Numerical Study of Heat Transfer Effects on Non-Newtonian Nanofluid Flow Between Two Parallel Plates in the Presence of Darcy Brinkman Forchheimer

Abstract

Due to its vast industrial applications and biological context, the investigation of the inconsistent heat and mass transfer that drives the flow of squeezing viscous nanofluids between two plates is a fascinating topic. In this study, we investigated the heat transfer analysis of unsteady viscous nanofluid between two parallel plates. The partial differential equations illustrating the flow model are converted to nonlinear ordinary differential equations by suggesting similarity transformations. The resulting dimensionless and nonlinear ODEs of temperature functions and velocity are solved using the well know numerical technique shooting method by transforming the problem into initial value problem from boundary value problem. The results found are consistent with this numerical solution. Graphically explore the impacts of different parameters on temperature profiles and velocity. The results are compared with the results solved by HPM. The results concurred with this numerical solution. These findings are considered much be useful in the application of polymer processing, power transmission, compression, temporary loading of mechanical parts, food processing, cooling water, gravity machinery, modeling of plastic transport in vivo, chemical processing instruments, and demolition due to freezing.

1 Introduction

Nanofluids are mainly found in industrial and technological applications, particularly in melts of polymer, paints, asphalts and glues, biological solutions, etc. It has been observed that Nanofluids have the potential to significantly enhance the rate of transfer of heat in various areas. The term Nanofluids is a liquid containing an array of nanometer-sized (less than 100 nm in diameter) particles mixed in a base liquid. Nanoparticles used in Nanofluids are typically metals (Al, Cu) while oxides are includes (${\text{CUO}}$, ${\text{Al}}_{2} {\text{O}}_{3}$, ${\text{SiO}}_{2}$, ${\text{TiO}}_{2}$), carbides (SiC), nitrites (SiN, AlN), and non-metals (graphite, carbon nanotubes). The base liquid is typically a conductive liquid for example ethylene glycol or water. Toluene, oils, biological fluids, other lubricants, and polymer solutions are also base fluids. Nanoparticles occur in nanofluids with volume fractions up to 5%. Conventional heat exchangers are considered to have poor thermal conductivity. In this regard, Nanofluids are better since it has the capability of a high heat transfer rate. In energy supply, these fluids play an important role to establish energy heat transfer equipment. These nano-sized conducting metals are used to increase the thermal conductivity of these fluids. Thus, a proper understanding of these particles is very important to make their efficient utilization. The greatest uses of nanofluids occur in electrochemical cells, microelectronics, and pharmaceutical procedures. Choi and Eastman [1] pioneered the use of the word nanofluid for nanoscale particles with low thermal conductivity.

Industrial applications of nanofluids include polymer processing, power transmission, compression, temporary loading of mechanical parts, food processing, cooling water, gravity machinery, modeling of plastic transport in vivo, chemical processing instruments, and demolition due to freezing. Due to its vast industrial applications and biological context, the investigation of the inconsistent heat and mass transfer that drives the flow of squeezing viscous nanofluids between two plates is a fascinating topic. Stefan [2] was the first to study squish flow under the lubrication approximation. Domairry and Hatami [3] studied the analytical flow of nanofluids between two parallel plates of copper–water-compressed nanofluids. Khan et al. [4] studied the flow of Cu-water (or kerosene) nanofluids between two parallel plates with the influence of viscous fluid and velocity sliding. On the other hand, Sheikholeslami and Ganji [5] performed an analytical study on heat transfer in compressed nanofluid flows between parallel plates with the help of the Homotopy Perturbation Method. They found a direct correlation between the Nusselt number and nanoparticles in percentage. It is reported in their studies that the Nusselt number is directly proportional to the volume fraction of nanoparticles, whereas when the two parallel plates are separated, the Nusselt number is inversely related to the number squeezed.

Most engineering-related problems are solved using numerical methods but are also resolved by using different well known analytical techniques such as Homotopy Perturbation Method, Variational Iteration Method, and Homotopy Analysis Method and to exclude small parameters, some new methods have recently been developed. HPM is a semi-precise method that does not require small parameters [6]. In most cases, this method produces very fast convergence of the solution series. HPM has demonstrated its ability to efficiently solve several nonlinear problems with accuracy, ease, and approximate convergence. By iteration, this leads to a highly accurate solution. It is mainly used in engineering research. HPM was developed to solve nonlinear problems such as the MHD Jeffery-Hamel problem by Moghimi et al. [7]. Further, to analyze the influence of nanofluid flow between parallel plates by Mustafa et al. [8]. A Similar method is used by Siddique et al. [9] to solve nonlinear problems including Newtonian and non-Newtonian fluids. For a detailed understanding of the useful use of the Homotopy Perturbation method in the solution of problems regarding fluid dynamics can be viewed in the available Refs. [10,11,12].

Keeping in view the above discussion, this current work aims to study nanofluid flow and heat transfer between two parallel plates the rough numerical technique. The impact of Darcy Brinkman Forchheimer number on the flow of Nanofluid between two parallel plates of infinite extent. Although the numerical solution of the Darcy Brinkman Forchheimer equation is valid for forced convective fluid through a porous medium, we will find the impact of Darcy Brinkman Forchheimer numbers on the Nanofluid flowing between parallel palates. The classical Darcy's law is a model used to describe fluid flow in porous media, but it becomes inaccurate for high flow rates and larger porosity. Forchheimer and Muskat expanded Darcy's law to account for the effects of inertia and boundary by adding a square velocity term called the Forchheimer term. To the expression of Darcy's velocity, the Darcy-Forchheimer theory has been applied in a number of different areas including mixed convective flow, hydro-magnetic flow, and heat transfer in porous media. Recent studies have also investigated the Darcy-Forchheimer flow in materials with temperature-dependent thermal conductivity.

2 Mathematical Model

In the current study, the configuration of the flow model is explained as the plates are separated by a separation $y = \pm h\left( t \right) = \pm l\sqrt {1 - \alpha t}$ where $l$ is the initial position between the two plates (when time $t = 0$). Also, if α > 0, it means plates are pushed together until t = 1/α, and if α < 0, the plates become far away. To illustrate this problem, consider a Cartesian coordinate system with the x-axis along the axial flow. The y-axis and direction are taken perpendicular to the axial direction. Clearly, Fig. 1 shows the geometrical structure and coordinate system of the problem in question.

Fig. 1

Geometry of present work

This type of fluid flow can be explained with the help of momentum, continuity, vorticity, and energy equation as below:

$$ \frac{\partial u}{{\partial x}} + \frac{\partial v}{{\partial y}} = 0 $$

(1)

$$ \rho_{{n_{f} }} \left( {\frac{\partial u}{{\partial t}} + u\frac{\partial u}{{\partial x}} + v\frac{\partial u}{{\partial y}}} \right) = - \frac{\partial p}{{\partial x}} + \mu_{{n_{f} }} \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right) $$

(2)

$$ \rho_{{n_{f} }} \left( {\frac{\partial v}{{\partial t}} + u\frac{\partial v}{{\partial x}} + v\frac{\partial v}{{\partial y}}} \right) = - \frac{\partial p}{{\partial y}} + \mu_{{n_{f} }} \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right) $$

(3)

$$\begin{aligned} \frac{\partial T}{{\partial t}} + u\frac{\partial T}{{\partial x}} + v\frac{\partial T}{{\partial y}} & = { }\frac{{K_{nf} }}{{\left( {\rho_{cp} } \right)_{nf} }}\left( {\frac{{\partial^{2} T}}{{\partial x^{2} }} + \frac{{\partial^{2} T}}{{\partial y^{2} }}} \right) \\& \quad + \frac{{\mu_{nf} }}{{\left( {\rho_{cp} } \right)}}\left[ {4\left( {\frac{\partial u}{{\partial x}}} \right)^{2} + \left( {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}}} \right)^{2} } \right]\end{aligned} $$

(4)

$$ u\frac{\partial w}{{\partial x}} + v\frac{\partial w}{{\partial y}} = \frac{{\mu_{nf} }}{{\rho_{nf} }}\left( {\frac{{\partial^{2} w}}{{\partial x^{2} }} + \frac{{\partial^{2} w}}{{\partial y^{2} }}} \right) $$

(5)

Equation (5) is known as the vorticity equation (equation of curl of v) is obtained by doing; $\frac{\partial (3)}{{\partial x}} - \frac{\partial (2)}{{\partial y}}$ with $\frac{\partial u}{{\partial x}} - \frac{\partial v}{{\partial y}}$ where $u$ and $v$ are the components of velocities respectively along x and y, $\rho_{{n_{f} }}$ is the “effective density” of Nanofluid, “dynamic effective viscosity” of nanofluid is represented by $\mu_{nf}$, $\left( {\rho c_{p} } \right)_{nf}$ denotes the “heat capacity” of the nanofluid and $K_{nf}$ represents the “nanofluid thermal conductivity” which are written as:

$$ \rho_{nf} = \left( {1 - \phi } \right)\rho_{f} + \phi \rho_{p} $$

(6)

$$ \left( {\rho Cp} \right)_{nf} = \left( {1 - \phi } \right)\left( {\rho Cp} \right)_{f} + \phi \left( {\rho Cp} \right)_{p} $$

(7)

$$ \mu_{nf} = \frac{{\mu_{f} }}{{\left( {1 - \phi } \right)^{2.5} }} $$

(8)

$$ \mu_{nf} = \frac{{K_{s} + 2K_{f} - 2\phi \left( {K_{f} - K_{s} } \right)}}{{K_{s} + 2K_{f} + 2\phi \left( {K_{f} - K_{s} } \right)}}K_{f} $$

(9)

For the time-dependent squeezing flow boundary conditions are given by:

$$ u = 0{ },\;v = 0,\;T = T_{1} \;and\;y = 0 $$

(10)

$$ u = 0,\;v = \frac{dh }{{dt}},\;T = T_{2} \;and\;y = h\left( t \right) $$

(11)

Equation (11) gives that $v = \frac{dh}{{dt}}$ indicates the velocity with which the relative motion of the upper plate with respect to the lower plate is constant at a distance $y = 0$ from the upper plate at $y = h\left( t \right)$. The parameter $v$ is defined as $\left[ {\frac{ - \alpha l}{{2(1 - \alpha t)^{\frac{1}{2}} }}} \right]$.

Whereas Eqs. (4)–(5) combined using the conditions (10) and (11) governing nanofluids squeeze flow are highly nonlinear in nature, this cannot be solved analytically. Therefore, the system of PDE, corresponding from (4) to (5) is reduced to ordinary ODE with help of the similarity transformation.

$$ \eta = \frac{y}{{l(1 - \alpha t)^{\frac{1}{2}} }} = \frac{y}{h\left( t \right)} $$

(12)

$$ u = \frac{\alpha x}{{2(1 - \alpha t)}}f^{\prime}(\eta ) $$

(13)

$$ v = \frac{ - \alpha l}{{2\left( {1 - \alpha t} \right)^{\frac{1}{2}} }}f\left( \eta \right) $$

(14)

$$ \varTheta = \frac{{T - T_{2} }}{{T_{1} - T_{2} }} $$

(15)

Substituting Eq. (12) into Eqs. (5) and (6) yields the reduced governing equations.

Fourth-order nonlinear ODE for the momentum equation.

$$ f^{{\prime} {\prime} {\prime} {\prime} } - S\frac{A_1 }{{A_2 }}\left( {3f^{{\prime} {\prime} } + \eta f^{{\prime} {\prime} {\prime} } + f^{\prime} f^{{\prime} {\prime} } - ff^{{\prime} {\prime} {\prime} } } \right) - Daf^{\prime} - Ff^{{\prime} 2} = 0 $$

(16)

$$ \theta^{{\prime} {\prime} } + SPr\frac{A_4 }{{A_3 }}\left( {f\theta^{\prime} - \eta \theta^{\prime} } \right) + \left( {P_r E_c \frac{A_2 }{{A_4 }}} \right)\left( {f^{{\prime} {\prime} 2} + 4\delta^2 f^{{\prime} 2} } \right) = 0 $$

(17)

The second-order differential equation for the energy equation. Where $A_{1} ,\,A_{2} \;{\text{and}}\;A_{3}$ are dimensionless constants defined as follows:

$$ A_{1} = \frac{{\rho_{nf} }}{{\rho_{f} }} $$

(18)

$$ A_{2} = \frac{{\mu_{nf} }}{{\mu_{f} }} $$

(19)

$$ A_{3} = \frac{{K_{nf} }}{{K_{f} }} $$

(20)

$$ A_{4} = \frac{{\left( {\rho c_{p} } \right)_{nf} }}{{\left( {\rho c_{p} } \right)_{f} }} $$

(21)

Using boundary condition (10) in the terms of similarity transformation (11) becomes:

$$ f^{\prime} \left( 0 \right) = 0,\;f\left( 0 \right) = 0,\;\theta = 1\;at\quad \eta = 0 $$

(22)

$$ f^{\prime} \left( 1 \right) = 0,\;f\left( 1 \right) = 1,\;\theta = 0\;at\quad \eta = 1 $$

(23)

where $S = \frac{{\alpha l^{2} }}{{2\upsilon_{f} }}$ is the squeeze number, $P_{r} = \frac{{\left( {\rho Cp} \right)_{f} }}{{K_{f} }}\upsilon_{f}$ is the Prandtl number, $E_{c} = \frac{{\alpha^{2} x^{2} }}{{\left( {T_{1} - T_{2} } \right)_{{c_{{p_{f} }} }} }}$, Darcy-Brinkman number is $Da = \frac{\mu d}{k}$ and Forchheimer parameter is $F = \frac{{C_{f} \rho u_{r} }}{{k^{\frac{1}{2}} }}d^{2}$.

The other required physical quantities under consideration are the Nusselt number $N_{\upsilon }$, and skin-friction coefficient $C_{f}$, are given by:

$$ c_{f} = \frac{{\tau_{w} }}{{\rho_{nf} \upsilon_{w} }} $$

(24)

$$ N_{\upsilon } = \frac{{lq_{w} }}{{k_{f} \left( {T_{1} - T_{2} } \right)}} $$

(25)

where

$$ \tau_{w} = \mu_{nf} \left( {\frac{\partial u}{{\partial y}}} \right) $$

(26)

$$ q_{w} = \left( {k_{nf} \frac{\partial T}{{\partial y}}} \right)_{y = 0}^{y = 0} $$

(27)

Using (7) and (11) in (10), we get:

$$ C_f^{*} = \frac{x^2 }{{l^2 }}\left( {1 - \alpha t} \right)R_{ex} C_f = \frac{{f^{{\prime} {\prime} } 0)}}{{\left( {1 - \phi } \right)^{2.5} A_1 }} $$

(28)

$$ N_{ux}^{*} = \sqrt {1 - \alpha t} N_{ux} = - A_{3} \Theta {^{\prime}}(0) $$

(29)

where $R_{ex} = \frac{{\alpha l^{5} }}{{2x^{3} (1 - \alpha t)^{\frac{1}{2}} \upsilon_{f} }}$ is the local Reynolds number.

3 Solution of the Problem

To convert the boundary value problem into initial value problems the best technique is the shooting method in numerical analysis. The goal is to solve the initial value problem by utilizing the given various initial conditions until an accurate solution is found which must be fulfil the boundary conditions of the problem. In layman’s terms, we “shoot” out trajectories from one boundary in different directions until it finds the trajectories that “shoot” into other boundary conditions.

In numerical analysis, Runge–Kutta (R K) method is a series of implicit and explicit iterative methods which together Euler method is used in time to make a discrete form of approximate solutions to simultaneous nonlinear equations.

Applying the Runge–Kutta method. Suppose the mathematical problem:

$$ \frac{dl}{{dt}} = f\left( {t,l} \right),\quad l\left( {t_{0} } \right) = l_{0} $$

Here $y$ is a function of time $t$ which is to be determined, it may be scalar or vector. To find the approximate solution of l using $\frac{dl}{{dt}}$. The initial time is represented by $t_{0}$ who’s corresponds to the $value\;is\;l_{0}$. At the initial condition $(t_{0} , l_{0} )$ the function $f$ is given.

Now we select a step-size of $h > 0$ and define it as:

$$ l_{n + 1} = l_{n} + \frac{1}{6}\left( {k_{1} + 2k_{2} + 2k_{3} + k_{4} } \right)h $$

$$ t_{n + 1} = t_{n} + h $$

For $n = 0,1,2,3,4, \ldots .$

$$ \begin{array}{*{20}c} {k_{1} = f\left( {t_{n} ,l_{n} } \right)} \\ {k_{2} = f\left( {t_{n} + \frac{h}{2}, l_{n} + h\frac{{k_{1} }}{2}} \right)} \\ {k_{3} = f \left( {t_{n} + \frac{h}{2}, l_{n} + h\frac{{k_{2} }}{2}} \right)} \\ {k_{4} = f \left( {t_{n} + h, l_{n} + hk_{3} } \right)} \\ \end{array} $$

The approximate value of $y\left( {t_{n + 1} } \right)$ is $y_{n + 1} { }$ in RK4 and the second term $\left( {l_{n + 1} } \right)$ is calculated using the first value of $(l_{n} )$ plus the weighted mean of four small increments where each small increment represents the product of interval size “h”. Finally, an estimated slope given by function $f$ on the right-hand side of the differential equation.

From Eqs. (16) and (17), it can be obtained the following systems of equations.

$$ f = f(1) $$

(30)

$$ f^{\prime} = f(2) $$

(31)

$$ f^{{\prime} {\prime} } = f(3) $$

(32)

$$ f^{{\prime} {\prime} {\prime} } = f(4) $$

(33)

$$ f^{{\prime} {\prime} {\prime} {\prime} } = f\left( 5 \right) = \frac{SA_1 }{{A_2 }}\left( {3f\left( 3 \right) + \eta f\left( 4 \right) - f\left( 2 \right)f\left( 3 \right) - f(1)f(4)} \right) $$

(34)

$$ \theta^{\prime} = f(6) $$

(35)

$$ \theta^{{^{\prime\prime}}} = f\left( 7 \right) = - \left( {SPr\frac{{A_{4} }}{{A_{3} }}\left( {f\left( 1 \right)f\left( 6 \right) - \eta f\left( 6 \right)} \right) + \left( {PrEc\frac{{A_{2} }}{{A_{4} }}} \right)\left( {f\left( 3 \right)^{2} + 4\delta^{2} f\left( 2 \right)^{2} } \right)} \right) $$

(36)

Now Eqs. (30)–(36) are solved by MATLAB software. The obtained graphical results of velocity and temperature have been discussed in the next section.

4 Results and Discussion

The unsteady squeezed nanofluid flow between the two parallel plates is solved by utilizing the numerical technique. To support the present numerical solution, we compared our results with results given by using HPM. They are in an excellent agreement as they have been demonstrated in Table 1. Further from the current study, the temperature distribution graphs are same response as obtained by [8] for the various parameters.

Table 1 Comparison the present results with previously published results using HP

The results are in outstanding agreement as they have been shown in Table 1. The graphical results also illustrated for various pertinent parameters. Figure 1 demonstrates that the axial velocity profile is inversely related to the Darcy number, i.e., by increasing the Darcy number velocity profile decreases. Figure 2 depicts that by increasing Darcy’s number the redial velocity behaves different way. Initially, it reduces however from the mid of the plates it is going larger for various values of Da. Figure 3 displayed the behavior of the Forchheimer number and axial velocity profile. It can be seen from the figure that for larger values of F the radial velocity decreases slightly. While Fig. 4, shows that by increasing the Forchheimer number the radial velocity is increasing. Figures 5 and 6 produces graphs of the temperature profile for different values of the Prandtl and Eckert number. It can see that the temperature description improves when the Prandtl number increases in the flow regime. The change in the temperature description is mostly dependent on a large decrease in thermal diffusivity with larger values of Pr and a decrease in the thickness of the thermal boundary layer. This increases the thermal profile of the flow regime. Further, Fig. 7 shows the impact of the Eckert number on the temperature profile. From Fig. 7 it can be viewed that the temperature description improves as the value of the Eckert number increases. Whereas this increase in temperature description is expected as it directly affects the heat loss process and increases the temperature field in the flow domain.

Fig. 2

The effects of Darcy-Brinkman number on $\user2{ f}({\varvec{\eta}})$

Fig. 3

The graph of Darcy-Brinkman number on $\user2{ f^{\prime}}({\varvec{\eta}})$

Fig. 4

The graph of Forchheimer parameter on $\user2{ f}({\varvec{\eta}})$

Fig. 5

The effects of Forchheimer parameter on $\user2{ f^{\prime}}({\varvec{\eta}})$

Fig. 6

The influence of Prandtl number on ${\varvec{\theta}}({\varvec{\eta}})$

Fig. 7

The effect of Eckert number on ${\varvec{\theta}}({\varvec{\eta}})$

From Table 1 we concluded that for promoting values of S, the coefficient of skin friction is upgrading while local Nusselt number is decreased at static values of Eckert and Prandtl numbers, i.e. $\left( {{\text{Ec}} = 0.01,{\text{Pr}} = 6.2} \right)$.

5 Conclusions

This paper investigates the effects of heat transfer in an unsteady flow of viscous and incompressible nanofluid in two dimensions between infinitely expanding parallel plates with collisions caused by viscous dissipation. The associated nonlinear PDEs are converted into a set of ODEs and are solved analytically by using the RK4. From the discussion above, we got the following notable results are:

• The temperature of Cu-water Nano fluids increases by increasing the Darcy-Brinkmann number Da and the Forchheimer parameter.

• The temperature of Cu-water Nano fluids increases when “Eckert number” Ec increases.

• By increasing Prandtl number Pr the temperature of Cu-water Nano fluids increases.

• The coefficient of Nusselt number and friction are related to the pinch number S.