Flow and heat transfer of nanofluid with injection through an expanding or contracting porous channel under magnetic force field

1. Introduction

Flow and heat transfer of fluids plays significant role in determining best ways to convey different categories of fluids with the aim of achieving high efficient transport system at economical cost. Due to their vast importance in industrial fields such as reservoir engineering, flow through porous industrial materials, heat exchange between fluid beds, ceramic processing, polymer solution, food technology, environmental protection, power plant operations, manufacturing, transportation and oil recovery amongst others.

In past efforts to study flow and heat transfer, numerical methods was adopted by Pourmehran et al. [1] in optimizing nanofluid flow in saturated porous medium while Tang and Jing [2] investigated sinusoidal wavy cavity effect of heat transfer under natural convection with phase deviation. High accuracy spherical motion particle was presented by Hatami et al. [3] in coquette fluid film flow. Hatami et al. [4] optimized circular wavy cavity nanofluid flow under natural convection. Ghasemi et al. [5] utilized least square methods of weighted residuals to analyze electrohydrodynamic effect of fluid flowing through a circular conduit, shortly after Ghasemi et al. [6] studied blood flow through porous arteries under the influence of magnetic force field with nanoparticles. Multi step differential transform method was applied by Hatami and Ganji [7]to analyze spherical particle motion of a rotating parabola while Hatami and Jing [8] optimized lid driven T-shaped porous cavity in the bid to improve mixed convective heat transfer. Non-Newtonian fluid flow under natural convection through vertical parallel plates was investigated by Karger and Akbarzade [9]. Fakour et al. [10] presented micropolar flow and heat transfer flowing in a channel with permeable walls. Nanoparticle migration around heated cylinder was studied by Hatami [11] considering wavy enclosure wavy.

In the bid to improve the thermal conductivity of viscous fluids such as water, oils, grease, ethylene. Choi [12] presented a novel approach to improving thermal conductivity of incompressible fluid through the addition of nanometersized particle into the base fluid, it was observed that upon the addition of metallic nanosized particle into base fluid thermal conductivity of fluid improves to about three times it present state . Therefore improving the overall transport capability of fluids making them potentially useful in fields such as biomedicine, manufacturing, fuel cells, and hybrid power generators amongst other practical application. These have created renaissance amongst researchers in science and engineering to explore the useful potential of nanofluid [13], [14], [15], [16], [17], [18], [19].

Since higher order nonlinear equations which describes the flow and heat transfer of the nanofluid requires the use of approximate analytical or numerical methods of solution to analyze the system of coupled equations. Approximate analytical methods of solutions applied by researchers in study of the heat transfer include the pertubation method (PM), adomian decomposition method (ADM), homotopy pertubation method (HPM), variational iteration method (VIM), differential transformation method (DTM) and methods of weighted residuals [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. Methods such as PM are limited owing to the problems of weak nonlinearities and artificial pertubation parameter which are non existence in real life. The need to find initial condition to satisfy the boundary condition makes methods such as VIM, DTM, HAM requires computational tools in handling a solution of large parameters resulting to large computational cost and time . Also the problem of finding the adomian polynomials makes the ADM not attractive to researchers. Whereas the methods of weighted residuals which includes the collocation method (CM), Garlerkin method (GM) and least square method (LSM) involves the need to determine weighing residuals to satisfy weighing functions which may be arbitrary. However the homotopy pertubation method been a relatively simplistic method of solving nonlinear, coupled equations due to its highly successive and accurate approximation makes it a favourable analytical technique to researchers.

Motivated by past research works, the homtopy pertubation method (HPM) is used to analyze the unsteady flow and heat transfer of nanofluid with injection through an expanding or contracting porous channel under the influence of uniform magnetic flux. Table 1.

Table 1. Thermo physical properties of nanofluid.

Empty Cell	Density (kg/m3)	Specific heat capacity (J/kgK)	Thermal conductivity (W/mk)
Water	997.1	4179	0.613
Copper	8933	385	401

2. Model development and analytical solution

Here nanofluid flows through parallel plates held horizontally against each other unsteadily under magnetic force field influence. The upper plate expands and contracts with time while the lower plate is stationary and is externally heated which is described in the physical model diagram Fig. 1. The heated wall is cooled by injecting cool fluid with uniform velocity vw from the upper plate which expands and contract at time rate a (t). The both plates are perpendicular to y axis. u and v are the velocity component of x and y direction. According to this view, fluid flow may be assumed to be stagnation flow. The models are developed assuming a two component nanofluid mix, incompressible fluid flow since fluid is liquid, negligible radiation effect owing to flow geometry, nanoparticle and base fluid are in thermal equilibrium since Nano mixture is thermodynamically compartible. With respect to these condition Navier – Stokes equation can be presented as:(1) $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$ (2) $\begin{matrix} ρ_{nf} (\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y}) = - \frac{\partial p^{*}}{\partial x} + \\ μ_{nf} (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}) - σ B^{2} (t) u \end{matrix}$ (3) $ρ_{nf} (\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial T}{\partial y}) = - \frac{\partial p^{*}}{\partial y} + μ_{nf} (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}})$ (4) $\frac{\partial T}{\partial t} + u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{k_{nf}}{{(ρ C_{p})}_{nf}} (\frac{\partial^{2} T}{\partial x^{2}} - \frac{\partial^{2} T}{\partial y^{2}})$ where the effective density $(ρ nf)$ , effective dynamic viscosity $(μ_{nf})$ , heat capacitance ${(ρ C_{p})}_{nf}$ and thermal conductivity ( $k_{nf})$ of the nanofluid are defined as follows Sobamowo and Akinshilo [19]:(5) $ρ_{nf} = (1 - ϕ) ρ_{f} + ϕ ρ_{s}$ (6) ${(ρ C_{p})}_{nf} = (1 - ϕ) {(ρ C_{p})}_{f} + ϕ {(ρ C_{p})}_{s}$ (7) $μ_{nf} = \frac{μ_{f}}{{(1 - ϕ)}^{2.5}}$ (8) $\frac{k_{nf}}{k_{f}} = \frac{k_{s} + 2 k_{f} - 2 ϕ (k_{f -} k_{s})}{k_{s} + 2 k_{f} - ϕ (k_{f -} k_{s})}$

With appropriate boundary condition stated as(9) $u = 0 v = - v_{w} = - aA T (a) = T_{0} at y = a (t) u = 0 v = 0 at y = 0$ where A = vw/a is the measure of wall permeability and To is the porous platetemperature which as the same temperature as the injected coolant. With respect to the above stream function is introduced as:(10) $ψ = \frac{vx}{a} F (η, t)$

Further expression of stream function in terms of fluid velocity yields:(11) $ψ = \frac{v_{f} x}{a} F (η, t)$

Upon substituting Eq. (11) into Eqs. (1), (2), (3), eliminating pressure term from momentum equation. The following expression is obtained:(12) $\begin{matrix} F_{η η η η} + \frac{A_{1}}{A_{2}} α (η F_{η η η} + 3 F_{η η}) - {FF}_{η η η} - \\ F_{η} F_{η η} - \frac{A_{1}}{A_{2}} α^{2} v_{f}^{- 1} F_{η η t} - {MF}_{η η} = 0 \end{matrix}$ where the wall expansion ratio is defined by α which is negative for contraction and positive for expansion.(13) $α = \frac{a \dot{a}}{v_{f}}$

Nanofluid constant parameters A1 and A2 are expressed as:(14) $A_{1} = \frac{ρ_{nf}}{ρ_{f}}$ (15) $A_{2} = \frac{μ_{nf}}{μ_{f}}$

Relevant boundary condition is given as(16) $F_{η} (0) = 0 F (0) = 0 F_{η} (1) = 0 F (1) = \frac{A_{1}}{A_{2}} R$ where R is the Reynolds number, which is positive for injection and negative for suction. Since injection is considered, R is positive.(17) $f = \frac{F}{R}, l = \frac{x}{a}$

A similar solution can be described for both time and space following the transformation developed by [36], [37] which leads to fηηt=0 which is obtained by specifying the initial value of the expansion ratio α.(18) $α = \frac{a \dot{a}}{v_{f}} = \frac{a_{0} {\dot{a}}_{0}}{v_{f}} = Cte$ where α0 and a0 connote initial channel height and expansion ratio respectively. Upon integrating Eq. (9) with respect to time, this is simply expressed as:(19) $\frac{a}{a_{0}} = \frac{v_{w} (0)}{v_{w} (t)} = \sqrt{1 + 2 v_{f} α {ta}_{0}^{- 2}}$

Since A is the coefficient for injection which is a constant. Therefore vw = Aa, which is an expression for velocity variation of injection. Based on the above, the velocity profile incorporating the Hartmann parameter which measures the influence of magnetism is presented as:(20) $\frac{d^{4} f}{d η^{4}} + \frac{A_{1}}{A_{2}} (α (η \frac{d^{3} f}{d η^{3}} + 3 \frac{d^{2} f}{d η^{2}}) + R (f \frac{d^{3} f}{d η^{3}} - \frac{df}{d η} \frac{d^{2} f}{d η^{2}})) - M^{2} \frac{d^{2} f}{d η^{2}} = 0$

With appropriate boundary condition introduced as(21) $f (0) = 0 \frac{df}{d η} (0) = 0 f (1) = 1 \frac{df}{d η} (1) = 1$

Fluid temperature from the wall at distance η is described as(22) $T = T_{0} + \sum C_{m} {(x / a)}^{m} q_{m}$

The stationery heated wall temperature is expressed as:(23) $\frac{d^{2} q_{m}}{d η^{2}} - \frac{A_{3}}{A_{4}} \Pr {Rmq}_{m} \frac{df}{d η} + \frac{A_{3}}{A_{4}} \Pr Rf \frac{{dq}_{m}}{d η} + \frac{A_{3}}{A_{4}} α {Prm q}_{m} + \frac{A_{3}}{A_{4}} \Pr η α \frac{{dq}_{m}}{d η}$

With dimensionless parameters stated as(23b) $R = \frac{{ah}^{2}}{v} M = \frac{σ B_{0}^{2} h^{2}}{ρ v} \Pr = \frac{μ}{ρ_{f} α} α = Cte$ where R is the Reynolds number which measures the significance of inertia compared to viscous fluid effect, M is the Hartmann or magnetic parameter which quantifies the magnetic field intensity, α is the expansion ratio which predict the influence of expansion or contraction on fluid transport and Pr is the Prandtl no which determines the effect of momentum diffusivity against thermal diffusivity on nanofluid flow.

Relevant constant parameters of the nanofluid are stated as:(24) $A_{3} = \frac{{(ρ C_{p})}_{nf}}{{(ρ C_{p})}_{f}} A_{4} = \frac{k_{nf}}{k_{f}}$

With appropriate boundary condition expressed as(25) $q_{m} (0) = 1, q_{m} (1) = 0$

Since single value for heat transfer coefficient cannot be obtained along the heated wall if wall temperature follows polynomial variation. Unless single term is used to express temperature along heated surface.(26) $T_{w} = T_{0} + C_{m} {(X / a)}^{m} q_{m} (0)$

Important characteristics of flow, heat and mass transfer having practical relevance are reduced to skin friction and Nusselt number which are defined as:(27) $C_{fr} = f^{″} (1) where C_{fr} = \frac{H^{2} \sqrt{1 - α t {Re}_{r}}}{r^{2}} C_{f}$ (28) $Nur = - A_{4} q_{m}^{'} (0) where - \frac{k_{nf} \partial T}{k_{f} \partial η} / (T_{w} - T_{0})$

where ${Re}_{r} = \frac{r α H}{2 v}$ is the local squeeze Reynolds number

2.1. Principles of homotopy perturbation method

The following equation is considered in explaining the fundamentals of the homotopy perturbation method Kargar and Akbazade [9]:(29) $A (u) - f (r) = 0 r \in Ω$

Utilizing the boundary condition(30) $B (u, \frac{\partial u}{\partial η}) = 0 r \in Γ$

A is the general differential operator, B is the boundary operator, f(r) is the analytical function and Γ is the boundary domain of Ω. Separating A into two components of linear and nonlinear terms L and N respectively. The Eq. (29) is reconstructed as(31) $L (u) + N (u) - f (r) = 0 r \in Ω$

Homotopy perturbation structure takes the form(32) $H (v, p) = (1 - p) [L (v) - L (u_{0})] + P [A (v) - f (r)] = 0$

where(33) $v (r, p) : Ω x [0, 1] - R$

P ∊ (0, 1) is the embedding parameter and U0 is taken as the initial term that satisfies boundary condition. The power series of Eq. (32) can be expressed as:(34) $v = v_{0} + {Pv}_{1} + P^{2} v_{2} + P^{3} v_{3} + \dots$

Most appropriate solution for the problem takes the form(35) $u = \lim_{p \to 1} (v_{0} + {Pv}_{1} + P^{2} v_{2} + P^{3} v_{3} + \dots)$

1. Introduction

2. Model development and analytical solution

2.1. Principles of homotopy perturbation method

2.2. Application of the homotopy perturbation method