scholarly journals Magnetohydrodynamic (MHD) Flow of Micropolar Fluid with Effects of Viscous Dissipation and Joule Heating Over an Exponential Shrinking Sheet: Triple Solutions and Stability Analysis

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 142 ◽  
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Ilyas Khan ◽  
Jawad Raza ◽  
El-Sayed M. Sherif ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Viscous Dissipation ◽  
Joule Heating ◽  
Micropolar Fluid ◽  
Numerical Study ◽  
Mhd Flow ◽  
Shrinking Sheet ◽  
Linear Ordinary Differential Equations ◽  
All Solutions

A numerical study was carried out to examine the magnetohydrodynamic (MHD) flow of micropolar fluid on a shrinking surface in the presence of both Joule heating and viscous dissipation effects. The governing system of non-linear ordinary differential equations (ODEs) was obtained from the system of partial differential equations (PDEs) by employing exponential transformations. The resultant equations were transformed into initial value problems (IVPs) by shooting technique and then solved by the Runge–Kutta (RK) method. The effects of different parameters on velocity, angular velocity, temperature profiles, skin friction coefficient, and Nusselt number were obtained and demonstrated graphically. We observed that multiple solutions occurred in certain assortments of the parameters for suction on a surface. The stability analysis of solutions was performed, and we noted that the first solution was stable while the remaining two solutions were not. The results also showed that the velocity of the fluid increased as the non-Newtonian parameter rose in all solutions. Furthermore, it was detected that the temperature of fluid rose at higher values of the Eckert number in all solutions.

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 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 Numerical Solution of Boundary Layer MHD Flow with Viscous Dissipation

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
S. R. Mishra ◽  
S. Jena
Keyword(s):  
Differential Equations ◽  
Viscous Dissipation ◽  
Graphical Representation ◽  
Mhd Flow ◽  
Transverse Magnetic ◽  
Shrinking Sheet ◽  
Shooting Technique ◽  
Electrically Conducting ◽  
Similarity Transformations ◽  
Runge Kutta Method

The present paper deals with a steady two-dimensional laminar flow of a viscous incompressible electrically conducting fluid over a shrinking sheet in the presence of uniform transverse magnetic field with viscous dissipation. Using suitable similarity transformations the governing partial differential equations are transformed into ordinary differential equations and then solved numerically by fourth-order Runge-Kutta method with shooting technique. Results for velocity and temperature profiles for different values of the governing parameters have been discussed in detail with graphical representation. The numerical evaluation of skin friction and Nusselt number are also given in this paper.

scholarly journals Mixed Convective Flow and Heat Transfer of a Dual Stratified Micropolar Fluid Induced by a Permeable Stretching/Shrinking Sheet

Entropy ◽  
2019 ◽  
Vol 21 (12) ◽  
pp. 1162 ◽  
Author(s):  
Najiyah Safwa Khashi’ie ◽  
Norihan Md Arifin ◽  
Roslinda Nazar ◽  
Ezad Hafidz Hafidzuddin ◽  
Nadihah Wahi ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Micropolar Fluid ◽  
Convective Flow ◽  
Mhd Flow ◽  
Similarity Solutions ◽  
Surface Velocity ◽  
Shrinking Sheet ◽  
Transfer Characteristics ◽  
Flow And Heat Transfer

The present study accentuates the magnetohydrodynamics (MHD) flow and heat transfer characteristics of a dual stratified micropolar fluid over a permeable stretching/shrinking sheet. Thermal and solutal buoyancy forces are also included to incorporate with the stratification effect. Similarity, transformation is applied to reduce the governing model (partial differential equations) into a set of nonlinear ordinary differential equations (ODEs) due to its complexity. Using bvp4c solver in the MATLAB software, numerical results for some limiting cases are in favorable agreement with the earlier published results. Both assisting and opposing buoyancy flows have dual similarity solutions within specific range of suction and stretching/shrinking parameters, whereas only a distinctive solution is observed for pure forced convective flow. The micropolar fluid shows a disparate pattern of flow, heat and mass transfer characteristics between stretching and shrinking cases. Unlike the shrinking flow, the surface velocity gradient, local Nusselt and Sherwood numbers for stretching flow intensify with the increment of the material parameter. The result from stability analysis reveals that the first solution is the real solution, whereas the second solution is virtual.

MHD FLOW OF AN EYRING-POWELL FLUID WITH THE EFFECT OF THERMAL RADIATION, JOULE HEATING AND VISCOUS DISSIPATION

2020 ◽  
Vol 9 (11) ◽  
pp. 9259-9271
Author(s):  
K.R. Babu ◽  
G. Narender ◽  
K. Govardhan
Keyword(s):  
Differential Equations ◽  
Thermal Radiation ◽  
Ordinary Differential Equations ◽  
Viscous Dissipation ◽  
Joule Heating ◽  
Mhd Flow ◽  
Physical Parameters ◽  
Shooting Technique ◽  
Similarity Transformations ◽  
The Magnetic Field

A two-dimensional stream of an magnetohydrodynamics (MHD) Eyring-Powell fluid on a stretching surface in the presence of thermal radiation, viscous dissipation and the Joule heating is analyzed. The flow model in the form of the Partial Differential Equations (PDEs) is transformed into a system of non-linear and coupled Ordinary Differential Equations (ODEs) by implementing appropriate similarity transformations. The resulting ordinary differential equations are solved numerically by the shooting technique with Adams-Moulton Method of fourth order. The numerical solution obtained for the velocity and temperature profiles has been presented through graphs for different choice of the physical parameters. The magnetic field is found to have a direct relation with the temperature profile and an inverse with the velocity profile. Increasing the thermal radiation, the temperature tends to rise.

scholarly journals MHD Flow and Heat Transfer of Double Stratified Micropolar Fluid over a Vertical Permeable Shrinking/Stretching Sheet with Chemical Reaction and Heat Source

2020 ◽  
Vol 21 (1) ◽  
pp. 1-14
Author(s):  
Ansab Azam Khan ◽  
Suliadi Firdaus Sufahani ◽  
Khairy Zaimi ◽  
Mohammad Ferdows
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Heat Source ◽  
Chemical Reaction ◽  
Sherwood Number ◽  
Micropolar Fluid ◽  
Numerical Solutions ◽  
Mhd Flow ◽  
Shrinking Sheet ◽  
Local Sherwood Number

The present study analyses the magnetohydrodynamic (MHD) flow of a double stratified micropolar fluid across a vertical stretching/shrinking sheet in the presence of suction, chemical reaction, and heat source effects. The governing equations in the form of partial differential equations are transitioned into coupled nonlinear ordinary differential equations by means of similarity transformation. The numerical solutions are obtained with the aid of the boundary value problem bvp4c solver in the MATLAB software. Numerical results have been confirmed with the previous results for a certain case and the comparison is found to be in an excellent agreement. Results for related profiles and heat transfer characteristics are displayed through plots and tabulated for the governing parameters involved. It is found that the reduced skin friction coefficient and the local Nusselt number increase with the increasing chemical reaction and heat source parameters. The rising values of the chemical reaction parameter have increased the magnitude of the local Sherwood number. In contrary, the heat source parameter has the tendency to decrease the magnitude of the local Sherwood number.

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.

scholarly journals Viscous Dissipation and Suction characteristics of Heat and Mass Transfer past a Stretching Sheet.

2021 ◽  
Vol 12 (1S) ◽  
pp. 623-630
Author(s):  
AlfunsaPrathiba, Et. al.
Keyword(s):  
Mass Transfer ◽  
Differential Equations ◽  
Heat And Mass Transfer ◽  
Viscous Dissipation ◽  
Schmidt Number ◽  
Stretching Sheet ◽  
Mhd Flow ◽  
Skin Friction Coefficient ◽  
Linear Ordinary Differential Equations ◽  
Similarity Transformations

in this paper, we analyzed the effect of a suction and Soret number on heat and mass transfer Magneto Hydrodynamics (MHD) flow past an exponentially stretching sheet with the heat source/sink. Appropriate similarity transformations were employed to convert the governing partial differential equations to a set of highly non-linear ordinary differential equations, which was then solved numerically by Runga kutta sixth order method together with shooting technique. The Numerical results are obtained for the skin friction coefficient, Nusselt and Sherwood numbers for selected values of the governing parameters, such as the suction, magnetic field parameter  , viscous dissipation parameter  , heat generation parameter  , Schmidt number  , and the chemical reaction rate parameter  . Besides, it is obtained that the concentration profile decreases with an increment of the Schmidt number. A comparison was made with a previous study available in the literature and we found that it is in a good agreement.

Heat and Mass Transfer of Thermophoretic MHD Flow of Powell–Eyring Fluid over a Vertical Stretching Sheet in the Presence of Chemical Reaction and Joule Heating

2015 ◽  
Vol 13 (1) ◽  
pp. 37-49 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Faqiha Sultan ◽  
Nadeem Alam Khan
Keyword(s):  
Mass Transfer ◽  
Differential Equations ◽  
Heat And Mass Transfer ◽  
Chemical Reaction ◽  
Joule Heating ◽  
Stretching Sheet ◽  
Mhd Flow ◽  
Radiation Chemical ◽  
Linear Ordinary Differential Equations ◽  
Non Linear

Abstract The present paper deals with the effect of surface heat and mass transfer on magnetohydrodynamic flow of Powell–Eyring fluid over a vertical stretching sheet. The effects of thermophoresis, Joule heating and chemical reaction are also considered. The governing non-linear partial differential equations of the model are transformed into coupled non-linear ordinary differential equations using a similarity transformation and solved numerically by Runge–Kutta method and analytically by homotopy analysis method (HAM). The convergence is carefully checked by plotting $$\hbar $$-curves. For different dimensionless parameters, numerical and analytical calculations are carried out and an investigation of the obtained results shows that the flow field is influenced considerably by the buoyancy ratio and thermal radiation, chemical reaction and magnetic field parameters. A totally analytical and consistently applicable solution is derived which agrees with numerical results.

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