MHD Stagnation Point Flow of Williamson Fluid over a Stretching Cylinder with Variable Thermal Conductivity and Homogeneous/Heterogeneous Reaction

Bilal, Muhammad; Sagheer, Muhammad; Hussain, Shafqat; Mehmood, Yasir

doi:10.1088/0253-6102/67/6/688

Cited by 26 publications

(12 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Cauchy stress tensor S used in this study for Williamson fluid is defined by Bilal et al:

S = - PI + τ_{1},

where,

τ_{1} = true[μ_{normal∞} + \frac{μ_{0} - μ_{normal∞}}{1 - Γ \hat{normalΥ}} true] \hat{normalΥ},

where P , I ,

μ_{0}, μ_{\infty}

τ_{1}

, and Γ > 0 are pressure, identity vector, limiting viscosity at 0, limiting viscosity at infinity, the extra stress tensor, a time constant, respectively, and shear rate

\overset{Υ}{true}

is written as:

\hat{normalΥ} = \sqrt{\frac{1}{2} \sum_{i} \sum_{j} {\overset{Υ}{true}}_{italicij} {\overset{Υ}{true}}_{italicji}} = \sqrt{\frac{1}{2} Π},

where,

normalΠ

is the second invariant of strain‐rate tensor and given by:

Π = \frac{1}{2} trace ({\nabla V + ({\nabla V)}^{normalT})}^{2} .

…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Under the above mentioned assumptions, here are the governing equations of the problems (Bilal et al and Mustafa et al):

\frac{\partial u}{\partial r} + \frac{u}{r} + \frac{\partial w}{\partial z} = 0,

u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} = \frac{μ}{ρ} \frac{\partial^{2} u}{\partial z^{2}} + Γ ν \sqrt{2} true(\frac{\partial u}{\partial z} true) true(\frac{\partial^{2} u}{\partial z^{2}} true) + g (β_{t} (T - T_{normal∞}) + β_{c} (C - C_{normal∞})) - \frac{σ B_{0}^{2}}{ρ} u,

u \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z} = \frac{k}{ρ c_{p}} \frac{\partial^{2} T}{\partial z^{2}} + τ true[{normalD}_{normalB} \frac{\partial C}{\partial z} \frac{\partial T}{\partial z} + \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial z})}^{2} true],

u \frac{\partial C}{\partial r} + w \frac{\partial C}{\partial z} = D_{B} \frac{\partial^{2} C}{\partial z^{2}} + \frac{D_{T}}{T_{\infty}} true(\partial^{2} T

…”

Section: Mathematical Formulationmentioning

confidence: 99%

See 1 more Smart Citation

Finite element method solution of mixed convection flow of Williamson nanofluid past a radially stretching sheet

Ibrahim

Gamachu

2019

Heat Trans

View full text Add to dashboard Cite

This computation reports the mixed convection flow of Williamson fluid past a radially stretching surface with nanoparticles under the effect of first-order slip and convective boundary conditions. The coupled nonlinear ordinary differential equations (ODEs) are obtained from the partial differential equations, which are derived from the conservation of momentum, energy, and species. Then, utilizing suitable resemblance transformation, these ODEs were changed into dimensionless form and then solved numerically by means of a powerful numerical technique called the Galerkin finite element method. The effect of different parameters on velocity, temperature, and concentration profiles is inspected and thrashed out in depth by graphs and tables. The upshots exhibit that the velocity profile augments as the values of concentration buoyancy and mixed convection parameters are engorged. Also, the results demonstrated that both temperature and concentration profiles are improved with an enhancement in values of thermophoresis parameters. The outcomes also indicate that for finer values of magnetic field parameter and thermophoresis parameter, the numerical value of local Nusselt and Sherwood numbers is reduced. K E Y W O R D S finite element method, mixed convection flow, nanofluid, radially stretching surface, Williamson fluid

show abstract

“…The Cauchy stress tensor S used in this study for Williamson fluid is defined by Bilal et al:

S = - PI + τ_{1},

where,

τ_{1} = true[μ_{normal∞} + \frac{μ_{0} - μ_{normal∞}}{1 - Γ \hat{normalΥ}} true] \hat{normalΥ},

where P , I ,

μ_{0}, μ_{\infty}

τ_{1}

, and Γ > 0 are pressure, identity vector, limiting viscosity at 0, limiting viscosity at infinity, the extra stress tensor, a time constant, respectively, and shear rate

\overset{Υ}{true}

is written as:

\hat{normalΥ} = \sqrt{\frac{1}{2} \sum_{i} \sum_{j} {\overset{Υ}{true}}_{italicij} {\overset{Υ}{true}}_{italicji}} = \sqrt{\frac{1}{2} Π},

where,

normalΠ

is the second invariant of strain‐rate tensor and given by:

Π = \frac{1}{2} trace ({\nabla V + ({\nabla V)}^{normalT})}^{2} .

…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Under the above mentioned assumptions, here are the governing equations of the problems (Bilal et al and Mustafa et al):

\frac{\partial u}{\partial r} + \frac{u}{r} + \frac{\partial w}{\partial z} = 0,

u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} = \frac{μ}{ρ} \frac{\partial^{2} u}{\partial z^{2}} + Γ ν \sqrt{2} true(\frac{\partial u}{\partial z} true) true(\frac{\partial^{2} u}{\partial z^{2}} true) + g (β_{t} (T - T_{normal∞}) + β_{c} (C - C_{normal∞})) - \frac{σ B_{0}^{2}}{ρ} u,

u \frac{\partial T}{\partial r} + w \frac{\partial T}{\partial z} = \frac{k}{ρ c_{p}} \frac{\partial^{2} T}{\partial z^{2}} + τ true[{normalD}_{normalB} \frac{\partial C}{\partial z} \frac{\partial T}{\partial z} + \frac{D_{T}}{T_{\infty}} {(\frac{\partial T}{\partial z})}^{2} true],

u \frac{\partial C}{\partial r} + w \frac{\partial C}{\partial z} = D_{B} \frac{\partial^{2} C}{\partial z^{2}} + \frac{D_{T}}{T_{\infty}} true(\partial^{2} T

…”

Section: Mathematical Formulationmentioning

confidence: 99%

Finite element method solution of mixed convection flow of Williamson nanofluid past a radially stretching sheet

Ibrahim

Gamachu

2019

Heat Trans

View full text Add to dashboard Cite

show abstract

“…Moreover, many researchers' works are based on heat and mass transportation, see Refs. [8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Multiple characteristics of three‐dimensional radiative Cross fluid with velocity slip and inclined magnetic field over a stretching sheet

et al. 2021

View full text Add to dashboard Cite

This article focuses on the three‐dimensional Cross fluid flow of a radiative nanofluid over an expanding sheet with aligned magnetic field, chemical reaction, and heat generation phenomenon. The stretching sheet has convective heat and slip boundary conditions. The similarity variables are properly used for the conversion of a dimensional mathematical model into a nondimensional one. The transformed ordinary differential equations are handled for the numerical outcomes of the suggested fluidic model by incorporating the shooting scheme. Furthermore, the numeric investigations are also compared by bvp4c MATLAB built‐in package. In a limited case, both the techniques are checked with already published articles, thereby revealing good agreement. Furthermore, the effects of few parameters like Prandtl number, Weissenberg number, heat generation, stretching rate parameter, magnetic parameter, thermal radiation, Brownian and thermophoresis parameters, and Lewis number on concentration, temperature, and velocity profiles have been presented using figures and numerical tables. The strong intensity of the magnetic field across the fluid and increment in the inclination angle (ϑ) result in a lower velocity profile. Temperature is more prominent for the higher slip mechanism. Furthermore, there in an increase in thermophoretic force, which pushes the nanoparticles, and this mixing of nanoparticles helps to increase the concentration profile. A higher Cross fluid index responds to a larger velocity.

show abstract

“…Acharya et al 23 studied the unsteady MHD boundary layer fluid flow over a stretching cylinder in the presence of a nonuniform heat source. Recently, many investigations including these type of problems with various aspects have been investigated by numerous authors 24,25 …”

Section: Introductionmentioning

confidence: 99%

Analysis of the velocity, thermal, and concentration MHD slip flow over a nonlinear stretching cylinder in the presence of outer velocity

Vinita

Poply²,

Goyal

et al. 2020

Heat Trans

View full text Add to dashboard Cite

The intention of the present work is to analyze the influence of magnetohydrodynamic slip flow with radiation effect toward a nonlinear stretching cylinder in the presence of outer velocity. Similarity variables are applied to convert the governing partial differential equations into ordinary differential equations. The Runge‐Kutta‐Fehlberg approach was adopted to numerically solve the modified equations by use of the Shooting method. The effect of prominent fluid parameters especially the velocity slip parameter, temperature slip parameter, concentration slip parameter, outer velocity, magnetic parameter, nonlinear stretching parameter, Schmidt number, and Eckert number on the velocity, temperature, and concentration have been examined and are displayed through graphs and tables. Numerical results of various parameters involved for the skin friction coefficient, the local Nusselt numbers, and the local Sherwood numbers are determined and also discussed in detail. In the present study, we used MATLAB for finding the final outcomes and relating the conclusive results for −θ0′(0) with those of already published papers. The outcomes reveal that with an increase in outer velocity, the fluid velocity increases while the temperature and concentration decrease. In the case of a higher nonlinear stretching parameter, the temperature as well as concentration decrease. The findings of the present study help to control the rate of heat transportation and highlight many applications in the insulation of wires, manufacturing of tetrapacks, production of glass fibers, fabrication of various polymers and plastic packs, rubber sheets, and so forth, where the quality of the desired product depends on the rate stretching, external magnetic field, and the composition of the material used, and manufacturing processes.

show abstract

MHD Stagnation Point Flow of Williamson Fluid over a Stretching Cylinder with Variable Thermal Conductivity and Homogeneous/Heterogeneous Reaction

Cited by 26 publications

References 32 publications

Finite element method solution of mixed convection flow of Williamson nanofluid past a radially stretching sheet

Finite element method solution of mixed convection flow of Williamson nanofluid past a radially stretching sheet

Multiple characteristics of three‐dimensional radiative Cross fluid with velocity slip and inclined magnetic field over a stretching sheet

Analysis of the velocity, thermal, and concentration MHD slip flow over a nonlinear stretching cylinder in the presence of outer velocity

Contact Info

Product

Resources

About