or very singular solution. [ABSTRACT FROM AUTHOR]

Main purpose here is to analysis magnetohydrodynamic (MHD) nonlinear convective flow of Jeffrey nanofluid by a nonlinear convectively heated stretching sheet of variable thickness. Nonlinear thermal radiation, heat generation/absorption and chemical reaction are considered. Brownian motion and thermophoresis have been utilized in energy and concentration expressions. Recently suggested condition employing volume fraction of nanoparticle at the surface to be controlled passively rather than actively is utilized. Appropriate variables lead to development of nonlinear ordinary differential systems. Effect of various pertinent variables for velocity, temperature and nanoparticle concentration have been displayed and explored. Skin friction coefficient and local Nusselt number are calculated and addressed. It is revealed that larger values of relaxation and retardation times ratio diminish velocity distribution. Moreover impact of heat generation/absorption on the temperature and heat transfer rate are quite reverse. [ABSTRACT FROM AUTHOR]

Mathematical analysis for magnetohydrodynamic (MHD) nonlinear convective flow of Walter-B nanofluid over a nonlinear stretching sheet with variable thickness is introduced. Heat transfer phenomenon is based through involvement of nonlinear thermal radiation and heat generation/absorption. Intention in present analysis is to develop a model for nanomaterial comprising Brownian motion and thermophoresis phenomena. Boundary layer approximation is applied to partial differential equations. Governing equations are then converted into ordinary differential equations by invoking appropriate variables. The transformed expressions are explored through homotopic algorithm. Plots for velocity, temperature and concentration fields are presented and their behavior is deliberated for several sets of values of governing variables. Skin friction coefficient and local Nusselt number are inspected through numerical values. It is concluded that viscoelasticity reduces the velocity field while reverse situation is observed due to wall thickness parameter. Thermal field and heat transfer rate are enhanced for temperature ratio parameter. Moreover effect of Brownian motion and thermophoresis parameter are quite reverse for the concentration field. [ABSTRACT FROM AUTHOR]

The present article investigates the magnetohydrodynamic (MHD) three-dimensional nonlinear convective flow of an Oldroyd-B nanofluid over a stretching sheet. Heat transfer analysis is reported in the presence of nonlinear thermal radiation and heat generation/absorption. The effects of Brownian motion and thermophoresis are considered in energy and concentration expressions. Meaningful solutions are established for the velocity, temperature and concentration. It is observed that both components of velocity show opposite behavior for mixed convection parameter, ratio of concentration to thermal buoyancy forces and ratio parameter. Local Nusselt and Sherwood numbers are analyzed for the pertinent parameters. Temperature and heat transfer rate are enhanced for thermal radiation and temperature ratio parameters. It is observed that the impact of Brownian motion on the temperature and nanoparticle concentration is quite reverse. [ABSTRACT FROM AUTHOR]

The current study compacts with effect of nonlinear convection and radiation on tangent hyperbolic fluid flow of through a convectively heated vertical surface. The converted set of boundary layer equations are solved numerically by Runge-Kutta-Fehlberg method. The effect of various pertinent parameters on flow and heat transfer characteristics are discussed with tabulated numerical values and deliberate figures. Additionally, the skin friction coefficient and Nusselt number are also presented. We noticed that, the skin friction factor and heat transfer rates are higher in presence of nonlinear convection than its absence. Further, velocity profile decreases by increasing power law index but establishes opposite results for skin friction. [ABSTRACT FROM AUTHOR]

Mathematical modeling for magnetohydrodynamic (MHD) nonlinear convective flow of Jeffrey nanofluid over a nonlinear stretching sheet is introduced. Effect of thermal radiation is considered. Phenomena of heat and mass transfer is based through involvement of convective conditions. The Brownian motion and thermophoresis effects are deliberated in energy and concentration expressions. Dimensional nonlinear systems for momentum, energy and concentration are converted into dimensionless systems by invoking appropriate variables. Series solutions are acquired through convergence domains. Behavior of various physical variables on velocity, temperature and nanoparticle concentration are scrutinized graphically. Skin friction coefficient, local Nusselt and Sherwood numbers are calculated through numerical values. It is concluded that velocity field enhances for Deborah number while reverse situation is noticed regarding power index. Temperature and heat transfer rate are enhanced via thermal radiation. Moreover impact of Brownian motion on the concentration and local Sherwood number are quite reverse. [ABSTRACT FROM AUTHOR]

Objective of present article is to describe magnetohydrodynamic (MHD) non-linear convective flow of Powell-Erying nanofluid over a stretching surface. Characteristics of Newtonian heat and mass conditions in this attempt is given attention. Heat and mass transfer analysis is examined in the frame of thermal radiation and chemical reaction. Brownian motion and thermophoresis concept is introduced due to presence of nanoparticles. Nonlinear equations of momentum, energy and concentration are transformed into dimensionless expression by invoking suitable variables. The series solutions are obtained through homotopy analysis method (HAM). Impact of embedded variables on the velocity, temperature and nanoparticles concentration is graphically presented. Numerical values of skin friction coefficient, local Nusselt and Sherwood numbers are computed and analyzed. It is concluded that velocity field enhances for fluid variable while reverse situation is noticed regarding Hartman number. Temperature and heat transfer rate behave quite reverse for Prandtl number. It is also noted that the concentration and local Sherwood number have opposite behavior in the frame of Brownian motion. [ABSTRACT FROM AUTHOR]

Mathematical analysis of magnetohydrodynamic (MHD) three-dimensional nonlinear convective flow of Maxwell nanofluid towards a stretching surface is made in this article. Characteristics of heat transfer are examined under thermal radiation, heat generation/absorption and prescribed heat flux condition. Nanofluid model includes Brownian motion and thermophoresis. Dimensional nonlinear expressions of momentum, energy and concentration are converted into dimensionless systems by invoking suitable similarity variables. A well-known homotopic technique is implemented for dimensionless expressions. Impact of different quantities on velocities, temperature and concentration are scrutinized graphically and discussed in detail. The expressions of Nusselt and Sherwood numbers are calculated and addressed comprehensively. It is also seen that thermal radiation parameter enhances the temperature field and heat transfer rate. [ABSTRACT FROM AUTHOR]

Nonlinear thermal radiation and chemical reaction in magnetohydrodynamic (MHD) flow of third grade nanofluid over a stretching sheet with variable thickness are addressed. Heat generation/absorption and nonlinear convection are considered. The sheet moves with nonlinear velocity. Sheet is convectively heated. In addition zero mass flux condition for nanoparticle concentration is imposed. Results for velocity, temperature, concentration, skin friction and local Nusselt number are presented and examined. It is found that velocity and boundary layer thickness are increasing for Reynolds number. Temperature is a increasing function of the heat generation/absorption parameter while it causes a decrease in the heat transfer rate. Moreover effect of Brownian motion and chemical reaction on the concentration are quite reverse. [ABSTRACT FROM AUTHOR]

We investigate the effects of momentum, thermal, and solute slip boundary conditions on nanofluid boundary layer flow along a permeable surface. The conventional no-slip boundary conditions at the surface are replaced by slip boundary conditions. At moderate to high temperatures, the temperature-concentration dependence relation is nonlinear and the Soret effect is significant. The governing partial differential equations are solved numerically. The influence of significant parameters on the fluid properties as well as on the skin friction, local Nusselt number, local Sherwood number, and the local nanoparticle Sherwood number are determined. We show, among other results, that the existence and uniqueness of the solutions depends on the slip parameters, and that the region of existence of the dual solution increases with the slip parameters. [ABSTRACT FROM AUTHOR]