Mixed convection flow of Casson fluid over a stretching sheet with convective boundary conditions and Hall effect
- M Bilal Ashraf^{1}Email author,
- T Hayat^{2, 3} and
- A Alsaedi^{3}
Received: 4 January 2017
Accepted: 2 May 2017
Published: 20 September 2017
Abstract
This learning pacts with the MHD mixed convection flow of Casson fluid over a stretching surface. Examination is achieved in presence of Hall and thermal radiation effects. Heat and mass transfer analysis is deliberated subject to convective boundary conditions. The boundary layer partial differential equations are concentrated into ordinary differential equations via similarity transformations. Series solutions of the causing problems are obtained. The effects of physical parameters on the velocity, temperature and concentration profiles are analyzed and deliberated. Numerical values of skin friction coefficients and local Nusselt and Sherwood numbers for different values of Casson fluid parameter, mixed convection parameter, Hall parameter, Hartman number, radiation parameter and Biot numbers are computed and inspected.
Keywords
1 Introduction
Recently the flow of an electrically conducting fluid in the presence of magnetic field has had importance in various areas of technology and engineering such as MHD power generation, drawing, annealing, in the purification of molten metals from non-metallic inclusions, electromagnetic pumps, MHD pumps etc. In particular, many metallurgical processes involve the cooling of continuous strips or filaments by drawing through a quiescent fluid. Some-times these strips or filaments are stretched. Actually, the properties of the final product depend upon the rate of cooling in these metallurgical processes. The rate of cooling can be controlled by drawing such strips or filaments in an MHD fluid, which permits to obtain the final product with desired characteristics. The magnetic field can enhance a Lorentz force even in a weak electric current and a magnetization force. It is well known that the influence of Hall current is very important in the presence of a strong magnetic field. In fact, in an ionized gas of low density and/or strong magnetic field, the conductivity normal to the magnetic field decreases by free growth of electrons and ions about the magnetic lines of force before suffering collisions. A current induced in a direction normal to the electric and magnetic fields is called Hall current. Gupta [1] analyzed the hydromagnetic flow past a porous flat plate with Hall effects. Hayat et al. [2] studied the effects of Hall current and heat transfer on the rotating flow of second grade fluid through a porous medium. Saleem and Aziz [3] explored hydromagnetic flow over a stretching surface with internal heat generation and Hall current. Aziz and Nabil [4] discussed the hydromagnetic mixed convection flow by an exponentially stretching surface with Hall current. Recently Pal [5] analyzed the influence of Hall current and thermal radiation on the unsteady flow of viscous fluid over a permeable stretching surface.
The study of stretching surfaces through various combinations is important in many practical applications. For instance, the production of sheeting material arises in a number of industrial manufacturing processes which include both metal and polymer sheets. After the initial contribution of Crane [6], various researchers extended the flow over a stretching surface in the directions of Newtonian and non-Newtonian fluid models under different geometries. The Casson fluid model is one of the non-Newtonian fluid models which reveal the characteristics of yield stress. Also Casson fluid acts like a solid when the shear stress less than the yield stress is applied, and it moves if the applied shear stress is greater than the yield stress. Jelly, soup, honey, tomato sauce, concentrated fruit juices and many others are the examples of Casson fluid. Some relevant studies about this fluid model can be seen in the refs [7–10].
Thermal and concentration convections in the boundary layer flow over a stretching surface subject to constant but different temperatures and concentrations at the boundaries have been considered extensively by the researchers. Such flows occur in a number of engineering, geophysical and energy storage applications. In nature most of the problems dealing with flow over a moving surface are developed by the movement of the boundary and buoyancy effects through thermal and concentration convections. Practical examples of such flows are solar central receivers exposed to wind currents, electronic devices cooled by fans, nuclear reactors cooled during emergency shutdown etc. The problem of steady laminar hydromagnetic heat transfer in the mixed convection flow by a vertical plate embedded in a uniform porous medium was studied by Makinde and Sibanda [11]. Turkyilmazoglu [12] constructed the analytical solution of mixed convection flow of MHD viscoelastic fluid over a permeable stretching surface. Recently Alsaadi et al. [13] studied the mixed convection flow of second grade fluid bounded by a permeable stretching surface with Soret and Dufour effects. Note that the aforementioned research disregarded the radiation effects. However, the technological processes at high temperatures involve the thermal radiation heat transfer. For example, in hypersonic flights, missile reentry rocket combustion chambers, gas cooled nuclear reactors and power plants for inter-planetary flight, the attention of researchers was focused on thermal radiation as a mode of energy transfer, and they emphasized the need for discussion on inclusion of radiative transfer in these processes [14, 15]. Mukhopadhyay [16] investigated the effects of thermal radiation in the unsteady boundary layer mixed convection flow by a vertical permeable stretching surface embedded in a porous medium. The fluid is assumed viscous and incompressible. Turkyilmazoglu [17] considered the impact of thermal radiation on the unsteady laminar convective MHD temperature-dependent viscosity flow over a rotating porous disk. Shehzad et al. [18] presented the magnetohydrodynamic (MHD) radiative flow of an incompressible Jeffrey fluid over a linearly stretched surface with Joule heating and thermophoretic effects. Recently Aziz [19] used the model of convective boundary condition for the investigation of Blasius flow. Some studies relevant to convective conditions can be read in the attempts [20–22].
The aim of present investigation is to study the Hall effects on the flow of Casson fluid in presence of thermal radiation and mixed convection. Convective heat and mass transfer at the boundaries is also considered. To our knowledge, there is no such study available in the literature. The resulting boundary layer partial differential equations are transformed into ordinary differential equations. Convergence criteria of series solutions are given by using the homotopy analysis method [23–30]. Impacts of embedding parameters on the flow, temperature and concentration fields are examined graphically. Numerical values of skin friction coefficient, transversal skin friction coefficient, local Nusselt and Sherwood numbers for secondary variables are obtained and discussed.
2 Mathematical modeling
Consider the mixed convection boundary layer flow of Casson fluid over a stretching surface in presence of thermal radiation. A uniform magnetic field \(B_{0}\) is imposed along the normal direction to a stretching surface. The effect of Hall current is taken into account. Heat and mass transfer in presence of convective boundary conditions is considered.
3 Series solutions
4 Convergence analysis and discussion
Merging of series results for altered order of calculations when \(\pmb{\beta = m = 0.5}\) , \(\pmb{\lambda = N = R = 0.3}\) , \(\pmb{M = \gamma _{1} = \gamma_{2} = 0.2}\) , \(\pmb{\Pr = 1.0}\) , \(\pmb{\mathrm {Sc}= 0.7}\) and \(\pmb{\hbar_{f} = \hbar _{w} = \hbar_{\theta } = \hbar_{\varphi } = - 0.3}\)
Order of approximations | \(\boldsymbol{- f^{\prime \prime } (0)}\) | \(\boldsymbol{h^{\prime} (0)}\) | \(\boldsymbol{- \theta '(0)}\) | \(\boldsymbol{- \varphi '(0)}\) |
---|---|---|---|---|
1 | 0.70525 | 0.015000 | 0.16222 | 0.16347 |
5 | 0.58605 | 0.027872 | 0.15239 | 0.15513 |
10 | 0.57654 | 0.030466 | 0.14810 | 0.15009 |
15 | 0.57496 | 0.031056 | 0.14688 | 0.14782 |
20 | 0.57468 | 0.031221 | 0.14661 | 0.14682 |
25 | 0.57466 | 0.031273 | 0.14659 | 0.14639 |
30 | 0.57467 | 0.031292 | 0.14662 | 0.14622 |
35 | 0.57467 | 0.031299 | 0.14663 | 0.14616 |
40 | 0.57467 | 0.031299 | 0.14663 | 0.14616 |
Mathematical values of physical constraints at the boundaries for altered values of factors β , R , M , \(\pmb{\gamma_{1}}\) , \(\pmb{\gamma_{2}}\) when \(\pmb{m = 0.5}\) , \(\pmb{\Pr = 1.0}\) , \(\pmb{\mathrm {Sc}= 0.7}\) and \(\pmb{\hbar_{f} = \hbar _{h} = \hbar_{\theta } = \hbar_{\varphi } = - 0.3}\)
β | M | λ | N | \(\boldsymbol{\gamma_{1}}\) | \(\boldsymbol{\gamma_{2}}\) | R | \(\boldsymbol{- ( 1 + \frac{1}{\beta } ) f''(0)}\) | \(\boldsymbol{( 1 + \frac{1}{\beta } ) h'(0)}\) | \(\boldsymbol{- \theta '(0)}\) | \(\boldsymbol{- \varphi '(0)}\) |
---|---|---|---|---|---|---|---|---|---|---|
0.3 | 0.2 | 0.3 | 0.3 | 0.2 | 0.2 | 0.3 | 2.0819 | 0.10957 | 0.14801 | 0.14777 |
0.5 | 1.7240 | 0.093799 | 0.14659 | 0.14645 | ||||||
0.7 | 1.4865 | 0.086024 | 0.14534 | 0.14569 | ||||||
0.5 | 0.2 | 1.7240 | 0.093799 | 0.14660 | 0.14670 | |||||
0.5 | 1.8292 | 0.22869 | 0.14613 | 0.14614 | ||||||
0.8 | 1.9390 | 0.35670 | 0.14560 | 0.14556 | ||||||
0.5 | 0.2 | −0.3 | 1.8792 | 0.092367 | 0.14594 | 0.14574 | ||||
0.0 | 1.8005 | 0.093237 | 0.14628 | 0.14596 | ||||||
0.5 | 1.6743 | 0.094184 | 0.14679 | 0.14656 | ||||||
0.5 | 0.2 | 0.3 | 0.0 | 1.7416 | 0.093717 | 0.14653 | 0.14590 | |||
0.3 | 1.7240 | 0.093775 | 0.14660 | 0.14610 | ||||||
0.6 | 1.7065 | 0.093952 | 0.14667 | 0.14640 | ||||||
0.5 | 0.2 | 0.3 | 0.3 | 0.2 | 1.7240 | 0.093775 | 0.14660 | 0.14610 | ||
0.5 | 1.6789 | 0.094190 | 0.26222 | 0.14667 | ||||||
0.7 | 1.6610 | 0.094277 | 0.30859 | 0.14701 | ||||||
0.5 | 0.2 | 0.3 | 0.3 | 0.2 | 0.3 | 1.7183 | 0.093792 | 0.14661 | 0.19390 | |
0.5 | 1.7102 | 0.093829 | 0.14664 | 0.26108 | ||||||
0.7 | 1.7047 | 0.093841 | 0.14668 | 0.30757 | ||||||
0.5 | 0.2 | 0.3 | 0.3 | 0.2 | 0.2 | 0.0 | 1.7391 | 0.093643 | 0.15413 | 0.14624 |
0.3 | 1.7240 | 0.093720 | 0.14659 | 0.14639 | ||||||
0.6 | 1.7098 | 0.093855 | 0.14038 | 0.14651 |
5 Conclusions
- (a)
Momentum boundary layer thickness and the velocity profile \(f'(\eta )\) decline for higher values of M and m.
- (b)
In the situation of assisting flow (\(\lambda > 0\)), both the velocity profile \(f^{\prime} (\eta )\) and momentum boundary layer thickness are increase, while opposite performance is perceived in the case of opposing flow (\(\lambda < 0\)).
- (c)
Transversal velocity \(h(\eta )\) upsurges with an upturn in Hartman number M and concentration buoyancy constraint N, while it declines with an upturn in Hall restriction m.
- (d)
Thermal boundary layer thickness improves with an upsurge in Biot number \(\gamma_{1}\) and thermal radiation factor R, while it lessens for superior Pr.
- (e)
Skin friction quantity, transversal skin friction coefficient, local Nusselt and Sherwood numbers are compact for greater Casson fluid parameter β.
- (f)
In the case of assisting flow (\(\lambda > 0\)), the skin friction coefficient reduces, although opposite performance for opposing flow (\(\lambda < 0\)) is perceived.
- (g)
Skin friction coefficient declines with a rise in concentration buoyancy factor N, while transversal skin resistance number, local Nusselt and Sherwood records drop.
- (h)
Local Nusselt and Sherwood figures upsurge for larger Biot quantities \(\gamma_{1}\) and \(\gamma_{2}\).
Declarations
Acknowledgements
This work was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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