scholarly journals Hybrid Nanofluid Flow over a Permeable Shrinking Sheet Embedded in a Porous Medium with Radiation and Slip Impacts

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 878
Author(s):  
Shahirah Abu Bakar ◽  
Norihan Md Arifin ◽  
Najiyah Safwa Khashi’ie ◽  
Norfifah Bachok
Keyword(s):  
Porous Medium ◽  
Differential Equations ◽  
Similarity Transformation ◽  
Lower Solution ◽  
Shrinking Sheet ◽  
Physical Parameters ◽  
Hybrid Nanofluid ◽  
Uniqueness Of Solutions ◽  
Nanofluid Flow ◽  
The Stability

The study of hybrid nanofluid and its thermophysical properties is emerging since the early of 2000s and the purpose of this paper is to investigate the flow of hybrid nanofluid over a permeable Darcy porous medium with slip, radiation and shrinking sheet. Here, the hybrid nanofluid consists of Cu/water as the base nanofluid and Al2O3–Cu/water works as the two distinct fluids. The governing ordinary differential equations (ODEs) obtained in this study are converted from a series of partial differential equations (PDEs) by the appropriate use of similarity transformation. Two methods of shooting and bvp4c function are applied to solve the involving physical parameters over the hybrid nanofluid flow. From this study, we conclude that the non-uniqueness of solutions exists through a range of the shrinking parameter, which produces the problem of finding a bigger solution than any other between the upper and lower branches. From the analysis, one can observe the increment of heat transfer rate in hybrid nanofluid versus the traditional nanofluid. The results obtained by the stability of solutions prove that the upper solution (first branch) is stable and the lower solution (second branch) is not stable.

scholarly journals Magnetized Flow of Cu + Al2O3 + H2O Hybrid Nanofluid in Porous Medium: Analysis of Duality and Stability

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1513
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Sumera Dero ◽  
Ilyas Khan ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Porous Medium ◽  
Differential Equations ◽  
Similarity Transformation ◽  
Fluid Model ◽  
Flow Characteristics ◽  
Stable Solution ◽  
Similarity Solutions ◽  
Medium Effect ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid

In this analysis, we aim to examine the heat transfer and flow characteristics of a copper-aluminum/water hybrid nanofluid in the presence of viscous dissipation, magnetohydrodynamic (MHD), and porous medium effect over the shrinking sheet. The governing equations of the fluid model have been acquired by employment of the model of Tiwari and Das, with additional properties of the hybrid nanofluid. The system of partial differential equations (PDEs) has been converted into ordinary differential equations (ODEs) by adopting the exponential similarity transformation. Similarity transformation is an essential class of phenomenon where the symmetry of the scale helps to reduce the number of independent variables. Note that ODE solutions demonstrate the PDEs symmetrical behavior for the velocity and temperature profiles. With BVP4C solver in the MATLAB program, the system of resulting equations has been solved. We have compared the present results with the published results and found in excellent agreements. The findings of the analysis are also displayed and discussed in depth graphically and numerically. It is discovered that two solutions occur in definite ranges of suction and magnetic parameters. Dual (no) similarity solutions can be found in the range of Sc≤S and Mc≤M (Sc>S and Mc>M). By performing stability analysis, the smallest values of eigenvalue are obtained, suggesting that a stable solution is the first one. Furthermore, the graph of the smallest eigenvalue shows symmetrical behavior. By enhancing the Eckert number values the temperature of the fluid is raised.

scholarly journals Magnetohydrodynamic Hybrid Nanofluid Flow Past an Exponentially Stretching Sheet with Slip Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3291
Author(s):  
Abdul Samad Khan ◽  
He-Yong Xu ◽  
Waris Khan
Keyword(s):  
Nusselt Number ◽  
Differential Equations ◽  
Skin Friction ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid ◽  
Slip Parameter ◽  
Nanofluid Flow ◽  
Slip Conditions ◽  
Exponentially Stretching ◽  
The Impact

This study presents the magnetized hybrid nanofluid flow with heat source/sink over an exponentially stretching/shrinking sheet. Slip conditions are implemented to analyze the hybrid nanofluid flow for both slip and no-slip conditions. Additionally, the hybrid nanofluid of alumina and copper (hybrid nanoparticles) with blood (base fluid) has been considered and discussed with both suction and injection parameters. The appropriate similarity variables are used to convert partial differential equations (PDEs) into ordinary differential equations (ODEs) and solved analytically with the help of the homotopy analysis method (HAM). The impact of different embedded parameters has been shown in the form of graphs and tables. The numerical values of skin friction and Nusselt number are presented in the form of Tables for both slip and no-slip cases. It is summarized that the upsurge of the velocity slip parameter and magnetic parameter increases the skin friction, while the rising of the thermal slip parameter and heat generation parameter decreases the Nusselt number.

scholarly journals Rotating 3D Flow of Hybrid Nanofluid on Exponentially Shrinking Sheet: Symmetrical Solution and Duality

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1637
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Sumera Dero ◽  
Dumitru Baleanu ◽  
Ilyas Khan
Keyword(s):  
Differential Equations ◽  
Transfer Rate ◽  
Three Dimensional ◽  
Rotation Parameter ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid ◽  
3D Flow ◽  
Exponentially Shrinking Sheet ◽  
Symmetrical Solution ◽  
The Stability

This article aims to study numerically the rotating, steady, and three-dimensional (3D) flow of a hybrid nanofluid over an exponentially shrinking sheet with the suction effect. We considered water as base fluid and alumina (Al2O3), and copper (Cu) as solid nanoparticles. The system of governing partial differential equations (PDEs) was transformed by an exponential similarity variable into the equivalent system of ordinary differential equations (ODEs). By applying a three-stage Labatto III-A method that is available in bvp4c solver in the Matlab software, the resultant system of ODEs was solved numerically. In the case of the hybrid nanofluid, the heat transfer rate improves relative to the viscous fluid and regular nanofluid. Two branches were obtained in certain ranges of the involved parameters. The results of the stability analysis revealed that the upper branch is stable. Moreover, the results also indicated that the equations of the hybrid nanofluid have a symmetrical solution for different values of the rotation parameter (Ω).

scholarly journals Mathematical Analysis of Magnetized Rotating Nanofluid Flow Over nonlinear shrinking surface: Duality and Stability

2021 ◽  
Vol 5 (2) ◽  
pp. 1-13
Author(s):  
Dr Sumera Dero ◽  
Ghulam Hyder Talpur ◽  
Abbas Ali Ghoto ◽  
Shokat Ali
Keyword(s):  
Differential Equations ◽  
Hartmann Number ◽  
Similarity Transformation ◽  
Flow Characteristics ◽  
Shrinking Sheet ◽  
Nanofluid Flow ◽  
Suction Parameter ◽  
Stable Branch ◽  
Shrinking Surface ◽  
Good Agreement

In this study, the MHD effect on boundary layer rotating flow of a nanofluid is investigated for the multiple branches case. The main focus of current research is to examine flow characteristics on a nonlinear permeable shrinking sheet. Moreover, the governing partial differential equations (PDEs) of the problem considered are reduced into coupled nonlinear ordinary differential equations (ODEs) with the appropriate similarity transformation.  Numerical results based on the plotted graphs are gotten by solving ODEs with help of the three-stage Labatto IIIA method in bvp4c solver in MATLAB. To confirm numerical outcomes, current results are compared with previously available outcomes and found in good agreement. Skin friction coefficients, Nusselt and Sherwood numbers, velocity profiles, temperature profiles, and concentration profiles are examined. The results show that dual (no) branches exist in certain ranges of the suction parameter i.e., SSc (SSc). Further, profiles of velocity decrease for rising values of Hartmann number in the upper branch, while reverse trend has been noticed in a lower branch. Profiles of temperature and concentration enhance for the increasing values of thermophoresis in both branches. stability analysis of the branches is also done and noticed that upper branch is a stable branch from both branches. Finally, it is noted that the stable branch has symmetrical behavior with regard to the parameter of rotation.

scholarly journals Thermal Diffusion and Diffusion Thermo Effects on the Viscous Fluid Flow with Heat and Mass Transfer through Porous Medium over a Shrinking Sheet

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Nabil Eldabe ◽  
Mahmoud Abu Zeid
Keyword(s):  
Mass Transfer ◽  
Porous Medium ◽  
Shear Stress ◽  
Differential Equations ◽  
Heat And Mass Transfer ◽  
Shrinking Sheet ◽  
Physical Parameters ◽  
Transfer Parameter ◽  
Mass Transfer Parameter ◽  
Permeability Parameter

The motion of the viscous, incompressible fluid through a porous medium with heat and mass transfer over a shrinking sheet is investigated. The cross-diffusion effect between temperature and concentration is considered. This phenomenon is modulated mathematically by a set of partial differential equations which govern the continuity, momentum, heat, and mass. These equations are transformed to a set of ordinary differential equations by using similarity solutions. The analytical solutions of these equations are obtained. The velocity, temperature, and concentration of the fluid as well as the heat and mass transfer with shear stress at the sheet are obtained as a function of the physical parameters of the problem. The effects of Prandtl number, mass transfer parameter, the wall shrinking parameter, the permeability parameter, and Dufour and Soret numbers on temperature and concentration are studied. Also, the effects of mass transfer parameter, permeability parameter, and shrinking strength on the velocity and shear stress are discussed. These effects are illustrated graphically through a set of figures.

A new similarity solution with stability analysis for the three-dimensional boundary layer of hybrid nanofluids

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Najiyah Safwa Khashi'ie ◽  
Norihan M. Arifin ◽  
Ioan Pop ◽  
Roslinda Nazar ◽  
Ezad Hafidz Hafidzuddin
Keyword(s):  
Boundary Layer ◽  
Stability Analysis ◽  
Differential Equations ◽  
Similarity Transformation ◽  
Three Dimensional ◽  
Physical Parameters ◽  
Hybrid Nanofluid ◽  
Content Type ◽  
Boundary Layer Equations ◽  
Dimensional Boundary

Purpose The purpose of this study is to implement a new class of similarity transformation in analyzing the three-dimensional boundary layer flow of hybrid nanofluid. The Cu-Al2O3/water hybrid nanofluid is formulated using the single-phase nanofluid model with modified thermophysical properties. Design/methodology/approach The governing partial differential equations are reduced to the ordinary (similarity) differential equations using the proposed similarity transformation. The resulting equations are programmed in Matlab software through the bvp4c solver to obtain their solutions. The features of the reduced skin frictions and the velocity profiles for different values of the physical parameters are analyzed and discussed. Findings The non-uniqueness of the solutions is observed for certain physical parameters. The dual solutions are perceived for both permeable and impermeable cases and being the main agenda of the work. The execution of stability analysis proves that the first solution is undoubtedly stable than the second solution. An increase in the mass transpiration parameter leads to the uniqueness of the solution. Oppositely, as the injection parameter increase, the two solutions remain. However, no separation point is detected in this problem within the considered parameter values. The present results are decisive to the pair of alumina and copper only. Originality/value The present findings are original and can benefit other researchers particularly in the field of fluid dynamics. This study can provide a different insight of the transformation that is applicable to reduce the complexity of the boundary layer equations.

scholarly journals Unsteady MHD Bionanofluid Flow in a Porous Medium with Thermal Radiation near a Stretching/Shrinking Sheet

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
M. Irfan ◽  
M. Asif Farooq ◽  
A. Mushtaq ◽  
Z. H. Shamsi
Keyword(s):  
Porous Medium ◽  
Friction Coefficient ◽  
Differential Equations ◽  
Thermal Radiation ◽  
Skin Friction ◽  
Internal Heat ◽  
Skin Friction Coefficient ◽  
Shrinking Sheet ◽  
Physical Parameters ◽  
The Impact

This research aims at providing the theoretical effects of the unsteady MHD stagnation point flow of heat and mass transfer across a stretching and shrinking surface in a porous medium including internal heat generation/absorption, thermal radiation, and chemical reaction. The fundamental principles of the similarity transformations are applied to the governing partial differential equations (PDEs) that lead to ordinary differential equations (ODEs). The transformed ODEs are numerically solved by the shooting algorithm implemented in MATLAB, and verification is done from MATLAB built-in solver bvp4c. The numerical data produced for the skin friction coefficient, the local Nusselt number, and the local Sherwood number are compared with the available result and found to be in a close agreement. The impact of involved physical parameters on velocity, temperature, concentration, and density of motile microorganisms profiles is scrutinized through graphs. It is analyzed that the skin friction coefficient enhances with increasing values of an unsteady parameter A , magnetic parameter M , and porosity parameter Kp . In addition, we observe that the density of a motile microorganisms profile enhances larger values of the bioconvection Lewis number Lb and Peclet number Pe and decreases with the increasing values of an unsteady parameter A .

scholarly journals Cu-Al2O3/Water Hybrid Nanofluid Stagnation Point Flow Past MHD Stretching/Shrinking Sheet in Presence of Homogeneous-Heterogeneous and Convective Boundary Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1237
Author(s):  
Nur Syazana Anuar ◽  
Norfifah Bachok ◽  
Ioan Pop
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Stagnation Point ◽  
Numerical Solutions ◽  
Heterogeneous Reactions ◽  
Stagnation Point Flow ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid ◽  
Nanofluid Flow ◽  
Convective Boundary

The intent of this research was to present numerical solutions to homogeneous–heterogeneous reactions of the magnetohydrodynamic (MHD) stagnation point flow of a Cu-Al2O3/water hybrid nanofluid induced by a stretching or shrinking sheet with a convective boundary condition. A proper similarity variable was applied to the system of partial differential equations (PDEs) and converted into a system of ordinary (similarity) differential equations (ODEs). These equations were solved using Matlab’s in-built function (bvp4c) for various values of the governing parameters numerically. The present investigation considered the effects of homogeneous–heterogeneous reactions and magnetic field in the hybrid nanofluid flow. It was observed that dual solutions were visible for the shrinking sheet, and an analysis of stability was done to determine the physically realizable in the practice of these solutions. It was also concluded that hybrid nanofluid acts as a cooler for some increasing parameters. The magnetohydrodynamic parameter delayed the boundary layer separation; meanwhile, the nanoparticle volume fraction quickened the separation of the boundary layer that occurred. In addition, the first solution of hybrid nanofluid was found to be stable; meanwhile, the second solution was not stable. This study is therefore valuable for engineers and scientists to get acquainted with the properties of hybrid nanofluid flow, its behavior and the way to predict it.

scholarly journals Flow and heat transfer past a permeable stretching/shrinking sheet in Cu−Al2O3/water hybrid nanofluid

2019 ◽  
Vol 30 (3) ◽  
pp. 1197-1222 ◽  
Author(s):  
Rusya Iryanti Yahaya ◽  
Norihan M. Arifin ◽  
Roslinda Nazar ◽  
Ioan Pop
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Skin Friction Coefficient ◽  
The Other ◽  
Shrinking Sheet ◽  
Physical Parameters ◽  
Hybrid Nanofluid ◽  
Content Type ◽  
Flow And Heat Transfer

Purpose The purpose of this paper is to study the flow and heat transfer of a hybrid nanofluid, Cu–Al2O3/water, past a permeable stretching/shrinking sheet. The effects of Brownian motion and thermophoresis are considered here. Design/methodology/approach Similarity transformations are used to reduce the governing partial differential equations to a system of ordinary (similarity) differential equations. A MATLAB solver called the bvp4c is then used to compute the numerical solutions of equations (12) to (14) subject to the boundary conditions of equation (15). Then, the effects of various physical parameters on the flow and thermal fields of the hybrid nanofluid are analyzed. Findings Multiple (dual) solutions are found for the basic boundary layer equations. A stability analysis is performed to see which solutions are stable and, therefore, applicable in practice and which are not stable. Besides that, a comparison is made between the hybrid nanofluid and a traditional nanofluid, Cu/water. The skin friction coefficient and Nusselt number of the hybrid nanofluid are found to be greater than that of the other nanofluid. Thus, the hybrid nanofluid has a higher heat transfer rate than the other nanofluid. However, the increase in the shrinking parameter reduces the velocity of the hybrid nanofluid. Originality/value The present results are original and new for the study of the flow and heat transfer past a permeable stretching/shrinking sheet in Cu–Al2O3/water hybrid nanofluid.

scholarly journals Flow towards a Stagnation Region of a Vertical Plate in a Hybrid Nanofluid: Assisting and Opposing Flows

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 448
Author(s):  
Iskandar Waini ◽  
Anuar Ishak ◽  
Ioan Pop
Keyword(s):  
Radiation Effects ◽  
Vertical Plate ◽  
Similarity Transformation ◽  
Fluid Motion ◽  
Hybrid Nanoparticles ◽  
Problem Solver ◽  
Hybrid Nanofluid ◽  
Al2o3 Nanoparticles ◽  
Nanofluid Flow ◽  
Stagnation Region

This study investigates a hybrid nanofluid flow towards a stagnation region of a vertical plate with radiation effects. The hybrid nanofluid consists of copper (Cu) and alumina (Al2O3) nanoparticles which are added into water to form Cu-Al2O3/water nanofluid. The stagnation point flow describes the fluid motion in the stagnation region of a solid surface. In this study, both buoyancy assisting and opposing flows are considered. The similarity equations are obtained using a similarity transformation and numerical results are obtained via the boundary value problem solver (bvp4c) in MATLAB software. Findings discovered that dual solutions exist for both opposing and assisting flows. The heat transfer rate is intensified with the thermal radiation (49.63%) and the hybrid nanoparticles (32.37%).

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