Energy transfer of Jeffery–Hamel nanofluid flow between non-parallel walls using Maxwell–Garnetts (MG) and Brinkman models

Abstract In this letter, energy transfer of Jeffery–Hamel nanofluid flow in non-parallel walls is investigated analytically using Galerkin method. The effective thermal conductivity and viscosity of nanofluid are calculated by the Maxwell–Garnetts (MG) and Brinkman models, respectively. The influence of the nanofluid volume friction, Reynolds number and angle of the channel on velocity and temperature profiles are investigated. Results show that Nusselt number increases with increase of Reynolds number and nanoparticle volume friction. Also it can be found that skin friction coefficient is an increasing function of Reynolds number, opening angle and nanoparticle volume friction.


Introduction
Nanotechnology suggests new kind of working fluid with higher thermal conductivity. Nanofluid can be used in various field of engineering. Fluid heating and cooling are important in many industries fields such as power, manufacturing and transportation. Effective cooling techniques are absolutely needed for cooling any sort of high energy device. Common heat transfer fluids such as water, ethylene glycol, and engine oil have limited heat transfer capabilities due to their low heat transfer properties. In contrast, metals thermal conductivities are up to three times higher than the fluids, so it is naturally desirable to combine the two substances to produce a heat transfer medium that behaves like a fluid, but has the thermal conductivity of a metal. Zin et al. (2017) investigated Jeffrey nanofluid free convection in a porous media under the effect of magnetic field. Abro and Khan (2017) investigated flow and heat transfer of Casson fluid in a porous medium. Sheikholeslami et al. (2018a) utilized nanoparticles for condensation process. They analyzed entropy generation and exergy loss of nano-refrigerant. Ullah et al. (2017) investigated slip effect on Casson fluid flow over a porous plate in existence of Lorentz forces. Sheikholeslami et al. (2018f) investigated exergy loss analysis for nanofluid forced convection heat transfer in a pipe with modified turbulators. Sheikholeslami et al. (2018d) presented nanofluid forced convection * Corresponding author.
E-mail address: ilyaskhan@tdt.edu.vn (I. Khan). turbulent flow in a pipe. Sheikholeslami et al. (2018g) studied the nanofluid natural convection in a porous cubic cavity by means of Lattice Boltzmann method. Sheikholeslami (2018e) simulated solidification process of nano-enhanced PCM in a thermal energy storage.
There are some simple and accurate approximation techniques for solving differential equations called the Weighted Residuals Methods (WRMs). Collocation, Galerkin and Least Square are examples of the WRMs. Hosseini et al. (2018) utilized Galerkin method to investigated Nanofluid heat transfer analysis in a microchannel heat sink (MCHS) under the effect of magnetic field. Vaferi et al. (2012) have studied the feasibility of applying of Orthogonal Collocation method to solve diffusivity equation in the radial transient flow system. Hendi and Albugami (2010) used Collocation and Galerkin methods for solving Fredholm-Volterra integral equation. Recently Least square method is introduced by A. Aziz and M.N. Bouaziz (Bouaziz and Aziz, 2010) and is applied for a predicting the performance of a longitudinal fin Aziz and Bouaziz (2011). They found that least squares method is simple compared with other analytical methods. Shaoqin and Huoyuan (2008) developed and analyzed least-squares approximations for the incompressible magneto-hydrodynamic equations.
In this study, the purpose is to solve nonlinear equations through the GM. The effect of active parameters such as nanoparticle volume friction, opening angle and Reynolds number on velocity and temperature boundary layer thicknesses have been examined.

Problem description
Consider a system of cylindrical polar coordinates (r, z, θ) which steady two-dimensional flow of an incompressible conducting viscous fluid from a source or sink at channel walls lie in planes, and intersect in the axis of z. Assuming purely radial motion which means that there is no change in the flow parameter along the z direction. The flow depends on r and θ (see Fig. 1).
The reduced forms of continuity, Navier-Stokes and energy equations are (Sheikholeslami et al., 2012): Where, u r is the velocity along radial direction, P is the fluid pressure, υ n f the coefficient of kinematic viscosity and ρ n f the fluid density. The effective density ρ n f , the effective dynamic viscosity µ n f and kinematic viscosity υ n f of the nanofluid are given as: Here, φ is the solid volume fraction. Considering u θ = 0 for purely radial flow, one can define the velocity parameter as: Introducing the η = θ α as the dimensionless degree, the dimensionless form of the velocity parameter can be obtained by dividing that to its maximum value as: Substituting Eq. (6) into Eqs. (2) and (3), and eliminating P, one can obtain the ordinary differential equation for the normalized function profile as: Where Re is Reynolds number, Pr is Prandtl number and Ec is the Eckert number. On introducing the following non-dimensional quantities, With the following reduced form of boundary conditions Physically these boundary conditions mean that maximum values of velocity are observed at centerline (η = 0) as shown in Fig. 1, and we consider fully develop velocity profile, thus rate of velocity is zero at (η = 0). Also, in fluid dynamics, the no-slip condition for fluid states that at a solid boundary, the fluid will have zero velocity relative to the boundary. The fluid velocity at all fluid-solid boundaries is equal to that of the solid boundary, so we can see that value of velocity is zero at (η = 1).

Weighted residual methods (WRMs)
There existed an approximation technique for solving differential equations called the Weighted Residual Methods (WRMs). Suppose a differential operator D is acted on a function u to produce a function p: It is considered that u is approximated by a functionũ, which is a linear combination of basic functions chosen from a linearly independent set. That is, Now, when substituted into the differential operator, D, the result of the operations generally is not p(x). Hence an error or  residual will exist: The notion in WRMs is to force the residual to zero in some average sense over the domain. That is: Where the number of weight functions Wi is exactly equal to the number of unknown constants ci inũ. The result is a set of n algebraic equations for the unknown constants ci. Two methods of WRMs are explained in the following subsections. Galerkin method may be viewed as a modification of the Least Square Method. Rather than using the derivative of the residual with respect to the unknown ci, the derivative of the approximating function or trial function is used. In this method weight functions are:

Results and discussions
In this paper flow and heat transfer between two diverging channel is investigated using Galerkin method (Fig. 1). Cu-water nanofluid is considered as working fluid.(See Table 1.) The numerical solution which is applied to solve the present case is the fourth order Runge-Kutta procedure. As shown in Fig. 2 and Table 2, GM has a good accuracy.
Effect of the Reynolds number on the velocity and temperature profiles are shown in Figs. 3 and 4. As Reynolds number increases both velocity and thermal boundary layer thicknesses decrease. Also it can be seen that back flow is stronger for higher  Reynolds number. Effect of the channel half angle on the velocity and temperature profiles are shown in Figs. 5 and 6. Back flow is observed for higher values of angle. Increasing this angle makes thermal boundary layer thickness to increase so Nusselt number decreases with increase of this parameter. Effect of the Eckert number on the temperature profile is shown in Fig. 7. As Eckert number increase, viscous dissipation increase and in turn thermal boundary layer thickness decreases. Effects of the volume fraction of nanofluid on the velocity and temperature profile are shown in Figs. 8 and 9. As nanofluid volume fraction increases velocity increases. Also thermal boundary layer thickness decreases with increase of nanofluid volume fraction. So, rate of heat transfer increases with increase of nanofluid volume fraction.

Conclusion
In this paper, Galerkin method is used to solve the problem of Jeffery-Hamel nanofluid flow and heat transfer. It can be found