Boundary-Layer Flow in a Porous Medium of a Nanofluid Past a Vertical Cone

The natural convection flow over a surface embedded in saturated porous media is encountered in many engineering problems such as the design of pebble-bed nuclear reactors, ceramic processing, crude oil drilling, geothermal energy conversion, use of fibrous material in the thermal insulation of buildings, catalytic reactors and compact heat exchangers, heat transfer from storage of agricultural products which generate heat as a result of metabolism, petroleum reservoirs, storage of nuclear wastes, etc.


Introduction
The natural convection flow over a surface embedded in saturated porous media is encountered in many engineering problems such as the design of pebble-bed nuclear reactors, ceramic processing, crude oil drilling, geothermal energy conversion, use of fibrous material in the thermal insulation of buildings, catalytic reactors and compact heat exchangers, heat transfer from storage of agricultural products which generate heat as a result of metabolism, petroleum reservoirs, storage of nuclear wastes, etc.
The derivation of the empirical equations which govern the flow and heat transfer in a porous medium has been discussed in [1][2][3][4][5]. The natural convection on vertical surfaces in porous media has been studied used Darcy's law by a number of authors [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Boundary layer analysis of natural convection over a cone has been investigated by Yih [21][22][23][24]. Murthy and Singh [25] obtained the similarity solution for non-Darcy mixed convection about an isothermal vertical cone with fixed apex half angle, pointing downwards in a fluid saturated porous medium with uniform free stream velocity, but a semi-similar solution of an unsteady mixed convection flow over a rotating cone in a rotating viscous fluid has been obtained Roy and Anilkumar [26]. The laminar steady nonsimilar natural convection flow of gases over an isothermal vertical cone has been investigated by Takhar et al. [27]. The development of unsteady mixed convection flow of an incompressible laminar viscous fluid over a vertical cone has been investigated by Singh and Roy [28] when the fluid in the external stream is set into motion impulsively, and at the same time the surface temperature is suddenly changed from its ambient temperature. An analysis has been carried out by Kumari and Nath [29] to study the non-Darcy natural convention flow of Newtonian fluids on a vertical cone embedded in a saturated porous medium with power-law variation of the wall temperature/concentration or heat/mass flux and suction/injection. Cheng [30][31][32][33][34] focused on the problem of natural convection from a vertical cone in a porous medium with mixed thermal boundary conditions, Soret and Dufour effects and with variable viscosity.
The conventional heat transfer fluids including oil, water and ethylene glycol etc. are poor heat transfer fluids, since the thermal conductivity of these fluids play an important role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface. An innovative technique for improving heat transfer by using ultra fine solid particles in the fluids has been used extensively during the last several years. Choi [35] introduced the term n a n o f l u i d r e f e r s t o t h e s e k i n d s o f f l u i d s b y suspending nanoparticles in the base fluid. Khanafer et al. [36] investigated the heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. The convective boundary-layer flow over vertical plate, stretching sheet and moving surface studied by numerous studies and in the review papers Buongiorno [37], Daungthongsuk and Wongwises [38], Oztop [39], Nield and Kuznetsov [40,41], Ahmad and Pop [42], Khan and Pop [43], Kuznetsov and Nield [44,45] and Bachok et al. [46].
From literature survey the base aim of this work is to study the free convection boundarylayer flow past a vertical cone embedded in a porous medium filled with a nanofluid, the basic fluid being a non-Newtonian fluid by using similarity transformations. The reduced coupled ordinary differential equations are solved numerically. The effects of the parameters governing the problem are studied and discussed.

Mathematical formulation of the problem
Consider the problem of natural convection about a downward-pointing vertical cone of half angle  embedded in a porous medium saturated with a non-Newtonian power-law nanofluid. The origin of the coordinate system is placed at the vertex of the full cone, with x being the coordinate along the surface of the cone measured from the origin and y being the coordinate perpendicular to the conical surface Fig (1). The temperature of the porous medium on the surface of the cone is kept at constant temperature w T , and the ambient porous medium temperature is held at constant temperature T  . The nanofluid properties are assumed to be constant except for density variations in the buoyancy force term. The thermo physical properties of the nanofluid are given in Table 1 (see Oztop and Abu-Nada [39]). Assuming that the thermal boundary layer is sufficiently thin compared with the local radius, the equations governing the problem of Darcy flow through a homogeneous porous medium saturated with power-law nanofluid near the vertical cone can be written in two-dimensional Cartesian coordinates (,) xy as: Where u and v are the volume-averaged velocity components in the x and y directions, respectively, T is the volume-averaged temperature. n is the power-law viscosity index of the power-law nanofluid and g is the gravitational acceleration. 0 m    corresponds to flow over a vertical flat plate and 1 m  corresponds to flow over a vertical cone. n is the viscosity index. For the case of 1 n  , the base fluid is Newtonian. We note that 1 n  and 1 n  represent pseudo-plastic fluid and dilatant fluid, respectively. Property nf  and nf  are the density and effective viscosity of the nanofluid, and K is the modified permeability of the porous medium. Furthermore, nf  and nf  are the equivalent thermal diffusivity and the thermal expansion coefficient of the saturated porous medium, which are defined as (see Khanafer et al. [36]): . 22 Here  is the solid volume fraction.
Substituting variables (6) into Eqs. (1)-(5) with Eq. (9), we obtain the following system of ordinary differential equations: along with the boundary conditions: where primes denote differentiation with respect to  , the quantity of practical interest, in this chapter is the Nusselt number x Nu which is defined in the form: where h denotes the local heat transfer coefficient.

Results and discussion
In this study we have presented similarity reductions for the effect of a nanoparticle volume fraction on the free convection flow of nanofluids over a vertical cone via similarity transformations. The numerical solutions of the resulted similarity reductions are obtained for the original variables which are shown in Eqs. (10) and (11) along with the boundary conditions (12) by using the implicit finite-difference method. The physical quantity of interest here is the Nusselt number x Nu and it is obtained and shown in Eqs. (13) and (14). In order to verify the accuracy of the present method, we have compared our results with those of Yih [22] for the rate of heat transfer (0) The comparisons in all the above cases are found to be in excellent agreement, as shown in Table 2. It is clear that as a geometry shape parameter m increases, the local Nusselt number increases. While Table 3  . It is noted that the velocity along the cone increases with the nanoparticle volume fraction in both of the two cases (i.e. Cu-water and Ag-water), moreover the velocity distribution in the case of Ag-water is larger than that for Cuwater. We can show that the change of the velocity distribution when we use different types of nanoparticles from Fig. 4, which depict the Ag-nanoparticles are the highest when the base fluid is water and when 0.1   . Thus the presence of the nanoparticles volume fraction increases the momentum boundary layer thickness.       Fig. 8 shows the variation of the reduced Nusselt number with the nanoparticles volume fraction  for the selected types of the nanoparticles. It is clear that the heat transfer rates decrease with the increase in the nanoparticles volume fraction  . The change in the reduced Nusselt number is found to be lower for higher values of the parameter  . It is observed that the reduced Nusselt number is higher in the case of Ag-nanoparticles and next Cunanoparticles, TiO2-nanoparticles and Al2O3-nanoparticles. Also, the Fig. 8 and Table 3 show that the values of (0)   change with nanofluid changes, namely we can say that the shear stress and heat transfer rate change by taking different types of nanofluid. Furthermore this depicts that the nanofluids will be very important materials in the heating and cooling processes.