Introduction

The cooling of miniature equipment in the micro electro mechanical and nano-electromechanical industries has increased the need of understanding the fluid flow and heat transfer in the micro- and nanogeometrics. The behavior of fluid flow and heat transfer in the miniature scales and by using nanofluid, due to the improvement in heat transfer mechanisms in nano- and microdimensions, comparing to the custom scales, is far different. Numerous numerical and empirical studies have been done for investigating the flow and heat transfer of custom fluids and nanofluid in the microchannels whose main purpose is increasing the heat transfer [1,2,3,4,5]. The investigation of heat transfer enhancement in different industrial and experimental fluids by using novel methods has been expanded as the study fields among the adherents of this issue [6]. The microchannel heat sink as an applicable miniature equipment has high importance in heat transfer of electronic industries. This equipment has been suggested by Tukerman and Pease [7] for cooling the electronic chips. In recent decades, this equipment has been investigated and optimized by researchers in different structures and arrangements for enhancing the cooling of electronic chips [8,9,10]. Kulkami et al. [11] numerically studied the multi-purposed optimization of double-layer microchannel heat sink with the cross-figured inlet section. Their results evidenced that the microchannel with narrower design has lower thermal resistance and higher pumping power and the pumping power by increasing the heat flux reduces significantly. Husain and Kim [12, 13] optimized the indented microchannel heat sink and indicated that the thermal resistance of microchannel heat sink by optimization reduces considerably. Xie et al. [14] studied the efficiency of double-layer microchannel heat sink with the wavy wall in the states of parallel and contrary flows. They investigated the effects of wavy wall limitation and the ratio of mass flow on the thermal resistance and pressure drop parameters. Seyf and Nikaaein [15] by using Al2O3, zinc and Cu nanoparticles in the ethylene glycol/water fluid numerically studied the effects of nanoparticles dimensions and Brownian motion of nanoparticles on the thermal performance of a rectangular microchannel heat sink. Their results showed that the amount of nanofluid conductivity without considering the Brownian motion reduces almost to 6.5%. Wu et al. [16] numerically studied the thermal resistance, pumping power and thermal distribution on the wall surface of double-layer microchannel heat sink (DL-MCHS). In their research, different parameters of microchannel dimensions and different flow conditions have been studied. The results of his study showed that the improvement in total efficiency of double-layer microchannel heat sink depends on the pumping power. Chen and Chung [17] used the water/Cu nanofluid. In their investigation, the absorbed energy by the nanofluid was more than the absorbed energy by water, and it has been observed that by enhancing volume fraction of nanoparticles, the high-temperature differences accomplish between the inlet and outlet sections of microchannel heat sink in a low flow rate. Jang and Choi [18] by using nanofluid numerically studied the cooling performance of a microchannel heat sink. They reported that the nanofluid causes the reduction in thermal resistance and dimensionless temperature difference in microchannel heated wall and cooling fluid. Sui et al. [19] numerically investigated the fluid flow in the wavy microchannels. Their numerical results indicated that with the uniform cross section, the thermal performance of wavy microchannel is higher than the rectangular flat one. Ho et al. [20] studied the forced convection cooling performance of a Copper microchannel heat sink with water/Al2O3 nanofluid as the cooling fluid. Their results showed that the heat sink cooled by nanofluid, comparing to the heat sink cooled by water, has more average heat transfer coefficient. Till now, numerous researches about the heat transfer in the microchannels and nanofluid have been presented, and sometimes, the slip velocity conditions, the effects of magnetic field and the forced heat transfer under the influence of constant temperature or constant heat flux have been investigated disparately [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. Nikkhah et al. [36] numerically studied the water nanofluid/functional multi-walled carbon nanotubes in a two-dimensional microchannel with slip and no-slip boundary conditions. They concluded that the augment of solid nanoparticles weight fraction and slip velocity coefficient cause the increase in Nusselt number, and in higher Reynolds numbers, this enhancement is more considerable. In their research, the computational fluid dynamics and laminar heat transfer of kerosene nanofluid/multi-walled carbon nanotubes in the double-layer microchannel heat sink are simulated in the two-dimensional domain. By considering the effect of slip boundary condition on the outcome results of numerical simulation, in this study, the slip velocity boundary condition on the solid walls is used. The results of this research are presented for different volume fractions of nanoparticles, slip velocity coefficients and different ranges of Reynolds numbers. The main purpose of this study is investigating the behavior of temperature domain and hydrodynamic of laminar flow of nanofluid in the two-dimensional double-layer microchannel.

Problem statement

In the present study, the laminar flow of kerosene nanofluid/multi-walled carbon nano tubes in volume fractions of 0, 4 and 8% of nanoparticles is investigated. Figure 1 indicates the studied geometrics of this paper. In this research, the material of microchannel is silicon. In Fig. 1, the bottom layer of microchannel is L = 3 mm and the height is H = 50 μm. The top layer of microchannel with the length of L 2 is equal to L 2 = 1.5 mm, and by placing on the bottom layer at the interface area, the heat transfers with it and in this region, the amount of heat generation is constant and is equal to 100 kw/m3. In each layer of microchannel, the silicon material with the thickness of t = 12.5 µm has surrounded the layers. The external areas of top layer with the length of L 2 are insulated, and the bottom area of microchannel, on the external wall with the length of L, is under the influence of sinusoidal flux with the equation of \(q^{{\prime \prime }} \left( X \right) = 2q_{0}^{{\prime \prime }} + q_{0}^{{\prime \prime }} \sin \left( {\frac{\pi X}{4}} \right)\) in which the amount is calculated from the equation of (q 0″). With the definition of dimensionless slip velocity coefficient as (B = β/H), the ratio of slip velocity coefficient to the height of microchannel, in this research, the numerical simulation is done for the dimensionless slip velocity coefficients (B = β/H) of 0.001, 0.01 and 0.1 and Reynolds numbers of 1, 10 and 100. The inlet fluid at the top and bottom layers enters with the temperature of 301 K as shown in Fig. 1. All of the internal walls which are in contact with fluid have the slip velocity boundary condition. The used nanofluid properties of this simulation and the material of microchannel wall are described, respectively, in Table 1.

Fig. 1
figure 1

The studied schematics of this research

Full size image
Table 1 The thermophysical properties of base fluid and nanoparticle of multi-walled carbon nanotubes and silicon [37, 38]
Full size table

In this simulation, the fluid flow and heat transfer are considered as laminar and fully developed. The nanofluid properties are considered as constant and independent from the temperature. The solid–liquid suspension in less densities is modeled as single-phased, and on the channel walls, the oscillating heat flux is applied. The slip boundary condition is used on the microchannel. The numerical simulation domain is two dimensional.

Governing equations

The dimensionless governing equations on the simulation domain are defined as follows [39, 40]:

Continuity equation:

$$\frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} = 0$$
(1)

Momentum equation:

$$U\frac{\partial U}{\partial X} + V\frac{\partial U}{\partial Y} = - \frac{\partial P}{\partial X} + \frac{{\mu_{\text{nf}} }}{{\rho_{\text{nf}} \nu_{\text{f}} }}\frac{1}{Re}\left( {\frac{{\partial^{2} U}}{{\partial X^{2} }} + \frac{{\partial^{2} U}}{{\partial Y^{2} }}} \right)$$
(2)
$$U\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y} = - \frac{\partial P}{\partial Y} + \frac{{\mu_{\text{nf}} }}{{\rho_{\text{nf}} \nu_{\text{f}} }}\frac{1}{Re}\left( {\frac{{\partial^{2} V}}{{\partial X^{2} }} + \frac{{\partial^{2} V}}{{\partial Y^{2} }}} \right)$$
(3)

Energy equation:

$$U\frac{\partial \theta }{\partial X} + V\frac{\partial \theta }{\partial Y} = \frac{{\mu_{\text{nf}} }}{{\alpha_{\text{f}} }}\frac{1}{Re\;Pr}\left( {\frac{{\partial^{2} \theta }}{{\partial X^{2} }} + \frac{{\partial^{2} \theta }}{{\partial Y^{2} }}} \right)$$
(4)

For non-dimensioning Eqs. (1)–(4), following parameters are used [36]:

$$\begin{array}{*{20}l} {X = \frac{x}{H}} \hfill & {Y = \frac{y}{H}} \hfill & {V = \frac{v}{{u_{\text{c}} }}} \hfill \\ {\theta = \frac{{T - T_{\text{c}} }}{\Delta T}} \hfill & {U = \frac{\upsilon }{{u_{\text{c}} }}} \hfill & {B = \frac{\beta }{H}} \hfill \\ {\Delta T = \frac{{q_{0}^{{\prime \prime }} H}}{{k_{\text{f}} }}} \hfill & {Pr = \frac{{\upsilon_{\text{f}} }}{{\alpha_{\text{f}} }}} \hfill & {P = \frac{{\bar{P}}}{{\rho_{\text{nf}} u_{\text{c}}^{2} }}} \hfill \\ \end{array}$$
(5)

Another parameter for investigating the microchannel performance is the friction coefficient which is calculated from the following equation [41]:

$$C_{\text{f}} = \frac{{2 \times \tau_{\text{w}} }}{{\rho u_{\text{in}}^{2} }}$$
(6)

The average Nusselt number can be obtained as follows [42, 43]:

$$Nu_{\text{x}} = \frac{h \times H}{{k_{\text{f}} }} \to Nu_{\text{ave}} = \frac{1}{L}\int_{0}^{L} {Nu_{\text{x}} \left( X \right){\text{d}}X}$$
(7)

The amounts of thermal resistance [44, 45] of bottom wall of microchannel and pressure drop are calculated from the following equation:

$$R = \frac{{T_{\hbox{max} } - T_{\hbox{min} } }}{{q_{0}^{{\prime \prime }} \times A}} = \frac{{T_{\hbox{max} } - T_{in} }}{{q_{0}^{{\prime \prime }} \times A}} \to A = W \times L \to R \times W = \frac{{T_{\hbox{max} } - T_{\hbox{min} } }}{{q_{0}^{{\prime \prime }} \times L}}$$
(8)
$$\Delta P = P_{\text{in}} - P_{\text{out}}$$
(9)

In Eq. (9), T max, T min, A and q 0″ are, respectively, the maximum temperature of bottom wall, the minimum temperature (the temperature of inlet fluid), cross section and the applied heat flux to the AB wall.

The governing boundary conditions on the problem-solving

The hydrodynamic and thermal boundary conditions used in this problem are as follows:

$$\begin{aligned} U = 1,\quad V = 0\quad {\text{and}}\quad \theta = 0\quad {\text{for}}\quad X = 0\quad {\text{and}}\quad 0.25 \le Y \le 1.25\quad {\text{and}}\quad X = 60,\;1.75 \le Y \le 2.75 \hfill \\ V = 0\quad {\text{and}}\quad \frac{\partial \theta }{\partial X} = \frac{\partial U}{\partial X}\quad {\text{for}}\quad X = 60\quad {\text{and}}\quad 0.25 \le Y \le 1.25\quad {\text{and}}\quad X = 30,\;1.75 \le Y \le 2.75 \hfill \\ V = 0,\quad U = 0\quad {\text{and}}\quad \frac{\partial \theta }{\partial Y} = 2q_{0}^{\prime \prime } + q_{0}^{\prime \prime } \sin \left( {\frac{\pi X}{4}} \right)\quad {\text{for}}\quad Y = 0\quad {\text{and}}\quad 0 \le X \le 60 \hfill \\ V = 0,\quad U_{\text{s}} = B\frac{\partial \theta }{\partial Y}\quad {\text{and}}\quad k_{\text{nf}} \frac{\partial \theta }{\partial Y} = k_{\text{s}} \frac{\partial \theta }{\partial Y}\quad {\text{for}}\quad Y = 0.25\quad {\text{and}}\quad 0 \le X \le 60 \hfill \\ V = 0,\quad U = 0\quad {\text{and}}\quad \frac{\partial \theta }{\partial Y} = 0\quad {\text{for}}\quad Y = 1.5\quad {\text{and}}\quad 0 \le X \le 30\quad {\text{and}}\quad Y = 3\quad {\text{and}}\quad 0 \le X \le 60 \hfill \\ V = 0,\quad U_{\text{s}} = B\frac{\partial U}{\partial Y}\quad {\text{and}}\quad k_{\text{nf}} \frac{\partial \theta }{\partial Y} = k_{\text{s}} \frac{\partial \theta }{\partial Y}\quad {\text{for}}\quad Y = 1.25\quad {\text{and}}\quad 0 \le X \le 60 \hfill \\ V = 0,\quad - U_{\text{s}} = B\frac{\partial U}{\partial Y}\quad {\text{and}}\quad k_{\text{nf}} \frac{\partial \theta }{\partial Y} = k_{\text{s}} \frac{\partial \theta }{\partial Y}\quad {\text{for}}\quad Y = 1.75\quad {\text{and}}\quad 30 \le X \le 60 \hfill \\ V = 0,\quad - U_{\text{s}} = B\frac{\partial U}{\partial Y}\quad {\text{and}}\quad k_{\text{nf}} \frac{\partial \theta }{\partial Y} = k_{\text{s}} \frac{\partial \theta }{\partial Y}\quad {\text{for}}\quad Y = 2.75\quad {\text{and}}\quad 30 \le X \le 60 \hfill \\ \end{aligned}$$
(10)

The mesh study and numerical solving procedure

In order to ensure the results independency of this research, the rectangular organized grids have changed from the number of 30,000 to 100,000. The studied parameters in the validation of present investigation are including Nusselt number along the AB wall and the amount of pressure drop. The changes in these two parameters are investigated in Reynolds numbers of 10 and 100 and volume fraction of 8% of nanoparticles in the slip velocity coefficient of 0.01. According to Table 2, by choosing grid number of 100,000, comparing to other grid numbers, more accurate results can be obtained. However, the grid number of 63,000, compared to the grid number of 100,000, has acceptable error and less demanded time for solving the numerical domain; therefore, in this numerical simulation, the grid number of 63,000 has been used. In this study, in order to enhance the solving accuracy, to couple velocity and pressure, SIMPLEC algorithm [46, 47] has been used, and the maximum loss for results convergence of this simulation has been chosen 10−6 [48,49,50].

Table 2 The changes in studied grid numbers in the present study
Full size table

Results and discussion

Validation

The results of the present study have been validated with the numerical study of Nikkhah et al. [36] in Reynolds number of 100 for the dimensionless temperature parameter at central section of flow. Nikkhah et al. [36] numerically investigated the laminar flow and heat transfer of water nanofluid/functional carbon nanotubes in a rectangular microchannel with the ratio of length to the height of channel equal to 32. Their investigation has been done in Reynolds numbers of 1–100 for volume fractions of 0–0.25% of nanoparticles. According to Fig. 2 and proper coincidence of the results of the present research with the study of Nikkhah et al. [36], it can be said that the solving procedure and the applied boundary conditions are accurate.

Fig. 2
figure 2

The validation with numerical study of Nikkhah et al. [36]

Full size image

Figures 3–5 demonstrate the dimensionless temperature contours in Reynolds number of 1, volume fractions of 0–8% of nanoparticles and different dimensionless slip coefficients at the dimensionless length of 1.2. By entering the fluid to the microchannel, by considering the maximum temperature of surface and fluid, the most changes in dimensionless temperature arise in this region. By more processing of fluid, due to the contact with hot surface on the direction of fluid motion in the microchannel, the changes in dimensionless temperature enhance in a way that this augment of dimensionless temperature causes the reduction in heat transfer and enhancement of hot areas in the microchannel. Due to the generation of uniform heat between the top and bottom layers of microchannel, this factor influences the top and bottom areas of microchannel. According to the existence of hot areas in the top layer of microchannel and increase in the dimensionless temperature in this area, the heat transfer reduces in these regions. The amount of dimensionless temperature changes and the existence of hot area at the bottom layers are completely obvious from the middle area of microchannel to the lateral in Reynolds number of 1 and volume fraction of 0%. In all figures, by enhancing the dimensionless slip velocity coefficient and volume fraction, the elimination of hot areas at the top and bottom layers has been approximately solved.

Fig. 3
figure 3

The changes in dimensionless temperature in Reynolds number of 1 and different dimensionless slip coefficients in volume fraction of 0%

Full size image
Fig. 4
figure 4

The changes in dimensionless temperature in Reynolds number of 1 and different dimensionless slip coefficients in volume fraction of 4%

Full size image
Fig. 5
figure 5

The changes in dimensionless temperature in Reynolds number of 1 and different dimensionless slip coefficients in volume fraction of 8%

Full size image

Figure 6 illustrates the average friction coefficient for the bottom and top layers, respectively, along the walls of (AB) and (CD). This study has been done for dimensionless slip velocity coefficient of 0.001–0.1 in volume factions of 0–8% of nanoparticles in Reynolds numbers of 1, 10 and 100. By enhancing Reynolds number, the contact of surface and fluid reduces which causes the reduction in friction coefficient. By increasing volume fraction of nanoparticles, due to the enhancement of viscosity and density of cooling fluid, the average friction coefficient increases. By augmenting the dimensionless slip velocity coefficient, due to the movement of fluid with less resistance and depreciation on the surface, the friction coefficient decreases significantly. In each of the studied Reynolds numbers, the amount of average friction coefficient in Reynolds number of 1 is remarkable. In Reynolds numbers of 10 and 100, this factor decreases considerably. In the investigation of the amount of average friction coefficient at the top and bottom layers of microchannel, it can be said that comparing the top layer of microchannel to the bottom layer in volume fraction and Reynolds number and dimensionless slip velocity in the same conditions, the amount of average friction coefficient is more. This behavior is due to the enhancement of velocity gradients at the top layer comparing to the bottom layer.

Fig. 6
figure 6

The amount of average friction coefficient for the bottom and top layers, respectively, along the walls of (AB) and (CD)

Full size image

Figure 7 shows the local Nusselt number for the bottom layer (a) and top layer (b), respectively, along the walls of (AB) and (CD). This study has been investigated for the slip velocity coefficient of 0.1, Reynolds numbers of 1–100 at the bottom layer and Reynolds numbers of 1–10 at the top layer and volume fractions of 0–8%. By increasing Reynolds number, the local Nusselt number enhances. According to the increase in thermal conductivity coefficient of nanofluid, by enhancing volume fraction and the heat transfer, Nusselt number enhances. The other reason of this augmentation of Nusselt number in higher volume fractions is due to the acceleration of energy exchanging process in fluid because of the random movement of nanoparticles inside it. This process causes more uniform temperature distribution inside the nanofluid, and consequently, the rate of heat transfer between the wall and nanofluid increases. Because the fluid is only conductor heat flux between the upper- and down-layer microchannels, and since heat flux penetrates into all layers of fluid and is mixed during the movement of the fluid, therefore, the oscillatory shape of the heat flux is gone and the heat flux reaches the top layer uniformly. Hence, the shape of the Nusselt number diagrams in the upper layer does not depend on the oscillatory shape of the heat flux.

Fig. 7
figure 7

The amounts of average Nusselt number for the bottom and top layers, respectively, along the walls of (AB) and (CD)

Full size image

Figure 8 demonstrates the changes in static pressure for the bottom layer (a) and top layer (b) along the central line of flow in the dimensionless slip coefficient of 0.01. According to the figures, the amounts of pressure changing continue from the inlet section of microchannel to the outlet section. In the investigation of this parameter, it can be observed that the amounts of pressure changing at the top layer are less than at the bottom layer which is due to the reduction in the length of top channel comparing to the bottom channel. By increasing volume fraction of nanoparticles, due to the enhancement of particles number and density and viscosity of cooling fluid, the amount of static pressure on the direction of movement decreases more than the base fluid. By enhancing the slip velocity coefficient, the penetration of the effect of solid wall in the microchannel boundaries to the central core of flow reduces; therefore, by increasing the slip velocity coefficient, the amount of pressure drop decreases. According to Fig. 9, the minimum amount of pressure drop for the base fluid in slip velocity coefficient of 0.1 can be observed in each Reynolds number.

Fig. 8
figure 8

The changes in static pressure along the microchannel center in top and bottom layers

Full size image
Fig. 9
figure 9

The pressure drop for a bottom layer and b the top layer

Full size image

Figure 10 indicates the changes in average maximum temperature at the bottom layer of microchannel along the wall of (AB) in volume fractions of 0–8% for Reynolds numbers of 1, 10 and 100 for different slip velocity coefficients. The reduction in maximum temperature of surface is tantamount with better thermal removal from the hot surface. Among the studied states, the minimum amount of this parameter accomplishes in the highest fluid velocity and volume fraction. The effect of dimensionless slip velocity coefficient on the reduction in maximum temperature is considerable, in a way that in the dimensionless slip velocity coefficient of 0.1, comparing to the states of 0.001 and 0.01 has remarkable reduction. According to these figures, by increasing Reynolds number and volume fraction of nanoparticles and due to better mixture of fluid flow and heat transfer enhancement and increase in convection heat transfer coefficient, the maximum temperature factor reduces.

Fig. 10
figure 10

The changes in maximum temperature at the bottom layer of microchannel along the wall of (AB)

Full size image

Figure 11 shows the changes in minimum temperature at the bottom layer of microchannel along the wall of (AB). According to the figures, the minimum temperature of surface accomplishes in a state in which the fluid has the highest heat transfer. The changes in this factor are the same as the changes in maximum temperature, and the reduction in minimum temperature entails the heat transfer enhancement.

Fig. 11
figure 11

The changes in minimum temperature at the bottom layer of microchannel along the wall of (AB)

Full size image

Figure 12 illustrates the changes in thermal resistance at the bottom layer of microchannel along the wall of (AB) in Reynolds numbers of 10 and 100 in different volume fractions and the dimensionless slip velocity coefficient. This factor investigates the amount of maximum and minimum differences (the temperature of inlet fluid) of bottom layer of microchannel wall. The enhancement of dimensionless slip velocity coefficient causes the reduction in thermal resistance which entails the increase in heat transfer. In the studied Reynolds numbers of this research and in all considered factors, the increase in Reynolds number and volume fraction of nanoparticles causes the reduction in thermal resistance. With the increase in the fluid velocity (Reynolds number), the amount of heat transfer increases; due to an increase in the heat transfer coefficient, the maximum of surface temperature will reduce. By increasing the slip speed, the fluid moves with less deterrence on solid surfaces. As a result, the temperature of the hot surfaces is better transferred and the thermal resistance is reduced.

Fig. 12
figure 12

The changes in thermal resistance at the bottom layer of microchannel along the wall of (AB)

Full size image

Figure 13 shows the changes in average Nusselt number at the bottom layer of microchannel along the wall of (AB). According to the amount of heat transfer in the investigated microchannel under the influence of velocity and volume fraction and different dimensionless slip velocity coefficients, it can be seen that the maximum amount of average Nusselt number arises in Reynolds number of 100. In the studied Reynolds numbers, the amount of heat transfer increases between 1.5 and 2.5 times.

Fig. 13
figure 13

The changes in average Nusselt number at the bottom layer of microchannel along the wall of (AB)

Full size image

Figure 14 illustrates the changes in dimensionless temperature at the bottom layer of microchannel along the wall of (AB) for the dimensionless slip velocity coefficients of 0.01 and 0.1. This investigation describes the differences in dimensionless temperature in different volume fractions and Reynolds numbers and the effect of slip length on this parameter. The reduction in dimensionless temperature on the hot wall is tantamount with heat transfer enhancement. It can be concluded from Fig. 14, the augment of Reynolds number and volume fraction of nanoparticles at the bottom layer of microchannel, the amount of dimensionless temperature reduces significantly.

Fig. 14
figure 14

The changes in dimensionless temperature at the bottom layer of microchannel along the wall of (AB)

Full size image

Conclusions

In this research, the numerical simulation of laminar heat transfer of kerosene nanofluid/multi-walled carbon nanotubes in the microchannel heat sink by using finite volume method has been investigated. The results evidenced that in different Reynolds numbers, applying oscillating heat flux considerably influences the profile figure of Nusselt number, and this impressionability is obvious in Reynolds numbers of 100 and 10. Also, by enhancing the slip velocity coefficient on the solid surfaces, the amount of minimum temperature of surface decreases significantly which causes remarkable increase in heat transfer. According to the existence of hot area at the top layer of microchannel and increase in dimensionless temperature in this region, the heat transfer of this area reduces. By enhancing the dimensionless slip velocity coefficient, because the fluid moves on the surface with less resistance and depreciation, the friction coefficient decreases remarkably. In each of the studied Reynolds numbers, the amount of average friction coefficients in Reynolds number of 1 is more significant. By increasing Reynolds number and volume fraction of nanoparticles and due to better mixture of fluid flow and heat transfer enhancement and augmentation of convection heat transfer, the maximum temperature factor reduces. The increase in dimensionless slip velocity coefficient entails the reduction in thermal resistance which causes the enhancement of heat transfer amount. Eventually, existence of incomplete upper layer on microchannel, due to the short path of fluid flow and lack of fully development fluid flow, better control of the down-layer temperature and increased heat transfer in microchannel, is created. It is recommended that in order to better distribute the temperature in a single-layer microchannel, the form of two-layer microchannel with incomplete upper layer is used. The extension of this paper for nanofluid according to previous works [51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113] affords engineers a good option for nanoscale and microscale simulation. According to some previous studies, to increase the produced power in some power plants, or to some upgrading, there is an emergency need to increase the heat transfer capacity in existing systems [114,115,116,117,118,119,120]. One of the best solutions for this problem is using nanofluids instead of water in these cooling systems.