scholarly journals Stability Analysis of the Magnetized Casson Nanofluid Propagating through an Exponentially Shrinking/Stretching Plate: Dual Solutions

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1162 ◽  
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Ilyas Khan ◽  
El-Sayed M. Sherif ◽  
Hany S. Abdo
Keyword(s):  
Brownian Motion ◽  
Stability Analysis ◽  
Mhd Flow ◽  
Dual Solutions ◽  
Convective Boundary ◽  
Suction Parameter ◽  
Casson Nanofluid ◽  
Stretching Surfaces ◽  
Stretching Plate ◽  
Biot Numbers

In this research, we intend to develop a dynamical system for the magnetohydrodynamic (MHD) flow of an electrically conducting Casson nanofluid on exponentially shrinking and stretching surfaces, in the presence of a velocity and concertation slip effect, with convective boundary conditions. There are three main objectives of this article, specifically, to discuss the heat characteristics of flow, to find multiple solutions on both surfaces, and to do stability analyses. The main equations of flow are governed by the Brownian motion, the Prandtl number, and the thermophoresis parameters, the Schmid and Biot numbers. The shooting method and three-stage Lobatto IIIa formula have been employed to solve the equations. The ranges of the dual solutions are f w c 1 ≤ f w and λ c ≤ λ , while the no solution ranges are f w c 1 > f w and λ c > λ . The results reveal that the temperature of the fluid increases with the extended values of the thermophoresis parameter, the Brownian motion parameter, and the Hartmann and Biot numbers, for both solutions. The presence of dual solutions depends on the suction parameter. In order to indicate that the first solution is physically relevant and stable, a stability analysis has been performed.

Stability analysis of triple solutions of Casson nanofluid past on a vertical exponentially stretching/shrinking sheet

2021 ◽  
Vol 13 (11) ◽  
pp. 168781402110596
Author(s):  
Hazoor Bux Lanjwani ◽  
Salman Saleem ◽  
Muhammad Saleem Chandio ◽  
Muhammad Imran Anwar ◽  
Nadeem Abbas
Keyword(s):  
Boundary Layer ◽  
Stability Analysis ◽  
Differential Equations ◽  
Shrinking Sheet ◽  
Physical Parameters ◽  
Temperature Profiles ◽  
Convective Boundary ◽  
Suction Parameter ◽  
Casson Nanofluid ◽  
Exponential Stretching

The MHD two dimensional boundary layer flow of Casson nanofluid on an exponential stretching/shrinking sheet is considered with effects of radiation parameter, nanoparticles volume fractions (i.e. Fe3O4 and Ti6Al4V) and thermal convective boundary condition. The partial differential equations are transformed into ordinary differential equations by means of similarity transformations. The solutions of the transferred equations are achieved numerically with the help of shooting technique in Maple software. At different ranges of involved physical parameters, triple solutions are found. Therefore, stability analysis is performed by bvp4c in MATLAB to find the stable and physically reliable solution. Impacts of the physical parameter are presented through graphs and tables. Mainly, it is found that an increase in Casson and suction parameters decrease the corresponding velocity profiles while increase in Prandtl number, stretching/shrinking, and suction parameter decrease the temperature profiles. Furthermore, an increase in nanoparticles volumetric fraction, radiation and magnetic parameters as well as Biot number increase the temperature profiles and their thermal boundary layer thicknesses.

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 Convective Effect on Magnetohydrodynamic (MHD) Stagnation Point Flow of Casson Fluid over a Vertical Exponentially Stretching/Shrinking Surface: Triple Solutions

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1238 ◽  
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Ilyas Khan ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Stagnation Point ◽  
Casson Fluid ◽  
Physical Parameters ◽  
Three Solutions ◽  
Convective Boundary ◽  
Suction Parameter ◽  
Stretching Surfaces ◽  
The Impact

In the current study, the characteristics of heat transfer of a steady, two-dimensional, stagnation point, and magnetohydrodynamic (MHD) flow of shear thickening Casson fluid on an exponentially vertical shrinking/stretching surface are examined in attendance of convective boundary conditions. The impact of the suction parameter is also considered. The system of governing partial differential equations (PDEs) and boundary conditions is converted into ordinary differential equations (ODEs) with the suitable exponential similarity variables of transformations and then solved using the shooting method with the fourth order Runge–Kutta method. Similarity transformation is an important class of phenomena in which scale symmetry allows one to reduce the number of independent variables of the problem. It should be noted that solutions of the ODEs show the symmetrical behavior of the PDES for the profiles of velocity and temperature. Similarity solutions are obtained for the case of stretching/shrinking and suction parameters. It is revealed that there exist two ranges of the solutions in the specific ranges of the physical parameters, three solutions depend on the opposing flow case where stagnation point (A) should be equal to 0.1, two solutions exist when λ1 = 0 where λ1 is a mixed convection parameter and A > 0.1, and a single solution exists when λ1 > 0. Moreover, the effects of numerous applied parameters on velocity, temperature distributions, skin friction, and local Nusselt number are examined and given through tables and graphs for both shrinking and stretching surfaces.

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 Micropolar Nanofluid Flow with Slip Effect on Stretching/Shrinking Surfaces

Energies ◽  
2019 ◽  
Vol 12 (23) ◽  
pp. 4529 ◽  
Author(s):  
Sumera Dero ◽  
Azizah Mohd Rohni ◽  
Azizan Saaban ◽  
Ilyas Khan ◽  
Asiful H. Seikh ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Angular Velocity ◽  
Thermal Radiation ◽  
Velocity Slip ◽  
Dual Solutions ◽  
Nanofluid Flow ◽  
Linear Ordinary Differential Equations ◽  
Stretching Surfaces ◽  
Slip Effects ◽  
Micropolar Nanofluid

The purpose of the present paper is to investigate the micropolar nanofluid flow on permeable stretching and shrinking surfaces with the velocity, thermal and concentration slip effects. Furthermore, the thermal radiation effect has also been considered. Boundary layer momentum, angular velocity, heat and mass transfer equations are converted to non-linear ordinary differential equations (ODEs). Then, the obtained ODEs are solved by applying the shooting method and in the results, the dual solutions are obtained in the certain ranges of pertinent parameters in both cases of shrinking and stretching surfaces. Due to the presence of the dual solutions, stability analysis is done and it was found that the first solution is stable and physically feasible. The results are also compared with previously published literature and found to be in excellent agreement. Moreover, the obtained results reveal the angular velocity increases in the first solution when the value of micropolar parameter increases. The velocity of nanofluid flow decreases in the first solution as the velocity slip parameter increases, whereas the temperature profiles increase in both solutions when thermal radiation, Brownian motion and the thermophoresis parameters are increased. Concentration profile increases by increasing N t and decreases by increasing N b .

Stability analysis on the kerosene nanofluid flow with hybrid zinc/aluminum-oxide (ZnO-Al2O3) nanoparticles under Lorentz force

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
LiJun Zhang ◽  
Tayyaba Nazar ◽  
M.M. Bhatti ◽  
Efstathios E. Michaelides
Keyword(s):  
Magnetic Field ◽  
Stability Analysis ◽  
Aluminum Oxide ◽  
Dual Solutions ◽  
Graphical Form ◽  
Al2o3 Nanoparticles ◽  
Lower Branch ◽  
Suction Parameter ◽  
Content Type ◽  
Zinc Aluminum Oxide

Purpose The flow and heat transfer of a hybrid nanofluid composed of kerosene and ZnO-Al2O3 nanoparticles (NPs) is investigated. The flow occurs over complex surfaces with stretching and shrinking features. The base fluid is electrically conducting, and an external magnetic field is added so that the nanofluid and the electric field are in equilibrium. Irrotational flow with viscous dissipation effects is considered. Design/methodology/approach The governing equations of the system are formulated, and a similarity transformation is used to convert the system of equations into ordinary differential equations, which are solved numerically. The friction coefficient of the flow and the Nusselt number are calculated for a wide range of parameters, and the results are presented in graphical form. In addition, dual solutions of the problem were noticed to occur for a certain range of the unsteadiness parameter. A stability analysis has been performed and presented to elucidate the behavior of these dual solutions. Findings For the solution of the upper branch, the velocity and temperature profiles of the nanofluid are enhanced by increasing the magnetic field parameter M, but the same variables decrease in the solution of the lower branch. The same trend is detected for the velocity of the fluid with the suction parameter. The temperature of the nanofluid decreases in both branches of the solution by increasing the Prandtl number. Similarly, they decrease with the suction parameter. The temperature of the nanofluid slightly increases in both branches of the solution by increasing the Eckert number. With the stability analysis the authors performed, it was determined that the solution is stable in the upper branch, but unstable in the lower branch. Originality/value The kerosene nanofluid with hybrid Zinc/Aluminum-oxide is presented for the first time in the literature.

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.

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