Lower Branch Solutions Research Articles

Hybrid nanofluid flow with quadratic velocity and thermal slip over a permeable stretching/shrinking surface

The current paper investigates the viscous flow and characteristics of heat transfer comprising water-based Fe3O4/Al2O3 hybrid nanofluid past a continuously porous stretching/shrinking sheet using quadratic thermal and velocity slip effects. The goal is to obtain analytical or exact solutions for flow and heat transfer under various physical situations. The influences of second-order slip flow conditions in the presence of hybrid nanofluid are given special consideration. The current results in the absence of the hybrid nanoparticles are in good correlation with those reported in the literature. The unique outcome is obtained for the case of the stretching sheet, while the two outcomes are observed for the phenomenon of the shrinking sheet. The temperature distribution uplifts owing to the nanoparticle volume fraction, while the velocity upsurges and declines in the upper and lower branch solutions, respectively.

Flow and heat transfer analysis of a special third grade fluid over a stretchable surface in a parallel free stream

This paper aims to investigate a boundary layer flow with heat and mass transfer of a special third grade fluid over a stretchable surface in a parallel free stream. Using the similarity variables generated by the Lie group analysis, the governing nonlinear partial differential equations (PDEs) are transformed into a system of nonlinear ordinary differential equations (ODEs). The transformed equations are then solved numerically using the shooting technique. In addition, an attempt is made to carry the asymptotic solution behavior for large stretching and suction parameters, and the asymptotic results of skin friction coefficient are then compared with the direct numerical solutions, which shows a good agreement. It is observed that the similarity equations exhibit dual solutions in a certain range of shrinking strength. These two solution branches show different behavior on the skin friction coefficient, local Nusselt number, Sherwood number, velocity, and temperature profiles. Thus, emphasis has been given to carrying out a stability analysis to determine the physically reliable solution. The stability analysis shows that the upper branch solution is stable. It is observed that the suction parameter increases the range of dual solutions and the magnitude of the critical point from where the dual solutions bifurcate; however, the non-Newtonian parameter shows the opposite behavior. The momentum, thermal and concentration boundary layer thicknesses in the upper branch solution are lower than the lower branch solution.

Dual solution of MHD mixed convection flow and heat transfer over a shrinking sheet subject to thermal radiation

In this paper, the steady-state two-dimensional incompressible, magnetohydrodynamics, stagnation point flow and heat transfer of viscous fluid having mixed convection over the linearly shrinking porous sheet is studied. The governing non-linear partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by exercising the pertinent similarity variables. Numerical results are obtained by using the bvp4c solver in MATLAB to solve the resulting ODEs. The dual solution is obtained for a certain range of shrinking and mass suction parameters. The resulting dual solution projects that local skin friction coefficient and Nusselt number escalates in the upper branch solution and decelerates in the lower branch solution with the increase in the mass suction and magnetic parameter. However, the heat transfer rate is diminishing in both the upper and lower branch solutions due to an escalation of the radiation parameter. In addition, temporal stability analysis has been performed by which the upper branch solution is physically trustworthy and stable while the lower branch solution is not physically stable (unstable).

Unsteady micropolar hybrid nanofluid flow past a permeable stretching/shrinking vertical plate

The current analysis aims to find the solution to the buoyancy effect on time-dependent flow and heat transfer induced by a hybrid micropolar nanofluid over a permeable shrinking or stretching vertical flat plate. A novel hybrid nanofluid is utilized, which consists of the agglomeration of water (pure fluid) and two dissimilar nanoparticles like silver (Ag) and titanium dioxide (TiO2). Initially, the model is developed in the form of the non-linear partial differential equations (PDEs) with three independent variables, which are transformed to the set of dimensionless ordinary differential equations (ODEs) using the appropriate similarity transformations. These dimensionless ODEs are solved numerically via the bvp4c package in MATLAB software. The consequence of various involved controlling parameters on the velocity, microrotation, friction drag, temperature, and heat transfer characteristics for the upper branch solution (UPBS) and the lower branch solution (LOBS) are thoroughly inspected. In physical engineering quantities of interest, it is deeply observed that for the case of stretching, the solution of the stable (upper) branch is possible for the entire negative and positive selected values of the stretching/shrinking parameter. In contrast, the lower branch solution exists only for negative values of the stretching/shrinking parameter for the shrinking case.

An effect of MHD fluid flow heat transfer using CNTs with thermal radiation and heat source/sink across a stretching/shrinking sheet

Current work studies the magnetohydrodynamics (MHD) flow of the boundary layer and heat transfer with carbon nanotubes. Furthermore, the velocity and temperature profiles are examined using different physical parameters. The governing equations are reduced to highly nonlinear partial differential equation and ordinary differential equation via similarity variables that can be solved analytically. This research focuses on the magnetohydrodynamics flow and heat transfer of carbon nanotubes across a flat plate. Water is utilised as a base fluid with two types of carbon nanotubes; SWCNT (single wall carbon nanotubes) and MWCNT (multi wall carbon nanotubes). Carbon nanotubes have a wide range of industrial, biomedical, energy production, and space cooling applications because they can improve the thermal properties of base materials. The present work of the magnetohydrodynamics fluid flow with mass transpiration using carbon nanotubes due to a stretching/shrinking surface is deliberated in the momentum and temperature profiles. Moreover, when compared to the lower branch solutions, the upper branch solutions of the magnetic field profiles are analytically stable. According to the results, in comparison to negative values, positive values of the magnetohydrodynamics flow give stable profiles. The new research is expected to serve as a foundation for the dual solution of different sorts of Newtonian fluid flows across diverse types of surfaces. The impact of the actual boundaries used in the issue is discovered, and the results are graphically shown and analyzed. The momentum and temperature profiles of water-SWCNTs are significantly higher than those of water-MWCNTs, while fluid movement slows in the nanofluid region due to nanoparticle temperature and the thickness of the thermal boundary layer increases as the volume fraction of carbon nanotubes increases.

MHD flow of a nanofluid due to a nonlinear stretching/shrinking sheet with a convective boundary condition: Tiwari–Das nanofluid model

PurposeNanofluid research has piqued the interest of scientists due to its intriguing applications in nanoscience, biomedical and electrical engineering, medication delivery, biotechnology, food processing, chemotherapy and other fields. This paper aims to inspect the behavior of the mixed convection magnetohydrodynamic flow and heat transfer induced by a nonlinear stretching/shrinking sheet in a nanofluid with a convective boundary condition. Tiwari and Das mathematical nanofluid model is incorporated in the analysis.Design/methodology/approachThe mathematical model is initially transformed to a nondimensional form by using dimensionless variables. Then the nondimensional partial differential equations are further transformed to a set of similarity equations by using the similarity technique. These equations are solved numerically by the bvp4c function in MATLAB software.FindingsFor a certain range of the stretching/shrinking parameter, two solutions are obtained. The friction factor and the heat transfer rate escalate due to suction parameter with adding nanoparticles volume fraction by almost 27.15% and 0.153% for the upper branch solution, while the friction factor declines by almost 30.10% but the heat transfer rate augments by 0.145% for the lower branch solution. Furthermore, the behavior of the nanoparticle volume fractions on the heat transfer rate behaves differently in the presence of the mixed convection effect. The temperature of fluid augments with increasing Biot number for both solutions.Originality/valueThe present work considers the flow and heat transfer induced by a stretching/shrinking sheet in a nanofluid using the Tiwari–Das nanofluid model with a convective boundary condition, where the effect of the buoyancy force is taken into consideration. It is shown that two solutions are found for a certain range of the shrinking strength, while the solution is unique for the stretching case. This study is important for scientists working in the growing field of nanofluids to become familiar with the flow properties and behaviors of such nanofluids.

Simulation of time-dependent radiative heat motion over a stretching/shrinking sheet of hybrid Nanofluid: Stability analysis for dual solutions

The heat transfer characteristics for the flow of a time-dependent hybrid nanofluid with thermal radiation and source/sink over a stretching/shrinking sheet are examined in the current investigation. We have transformed the governing equations of the presented study into the similarity equations utilizing similarity variables. However, a numerical solution is obtained by using in-build MATLAB code bvp5c. The mass and energy profiles for diverse values of thermophysical parameters are studied together with their physical quantities. It is observed that dual solutions exist, that is, one is upper, and the other is lower branch solution for a definite choice of the unsteadiness parameter. Also, stability analysis is executed to determine the long-term stability of dual solutions, indicating that out of the two, only one is stable and the other is unstable. It is revealed that comparatively, the first solution shows stability, while the second solution shows instability. There is a considerable influence of second-order slip on the problem’s respective flow and heat transfer characteristics. Further, major outcomes also show the dimensionless frictional stress and the magnitude of conventional heat transfer enhancement with growing suction parameter values.

Flow and Heat Transfer Past a Stretching/Shrinking Sheet Using Modified Buongiorno Nanoliquid Model

This paper studies the boundary layer flow and heat transfer characteristics past a permeable isothermal stretching/shrinking surface using both nanofluid and hybrid nanofluid flows (called modified Buongiorno nonliquid model). Using appropriate similarity variables, the PDEs are transformed into ODEs to be solved numerically using the function bvp4c from MATLAB. It was found that the solutions of the resulting system have two branches, upper and lower branch solutions, in a certain range of the suction, stretching/shrinking and hybrid nanofluids parameters. Both the analytic and numerical results are obtained for the skin friction coefficient, local Nusselt number, and velocity and temperature distributions, for several values of the governing parameters. It results in the governing parameters considerably affecting the flow and heat transfer characteristics.

Insights into the dynamics of blood conveying gold nanoparticles on a curved surface when suction, thermal radiation, and Lorentz force are significant: The case of Non-Newtonian Williamson fluid

The motion of blood conveying gold nanoparticles on a curved surface when suction, thermal radiation, and Lorentz force are significant is explored in this report with the aim to announce the increasing effects of Williamson fluid parameter, volume fraction, radius of curvature, thermal radiation, and Lorentz force on such a transport phenomenon. This report was designed to explore the upper and lower solutions of the model suitable to study the enhancement of the aforementioned variables. The similarity solution of the dimensional governing equation was sought using the appropriate similarity variables. These dimensionless forms of ODEs are numerically solved using the 3-stage Lobatto formula, also known as bvp4c. The validation of the numerical scheme was considered. The drag force decelerates and then upsurges owing to the volume fraction of nanoparticles in the corresponding UBS and reduces in LBS, while the rate of heat transfer drastically decreases. The temperature and velocity gradient escalate and decelerate, respectively for both branches of results owing to the effect of higher curvature parameter. The temperature distribution decelerates in both outcomes due to the strength of mass suction while the velocity is​ weakened in the lower branch solution (LBS) and augments in the upper branch solution (UBS).

Mixed convection stagnation-point flow of Cross fluid over a shrinking sheet with suction and thermal radiation

This work focuses on numerically investigating the mixed convection stagnation-point flow and heat transfer in an incompressible Cross fluid over a permeable shrinking sheet, with suction and thermal radiation effects. The governing equations defining the current problem have been first transformed into the ordinary differential equations through similarity transformations. The resulting equations obtained are then solved numerically using the bvp4c programmed in MATLAB, revealing dual (upper and lower branch) solutions for both assisting and opposing flow regions. Stability analysis has been employed afterwards to determine a stable solution between them. Following from there, the upper branch solution has been found to be stable, while the lower branch solution is unstable. In addition, the study has established that the Weissenberg number, the Prandtl number and the suction parameter are strongly influenced to uplift the range of existence for dual solutions.

Mixed convection flow of a hybrid nanofluid past a vertical wedge with thermal radiation effect

PurposeThe purpose of this paper is to numerically study the problem of mixed convection flow of a hybrid nanofluid past a vertical wedge with thermal radiation effect.Design/methodology/approachThe governing nonlinear partial differential equations are transformed into a system of ordinary differential equations by a similarity transformation, which is then solved numerically through the function bvp4c from MATLAB for different values of the governing parameters. The solutions contain a mixed convection parameter λ that has a considerable impact on the flow fields.FindingsIt is found that the solutions of the ordinary (similarity) differential equations have two branches, upper and lower branch solutions, in a certain range of the mixed convection and several other parameters. To establish which of these solutions are stable and which are not, a stability analysis has been performed. The effects of the governing parameters on the fluid flow and heat transfer characteristics are illustrated in tables and figures. It is found that dual (upper and lower branch) solutions exist for both the cases of assisting and opposing flow situations. A stability analysis has also been conducted to determine the physical meaning and stability of the dual solutions.Practical implicationsThis theoretical study is significantly relevant to the applications of the heat exchangers placed in a low-velocity environment and electronic devices cooled by fans.Originality/valueThe case of mixed convection flow of a hybrid nanofluid past a vertical wedge with thermal radiation effects has not been studied before, and hence all generated numerical results are claimed to be original and novel.

Magneto-nanofluid coolants past heated shrinking/stretching surfaces: Dual solutions and stability analysis

Recent advancement in nanotechnology has provides a veritable platform for the emergence of a better ultrahigh-performance coolant known as nanofluid for many engineering and industrial technologies. In this study, we examine the influence of magnetic field on heat transfer enhancement of nanofluids coolants consisting of Cu-water, Al2O3-water, and Fe3O4-water over a slippery but convectively heated shrinking and stretching surfaces. Based on some realistic assumptions, the nonlinear model differential equations are obtained and numerically tackled using shooting procedure with Runge-Kutta-Fehlberg integration scheme. The existence dual solutions in the specific range of shrinking surface parameters is found. Temporal stability analysis to small disturbances is performed on these dual solutions. It is detected that the upper solution branch is stable, substantially realistic with positive smallest eigenvalues while the lower solution branch is unstable with negative smallest eigenvalues. Influence of numerous emerging parameters on the momentum and thermal boundary layer profiles, skin friction, Nusselt number and heat transfer enhancement rate are depicted graphically and quantitatively discussed.

Stability analysis and modeling for the three-dimensional Darcy-Forchheimer stagnation point nanofluid flow towards a moving surface

In this research, the three-dimensional (3D) steady and incompressible laminar Homann stagnation point nanofluid flow over a porous moving surface is addressed. The disturbance in the porous medium has been characterized by the Darcy-Forchheimer relation. The slip for viscous fluid is considered. The energy equation is organized in view of radiative heat flux which plays an important role in the heat transfer rate. The governing flow expressions are first altered into first-order ordinary ones and then solved numerically by the shooting method. Dual solutions are obtained for the velocity, skin friction coefficient, temperature, and Nusselt number subject to sundry flow parameters, magnetic parameter, Darcy-Forchheimer number, thermal radiation parameter, suction parameter, and dimensionless slip parameter. In this research, the main consideration is given to the engineering interest like skin friction coefficient (velocity gradient or surface drag force) and Nusselt number (temperature gradient or heat transfer rate) and discussed numerically through tables. In conclusion, it is noticed from the stability results that the upper branch solution (UBS) is more reliable and physically stable than the lower branch solution (LBS).

Linear Stability on the Local Thermal Nonequilibrium Model of Mixed Convection Boundary Layer Flow over a Moving Wedge in a Porous Medium: Viscous Dissipation and Radiation Effects

Abstract This paper studies the local thermal nonequilibrium (LTNE) model for two-dimensional mixed convection boundary-layer flow over a wedge, which is embedded in a porous medium in the presence of radiation and viscous dissipation. It is considered that the temperature of the fluid and solid phases is not identical; hence, we require two energy equations: one for each phase. The motion of the mainstream and wedge is approximated by the power of distance from the leading boundary layer. The flow and heat transfer in the LTNE phase is governed by the coupled partial differential equations, which are then reduced to nonlinear ordinary differential equations via suitable similarity transformations. Numerical simulations show that when the interphase rate of heat transfer is large, the system attains the local thermal equilibrium (LTE) state and so is for porosity scaled conductivity. When LTNE is strong, the fluid phase reacts faster to the mainstream temperature than the corresponding solid phase. The state of LTE rather depends on radiation and viscous dissipation of the model. Further, numerical solutions successfully predicted the upper and lower branch solutions when the velocity ratio is varied. To assess which of these solutions is practically realizable, an asymptotic analysis on unsteady perturbations for a large time leading to linear stability needs to be performed. This shows that the upper branch solutions are always stable and practically realizable. The physical dynamics behind these results are discussed in detail.

Dual solutions of nanomaterial flow comprising titanium alloy (Ti6Al4V) suspended in Williamson fluid through a thin moving needle with nonlinear thermal radiation: stability scrutinization

Titanium alloy nanoparticle has a variety of applications in the manufacturing of soap and plastic, microsensors, aerospace design material, nano-wires, optical filters, implantation of surgical, and many biological treatments. Therefore, this research article discussed the influence of nonlinear radiation on magneto Williamson fluid involving titanium alloy particles through a thin needle. The arising system of partial differential equations is exercised by the similarity transformations to get the dimensional form of ordinary differential equations. The dual nature of solutions is obtained by implementing bvp4c. The study of stability has been carried out to check which of the results are physically applicable and stable. Influences of pertinent constraints on the flow field are discussed with the help of graphical representations and the method validation is shown in Table 1. The results imply that more than one result is established when the moving needle and the free-stream travel in the reverse directions. Moreover, the magnetic parameter accelerates the severance of boundary-layer flow, while the separation delays in the absence of the nanoparticle. The velocity gradient of nanofluid decays owing to the Williamson parameter in both branches of the outcome, while the temperature shrinks in the first or upper branch solution (stable one) and uplifts in the second or lower branch solution (unstable one). The size of the needle decreases the velocity in the upper solution and accelerates in the lower solution. The patterns of streamlines are more complicated due to the reverse direction of the free stream and thin needle.

Axisymmetric flow of hybrid nanofluid due to a permeable non-linearly stretching/shrinking sheet with radiation effect

Purpose This paper aims to report theoretical and numerical results for the problem of laminar axisymmetric flow of hybrid nanofluid over a permeable non-linearly stretching/shrinking sheet with radiation effect. Design/methodology/approach The numerical solutions of the arising boundary value problem are obtained using the function bvp4c from MATLAB for different values of the governing parameters. Findings It is found that the solutions of the ordinary (similarity) differential equations have two branches, upper and lower branch solutions, in a certain range of the stretching/shrinking and suction parameters. To establish which of these solutions are stable and which are not, a stability analysis has been performed. Originality/value To the best of the authors’ knowledge, present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface, as they successfully extend the problem considered by Mustafa et al. (2015) to the case of hybrid nanofluids.

Flow and Wall Heat Transfer due to a Continuously Moving Slender Needle in Hybrid Nanofluid with Stability Analysis

Present work deals with the numerical study of flow due to a continuously moving slender needle in a hybrid nanoliquid. The mathematical model of this work is developed in terms of nonlinear partial differential equations. By adopting the relevant similarity transformations, these equations are reduced to a system of nonlinear ordinary differential equations. Afterward, the solution is determined computationally via a bvp4c solver in MATLAB software. The influences of nanoparticle volume fraction, needle thickness and velocity ratio parameter on the rate of heat transfer, coefficient of skin friction, velocity as well as temperature distributions are illustrated in graphical form to describe the important features of the solution. The multiple solutions seem to appear when the needle opposes the free stream flow. It is revealed from the study that the composite (hybrid) nanoparticles augment the heat transfer rate between the flow and the needle in a certain domain of the velocity ratio parameter. The analysis of stability has proved that the upper branch solution represents stable flow, whereas the lower branch solution represents unstable flow.

Numerical and Computer Simulations of Cross-Flow in the Streamwise Direction through a Moving Surface Comprising the Significant Impacts of Viscous Dissipation and Magnetic Fields: Stability Analysis and Dual Solutions

The inspiration for this study is to explore the crucial impact of viscous dissipation (VISD) on magneto flow through a cross or secondary flow (CRF) in the way of streamwise. Utilizing the pertinent similarity method, the primary partial differential equations (PDEs) are changed into a highly nonlinear dimensional form of ordinary differential equations (ODEs). These dimensionless forms of ODEs are executed numerically by the aid of bvp4c solver. The impact of pertinent parameters such as the suction parameter, magnetic parameter, moving parameter, and viscous dissipation parameter is discussed with the help of plots. Dual solutions are obtained for certain values of a moving parameter. The velocities in the direction of streamwise, as well as cross-flow, decline in the upper branch solution, while the contrary impact is seen in the lower branch solution. However, the influence of suction on the velocities in both directions uplifts in the upper branch solution and shrinks in the lower branch solution. The analysis is also performed in terms of stability to inspect which solution is stable or unstable, and it is observed that the lower branch solution is unstable, whereas the upper branch one is stable.

Cross flow and heat transfer past a permeable stretching/shrinking sheet in a hybrid nanofluid

PurposeThe purpose of this paper is to study the laminar boundary layer cross flow and heat transfer on a rotational stagnation-point flow over either a stretching or shrinking porous wall submerged in hybrid nanofluids. The involved boundary layers are of stream-wise type with stretching/shrinking process along the surface.Design/methodology/approachUsing appropriate similarity variables the partial differential equations are reduced to ordinary (similarity) differential equations. The reduced system of equations is solved analytically (by high-order perturbed field propagation for small to moderate stretching/shrinking parameter and low-order perturbation for large stretching/shrinking parameter) and numerically using the function bvp4c from MATLAB for different values of the governing parameters.FindingsIt was found that the basic similarity equations admit dual (upper and lower branch) solutions for both stretching/shrinking surfaces. Moreover, performing a linear stability analysis, it was confirmed that the upper branch solution is realistic (physically realizable), while the lower branch solution is not physically realizable in practice. These dual solutions will be studied in the present paper.Originality/valueThe authors believe that all numerical results are new and original and have not been published before for the present problem.

On dual solutions of the unsteady MHD flow on a stretchable rotating disk with heat transfer and a linear temporal stability analysis

The present paper is an extension of the work done by Fang and Tao (2012) in the presence of a magnetic field. The external magnetic field is applied normal to a stretchable disk surface and heat is transferred from the disk surface to the ambient fluid which gives a new family of unsteady viscous flow over a stretchable rotating disk with deceleration. The Navier–Stokes equations and the energy equation admit similarity solutions depending on unsteadiness parameter. The resulting set of highly nonlinear ordinary differential equations are solved numerically. Two solution branches are found due to the effect of unsteadiness parameter for several values of magnetic parameter and stretching parameter. The effects of pertinent flow parameters on the shear stresses on radial and tangential directions, Nusselt numbers, torque, dimensionless velocity and temperature profiles are shown graphically for both the solution branches and their physical interpretation are discussed in detail. Emphasis has been laid to check the stability of dual solution branches. The stability is tested by the sign of the smallest eigenvalue, and our analysis reveals that the lower solution branches are unstable.