Heat Transport Phenomena for the Darcy–Forchheimer Flow of Casson Fluid over Stretching Sheets with Electro-Osmosis Forces and Newtonian Heating

In this study, an investigation has been carried out to analyze the impact of electro-osmotic effects on the Darcy–Forchheimer flow of Casson nanofluid past a stretching sheet. The energy equation was modelled with the inclusion of electro-osmotic effects with viscous and Joule dissipations. The governing system of partial differential equations were transformed by using the suitable similarity transformations to a system of ordinary differential equations and then numerically solved by using the Runge–Kutta–Fehlberg method with a shooting scheme. The effects of various parameters of interest on dimensionless velocity and temperature distributions, as well as skin friction and heat transfer coefficient, have been adequately delineated via graphs and tables. A comparison with previous published results was performed, and good agreement was found. The results suggested that the electric and Forchheimer parameters have the tendency to enhance the fluid velocity as well as momentum boundary layer thickness. Enhancements in temperature distribution were observed for growing values of Eckert number. It was also observed that higher values of electric field parameter diminished the wall shear stress and local Nusselt number.

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Magnetized and non-magnetized Casson fluid flow with gyrotactic microorganisms over a stratified stretching cylinder

Scientific Reports ◽

10.1038/s41598-021-95878-8 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Abdullah Dawar ◽

Zahir Shah ◽

Hashim M. Alshehri ◽

Saeed Islam ◽

Poom Kumam

Keyword(s):

Fluid Flow ◽

Differential Equations ◽

Fluid Velocity ◽

Lewis Number ◽

Casson Fluid ◽

Stretching Cylinder ◽

Concentration Difference ◽

Gyrotactic Microorganisms ◽

Similarity Transformations ◽

Homotopy Analysis

AbstractThis study presents the magnetized and non-magnetized Casson fluid flow with gyrotactic microorganisms over a stratified stretching cylinder. The mathematical modeling is presented in the form of partial differential equations and then transformed into ordinary differential equations (ODEs) utilizing suitable similarity transformations. The analytical solution of the transformed ODEs is presented with the help of homotopy analysis method (HAM). The convergence analysis of HAM is also presented by mean of figure. The present analysis consists of five phases. In the first four phases, we have compared our work with previously published investigations while phase five is consists of our new results. The influences of dimensionless factors like a magnetic parameter, thermal radiation, curvature parameter, Prandtl number, Brownian motion parameter, Schmidt number, heat generation, chemical reaction parameter, thermophoresis parameter, Eckert number, and concentration difference parameter on physical quantities of interests and flow profiles are shown through tables and figures. It has been established that with the increasing Casson parameter (i.e. $$\beta \to \infty$$ β → ∞ ), the streamlines become denser which results the increasing behavior in the fluid velocity while on the other hand, the fluid velocity reduces for the existence of Casson parameter (i.e. $$\beta = 1.0$$ β = 1.0 ). Also, the streamlines of stagnation point Casson fluid flow are highly wider for the case of magnetized fluid as equated to non-magnetized fluid. The higher values of bioconvection Lewis number, Peclet number, and microorganisms’ concentration difference parameter reduces the motile density function of microorganisms while an opposite behavior is depicted against density number.

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Numerical study of chemically reacting unsteady Casson fluid flow past a stretching surface with cross diffusion and thermal radiation

Open Engineering ◽

10.1515/eng-2017-0013 ◽

2017 ◽

Vol 7 (1) ◽

pp. 69-76 ◽

Cited By ~ 12

Author(s):

K. Pushpalatha ◽

J.V. Ramana Reddy ◽

V. Sugunamma ◽

N. Sandeep

Keyword(s):

Fluid Flow ◽

Differential Equations ◽

Numerical Study ◽

Fluid Velocity ◽

Casson Fluid ◽

Stretching Surface ◽

Cross Diffusion ◽

Similarity Transformations ◽

Chemically Reacting ◽

Fourth Order Method

AbstractThe problem of an unsteady MHD Casson fluid flow towards a stretching surface with cross diffusion effects is considered. The governing partial differential equations are converted into a set of nonlinear coupled ordinary differential equations with the help of suitable similarity transformations. Further, these equations have been solved numerically by using Runge-Kutta fourth order method along with shooting technique. Finally, we studied the influence of various non-dimensional governing parameters on the flow field through graphs and tables. Results indicate that Dufour and Soret numbers have tendency to enhance the fluid velocity. It is also found that Soret number enhances the heat transfer rate where as an opposite result is observed with Casson parameter. A comparison of the present results with the previous literature is also tabulated to show the accuracy of the results.

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The Approximate Analytical Solution of the MHD Flow of a Power-Law Fluid

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.431.198 ◽

2013 ◽

Vol 431 ◽

pp. 198-201

Author(s):

Jing Zhu ◽

Lian Cun Zheng

Keyword(s):

Differential Equations ◽

Power Law ◽

Velocity Ratio ◽

Mhd Flow ◽

Convergent Series ◽

Analysis Method ◽

Governing System ◽

Homotopy Analysis ◽

Incompressible Mhd ◽

Good Agreement

This paper presents a theoretical analysis for the incompressible MHD stagnation-point flows of a Non-Newtonian Fluid over stretching sheets.The governing system of partial differential equations is first transformed into a system of dimensionless ordinary differential equations. By using the homotopy analysis method, a convergent series solution is obtained. The reliability and efficiency of series solutions are illustrated by good agreement with numerical results in the literature.Besides, the effects of the power-law indexthe magnetic field parameter and velocity ratio parameter on the flow are investigated.

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Thermally and chemically reacted MHD Maxwell nanofluid flow past an inclined permeable stretching surface

Proceedings of the Institution of Mechanical Engineers Part E Journal of Process Mechanical Engineering ◽

10.1177/09544089211050715 ◽

2021 ◽

pp. 095440892110507

Author(s):

Amar B. Patil ◽

Vishwambhar S. Patil ◽

Pooja P. Humane ◽

Nalini S. Patil ◽

Govind R. Rajput

Keyword(s):

Differential Equations ◽

Energy Transport ◽

Stretching Surface ◽

Nanofluid Flow ◽

Linear Ordinary Differential Equations ◽

Maxwell Nanofluid ◽

Similarity Transformations ◽

Flow Past ◽

Chemically Reacting ◽

The Impact

The present work deals with chemically reacting unsteady magnetohydrodynamic Maxwell nanofluid flow past an inclined permeable stretching surface embedded in a porous medium with thermal radiation. The formulated governing partial differential equations conveying the flow model of Maxwell with Buongiorno modeled nanofluid is transformed into the system of highly non-linear ordinary differential equations via suitable similarity transformations; those equations are transmuted into an initial value problem and then solved numerically by a shooting approach with Runge–-Kutta fourth-order schema. To obtain the physical insight of the flow situation, the influence of associated parameters on the velocity, temperature, and concentration profiles is sketched graphically with the aid of MATLAB software. Furthermore, engineering quantities of interest are interpreted graphically. The computed numerical results are compared to estimate the validity of the achieved results; it has been found out that the computed results are highly accurate. The impact of the Maxwell parameter and inclination angle of the sheet on the velocity field is observed in decaying. Both thermal and solutal energy transport are progressive in nature as the Maxwell parameter and thermophoresis parameter grows, and a reverse trend is observed for Prandtl number.

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

Symmetry ◽

10.3390/sym12081238 ◽

2020 ◽

Vol 12 (8) ◽

pp. 1238 ◽

Cited By ~ 1

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.

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On the MHD Casson Axisymmetric Marangoni Forced Convective Flow of Nanofluids

Mathematics ◽

10.3390/math7111087 ◽

2019 ◽

Vol 7 (11) ◽

pp. 1087 ◽

Cited By ~ 18

Author(s):

Anum Shafiq ◽

Islam Zari ◽

Ghulam Rasool ◽

Iskander Tlili ◽

Tahir Saeed Khan

Keyword(s):

Differential Equations ◽

Convective Flow ◽

Flow Behavior ◽

Marangoni Effect ◽

Casson Fluid ◽

Linear Ordinary Differential Equations ◽

Metallic Particles ◽

Flow Parameters ◽

Forced Convective Flow ◽

The Impact

The proposed investigation concerns the impact of inclined magnetohydrodynamics (MHD) in a Casson axisymmetric Marangoni forced convective flow of nanofluids. Axisymmetric Marangoni convective flow has been driven by concentration and temperature gradients due to an infinite disk. Brownian motion appears due to concentration of the nanosize metallic particles in a typical base fluid. Thermophoretic attribute and heat source are considered. The analysis of flow pattern is perceived in the presence of certain distinct fluid parameters. Using appropriate transformations, the system of Partial Differential Equations (PDEs) is reduced into non-linear Ordinary Differential Equations (ODEs). Numerical solution of this problem is achieved invoking Runge–Kutta fourth-order algorithm. To observe the effect of inclined MHD in axisymmetric Marangoni convective flow, some suitable boundary conditions are incorporated. To figure out the impact of heat/mass phenomena on flow behavior, different physical and flow parameters are addressed for velocity, concentration and temperature profiles with the aid of tables and graphs. The results indicate that Casson fluid parameter and angle of inclination of MHD are reducing factors for fluid movement; however, stronger Marangoni effect is sufficient to improve the velocity profile.

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Radiation and Partial Slip Effects on Magnetohydrodynamic Jeffrey Nanofluid Containing Gyrotactic Microorganisms Over a Stretching Surface

Journal of Thermal Science and Engineering Applications ◽

10.1115/1.4048213 ◽

2020 ◽

Vol 13 (3) ◽

Author(s):

K. Kumaraswamy Naidu ◽

D. Harish Babu ◽

S. Harinath Reddy ◽

P. V. Satya Narayana

Keyword(s):

Nusselt Number ◽

Differential Equations ◽

Thermal Radiation ◽

Coupled System ◽

Partial Slip ◽

Stretching Surface ◽

Gyrotactic Microorganisms ◽

Similarity Transformations ◽

Jeffrey Nanofluid ◽

The Impact

Abstract In this study, the impact of thermal radiation and partial slip on magnetohydrodynamic flow of the Jeffrey nanofluid comprising motile gyrotactic microorganisms via vertical stretching surface is analyzed. The governing partial differential equations are reformed to a system of coupled ordinary differential equations by utilizing the similarity transformations. The transformed equations are of order four, which are complex to solve analytically and hence, the coupled system is solved computationally by using the shooting technique along the Runge–Kutta integrated scheme. The ramifications of different thermophysical parameters on the density of gyrotactic microorganisms, Jeffrey nanofluid velocity, nanoparticles concentration, temperature, Sherwood number, and Nusselt number are illustrated graphically. Comparing this study with the results already published favors the validity of this study. It is established that the Nusselt number is boosted on enhancing the thermal radiation parameter, and the reverse trend has been observed on increasing the Richardson number, whereas the gyrotactic microorganisms density is more in case of viscous nanofluid compared to the Jeffrey nanofluid.

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Insights into the Stability of Mixed Convective Darcy–Forchheimer Flows of Cross Liquids from a Vertical Plate with Consideration of the Significant Impact of Velocity and Thermal Slip Conditions

Mathematics ◽

10.3390/math8010031 ◽

2019 ◽

Vol 8 (1) ◽

pp. 31 ◽

Cited By ~ 1

Author(s):

Umair Khan ◽

Aurang Zaib ◽

Ilyas Khan ◽

Kottakkaran Sooppy Nisar ◽

Dumitru Baleanu

Keyword(s):

Heat Transfer ◽

Differential Equations ◽

Vertical Plate ◽

Thermal Slip ◽

Slip Conditions ◽

Drag Forces ◽

Similarity Transformations ◽

Mixed Convective Flow ◽

The Stability ◽

The Impact

This paper reflects the effects of velocity and thermal slip conditions on the stagnation-point mixed convective flow of Cross liquid moving over a vertical plate entrenched in a Darcy–Forchheimer porous medium. A Cross liquid is a type of non-Newtonian liquid whose viscosity depends on the shear rate. The leading partial differential equations (PDEs) are altered to nonlinear ordinary differential equations (ODEs) via feasible similarity transformations. These transmuted equations are computed numerically through the bvp4c solver. The authority of sundry parameters on the temperature and velocity distributions is examined graphically. In addition, the characteristics of heat transfer are analyzed in the presence of the impact of drag forces. The outcomes reveal that the permeability parameter decelerates the drag forces and declines the rate of heat transfer in both forms of solutions. Moreover, it is found that the drag forces decline with the growing value of the Weissenberg parameter in the upper branch solutions, while a reverse trend is revealed in the lower branch solutions. However, the rate of heat transfer shows a diminishing behavior with an increasing value of the Weissenberg parameter.

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Hydromagnetic stagnation point flow of a micropolar viscoelastic fluid towards a stretching/shrinking sheet in the presence of heat generation

Canadian Journal of Physics ◽

10.1139/cjp-2013-0329 ◽

2014 ◽

Vol 92 (10) ◽

pp. 1113-1123 ◽

Cited By ~ 6

Author(s):

Z. Abbas ◽

Mariam Sheikh ◽

M. Sajid

Keyword(s):

Heat Generation ◽

Fluid Velocity ◽

Shrinking Sheet ◽

Series Solutions ◽

Flow Equations ◽

Electrically Conducting ◽

Similarity Transformations ◽

Governing System ◽

Homotopy Analysis ◽

Analytical Series

In this article, the boundary layer two-dimensional Hiemenz flow and heat transfer for a micropolar, viscoelastic electrically conducting fluid over a shrinking/stretching sheet with heat generation is investigated. The governing system of flow equations is first reduced to coupled nonlinear ordinary differential equations by means of similarity transformations. A purely analytical series solutions of these equations is obtained in which domain (0 ≤ η ≤ ∞) using an analytic technique, namely, the homotopy analysis method. The convergence of the series solutions is discussed explicitly. The influences of an emerging parameters on the fluid velocity, microrotation velocity and temperature field are shown graphically and discussed in detail. The comparison of present results with the existing results is also given and found in excellent agreement.

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A Numerical Approach for an Unsteady Tangent Hyperbolic Nanofluid Flow past a Wedge in the Presence of Suction/Injection

Mathematical Problems in Engineering ◽

10.1155/2021/8653091 ◽

2021 ◽

Vol 2021 ◽

pp. 1-15

Author(s):

U. Shahzad ◽

M. Mushtaq ◽

S. Farid ◽

K. Jabeen ◽

R.M.A. Muntazir

Keyword(s):

Differential Equations ◽

Graphical Representation ◽

Numerical Technique ◽

Darcy Number ◽

Numerical Approach ◽

Nanofluid Flow ◽

Similarity Transformations ◽

Flow Past ◽

Layer Flow ◽

The Impact

The analysis of unsteady tangent hyperbolic nanofluid flow past a wedge with injection-suction, because of its beneficial uses, has gained a lot of attention. The present study is mainly concerned with tangent hyperbolic nanofluid (non-Newtonian nanofluid). First, we have converted the system of partial differential equations (PDEs) to a system of ordinary differential equations (ODEs) with the help of appropriate similarity transformations. Boundary conditions are also transformed by utilizing suitable similarity transformation. Now, for the obtained ODEs, we have used the numerical technique bvp 4 c and investigated the velocity, temperature, and concentration profiles. The accuracy of the flow model is validated by applying MAPLE d-solve command having good agreement while comparing the numerical results obtained by bvp4c for both suction and injection cases. The effects of distinct dimensionless parameters on the various profiles are being analyzed. The novel features such as thermophoresis and Brownian motion are also discussed to investigate the characteristics of heat and mass transfer. Graphical representation of the impact of varying parameters and the solution method for the abovementioned model is thoroughly discussed. It was observed that suction or injection can play a key role in controlling boundary layer flow and brings stability in the flow. It was also noticed that by increasing the Darcy number, velocity profile increases in both injection-suction cases.

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