scholarly journals Multiple Solutions for Stagnation-Point Flow of Unsteady Carreau Fluid along a Permeable Stretching/Shrinking Sheet with Non-Uniform Heat Generation

Coatings ◽  
2021 ◽  
Vol 11 (9) ◽  
pp. 1012
Author(s):  
Dezhi Yang ◽  
Muhammad Israr Ur Rehman ◽  
Aamir Hamid ◽  
Saif Ullah
Keyword(s):  
Nusselt Number ◽  
Differential Equations ◽  
Stagnation Point ◽  
Local Nusselt Number ◽  
Nonlinear Partial Differential Equations ◽  
Stagnation Point Flow ◽  
Shrinking Sheet ◽  
Carreau Fluid ◽  
Similarity Relations ◽  
Uniform Heat

The aim of the present study was to explore the effect of a non-uniform heat source/sink on the unsteady stagnation point flow of Carreau fluid past a permeable stretching/shrinking sheet. The novelty of the flow model was enhanced with additional effects of magnetohydrodynamics, joule heating, and viscous dissipation. The nonlinear partial differential equations were converted into ordinary differential equations with the assistance of appropriate similarity relations and were then tackled by employing the Runge-Kutta-Fehlberg technique with the shooting method. The impacts of pertinent parameters on the dimensionless velocity and temperature profiles along with the friction factor and local Nusselt number were extensively discussed by means of graphical depictions and tables. The current results were compared to the previous findings under certain conditions to determine the precision and validity of the present study. The fluid flow velocity of Carreau fluid increased with the value of the magnetic parameter in the case of the first solution, and the opposite behavior was noticed for the second solution. It was seen that temperature of the Carreau fluid expanded with the higher values of unsteadiness and magnetic parameters. It was visualized from multiple branches that the local Nusselt number declined with the Eckert number parameter for both the upper and lower branch.

THREE-DIMENSIONAL STAGNATION-POINT FLOW OVER A PERMEABLE STRETCHING/SHRINKING SHEET WITH A SECOND ORDER SLIP FLOW

2021 ◽  
Vol 10 (9) ◽  
pp. 3273-3282
Author(s):  
M.E.H. Hafidzuddin ◽  
R. Nazar ◽  
N.M. Arifin ◽  
I. Pop
Keyword(s):  
Differential Equations ◽  
Stagnation Point ◽  
Flow Model ◽  
Nonlinear Partial Differential Equations ◽  
Slip Flow ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Second Order ◽  
Stagnation Point Flow ◽  
Shrinking Sheet

The problem of steady laminar three-dimensional stagnation-point flow on a permeable stretching/shrinking sheet with second order slip flow model is studied numerically. Similarity transformation has been used to reduce the governing system of nonlinear partial differential equations into the system of ordinary (similarity) differential equations. The transformed equations are then solved numerically using the \texttt{bvp4c} function in MATLAB. Multiple solutions are found for a certain range of the governing parameters. The effects of the governing parameters on the skin friction coefficients and the velocity profiles are presented and discussed. It is found that the second order slip flow model is necessary to predict the flow characteristics accurately.

scholarly journals MHD stagnation-point flow and heat transfer past a stretching/shrinking sheet in a hybrid nanofluid with induced magnetic field

2019 ◽  
Vol 30 (3) ◽  
pp. 1345-1364 ◽  
Author(s):  
Mohamad Mustaqim Junoh ◽  
Fadzilah Md Ali ◽  
Norihan Md Arifin ◽  
Norfifah Bachok ◽  
Ioan Pop
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Differential Equations ◽  
Stagnation Point ◽  
Nonlinear Partial Differential Equations ◽  
Stagnation Point Flow ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid ◽  
Induced Magnetic Field ◽  
Content Type

Purpose The purpose of this paper is to investigate the steady magnetohydrodynamics (MHD) boundary layer stagnation-point flow of an incompressible, viscous and electrically conducting fluid past a stretching/shrinking sheet with the effect of induced magnetic field. Design/methodology/approach The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations via the similarity transformations before they are solved numerically using the “bvp4c” function in MATLAB. Findings It is found that there exist non-unique solutions, namely, dual solutions for a certain range of the stretching/shrinking parameters. The results from the stability analysis showed that the first solution (upper branch) is stable and valid physically, while the second solution (lower branch) is unstable. Practical implications This problem is important in the heat transfer field such as electronic cooling, engine cooling, generator cooling, welding, nuclear system cooling, lubrication, thermal storage, solar heating, cooling and heating in buildings, biomedical, drug reduction, heat pipe, space aircrafts and ships with better efficiency than that of nanofluids applicability. The results obtained are very useful for researchers to determine which solution is physically stable, whereby, mathematically more than one solution exist. Originality/value The present results are new and original for the problem of MHD stagnation-point flow over a stretching/shrinking sheet in a hybrid nanofluid, with the effect of induced magnetic field.

scholarly journals Magnetohydrodynamics Stagnation-Point Flow of a Nanofluid Past a Stretching/Shrinking Sheet with Induced Magnetic Field: A Revised Model

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1078 ◽  
Author(s):  
Mohamad Mustaqim Junoh ◽  
Fadzilah Md Ali ◽  
Ioan Pop
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Stagnation Point ◽  
Nonlinear Partial Differential Equations ◽  
Stagnation Point Flow ◽  
Shrinking Sheet ◽  
Governing Equations ◽  
Induced Magnetic Field ◽  
The Stability ◽  
Stretching Or Shrinking Sheet

The revised Buongiorno’s nanofluid model with the effect of induced magnetic field on steady magnetohydrodynamics (MHD) stagnation-point flow of nanofluid over a stretching or shrinking sheet is investigated. The effects of zero mass flux and suction are taken into account. A similarity transformation with symmetry variables are introduced in order to alter from the governing nonlinear partial differential equations into a nonlinear ordinary differential equations. These governing equations are numerically solved using the bvp4c function in Matlab solver, a very adequate finite difference method. The influences of considered parameters ( P r , M, χ , L e , N b , N t , S, and λ ) on velocity, induced magnetic, temperature, and concentration profiles together with the reduced skin friction and heat transfer rate are discussed. Results from these criterion exposed the existence of dual solutions when magnetic field and suction are applied for a specific range of λ . The stability of the solutions obtained is carried out by performing a stability analysis.

Mixed Convection Boundary Layer Stagnation-Point Flow of a Jeffery Fluid Past a Permeable Vertical Flat Plate

2014 ◽  
Vol 69 (12) ◽  
pp. 687-696 ◽  
Author(s):  
Mohammad M. Rahman ◽  
Pop Ioan
Keyword(s):  
Nusselt Number ◽  
Differential Equations ◽  
Mixed Convection ◽  
Skin Friction ◽  
Stagnation Point ◽  
Mass Flux ◽  
Nonlinear Partial Differential Equations ◽  
Stagnation Point Flow ◽  
Convection Boundary Layer ◽  
Variable Surface

AbstractThis paper analyzes the combined effects of buoyancy force, mass flux, and variable surface temperature on the stagnation point flow and heat transfer due to a Jeffery fluid over a vertical surface. The governing nonlinear partial differential equations are transformed into a system of coupled nonlinear ordinary differential equations using similarity transformations and then solved numerically using the function bvp4c from computer algebra software Matlab. Numerical results are obtained for skin friction coefficient, Nusselt number as well as dimensionless velocity and temperature profiles for various values of the controlling parameters namely mixed convection parameter λ, mass flux parameter s, elastic parameter (Deborah number) γ, and the ratio of relaxation and retardation time parameter λ1. The results indicate that dual solutions exist in a certain range of the mixed convection and mass flux parameters. In order to establish the physically realizable of these solutions, a stability analysis has also been performed. The results indicate that mixed convection and mass flux significantly affects the nature of the solutions, skin friction, and Nusselt number of a Jeffery fluid.

Unsteady separated stagnation-point flow and heat transfer past a stretching/shrinking sheet in a copper-water nanofluid

2019 ◽  
Vol 29 (8) ◽  
pp. 2588-2605 ◽  
Author(s):  
Natalia C. Roşca ◽  
Alin V. Roşca ◽  
Ioan Pop
Keyword(s):  
Heat Transfer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Stagnation Point ◽  
Design Methodology ◽  
Similarity Transformation ◽  
Stagnation Point Flow ◽  
Shrinking Sheet ◽  
Content Type ◽  
Flow And Heat Transfer

Purpose The purpose of this paper is to theoretically investigate the unsteady separated stagnation-point flow and heat transfer past an impermeable stretching/shrinking sheet in a copper (Cu)-water nanofluid using the mathematical nanofluid model proposed by Tiwari and Das. Design/methodology/approach A similarity transformation is used to reduce the governing partial differential equations to a set of nonlinear ordinary (similarity) differential equations which are then solved numerically using the function bvp4c from Matlab for different values of the governing parameters. Findings It is found that the solution is unique for stretching case; however, multiple (dual) solutions exist for the shrinking case. Originality/value The authors believe that all numerical results are new and original, and have not been published elsewhere.

scholarly journals Magnetohydrodynamic Flow and Heat Transfer Induced by a Shrinking Sheet

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1175
Author(s):  
Nor Ain Azeany Mohd Nasir ◽  
Anuar Ishak ◽  
Ioan Pop
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Differential Equations ◽  
Prandtl Number ◽  
Heat Generation ◽  
Local Nusselt Number ◽  
Magnetic Parameter ◽  
Skin Friction Coefficient ◽  
Shrinking Sheet ◽  
Flow And Heat Transfer

The magnetohydrodynamic (MHD) stagnation point flow over a shrinking or stretching flat sheet is investigated. The governing partial differential equations (PDEs) are reduced into a set of ordinary differential equations (ODEs) by a similarity transformation and are solved numerically with the help of MATLAB software. The numerical results obtained are for different values of the magnetic parameter M, heat generation parameter Q, Prandtl number Pr and reciprocal of magnetic Prandtl number ε. The influences of these parameters on the flow and heat transfer characteristics are investigated and shown in tables and graphs. Two solutions are found for a certain rate of the shrinking strength. The stability of the solutions in the long run is determined, and shows that only one of them is stable. It is found that the skin friction coefficient f ″ ( 0 ) and the local Nusselt number − θ ′ ( 0 ) decrease as the magnetic parameter M increases. Further, the local Nusselt number increases as the heat generation increases.

Effect of Porous Medium in Stagnation Point Flow of Ferrofluid Due to a Variable Convected Thicked Sheet

2019 ◽  
Vol 141 (11) ◽  
Author(s):  
Maria Imtiaz ◽  
Hira Nazar ◽  
Tasawar Hayat ◽  
Ahmed Alsaedi
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Nusselt Number ◽  
Differential Equations ◽  
Stagnation Point ◽  
Skin Friction Coefficient ◽  
Stagnation Point Flow ◽  
Heat Transfer Analysis ◽  
Series Solutions ◽  
The Impact

Abstract The focus of this paper is to study the effects of stagnation point flow and porous medium on ferrofluid flow over a variable thicked sheet. Heat transfer analysis is discussed by including thermal radiation. Suitable transformations are applied to convert partial differential equations to ordinary differential equations. Convergent results for series solutions are calculated. The impact of numerous parameters on velocity and temperature is displayed for series solutions. Graphical behavior for skin friction coefficient and Nusselt number is also analyzed. Numerical values of Nusselt number are tabulated depending upon various parameters

scholarly journals Stagnation-point flow past a shrinking sheet in a nanofluid

Open Physics ◽  
2011 ◽  
Vol 9 (5) ◽  
Author(s):  
Roslinda Nazar ◽  
Mihaela Jaradat ◽  
Norihan Arifin ◽  
Ioan Pop
Keyword(s):  
Heat Transfer ◽  
Friction Coefficient ◽  
Nonlinear System ◽  
Differential Equations ◽  
Skin Friction ◽  
Stagnation Point ◽  
Volume Fraction ◽  
Skin Friction Coefficient ◽  
Stagnation Point Flow ◽  
Shrinking Sheet

AbstractIn this paper, the stagnation-point flow and heat transfer towards a shrinking sheet in a nanofluid is considered. The nonlinear system of coupled partial differential equations was transformed and reduced to a nonlinear system of coupled ordinary differential equations, which was solved numerically using the shooting method. Numerical results were obtained for the skin friction coefficient, the local Nusselt number as well as the velocity and temperature profiles for some values of the governing parameters, namely the nanoparticle volume fraction φ, the shrinking parameter λand the Prandtl number Pr. Three different types of nanoparticles are considered, namely Cu, Al2O3 and TiO2. It was found that nanoparticles of low thermal conductivity, TiO2, have better enhancement on heat transfer compared to nanoparticles Al2O3 and Cu. For a particular nanoparticle, increasing the volume fraction φ results in an increase of the skin friction coefficient and the heat transfer rate at the surface. It is also found that solutions do not exist for larger shrinking rates and dual solutions exist when λ < −1.0.

Impact of Entropy Generation on Stagnation-Point Flow of Sutterby Nanofluid: A Numerical Analysis

2016 ◽  
Vol 71 (9) ◽  
pp. 837-848 ◽  
Author(s):  
Ehtsham Azhar ◽  
Z. Iqbal ◽  
E.N. Maraj
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Stagnation Point ◽  
Entropy Generation ◽  
Computational Analysis ◽  
Similarity Transformation ◽  
Nonlinear Partial Differential Equations ◽  
Stagnation Point Flow ◽  
Physical Parameters ◽  
Stretching Plate

AbstractThe present article dicusses the computational analysis of entropy generation for the stagnation-point flow of Sutterby nanofluid over a linear stretching plate. The Sutterby fluid is chosen to study the effect for three major classes of non-Newtonian fluids, i.e. pseudoplastic, Newtonian, and dilatant. The effects of pertinent physical parameters are examined under the approximation of boundary layer. The system of coupled nonlinear partial differential equations is simplified by incorporating suitable similarity transformation into a system of non-linear-coupled ordinary differential equations. Entropy generation analysis is conducted numerically, and the results are displayed through graphs and tables. Significant findings are listed in the closing remarks.

scholarly journals A New Numerical Solution of Maxwell Fluid over a Shrinking Sheet in the Region of a Stagnation Point

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
S. S. Motsa ◽  
Y. Khan ◽  
S. Shateyi
Keyword(s):  
Differential Equations ◽  
Stagnation Point ◽  
Nonlinear Partial Differential Equations ◽  
Maxwell Fluid ◽  
Shrinking Sheet ◽  
Two Dimensional ◽  
Similarity Transformations ◽  
The Mathematical Model ◽  
Successive Linearisation Method ◽  
Fluid Parameters

The mathematical model for the incompressible two-dimensional stagnation flow of a Maxwell fluid towards a shrinking sheet is proposed. The developed equations are used to discuss the problem of being two dimensional in the region of stagnation point over a shrinking sheet. The nonlinear partial differential equations are transformed to ordinary differential equations by first-taking boundary-layer approximations into account and then using the similarity transformations. The obtained equations are then solved by using a successive linearisation method. The influence of the pertinent fluid parameters on the velocity is discussed through the help of graphs.

Sign in / Sign up

Export Citation Format

Share Document