Dual solutions in magnetohydrodynamic (MHD) flow on a nonlinear porous shrinking sheet: A stability analysis

2018 ◽  
Author(s):  
Mohamad Mustaqim Junoh ◽  
Fadzilah Md Ali ◽  
Norihan Md Arifin ◽  
Norfifah Bachok
Keyword(s):  
Stability Analysis ◽  
Mhd Flow ◽  
Dual Solutions ◽  
Shrinking Sheet

scholarly journals Stability analysis of heat transfer in nanomaterial flow of boundary layer towards a shrinking surface: Hybrid nanofluid versus nanofluid

2021 ◽  
Author(s):  
Aqeel ur Rehman ◽  
Zaheer Abbas
Keyword(s):  
Heat Transfer ◽  
Stability Analysis ◽  
Differential Equations ◽  
Stable Solution ◽  
Mhd Flow ◽  
Least Square Method ◽  
Dual Solutions ◽  
Least Square ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid

Many boundary value problems (BVPs) have dual solutions in some cases containing one stable solution (upper branch) while other unstable (lower branch). In this paper, MHD flow and heat transfer past a shrinking sheet is studied for three distinct fluids: kerosene hybrid nanofluid, kerosene nanofluid, and kerosene nanofluid. The partial differential equations (PDEs) are turned into ordinary differential equations (ODEs) using an appropriate transformation and then dual solutions are obtained analytically by employing the Least Square method (LSM). Moreover, stability analysis is implemented on the time-dependent case by calculating the least eigenvalues using Matlab routine bvp4c. It is noticed that negative eigenvalue is related to unstable solution i.e., it provides initial progress of disturbance and positive eigenvalue is related to stable solution i.e., the disturbance in solution decline initially. The impacts of various parameters, skin friction coefficient, and local Nusselt number for dual solutions are presented graphically. It is also noted that the results obtained for hybrid nanofluids are better than ordinary nanofluids.

scholarly journals Dual Solutions and Stability Analysis of a Hybrid Nanofluid over a Stretching/Shrinking Sheet Executing MHD Flow

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 276 ◽  
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Ilyas Khan ◽  
El-Sayed M. Sherif
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Temporal Stability ◽  
Mhd Flow ◽  
Numerical Results ◽  
Dual Solutions ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid

In this paper, the unsteady magnetohydrodynamic (MHD) flow of hybrid nanofluid (HNF) composed of C u − A l 2 O 3 /water in the presence of a thermal radiation effect over the stretching/shrinking sheet is investigated. Using similarity transformation, the governing partial differential equations (PDEs) are transformed into a system of ordinary differential equations (ODEs), which are then solved by using a shooting method. In order to validate the obtained numerical results, the comparison of the results with the published literature is made numerically as well as graphically and is found in good agreements. In addition, the effects of many emerging physical governing parameters on the profiles of velocity, temperature, skin friction coefficient, and heat transfer rate are demonstrated graphically and are elucidated theoretically. Based on the numerical results, dual solutions exist in a specific range of magnetic, suction, and unsteadiness parameters. It was also found that the values of f ″ ( 0 ) rise in the first solution and reduce in the second solution when the solid volume fraction ϕ C u is increased. Finally, the temporal stability analysis of the solutions is conducted, and it is concluded that only the first solution is stable.

Analysis of Hiemenz flow of Reiner-Rivlin fluid over a stretching/shrinking sheet

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Golam Mortuja Sarkar ◽  
Suman Sarkar ◽  
Bikash Sahoo
Keyword(s):  
Nusselt Number ◽  
Stability Analysis ◽  
Friction Coefficient ◽  
Skin Friction Coefficient ◽  
Dual Solutions ◽  
Shrinking Sheet ◽  
Content Type ◽  
Hiemenz Flow ◽  
Self Similar ◽  
The Stability

Purpose This paper aims to theoretically and numerically investigate the steady two-dimensional (2D) Hiemenz flow with heat transfer of Reiner-Rivlin fluid over a linearly stretching/shrinking sheet. Design/methodology/approach The Navier–Stokes equations are transformed into self-similar equations using appropriate similarity transformations and then solved numerically by using shooting technique. A simple but effective mathematical analysis has been used to prove the existence of a solution for stretching case (λ> 0). Moreover, an attempt has been laid to carry the asymptotic solution behavior for large stretching. The obtained asymptotic solutions are compared with direct numerical solutions, and the comparison is quite remarkable. Findings It is observed that the self-similar equations exhibit dual solutions within the range [λc, −1] of shrinking parameter λ, where λc is the turning point from where the dual solutions bifurcate. Unique solution is found for all stretching case (λ > 0). It is noticed that the effects of cross-viscous parameter L and shrinking parameter λ on velocity and thermal fields show opposite character in the dual solution branches. Thus, a linear temporal stability analysis is performed to determine the basic feasible solution. The stability analysis is based on the sign of the smallest eigenvalue, where positive or negative sign leading to a stable or unstable solution. The stability analysis reveals that the first solution is stable that describes the main flow. Increase in cross-viscous parameter L resulting in a significant increment in skin friction coefficient, local Nusselt number and dual solutions domain. Originality/value This work’s originality is to examine the combined effects of cross-viscous parameter and stretching/shrinking parameter on skin friction coefficient, local Nusselt number, velocity and temperature profiles of Hiemenz flow over a stretching/shrinking sheet. Although many studies on viscous fluid and nanofluid have been investigated in this field, there are still limited discoveries on non-Newtonian fluids. The obtained results can be used as a benchmark for future studies of higher-grade non-Newtonian flows with several physical aspects. All the generated results are claimed to be novel and have not been published elsewhere.

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

2022 ◽  
pp. 239779142110690
Author(s):  
Seema Tinker ◽  
SR Mishra ◽  
PK Pattnaik ◽  
Ram Prakash Sharma
Keyword(s):  
Heat Transfer ◽  
Stability Analysis ◽  
Dual Solutions ◽  
The Other ◽  
Time Dependent ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid ◽  
Heat Transfer Characteristics ◽  
Suction Parameter ◽  
Transfer Characteristics

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.

scholarly journals Effect of Viscous Dissipation in Heat Transfer of MHD Flow of Micropolar Fluid Partial Slip Conditions: Dual Solutions and Stability Analysis

Energies ◽  
2019 ◽  
Vol 12 (24) ◽  
pp. 4617 ◽  
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Ilyas Khan ◽  
Seifedine Kadry ◽  
Seungmin Rho ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Thermal Radiation ◽  
Viscous Dissipation ◽  
Micropolar Fluid ◽  
Fluid Velocity ◽  
Dual Solutions ◽  
Critical Values ◽  
Partial Slip ◽  
Shrinking Sheet

In this study, first-order slip effect with viscous dissipation and thermal radiation in micropolar fluid on a linear shrinking sheet is considered. Mathematical formulations of the governing equations of the problem have been derived by employing the fundamental laws of conservations which then converted into highly non-linear coupled partial differential equations (PDEs) of boundary layers. Linear transformations are employed to change PDEs into non-dimensional ordinary differential equations (ODEs). The solutions of the resultant ODEs have been obtained by using of numerical method which is presented in the form of shootlib package in MAPLE 2018. The results reveal that there is more than one solution depending upon the values of suction and material parameters. The ranges of dual solutions are S ≥ S c i , i = 0 , 1 , 2 and no solution is S < S c i where S c i is the critical values of S . Critical values have been obtained in the presence of dual solutions and the stability analysis is carried out to identify more stable solutions. Variations of numerous parameters have been also examined by giving tables and graphs. The numerical values have been obtained for the skin friction and local Nusselt number and presented graphically. Further, it is observed that the temperature and thickness of the thermal boundary layer increase when thermal radiation parameter is increased in both solutions. In addition, it is also noticed that the fluid velocity increases in the case of strong magnetic field effect in the second solution.

scholarly journals Effects of Stefan Blowing and Slip Conditions on Unsteady MHD Casson Nanofluid Flow Over an Unsteady Shrinking Sheet: Dual Solutions

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 487 ◽  
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Jawad Raza ◽  
Ilyas Khan ◽  
El-Sayed M. Sherif
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Mhd Flow ◽  
Dual Solutions ◽  
Partial Slip ◽  
Shrinking Sheet ◽  
Physical Parameters ◽  
Casson Nanofluid ◽  
Slip Conditions ◽  
Similarity Transformations

In this article, the magnetohydrodynamic (MHD) flow of Casson nanofluid with thermal radiation over an unsteady shrinking surface is investigated. The equation of momentum is derived from the Navier–Stokes model for non-Newtonian fluid where components of the viscous terms are symmetric. The effect of Stefan blowing with partial slip conditions of velocity, concentration, and temperature on the velocity, concentration, and temperature distributions is also taken into account. The modeled equations of partial differential equations (PDEs) are transformed into the equivalent boundary value problems (BVPs) of ordinary differential equations (ODEs) by employing similarity transformations. These similarity transformations can be obtained by using symmetry analysis. The resultant BVPs are reduced into initial value problems (IVPs) by using the shooting method and then solved by using the fourth-order Runge–Kutta (RK) technique. The numerical results reveal that dual solutions exist in some ranges of different physical parameters such as unsteadiness and suction/injection parameters. The thickness of the velocity boundary layer is enhanced in the second solution by increasing the magnetic and velocity slip factor effect in the boundary layer. Increment in the Prandtl number and Brownian motion parameter is caused by a reduction of the thickness of the thermal boundary layer and temperature. Moreover, stability analysis performed by employing the three-stage Lobatto IIIA formula in the BVP4C solver with the help of MATLAB software reveals that only the first solution is stable and physically realizable.

Stability analysis of Cu–H2O nanofluid over a curved stretching–shrinking sheet: existence of dual solutions

2019 ◽  
Vol 97 (8) ◽  
pp. 911-922 ◽  
Author(s):  
Usama ◽  
S. Nadeem ◽  
A.U. Khan
Keyword(s):  
Stability Analysis ◽  
Volume Fraction ◽  
Reverse Flow ◽  
Velocity Slip ◽  
Coupled System ◽  
Dual Solutions ◽  
Shrinking Sheet ◽  
Flow Situation ◽  
The Stability ◽  
Parameter Values

The effect of mass suction with temperature jump and velocity slip of viscous, unsteady nanofluid flow past a curved shrinking–stretching surface is analyzed in this work. Copper (Cu) and water are considered nanoparticles and base fluids, respectively. The complicated coupled system of differential equations is converted into non-dimensional form with some suitable similarity variables. The solution of the nonlinear problem is produced by use of numerical scheme available in the form of bvp4c package in MATLAB. In the case of shrinking towards the surface, a reverse flow situation is also developed and requires careful selection of solution by examining the stability of the solution. Detailed stability analysis is done and critical values are determined for the possible existence of dual solutions. Variation in parameters is analyzed by plotting graphs and tables. The numerical values are also calculated for the reduced Nusselt number and skin friction due to variation in values of different flow parameters. Results have shown that for the curved shrinking surfaces, one should expect multiple solutions for a set of parameter values such as mass suction, curvature, nanoparticles volume fraction, and unsteadiness.

scholarly journals Numerical Investigation of Multiple Solutions for Caputo Fractional-Order-Two Dimensional Magnetohydrodynamic Unsteady Flow of Generalized Viscous Fluid over a Shrinking Sheet Using the Adams-Type Predictor-Corrector Method

Coatings ◽  
2019 ◽  
Vol 9 (9) ◽  
pp. 548 ◽  
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Sayer O. Alharbi ◽  
Ilyas Khan ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Boundary Layer ◽  
Similarity Transformation ◽  
Mhd Flow ◽  
Dual Solutions ◽  
Shrinking Sheet ◽  
Governing Equations ◽  
Boundary Layer Equations ◽  
Velocity Boundary Layer ◽  
Fractional Differential ◽  
Predictor Corrector

In this paper, magnetohydrodynamic (MHD) flow over a shrinking sheet and heat transfer with viscous dissipation has been studied. The governing equations of the considered problem are transformed into ordinary differential equations using similarity transformation. The resultant equations are converted into a system of fractional differential boundary layer equations by employing a Caputo derivative which is then solved numerically using the Adams-type predictor-corrector method (APCM). The results show the existence of two ranges of solutions, namely, dual solutions and no solution. Moreover, the results indicate that dual solutions exist for a certain range of specific parameters which are in line with the results of some previously published work. It is also observed that the velocity boundary layer decreases as the suction and magnetic parameters increase.

The effect of vertical throughflow on the boundary layer flow of a nanofluid past a stretching/shrinking sheet

2017 ◽  
Vol 27 (9) ◽  
pp. 1910-1927 ◽  
Author(s):  
Ioan Pop ◽  
Kohilavani Naganthran ◽  
Roslinda Nazar ◽  
Anuar Ishak
Keyword(s):  
Boundary Layer ◽  
Stability Analysis ◽  
Differential Equations ◽  
Boundary Layer Flow ◽  
Dual Solutions ◽  
Shrinking Sheet ◽  
Content Type ◽  
Boundary Layer Equations ◽  
Vertical Throughflow ◽  
Layer Flow

Purpose The purpose of this paper is to study the effects of vertical throughflow on the boundary layer flow and heat transfer of a nanofluid driven by a permeable stretching/shrinking surface. Design/methodology/approach Similarity transformation is used to convert the system of boundary layer equations into a system of ordinary differential equations. The system of governing similarity equations is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in Matlab software. The generated numerical results are presented graphically and discussed based on some governing parameters. Findings It is found that dual solutions exist in both cases of stretching and shrinking sheet situations. Stability analysis is performed to determine which solution is stable and valid physically. Originality/value Dual solutions are found for positive and negative values of the moving parameter. A stability analysis has also been performed to show that the first (upper branch) solutions are stable and physically realizable, while the second (lower branch) solutions are not stable and, therefore, not physically possible.

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