Unsteady stagnation flow and heat transfer towards a shrinking sheet

The unsteady stagnation flow towards a shrinking sheet is investigated. With assumptions that the sheet is shrunk impulsively from rest, and simultaneously the surface temperature is suddenly increased from that of surrounding fluid, the boundary layer equations are transformed to a set of nonlinear partial differential equations by means of a similarity transformation. The highly accurate analytical approximations are given, which match the numerical results given by the Keller–Box scheme.     

MHD stagnation point flow of a Maxwell nanofluid over a shrinking sheet (multiple solution)

In the current study, a realistic approach is used to investigate the MHD stagnation point flow of a Maxwell nanofluid past a shrinking sheet with a chemical reaction. First, the flow model is made non dimensionalized via an appropriate transformation. The non dimensionalized equations are numerically tackled by adopting the bvp4c technique. It is also analyzed that the dual solutions are obtained for a particular choice of shrinking parameter. A detailed analysis of the impact of several parameters on the velocity field, temperature distribution, and concentration distribution is carried out graphically. The computed result shows that the first solution significantly increases for higher values of the magnetic parameter, whereas the second solution decreases. Furthermore, it is noted that the first and second solutions decreases for the relaxation parameter. The physical quantities are observed graphically. It is exhibited that the Nusselt number shows a decreasing behavior for the both solutions via relaxation parameter.     

Radiation and slip effects on MHD point flow of nanofluid towards a stretching sheet with melting heat transfer

In the present paper, the melting heat transfer of a nanofluid over a stretching sheet is investigated. Magnetohydrodynamic stagnation point flow with thermal radiation and slip effects is considered for this study. The governing model of the flow is solved by Runge–Kutta fourth-order method using appropriate similarity transformations. Temperature and velocity fields are presented for various flow pertinent parameters. Nondimensional physical parameters such as Prandtl number, radiation parameter, Brownian motion parameter, Lewis number, thermophoresis parameter, magnetic parameter, and melting parameter on fluid velocity, heat, concentration, skin friction, Sherwood number, and Nusselt number are presented graphically and discussed numerically. Heat transfer rate can be increased by increasing slip, melting, or radiation parameter. Mass transfer increases for greater values of melting parameter or slip parameter while radiation parameter shows the opposite impact on mass transfer.     

Mixed convection of transient MHD stagnation point flow over a stretching sheet with quadratic convection and thermal radiation

This study investigates the influence of quadratic (nonlinear) convection in transient magnetohydrodynamic (MHD) combined convection over a two-dimensional stretching sheet. It explores the effects of thermal radiation, suction, and heat sources or sinks. The Crank–Nicholson implicit finite difference method is employed for numerical computations. The significance lies in considering secondary convection, which is crucial in understanding nonlinear convective effects affecting flow and heat transfer properties. The study aims to advance our comprehension of how secondary convection impacts the overall system behavior. Through numerical calculations validated against existing literature, strong agreement is demonstrated. The study evaluates secondary convection effects, magnetic, buoyancy, gravitational parameters, Prandtl number, and radiation parameters. Notably, strong quadratic convection alters flow patterns, affecting velocity and temperature profiles. Moreover, it is observed that when increasing the nonlinear (quadratic) convective factors, the temperature profile increases for $$ and decreases for $$. The prevalence of nonlinear or second-order convection highlights magnetic dominance. In essence, this research enhances our understanding of complex convection interactions in MHD flow, shedding light on the role of secondary convection in shaping system behavior.     

Unsteady natural convection magnetohydrodynamic stagnation point flow with assisting and opposing characteristics: Buongiorno model

In this investigation, the flow of an unsteady mixed convection boundary layer viscous nanofluid on a stretchable sheet is considered. The flow examination is affected by a magnetic field. The reason for the examination exhibited is to create models for nanomaterials that incorporate the Brownian movement and thermophoresis phenomena. The created nonlinear standard differential condition is illuminated numerically utilizing the Runge-Kutta-Gill technique and the start program. The different factors of speed, temperature, and concentration are reported and discussed. The examination shows that the speed, temperature, and concentration are lower in contrast with the consistent stream on account of an assisting flow, whereas the opposite situation is noticed in the opposing flow case. The effects of Brownian movement and thermophoresis in the concentration case are totally different.     

Stagnation point flow and heat transfer over a stretching/shrinking sheet in a porous medium

The steady stagnation point flow and heat transfer over a shrinking sheet in a porous medium is studied. A similarity transformation is used to reduce the governing system of partial differential equations to a set of nonlinear ordinary differential equations which are then solved numerically using the Keller-box method. The behavior of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. The results indicate that dual solutions exist for the shrinking case.     

Analytical solutions of triple-diffusive magnetohydrodynamic fluid flowing through a stretching/shrinking sheet

The aim of this article is to investigate the dual nature solutions of the triple diffusive magnetohydrodynamic flow due to stretching/shrinking surfaces. The system of nonlinear partial differential equations is transformed into nonlinear ordinary differential equations with the help of compatible transforms. Analytical dual solutions are obtained for every unknown velocity, temperature, and concentration profile in terms of known physical parameters. Heat and mass transfer analyses have been carried out in the presence of convective boundary conditions. The graphic interpretation of the possible dual solutions of dimensionless velocity, temperature, concentration, skin-friction coefficient, and Nusselt and Sherwood numbers is analyzed under the influence of different known physical parameters. The obtained results are validated against previously published results for a special case of the problem.     

Dual solutions in boundary layer stagnation-point flow and mass transfer with chemical reaction past a stretching/shrinking sheet

In this paper, an analysis is presented to study dual nature of solution of mass transfer with first order chemical reaction in boundary layer stagnation-point flow over a stretching/shrinking sheet. The governing equations are transformed into a set of self-similar ordinary differential equations by similarity transformations. The transformed equations are solved numerically using very efficient shooting method. The study reveals that the dual solutions of velocity and concentration exist for certain values of velocity ratio parameter (the ratio of stretching/shrinking rate and straining rate). The concentration boundary layer thickness decreases with increasing values of Schmidt number and reaction-rate parameter for both solutions.     

Impact of the Cattaneo‐Christov thermal and solutal diffusion models on the stagnation point slip flow of Walters' B nanofluid past an electromagnetic sheet

The present study investigates the effect of Cattaneo‐Christov thermal and solutal diffusion on the stagnation point flow of Walters' B nanofluid past an electromagnetic sheet subject to velocity, thermal, and solutal slips. The study analyzed the role of electromagnetic fields. In addition, the authors introduced the heat transfer aspect due to Brownian motion and thermophoretic force. The numerical solution of the transformed governing equations employed the spectral local linearization method. Comparisons showed an excellent agreement with the numerical data presented in previous notable works. The study reveals that the developed electromagnetic field due to the arrangement of the sheet causes accelerated fluid motion, and diminution of nondimensional temperature and concentration. In addition, augmented velocity, thermal, and solutal slips develop the corresponding descending boundary layers. The augmented short memory coefficient enhances the skin friction coefficient. The Cattaneo‐Christov thermal and solutal diffusion upsurge the heat and mass transfer rates from the electromagnetic sheet, respectively.     

Unsteady MHD flow and heat transfer with viscous dissipation past a stretching sheet

In this paper a study is carried out to analyze the unsteady heat transfer effects of viscous dissipation on the steady boundary layer flow past a stretching sheet with prescribed constant surface temperature in the presence of a transverse magnetic field. The sheet is assumed to stretch linearly along the direction of the fluid flow. The assumed initial steady flow and temperature field neglecting dissipation effects becomes transient by accounting dissipation effects when time t′ > 0. The temperature and the Nusselt number are computed numerically using an implicit finite difference method. The obtained steady temperature field with dissipation is of practical importance.     

Significance of heat source/sink on the radiative flow of Cross nanofluid across an exponentially stretching surface towards a stagnation point with chemical reaction

A numerical review on magnetohydrodynamics radiative motion of Cross nanofluid across an exponentially stretchable surface near stagnation point with varying heat source/sink is addressed. Brownian movement and thermophoretic impacts are assumed. The governing equations for this study are first altered as a system of ordinary differential equations by similarity transformation. With an aid of the Runge–Kutta 4th order mechanism together with the shooting procedure, the impacts of several pertinent parameters including chemical reaction on regular profiles (velocity, temperature, and concentration) are explicated. The consequences of the same parameters on surface drag force, transfer rates of heat, and mass are visualized in tables. From the analysis, it was noticed that the magnetic field parameter enhances the temperature and decreases the velocity of the Cross nanofluid. Also, fluid temperature is an increasing function with thermal radiation and nonuniform heat source/sink. The rate of heat transfer is increased with thermophoresis and diminished with Brownian motion. Sherwood's number is diminished with Brownian motion but it was boosted up with thermophoresis. The present results are compared with published results and those are in agreement.     

Effect of thermophoresis and Brownian motion on the melting heat transfer of a Jeffrey fluid near a stagnation point towards a stretching surface: Buongiorno's model

This analysis intends to address the coupled effect of phase change heat transfer, thermal radiation, and viscous heating on the MHD flow of an incompressible chemically reactive nanofluid in the vicinity of the stagnation point toward the stretching surface, taking a Jeffrey fluid as the base fluid. Convergent analytical solutions for the nonlinear boundary layer equations are obtained by the successive application of scaling variables and the highly efficacious homotopy analysis method. Error analysis is implemented to endorse the convergence of the solutions. Through parametric examination, influence of various physical parameters occurring in analysis of the profiles of velocity, temperature, and nanoparticle concentration, coefficient of surface drag, rates of mass and heat transfer is explored pictorially. The Deborah number and the melting parameter are found to enhance velocity, and the associated momentum boundary layers are thicker, whereas the magnetic field depreciates the flow rate. Temperature is observed to enhance with the thermophoresis parameter, Prandtl number and Eckert number, whereas a reduction is seen with the thermal radiation parameter and Brownian motion parameter. Nanoparticle concentration is depleted by the chemical reaction parameter, the thermophoresis parameter, and the Lewis number.     

Influence of heat transfer on the nonorthogonal stagnation point flow of a third‐order fluid towards a stretching surface

The present article looks at the theoretical analysis of a steady stagnation‐point flow with heat transfer of a third‐order fluid towards a stretching surface. The formulation of the problem has been carried out for a third order fluid and constructed partial differential equations are rehabilitated into ordinary differential equations. The consequential ordinary differential equations are solved analytically using the homotopy analysis method (HAM). Graphical illustrations are shown for various parameters involved in the flow equations. Numerical values of skin friction coefficients and heat flux are computed and presented through tables. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21042     

Computational study on heat transfer and MHD-electrified flow of fractional Maxwell nanofluids suspended with SWCNT and MWCNT

Recent developments in fluid dynamics have been focusing on nanofluids, which preserve significant thermal conductivity properties and magnify heat transport in fluids. Classical nanofluid studies are generally confined to models described by partial differential equations of an integer order, where the memory effect and hereditary properties of materials are neglected. To overcome these downsides, the present work focuses on studying nanofluids with fractional derivatives formed by differential equations with Caputo time derivatives that provide memory effect on nanofluid characteristics. Further, heat transfer enhancement and boundary layer flow of fractional Maxwell nanofluid with single-wall and multiple walls carbon nanotubes are investigated. The Maxwell nanofluid saturates the porous medium. Also, buoyancy, magnetic, electric, and heating effects are considered. Governing continuity, momentum, and energy equations involving Caputo time-fractional derivatives reduced nondimensional forms using suitable dimensionless quantities. Numerical solutions for arising nonlinear problems are developed using finite difference approximation combined with L1 algorithm. The influence of involved physical parameters on flow and heat transfer characteristics is analyzed and depicted graphically. Our simulations found out that surface drag of Maxwell nanofluid with single-walled carbon nanotubes dominates nanofluids with multiple walls carbon nanotubes, but the reverse trend is noticed for larger Grashof number values.     

Mixed convection flow near a non-orthogonal stagnation point towards a stretching vertical plate

An analysis of the steady two-dimensional mixed convection flow of an incompressible viscous fluid near an oblique stagnation point on a heated or cooled stretching vertical flat plate has been studied. It is assumed that the plate is stretched with a velocity proportional to the distance from a fixed point and the temperature of the plate is constant. Both the cases of the assisting and opposing flows are considered. It is shown that the velocity increases as the shear parameter γ increases with the increase of the straining parameter a/c. These flows have a boundary layer structure near the stagnation region. It is also found that the flow has an inverted boundary layer structure when the stretching velocity of the surface exceeds the stagnation velocity of the free stream (a/c < 1). It is shown that the position of the point x_s of zero skin friction (shear stress on the wall) is shifted to the left or to the right of the origin and it depends upon the balance between obliqueness, straining motion and buoyancy effects.     

Features of Cattaneo‐Christov heat flux model for Stagnation point flow of a Jeffrey fluid impinging over a stretching sheet: A numerical study

The present work focuses on a two‐dimensional steady incompressible stagnation point flow of a Jeffery fluid over a stretching sheet. The Cattaneo‐Christov heat flux model is incorporated into this study. Simulation is conducted via the Runge‐Kutta fourth‐order cum shooting method for the transformed system of nonlinear equations. The influence of the governing parameters on the dimensionless velocity, temperature, skin friction, streamlines, and isotherms is incorporated. A significant outcome of the current investigation is that an increase in the relaxation time parameter uplifts temperature; however, a gradual decrease is observed in the velocity field. Another important outcome of the present analysis is that the momentum boundary layer augments due to an increase in the Deborah number; however, a decrease is observed in the temperature. Furthermore, it is also observed that the skin friction coefficient escalates with an increase in the relaxation time parameter for the assisting flow, but a reverse trend is observed for the opposing flow.     

On impulsively started convection: The case of stagnation point flow

Transient convection in incompressible planar and axisymmetric point flow is analyzed numerically in this work, and the thermal boundary layer response to surface sudden heating and cooling in the two settings is presented and compared over a range of Prandtl number between 0.5 and 100. A comparison between surface sudden cooling and heating is performed and different criteria are established as to when surface sudden heating and cooling are equivalent in terms of the transition time. With no initial thermal boundary layer (surface and fluid are at the same temperature), the transition time from the initial steady state to the final steady state upon surface sudden cooling or heating is found to be a constant regardless of the surface heating or cooling extent above or below the initial surface temperature, and is dependent only on the Prandtl number. With the existence of an initial thermal boundary layer, the transition time is dependent upon the heating or cooling extent, the initial surface temperature, the Prandtl number and whether heating/cooling is towards building-up or demolishing the thermal boundary later. It takes longer time when surface sudden heating or cooling is towards demolishing the thermal boundary layer than building it up. With symmetric surface sudden cooling or heating above or below the far-field fluid temperature, the transition time is independent on the surface cooling or heating extent and is a function of only the Prandtl number. A considerable difference in the thermal boundary layer response in the two settings is found. The transition time from the initial to the final steady state in axisymmetric stagnation point flow is less than that in plane stagnation flow under the same conditions.     

Effect of thermal radiation on magnetohydrodynamic three‐dimensional motion of a nanofluid past a shrinking surface under the influence of a heat source

An analytical technique known as the homotopy analysis method is used to acquire solutions for magnetohydrodynamic 3‐D motion of a viscous nanofluid over a saturated porous medium with a heat source and thermal radiation. The governing nonlinear partial differential equations are changed to ordinary differential equations employing appropriate transformations. Validation of the present result is done with the help of error analysis for flow and temperature. The influences of pertinent parameters on momentum, energy, and Nusselt number are studied and discussed. The major findings are: the velocity of the nanofluid is affected by the nanoparticle volume fraction and the thickness of the thermal boundary layer becomes thinner and thinner subject to sink, whereas the effect is revered in case of the source.     

Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet

In the present study, we analyze the effects of partial slip on steady boundary layer stagnation-point flow of an incompressible fluid and heat transfer towards a shrinking sheet. Similarity transformation technique is adopted to obtain the self-similar ordinary differential equations and then the self-similar equations are solved numerically using shooting method. This investigation explores the conditions of the non-existence, existence, uniqueness and duality of the solutions of self-similar equations numerically. Due to the increase of slip parameter (δ), the range of velocity ratio parameter (c/a) where the similarity solution exists, increases.