scholarly journals Influence of a Darcy-Forchheimer porous medium on the flow of a radiative magnetized rotating hybrid nanofluid over a shrinking surface

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Sumera Dero ◽  
Hisamuddin Shaikh ◽  
Ghulam Hyder Talpur ◽  
Ilyas Khan ◽  
Sayer O. Alharbim ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Stability Analysis ◽  
Differential Equations ◽  
Shooting Method ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Dual Solutions ◽  
Hybrid Nanofluid ◽  
Solution Stability ◽  
Shrinking Surface

AbstractIn this paper, the heat transfer properties in the three-dimensional (3D) magnetized with the Darcy-Forchheimer flow over a shrinking surface of the $$Cu + Al_{2} O_{3} /$$ C u + A l 2 O 3 / water hybrid nanofluid with radiation effect were studied. Valid linear similarity variables convert the partial differential equations (PDEs) into the ordinary differential equations (ODEs). With the help of the shootlib function in the Maple software, the generalized model in the form of ODEs is numerically solved by the shooting method. Shooting method can produce non-unique solutions when correct initial assumptions are suggested. The findings are found to have two solutions, thereby contributing to the introduction of a stability analysis that validates the attainability of first solution. Stability analysis is performed by employing if bvp4c method in MATLAB software. The results show limitless values of dual solutions at many calculated parameters allowing the turning points and essential values to not exist. Results reveal that the presence of dual solutions relies on the values of the porosity, coefficient of inertia, magnetic, and suction parameters for the specific values of the other applied parameters. Moreover, it has been noted that dual solutions exist in the ranges of $$F_{s} \le F_{sc}$$ F s ≤ F sc , $$M \ge M_{C}$$ M ≥ M C ,$$S \ge S_{C} ,$$ S ≥ S C , and $$K_{C} \le K$$ K C ≤ K whereas no solution exists in the ranges of $$F_{s} > F_{sc}$$ F s > F sc , $$M < M_{c}$$ M < M c , $$S < S_{c}$$ S < S c , and $$K_{C} > K$$ K C > K . Further, a reduction in the rate of heat transfer is noticed with a rise in the parameter of the copper solid volume fraction.

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 Unsteady Three-Dimensional MHD Non-Axisymmetric Homann Stagnation Point Flow of a Hybrid Nanofluid with Stability Analysis

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 784 ◽  
Author(s):  
Nurul Amira Zainal ◽  
Roslinda Nazar ◽  
Kohilavani Naganthran ◽  
Ioan Pop
Keyword(s):  
Heat Transfer ◽  
Mathematical Model ◽  
Stability Analysis ◽  
Differential Equations ◽  
Stagnation Point ◽  
Three Dimensional ◽  
Industrial Sector ◽  
Dual Solutions ◽  
Stagnation Point Flow ◽  
Hybrid Nanofluid

The hybrid nanofluid under the influence of magnetohydrodynamics (MHD) is a new interest in the industrial sector due to its applications, such as in solar water heating and scraped surface heat exchangers. Thus, the present study accentuates the analysis of an unsteady three-dimensional MHD non-axisymmetric Homann stagnation point flow of a hybrid Al2O3-Cu/H2O nanofluid with stability analysis. By employing suitable similarity transformations, the governing mathematical model in the form of the partial differential equations are simplified into a system of ordinary differential equations. The simplified mathematical model is then solved numerically by the Matlab solver bvp4c function. This solving approach was proficient in generating more than one solution when good initial guesses were provided. The numerical results presented significant influences on the rate of heat transfer and fluid flow characteristics of a hybrid nanofluid. The rate of heat transfer and the trend of the skin friction coefficient improve with the increment of the nanoparticles’ concentration and the magnetic parameter; however, they deteriorate when the unsteadiness parameter increases. In contrast, the ratio of the escalation of the ambient fluid strain rate to the plate was able to adjourn the boundary layer separation. The dual solutions (first and second solutions) are obtainable when the surface of the sheet shrunk. A stability analysis is carried out to justify the stability of the dual solutions, and hence the first solution is seen as physically reliable and stable, while the second solution is unstable.

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.

Hybrid nanofluid flow and heat transfer over a permeable biaxial stretching/shrinking sheet

2019 ◽  
Vol 30 (7) ◽  
pp. 3497-3513 ◽  
Author(s):  
Iskandar Waini ◽  
Anuar Ishak ◽  
Ioan Pop
Keyword(s):  
Heat Transfer ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Dual Solutions ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid ◽  
Content Type ◽  
Volume Fractions ◽  
Biaxial Stretching ◽  
Flow And Heat Transfer

Purpose The purpose of this paper is to examine the axisymmetric flow and heat transfer of a hybrid nanofluid over a permeable biaxial stretching/shrinking sheet. Design/methodology/approach In this study, 0.1 solid volume fraction of alumina (Al2O3) is fixed, then consequently, various solid volume fractions of copper (Cu) are added into the mixture with water as the base fluid to form Cu-Al2O3/water hybrid nanofluid. The hybrid nanofluid equations are converted to the similarity equations by using the similarity transformation. The bvp4c solver, which is available in the Matlab software is used for solving the similarity equations numerically. The numerical results for selected values of the parameters are presented in tabular and graphical forms, and are discussed in detail. Findings It is found that dual solutions exist up to a certain value of the stretching/shrinking and suction parameters. The critical value λc < 0 for the existence of the dual solutions decreases as nanoparticle volume fractions for copper increase. The temporal stability analysis is performed to analyze the stability of the dual solutions, and it is revealed that only one of them is stable and physically reliable. Originality/value The present problem is new, original with many important results for practical problems in the modern industry.

Dual similarity solutions because of mixed convective flow of a double-nanoparticles hybrid nanofluid: critical points and stability analysis

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Ioan Pop ◽  
Mohammadreza Nademi Rostami ◽  
Saeed Dinarvand
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Flow Regime ◽  
Magnetic Parameter ◽  
Volume Fraction ◽  
Similarity Solutions ◽  
Hybrid Nanofluid ◽  
Content Type ◽  
Buoyancy Parameter ◽  
Mixed Convective Flow

Purpose The purpose of this article is to study the steady laminar magnetohydrodynamics mixed convection stagnation-point flow of an alumina-graphene/water hybrid nanofluid with spherical nanoparticles over a vertical permeable plate with focus on dual similarity solutions. Design/methodology/approach The single-phase hybrid nanofluid modeling is based on nanoparticles and base fluid masses instead of volume fraction of first and second nanoparticles as inputs. After substituting pertinent similarity variables into the basic partial differential equations governing on the problem, the authors obtain a complicated system of nondimensional ordinary differential equations, which has non-unique solution in a certain range of the buoyancy parameter. It is worth mentioning that, the stability analysis of the solutions is also presented and it is shown that always the first solutions are stable and physically realizable. Findings It is proved that the magnetic parameter and the wall permeability parameter widen the range of the buoyancy parameter for which the solution exists; however, the opposite trend is valid for second nanoparticle mass. Besides, mass suction at the surface of the plate as well as magnetic parameter leads to reduce both hydrodynamic and thermal boundary layer thicknesses. Moreover, the assisting flow regime always has higher values of similarity skin friction and Nusselt number relative to opposing flow regime. Originality/value A novel mass-based model of the hybridity in nanofluids has been used to study the foregoing problem with focus on dual similarity solutions. The results of this paper are completely original and, to the best of the authors’ knowledge, the numerical results of the present paper were never published by any researcher.

Numerical solutions of non-alignment stagnation-point flow and heat transfer over a stretching/shrinking surface in a nanofluid

2016 ◽  
Vol 26 (6) ◽  
pp. 1747-1767 ◽  
Author(s):  
Ioan Pop ◽  
Kohi Naganthran ◽  
Roslinda Nazar
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Stagnation Point ◽  
Numerical Solutions ◽  
Dual Solutions ◽  
Stagnation Point Flow ◽  
Content Type ◽  
Boundary Layer Equations ◽  
Flow And Heat Transfer ◽  
Shrinking Surface

Purpose – The purpose of this paper is to analyse numerically the steady stagnation-point flow of a viscous and incompressible fluid over continuously non-aligned stretching or shrinking surface in its own plane in a water-based nanofluid which contains three different types of nanoparticles, namely, Cu, Al2O3 and TiO2. Design/methodology/approach – Similarity transformation is used to convert the system of boundary layer equations which are in the form of partial differential equations into a system of ordinary differential equations. The system of similarity governing equations is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in Matlab software. Findings – Unique solution exists when the surface is stretched and dual solutions exist as the surface shrunk. For the dual solutions, stability analysis has revealed that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable. The effect of non-alignment is huge for the shrinking surface which is in contrast with the stretching surface. Practical implications – The results obtained can be used to explain the characteristics and applications of nanofluids, which are widely used as coolants, lubricants, heat exchangers and micro-channel heat sinks. This problem also applies to some situations such as materials which are manufactured by extrusion, production of glass-fibre and shrinking balloon. In this kind of circumstance, the rate of cooling and the stretching/shrinking process play an important role in moulding the final product according to preferable features. Originality/value – The present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface for the problem considered by Wang (2008) in a viscous fluid and extends to nanofluid by using the Tiwari and Das (2007) model.

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 Unsteady flow over an expanding cylinder in a nanofluid containing gyrotactic microorganisms

2016 ◽  
Vol 94 (5) ◽  
pp. 466-473 ◽  
Author(s):  
Hui Chen ◽  
Hongxing Liang ◽  
Tianli Xiao ◽  
Heng Du ◽  
Ming Shen
Keyword(s):  
Differential Equations ◽  
Unsteady Flow ◽  
Shooting Method ◽  
Schmidt Number ◽  
Volume Fraction ◽  
Dual Solutions ◽  
Physical Parameters ◽  
Gyrotactic Microorganisms ◽  
Similarity Transformations ◽  
Fifth Order

In this paper, an analysis is made for the unsteady flow due to an expanding cylinder in a nanofluid that contains both nanoparticles and gyrotactic microoganisms with suction. The nonlinear system of partial differential equations is transformed into high-order nonlinear ordinary differential equations using similarity transformations, and then solved numerically using a shooting method with fourth-fifth-order Runge–Kutta integration technique. The influences of significant physical parameters on the distributions of the velocity, temperature, nanoparticle volume fraction, as well as the density of motile microorganisms are graphically presented and discussed in detail. It is found that dual solutions exist for both stretching and shrinking cases and the range of dual solutions increases with the strength of the expansion. The results also indicate that larger bioconvection Peclet number and smaller Schmidt number lead to an increased concentration of microorganisms and thicker boundary layer thickness.

scholarly journals Mixed convective hybrid nanofluid flow in lid-driven undulated cavity: effect of MHD and Joule heating

2019 ◽  
Vol 16 (2) ◽  
pp. 109-126 ◽  
Author(s):  
Ishrat Zahan ◽  
R Nasrin ◽  
M A Alim
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Heat Transfer Rate ◽  
Joule Heating ◽  
Transfer Rate ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Hybrid Nanoparticles ◽  
Wave Number ◽  
Hybrid Nanofluid

A numerical analysis has been conducted to show the effects of magnetohydrodynamic (MHD) and Joule heating on heat transfer phenomenon in a lid driven triangular cavity. The heat transfer fluid (HTF) has been considered as water based hybrid nanofluid composed of equal quantities of Cu and TiO2 nanoparticles. The bottom wall of the cavity is undulated in sinusoidal pattern and cooled isothermally. The left vertical wall of the cavity is heated while the inclined side is insulated. The two dimensional governing partial differential equations of heat transfer and fluid flow with appropriate boundary conditions have been solved by using Galerkin's finite element method built in COMSOL Multyphysics. The effects of Hartmann number, Joule heating, number of undulation and Richardson number on the flow structure and heat transfer characteristics have been studied in details. The values of Prandtl number and solid volume fraction of hybrid nanoparticles have been considered as fixed. Also, the code validation has been shown. The numerical results have been presented in terms of streamlines, isotherms and average Nusselt number of the hybrid nanofluid for different values of governing parameters. The comparison of heat transfer rate by using hybrid nanofluid, Cu-water nanofluid,  TiO2 -water nanofluid and clear water has been also shown. Increasing wave number from 0 to 3 enhances the heat transfer rate by 16.89%. The enhanced rate of mean Nusselt number for hybrid nanofluid is found as 4.11% compared to base fluid.

scholarly journals Unsteady Stagnation Point Flow of Hybrid Nanofluid Past a Convectively Heated Stretching/Shrinking Sheet with Velocity Slip

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1649
Author(s):  
Nurul Amira Zainal ◽  
Roslinda Nazar ◽  
Kohilavani Naganthran ◽  
Ioan Pop
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Skin Friction ◽  
Stagnation Point ◽  
Volume Fraction ◽  
Velocity Slip ◽  
Slip Condition ◽  
Stagnation Point Flow ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid

Unsteady stagnation point flow in hybrid nanofluid (Al2O3-Cu/H2O) past a convectively heated stretching/shrinking sheet is examined. Apart from the conventional surface of the no-slip condition, the velocity slip condition is considered in this study. By incorporating verified similarity transformations, the differential equations together with their partial derivatives are changed into ordinary differential equations. Throughout the MATLAB operating system, the simplified mathematical model is clarified by employing the bvp4c procedure. The above-proposed approach is capable of producing non-uniqueness solutions when adequate initial assumptions are provided. The findings revealed that the skin friction coefficient intensifies in conjunction with the local Nusselt number by adding up the nanoparticles volume fraction. The occurrence of velocity slip at the boundary reduces the coefficient of skin friction; however, an upward trend is exemplified in the rate of heat transfer. The results also signified that, unlike the parameter of velocity slip, the increment in the unsteady parameter conclusively increases the coefficient of skin friction, and an upsurge attribution in the heat transfer rate is observed resulting from the increment of Biot number. The findings are evidenced to have dual solutions, which inevitably contribute to stability analysis, hence validating the feasibility of the first solution.

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