INTRODUCTION

Convection is the heat transfer due to bulk movement of molecules within fluids. Heat transfer by convection naturally occurs in fluid flow, such as wind and oceanic currents. Convection is also used in engineering practices of home, industrial processes, cooling of equipment etc. Heat transfer applications in a channel where both forced and free convection play a role in determining the velocity and temperature fields arise in many practical applications. A frequently encountered configuration in thermal engineering equipment is the vertical channel. This configuration is used to model, solar energy and cooling devices of electronic and microelectronic equipments. The concept of nanofluids was introduced by Choi. where he proposed the suspension of nanoparticles in a base fluid such as water, oil and ethylene glycol. Buongiorno attempted to explain the increase in the thermal conductivity of such fluids and developed a model that took into account the particle Brownian motion and Thermophoresis.

Nowadays nanofluids have been utilized as the working fluids, instead of the base fluids due to their high thermal conductivity. In 1959, Casson introduced Casson fluid model. If the shear stress is less than the applied yield stress on the fluid then casson fluid act as a solid. If the shear stress is greater than the applied yield stress then it act as a liquid. Casson fluid model is one of the non-Newtonian models which exhibits yield stress. However such fluid behaves like a solid. Examples of Casson fluid include jelly, soup, honey, tomato sauce, concentrated fruit juices, blood and many others. Literature review of the fully developed mixed convection flow in channels has been presented by Tao [1], Aung and Worku [2], Cheng et al. [3], Hamadah and Wirtz [4], Chen and Chung [5], Barletta [6], Barletta et al. [7,8], Boulama and Galanis [9], etc. It is observed that conventional heat transfer fluids such as ethylene glycol mixture, etc. are poor heat transfer fluids because of their low thermal conductivity. In order to increase the thermal conductivity of the base fluids, researchers have tried to suspend solid particles in fluids, since the thermal conductivity of solids is typically higher than that of liquids (see Ref.[10]) Fluids with particles of the order of nanometers suspended in liquids are called nanofluids (see Ref. [11]). It is reported that they have superior properties compared to usual heat transfer fluids. Successful application of nanofluids will support the current trend towards component miniaturization by enabling the design of smaller and lighter heat exchanger systems. The convective heat transfer characteristics of nanofluids depend on the thermophysical properties of the base fluid, the flow pattern and the volume fraction of the suspended particles and their dimensions. and the shape of these particles. Nanofluids have many important industrial applications also (for example nuclear reactors, nanodrug delivery, cancer therapeutics, sensing and imaging). Buongiorno [12] reported that the nanoparticle absolute velocity can be viewed as the sum of the base fluid velocity and a relative velocity. He calls it as the slip velocity. He has concluded that in the absence of turbulent effects, it is the Brownian diffusion and the thermophoresis that are important, and he has suggested conservation equations based on these two effects. This model has been used by Nield and Kuznetzov [13], Kuznetov and Nield [14], Khan and Pop [15] and several authors to study convective flows of nanofluids. Numerical and experimental studies on nanofluids inside cavities are made by Kang et al., Khanafer et al., Tiwari et al. and Aminossadati et al. [16-19]. Important reviews on nanofluid flows are given by Daungthongsuk and Wongwises [20], Wang and Mujumdar [21], and Kakac and Pramuanjaroenkij [22]. Fully Developed Mixed Convection in a Vertical Channel Filled by a Nanofluid is studied by Grosan and Pop [23]. Natural convective boundary-layer flow of a nanofluid past a vertical plate is discussed by Kuznetsov and Nield [24]. Effects of Heat Generation or Absorption on free convection flow of a Nanofluid past an isothermal inclined plate are analysed by Akilu and Narahari [25]. MHD free convection flow of a Nanofluid past a vertical Plate in the presence of heat generation/ absorption effects is studied by Chamkha and Aly [26]. Analytical solution for peristaltic flow of conducting nanofluids in an asymmetric channel with slip effect of velocity, temperature and concentration is studied by Sreenadh et al. [27]. Mixed convection flow of Casson nanofluid over a stretching sheet with convectively heated chemical reaction and heat source/ sink is studied by Hayat et al.[28]. MHD stagnation point flow of Casson nanofluid over a stretching sheet with effect of viscous dissipation is investigated by Srinivasulu et al.[29]. Unsteady Casson nanofluid flow over a stretching sheet with thermal radiation, convective and slip boundary conditions is discussed by Ibukum Sarah Oyelakin et al.[30]. The influence of slip boundary condition on Casson nanofluid flow over a stretching sheet in the presence of viscous dissipation and chemical reaction is studied by Afify et al.[31].

Motivated by the above investigations on Casson nanofluid and its wide applications, the objective of the present study is to analyze the convection in a vertical channel filled with a Casson nanofluid. The expressions for the velocity field, temperature and nanoparticle concentration are obtained. The results are deduced and discussed.

NOMENCLATURE

A to G Constants

C Nanoparticles volume fraction

Thermophoretic diffusion coefficient

Distance between parallel walls

Grash of number

Brownian motion parameter

Thermophoresis parameter

Sherwood number

velocity components in the - and - directions

heat capacity at constant pressure

Gravity acceleration vector

Brownian diffusion coefficient

Thermal conductivity

Buoyancy-ratio parameter

Nusselt number

Pressure

Fluid temperature

Cartesian coordinates

Greek Symbols

Pressure gradient,

Thermal expansion coefficient

Casson parameter

Rescaled nanoparticle volume fraction

Dynamic viscosity

Kinematic viscosity

Dimensionless temperature

Density

Subscripts

f Base fluid

p Solid particle

MATHEMATICAL FORMULATION:

Consider a Casson nanofluid that steadily flows between two vertical and parallel plane walls apportioned by a distance L. Let x-axis be aligned parallel to the gravitational acceleration vector, but with the positive direction and y-axis be taken orthogonal to the channel walls. Let y=0 and y=L act as left and right vertical walls respectively. It is assumed that the temperature and the nanoparticles concentration at the wall at y = 0 are T₁and C₁ and at the wall at y = L are T₂and C₂, respectively. We also assume that T=T(y), C=C(y) and the flow is fully developed with constant pressure gradient.

Fig.1 Physical model

In keeping with the Oberbeck-Boussinesq approximation and the assumptions stated above, the governing equations reduce to

(1)

(2)

(3)

(4)

The boundary conditions are given by

at y=0

at y=L (5)

In order to determine the pressure gradient from equation (2), the mass flux conservation Q is required. That is,

(6)

We introduce now the following dimensionless variables:

(7)

where, following Barleta and Zanchini[32], we assume that and .Substituting these variables into Eqs. (2)-(4),we get the following ordinary differential equations:

(8)

(9)

(10)

The boundary conditions (5) become

(11)

The mass flux conservation relation (6), becomes

(12)

where we have taken

In the above equations, is the pressure parameter, Gr is the Grashof number, Re is the Reynolds number , and Gr/Re is the mixed convection parameter, Nb is the Brownian motion parameter, and Nt is the thermophoresis parameter . These parameters are given by

(13)

The physical quantities of interest are the Nusselt and the Sherwood numbers. They are defined as

(14)

Substituting Eq.(7) into Eq.(4),we get

(15)

SOLUTION OF THE PROBLEM:

Equations (8)-(10) along with the boundary conditions (11) and the mass flux conservation equation (12) have been solved analytically. The expressions for velocity, temperature, concentration and the pressure gradient are given by

(16)

(17)

(18)

(19)

where

, (20)

Using Eqs.(17) and (18), the expressions for the Nusselt and Sherwood numbers defined by Eq. (15) become

(21)

RESULTS AND DISCUSSION:

Fig.2:Variation of dimensionless Velocity when Nr=0,10,100,500,1000 and Nt=Nb=0.5, =1.5.

Fig.3: Variation of dimensionless velocity When Nb=0.025,1,2.5,5,Nt=0.5,Nr=100

=1.5.

Fig.4: Variation of dimensionless velocity when Nt=0.1,2.5,5, Nr=100, Nb=0.5 and =1.5

Fig.5: Variation of dimensionless velocity when Nt= Nb=0.5 , Nr=100, and =1,1.25,1.5,1.75,2

Fig.6: Variation of dimensionless temparature (full line) and dimensionless concentration (dot line) when Nb=0.025,0.25,0.5,0.75,1,Nr=100 and Nt=0.5

Fig.7: Variation of dimensionless temparature (full line) and dimensionless concentration (dot line) when Nt=0,0.25,0.5,0.75,1 and Nb=0.5,Nr=100

Fig.8: Variation of reduced Nusselt number – (full line)and reduced Sherwood number (dot line)with respect to Nb when Nt=0.1,0.3,0.5.

Fig.9: Variation of reduced Nusselt number – (full line) and reduced Sherwood number (dot line)with respect to Nt when Nb=0.1,0.3,0.5.

In this paper, steady flow of Casson nanofluid in a vertical channel is examined and the results are discussed for various physical parameters such as the buoyancy ratio parameter Nr, the Brownonian motion parameter Nb, the Thermophoresis parameter Nt, Casson parameter . In this analysis, for numerical calculation we used

Table 1 Comparison between analytical and numerical results

	Nr	Nt	Nb			Pressure Gradient		Nusselt Number		Sherwood Number
	Nr	Nt	Nb			Analytic	Numeric	Analytic	Numeric	Analytic	Numeric
0	0	0	0.2	0.3	0.05	9.2308	9.2308	2. 4533	2.45326	2	2
	0	0.2	0.2	0.3	0.05	9.2308	9.2308	2.9338	2.93626	1.0662	1.08874
	5	0.2	0.2	0.3	0.05	8.4151	8.4151	2.9338	2.93626	1.0662	1.08874
1000	0	0	0.2	0.3	0.05	-75.5273	-75.5273	2.4533	2.45326	2	2
	0	0.2	0.2	0.3	0.05	-153.9121	-153.9121	2.9338	2.93626	1.0662	1.08874
	5	0.2	0.2	0.3	0.05	-154.7278	-154.7278	2.933	2.93626	1.0662	1.08874

For a fixed value of the mixed convection parameter Gr/Re=1000, these assigned values are kept as common in the entire study except for discrete values as displayed in Figures 2-9.

The variation of velocity with y is computed from equation (16) and is shown in Figures 2 to 5 for different values of Nt, Nb, Nr, and γ. It depicts that the velocity increases at cold wall and decreases at hot wall with increasing Nr, Nt, Nb, but the velocity decreases at both walls(hot and cold walls) with increasing Casson Parameter γ. The variation of temperature and nanoparticle volume fraction with y are computed from equations (17) and (18) for different values of Nt, Nb. Figs. 6 and 7 display the temperature and the nanoparticle volume fraction profiles for

different values of Nb and Nt. Thus, we notice from Figs. 6 and 7 that both temperature and nanoparticle volume fraction profiles increase with the increasing of the parameter Nb, where as temperature profile increases and nanoparticle volume fraction decreases with the increasing of Nt.

The variation of Nusselt and Sherwood numbers with Nt and Nb is computed from equation (21) and is shown in Figures 8 to 9 for different values of Nt and Nb. It is seen that the Nusselt number and Sherwood number both decreases with increasing of Nb where as Nusselt number decreases and Sherwood number increases with increasing of Nt.

In order to check the analytical solution (16), (17) and (18) with numerical solution, we applied RK method of fourth order along with Shooting technique. The comparison is presented in Table1. A very good agreement is seen between exact and numerical results.

CONCLUSION:

In this paper, we have studied the convection flow in a vertical channel filled by a Casson nanofluid, using a model in which Brownian motion and thermophoresis effects are accounted for. This model authorizes a simple analytical solution which depends on six dimensionless parameters, namely the mixed convection parameter Gr/Re, the buoyancy ratio parameter Nr, the Brownian motion parameter Nb, the thermophoresis parameter Nt and the Casson parameter .

The obtained results show that the reduced Nusselt number is a decreasing function of Nb and Nt. On the other hand, the reduced Sherwood number is a decreasing function of Nb and an increasing function of Nt. Also, we observe that the velocity decreases at the both walls (hot and cold walls) due to increasing Casson parameter. Similar behavior is reported by Vajravelu et al [33], for the mixed convective flow of a Casson Fluid over a vertical stretching sheet.

ACKNOWLEDGEMENT:

One of the authors (K.Sushma) is grateful to the Department of Science and Technology (DST) for providing INSPIRE Fellowship to undertake this work.

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Received on 19.06.2017 Modified on 21.09.2017

Research J. Science and Tech. 2017; 9(3):345-352.

DOI: 10.5958/2349-2988.2017.00060.2