Analysis of thixotropic nanomaterial in a doubly stratified medium considering magnetic field effects

doi:10.1016/j.ijheatmasstransfer.2016.06.090

International Journal of Heat and Mass Transfer

Volume 102, November 2016, Pages 1123-1129

https://doi.org/10.1016/j.ijheatmasstransfer.2016.06.090 Get rights and content

Highlights

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Flow of thixotropic nanofluid is modeled.
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Heat and mass transfer analysis are considered in presence of temperature and concentration stratifications.
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Applied magnetic field is accounted.
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Nanofluid model consists of Brownian motion and thermophoresis.

Abstract

Here characteristics of double stratification in magnetohydrodynamic (MHD) boundary layer flow of thixotropic nanofluid in presence of mixed convection are reported. Heat generation/absorption, Brownian motion and thermophoresis effects are also present. Dimensional nonlinear equations are converted into dimensionless expressions by employing suitable transformations. Homotopic procedure is implemented to solve the governing dimensionless problems. The variations in dimensionless temperature and nanoparticle concentration profiles corresponding to multiple values of physical parameters are shown and analyzed through graphs. Numerical values of local Nusselt and Sherwood numbers are tabulated and interpreted for several values of sundry parameters. It is found that thermal and concentration stratified parameters result in the reduction of temperature and concentration distributions respectively.

Introduction

In view of growing demands of new technologies such as microelectronics, chemical production and power station there is a necessity to develop novel types of fluids which transfer heat more effectively. Heat transfer efficiency of working liquid can likewise be improved by enhancing the thermal conductivity. In general commonly used heat transfer liquids (e.g. water, ethylene glycol and engine oil) have low thermal conductivities when compared with the thermal conductivity of solids. The addition of small particles of solids possessing high thermal conductivities to fluids can enhance the thermal conductivity of fluid. The usage of such suspensions of solid particles has been investigated by several researchers and huge points of interest are observed [1]. Researchers are permitted in this direction due to recent advances in nanotechnology to study the next generation heat transfer nanofluids (a term first introduced by Choi [2]). There are several liquids including water, ethylene glycol, engine oil, pump oil and glycerol that have been used as host liquids in nanofluids. The nanoparticles used in nanofluids are of different materials having better thermal conductivity than base fluid. Owing to this many researchers are persuaded and revolutionary works has been done by them (see [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]). Moreover the canvas of magnetic field has several applications in engineering, medicine and physics. Several equipments including pumps, bearings, MHD generators and boundary layer control are influenced due to the interaction between the electrically conducting liquid and magnetic field. The characteristics of the flow strongly depend on the orientation and intensity of the applied magnetic field. The applied magnetic field controls the suspended particles and adjusts their concentration in the liquid which firmly changes characteristics of heat transfer in the flow. The magnetic nanomaterial has both the magnetic and liquid characteristics. These materials have several applications in magneto-optical wavelength filters, optical switches, optical modulators and optical gratings. Moreover such materials have significance in cancer therapy, tumor analysis, designing of loudspeakers, drug delivery and sink-float separation. Thermal characteristics of the magnetic nanoparticles are additionally tunable by giving the changes in the magnetic field strength. Recently it is reported that magnetic nanoparticles are infused to the veins adjacent to cancerous tissues [15].

Heat and mass transfer analysis in presence of stratification is an important characteristic and it has been analyzed by various researchers. It emerges because of variations in concentration and temperature or by consolidating the liquids of distinctive densities. This phenomenon incorporates thermal stratification in heterogeneous mixtures in atmosphere, oceans and reservoirs, ground water reservoirs and energy storage. Concentration of the oxygen level turns out to be low in the lower base of the reservoirs due to biological procedures. Such difficulty can be controlled by the implication of thermal stratification. It is imperative to investigate the impact of double stratification on the convective flows when heat and mass transfer are present simultaneously. The investigation of mixed convection in a doubly stratified medium is an important problem. This is due to its importance in geophysical flows [16]. Such type of flows involve in the lakes, seas and rivers, solar ponds and thermal energy storage systems. Impact of double stratification on mixed convection flow of nanofluid in a porous medium is explored by Srinivasacharya and Surender [17]. Srinivasacharya and Reddy [18] further reported the behavior of double stratification in mixed convection flow of micropolar fluid. Srinivasacharya and Upendar [19] also studied the impact of double stratification in MHD flow of micropolar fluid with mixed convection. Effects of double stratification over an unsteady stretching sheet and cylinder in presence of thermal radiation and mixed convection are reported by Hayat et al. [20], [21].

The investigation of heat generation/absorption effects in moving liquids is imperative in problems dealing with chemical reactions and those concerned with dissociating liquids. These effects adjust the temperature distribution which influences the particle deposition rate in electronic chips, semi-conductor wafers and nuclear reactors. Although exact modeling of internal heat generation or absorption is quite difficult however some simple mathematical models can be utilized to express its general characteristics for most physical situations. Heat generation or absorption can be assumed constant, space-dependent or temperature-dependent. MHD unsteady flow of pseudoplastic nanofluid and heat transfer in a finite thin film over a stretching surface with internal heat generation is analyzed analytically by Lin et al. [22]. Ramesh et al. [23] investigated the MHD flow of dusty fluid near the stagnation point towards a permeable stretching sheet with non-uniform heat source/sink. Effect of inclined magnetic field in flow of third grade fluid with variable thermal conductivity is conducted by Hayat et al. [24]. Farooq et al. [25] reported the MHD flow of Jeffrey fluid with Newtonian heating and heat source/sink.

Flows of non-Newtonian materials have attracted the continuous attention of researchers. Particularly such materials are involved in geophysics, oil reservoir engineering, bioengineering, chemical and nuclear industries, polymer solution, cosmetic processes, paper production etc. Obviously all non-Newtonian materials on the basis of their behavior in shear are not predicted by one constitutive relationship. This fact of non-Newtonian material is different than the viscous liquids. Thus various models of non-Newtonian fluids have been suggested. Amongst these there is a thixotropic fluid model. The difference between thixotropic and shear thinning fluid is that a shear thinning fluid shows a decrease in viscosity with increasing shear rate while thixotropic fluid displays a decrease in viscosity over time at constant shear rate. Few studies relevant to thixotropic fluid can be seen in the Refs. [26], [27], [28], [29], [30].

The main motto of present research is to explore the characteristics of mixed convection in flow of thixotropic nanofluid over a stretching surface with double stratification effects. Effects of heat generation/absorption are also taken into account. Homotopic approach [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45] is implemented to solve the resulting differential systems. Convergence of the obtained solutions is verified. Results for different sundry variables on physical quantities are examined.

Section snippets

Formulation

Consider the steady magnetohydrodynamic (MHD) mixed convection flow of thixotropic nanofluid over a stretching sheet. Simultaneous effects of thermal and concentration stratifications with heat generation/absorption are considered. We denote ( $T_{w}, C_{w}$ ) and ( $T_{\infty}, C_{\infty}$ ) as the variable temperature and concentration at and away from the sheet respectively. Brownian motion and thermophoresis effects are present. Moreover x and y-axes are selected along and normal to the sheet. The boundary layer equations

Homotopic solutions and convergence analysis

The self-similar coupled ordinary differential Eqs. (7), (8), (9) are of third order in f(η) and second order in θ(η) and ϕ(η). To solve these equations, we employed homotopic procedure suggested by Liao [34]. The initial guesses (f₀, θ₀, ϕ₀) and auxiliary linear operators (L_f, L_θ, L_ϕ) for the dimensionless equations of momentum, energy and concentration are $f_{0} (η) = (1 - e^{- η}), θ_{0} (η) = (1 - S_{1}) \exp (- η), ϕ_{0} (η) = (1 - S_{2}) \exp (- η),$ with $L_{f} = f^{‴} - f^{'}, L_{θ} = θ^{″} - θ, L_{ϕ} = ϕ^{″} - ϕ,$ that satisfies the properties as follows: $L_{f} (C_{1} + C_{2} e^{η} + C_{3} e^{- η}$

Analysis

The analytical computations are performed for different values of the parameters appearing in the problem namely the Prandtl number Pr, thermophoretic parameter N_t, Brownian motion parameter N_b, thermal stratified parameter S₁, heat generation/absorption parameter δ, Schmidt number Sc and concentration stratified parameter S₂. To get a definite perception of the problem, the temperature and concentration profiles are presented graphically through Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8,

Conclusions

Here analysis is performed to investigate the simultaneous impacts of double stratification in magnetohydrodynamic (MHD) mixed convection flow of thixotropic nanofluid through heat generation/absorption. The worthmentioning outcomes of the presented analysis are listed below:

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Behaviors of Pr and S₁ on temperature are quite reverse when compared with N_t and N_b.
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The larger values of thermal stratification parameter S₁ show higher temperature θ(η) and thermal boundary layer thickness.
•
Influences of

References (45)

K. Khanafer et al.
Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids
Int. J. Heat Mass Transfer
(2003)
M. Turkyilmazoglu
Exact analytical solutions for heat and mass transfer of MHD slip flow in nanofluids
Chem. Eng. Sci.
(2012)
M. Sheikholeslami et al.
Simulation of MHD CuO-water nanofluid flow and convective heat transfer considering Lorentz forces
J. Magn. Magn. Mater.
(2014)
C. Zhang et al.
MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction
Appl. Math. Comput.
(2015)
M. Sheikholeslami et al.
Three dimensional mesoscopic simulation of magnetic field effect on natural convection of nanofluid
Int. J. Heat Mass Transfer
(2015)
R. Ellahi et al.
Shape effects of nanosize particles in Cu-H₂O nanofluid on entropy generation
Int. J. Heat Mass Transfer
(2015)
S.U. Rahman et al.
Simultaneous effects of nanoparticles and slip on Jeffrey fluid through tapered artery with mild stenosis
J. Mol. Liq.
(2016)
M. Khan et al.
Non-linear radiative flow of three-dimensional Burgers nanofluid with new mass flux effect
Int. J. Heat Mass Transfer
(2016)
W. Ibrahim et al.
The effect of double stratification on boundary-layer flow and heat transfer of nanofluid over a vertical plate
Comp. Fluids
(2013)
D. Srinivasacharya et al.
Effect of double stratification on MHD free convection in a micropolar fluid
J. Egypt. Math. Soc.
(2013)

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Analysis of thixotropic nanomaterial in a doubly stratified medium considering magnetic field effects

Highlights

Abstract

Introduction

Section snippets

Formulation

Homotopic solutions and convergence analysis

Analysis

Conclusions

Int. J. Heat Mass Transfer

Chem. Eng. Sci.

J. Magn. Magn. Mater.

Appl. Math. Comput.

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

J. Mol. Liq.

Int. J. Heat Mass Transfer

Comp. Fluids

J. Egypt. Math. Soc.

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

Int. J. Heat Mass Transfer

J. Mol. Liq.

J. Mol. Liq.

Int. J. Heat Mass Transfer

J. Mol. Liq.

J. Mol. Liq.

Int. J. Heat Mass Transfer

J. Mol. Liq.

J. Magn. Magn. Mater.

Int. J. Heat Mass Transfer