Theoretical Study of Convection Heat Transfer and Fluid Dynamics in Microchannels with Arrayed Microgrooves

)e analysis of heat transfer and fluid dynamics in a microchannel is currently one of numerous important research areas. )e present study investigated the heat transfer and fluid dynamics in a microchannel with arrayed microgrooves. )e boundary conditions in the microchannel and microgrooves were the velocity slip condition and temperature jump condition. )e numerical calculations in the present study were based on a new numerical analysis scheme that was used to name modified pointmatching methods. )e methods are applied to calculate the two items coefficients of the series expressions of velocity in the microchannel with microgrooves. )e methods analyzed four items coefficients of the series expressions of temperature in the microchannel with microgrooves. In addition, the fluid velocity, temperature distribution, Knudsen number effect, fluid friction factor in the microchannel, and heat transfer were discussed.


Introduction
e significance of high heat fluxes in micro-electromechanical system (MEMS), microdevice fabrication, bioengineering, and other devices has caused microscale heat transfer to be the subject of numerous recent studies. Research on microchannel flow behavior in microregions has been fundamental for developing this important knowledge and is of value in heat transfer engineering. Microfluid systems typically have characteristic lengths in the order of 1-100 μm. e Knudsen number (Kn), which characterizes the effect of rarefaction, is defined as the ratio of the molecular mean free path in the order of 100 nm. Rarefied gaseous flows in these microgeometries have frequently been observed, i.e., for a microchannel system, the Knudsen number, which is under the range from 10 −3 ≤ Kn ≤ 10 −1 belonging to the velocity-slip and temperature-jump regime. Consequently, a microchannel system should further consider the effects of the slip flow and temperature jump of the surface. e pioneer research and analyses of the laminar heat transfer in a slip-flow regime were performed by Sparrow and Lin [1] and Inman [2] for tubes with constant heat flux and a parallel plate channel or a circular tube with isothermal conditions. Next, Larrode et al. [3] considered the temperature jump condition and found that the effect of the fluid-wall interaction is also important. Natural convection of an enclosed fluid has received considerable attention in recent years due to its wide application in engineering problems. Earlier work on the isothermal parallel-plate channels was reported by Elenbaas [4]. Further investigations have been carried out for different thermal boundary conditions. An excellent review is given in the literature by Gebhart et al. [5]. All of the previous investigations were conducted at the macroscale; therefore, natural convection flow in microscale systems should be studied extensively. Aung [6] describes the influences of rarefaction and fluid-wall interaction on steady and fully-developed natural convection in microchannels.
e results of Aung's study can be applied to microheat exchangers.
Furthermore, Yu and Ameel [7] studied slip flow heat transfer in microchannels and found that the heat transfer increases, decreases, or remains unchanged when compared to nonslip flow conditions. Kavehpour et al. [8] investigated the effects of compressibility and rarefaction in microchannel heat transfer under slip flow and discussed the compressibility and rarefaction under different Reynolds numbers. Vasudeviah and Balamurugan [9] investigated the convective heat transfer in a microchannel with respect to rarefied gas flows and obtained analytical solutions for the mean Nusselt number. Tunc and Bayazitoglu [10] researched the microchannel heat transfer under the slip flow and constant heat flux problem and stated that increasing the rarefaction has a decreasing effect on the heat transfer. Ghodoossi and Egrican [11] studied the heat transfer in a rectangular microchannel for a slip flow regime. In the present paper, the integral transform method was applied to derive the velocity and temperature distributions and find the average Nusselt number. Zhu and Liao [12] studied the laminar forced convective heat transfer of a gas flowing through a microchannel with an axially-constant heat flux and circumferentially-varied wall temperature boundary condition in the slip-flow and the temperature-jump regime. Bahrami et al. [13] investigated the fully-developed laminar flow in smooth channels of arbitrary cross-sections of microchannel using existing analytical solutions for fluid flow to derive the fRe values. Zhang et al. [14] researched the experimental study on liquid flow and heat transfer in rough microchannels.
e results indicate that the surface roughness plays a significant role in the process of laminar flow and heat transfer in microchannels. In microchannels with roughness characteristics, the Poiseuille number of liquid laminar flow relies not only on the cross section shape of the rough microchannels but also on the Reynolds number of liquid flow. Elguennouni et al. [15] investigated the thermal lattice Boltzmann model for nonisothermal gas flow in a two-dimensional microchannel. e thermal Lattice Boltzmann method is used for the simulation of a gas microflow. A two-dimensional heated microchannel flow driven by a constant inlet velocity profile and nonisothermal walls is investigated numerically.
Important numerical and experimental studies with hydrothermal analyses under different boundary conditions and various high-performance obstacles are addressed by Peiravi and Alinejad [16]. Alinejad and Peiravi [17] investigated three-dimensional numerical simulation of a Newtonian droplet impact and break on two square cylinders based on dynamic contact angle of droplet at the spatial interface between two solid-fluid phases. Asshaari et al. [18] investigated nanofluids containing carbon nanotubes with radiation and velocity slip effects, in case of a moving plate containing both single-walled and multiwalled carbon nanotubes, in the presence of thermal radiation and the slip effect. In the recent research, Yuan et al. [19] created the computational simulation of MHD nanofluid which flows past a baffled U-shaped enclosure by using Koo-Kleinstreuer-Li (KKL) method. e U-shaped enclosure, nanofluid, magnetic field, and baffles are all included in the investigation. Mohebbi and Rashidi [20] investigated natural convection of a nanofluid inside of an L-shaped enclosure.
e results of the study presented that fluid flow field and convection heat transfer were significantly influenced by the heat source position. For all aspect ratios, when a heat source is positioned in the lower and beside the left wall inside the cavity, the Nusselt number is obtained.
Although numerous researches about microchannel or nanofluids under different conditions have been discussed, the microchannel heat and flow of arrayed microgrooves with different aspect ratios and cross-sectional areas are also worth discussing, particularly in thermal and flow analysis. e present paper thus investigated the heat and fluid flow of a microchannel with microgrooves, i.e., a microchannel with microgrooves positioned along the X-direction with a flow moving in the Z-direction. e characteristics of the heat transfer and fluid dynamics in the microchannel, e.g., the Knudson number and the boundary conditions of velocity slip and temperature jump, were considered in the present research.

Description of the Physical Model
e present study considered a steady-state fully developed laminar fluid flow in a two-dimensional microchannel. e height of the channel was aH, and it was assumed that the left side and the right side of the microchannel were infinite. e microchannel included parallel grooves arrayed along the X direction. e height of the grooves was bH, and the width of the grooves was 2(1 − c)H. e cross section of the microchannel and the physical model are shown in Figure 1. e direction of the flow was normal in this paper (Z-dimensional). Since the velocity of a fluid is very slow, it was assumed that the inertia force was negligible. e flow is hydraulically and thermal fully developed laminar, and it is incompressible and has constant properties. For such a flow, the pressure gradient along the Z-dimension is a constant. e momentum equation for the velocity of the microfluid could be expressed as Energy equation: where X is the horizontal coordinate axis; Y is the vertical coordinate axis; W is the 2D velocity distribution in the Zdirection; W(X, Y); Z is the direction normal to the paper surface; and P is the pressure. e Knudsen number (Kn) is a dimensionless parameter used in discussions on the rarefaction effect in a microchannel. is physical parameter is defined as Kn ≡ lL, where l is the molecular mean free path, defined as the average distance moved in two collisions of gas molecules, and L is the characteristic length of the microchannel. e stereogram shown in Figure 1 is represented as a 2D diagram.
Considering the factor of geometric symmetry, this study focused on the L-shaped region, represented as slashes in Figure 2. is L-shaped region (W z (X, Y)) was decomposed into two connected rectangular regions using the domain decomposition method, and the two regions were computed simultaneously. e dimensionless parameters were imported, in which w � −W/(H 2 dp/μ dZ) and θ(x, y) � (T(x, y) −T wall /Hq m L/kw). e L-shaped region was decomposed, and the noncoherent boundary conditions of the dimensionless slip velocity and the dimensionless temperature after dimensionless parameter displacement are as shown in the following diagram ( Figure 3), in which H is the width, μ is the viscosity coefficient, q m is the average heat flux, L is the wall surface circumference, k is the thermal conductivity, and w is the dimensionless average flow velocity.
L-shaped region: the governing equation of the left rectangle is expressed as follows: Momentum equation: where W 1 is the velocity of microfluid in the left rectangle. Boundary conditions: where β w λ is the velocity slip coefficient, β w � (2 − F/F), F is the tangential momentum accommodation coefficient, and the Knudsen number is kn � λ/a. For the L-shaped region, the governing equation of the right rectangle is expressed as follows: Momentum equation: where W 2 is the velocity of microfluid in right rectangle. Boundary conditions: Moreover, at the juncture of two rectangles, the two regions have equal velocity and shear force, and the following boundary conditions are obtained:

Mathematical Problems in Engineering
e velocity dimensionless parameters are defined as w � −W/(H 2 dp/μ dZ), x � X/H, and y � Y/H. e microchannel governing equation is nondimensionalized and expressed as follows.
e dimensionless momentum equation of Region 1 is expressed as follows: where w 1 is the dimensionless velocity of the microfluid in the left rectangle. Boundary conditions: e dimensionless momentum equation of Region 2 is expressed as follows: where W 2 is the dimensionless velocity of the microfluid in the right rectangle. Boundary conditions: e analytic solution of dimensionless velocity distribution of the aforesaid two regions is expressed as follows: e eigenvalue α n can be expressed as follows:  Mathematical Problems in Engineering e eigenvalue η m of the above equation can be expressed as follows: According to the velocity slip boundary conditions of microgroove, e velocity of the left region can be expressed as follows: Next, the velocity and shear stress of the two regions in the L-shaped domain can be matched along the common boundary. e conditions are given as e above two equations are changed to where in equations (17), (19), and (20), we choose N points along the boundary at x � c and obtain We truncate A n to N terms and B m to M terms. Note that M can be calculated from the following equation: where N � 40. e linear system of the M + N equations with M + N unknowns is solved for the coefficients of A n and B m .
Substituting A n and B m into the velocity profiles (equations (12) and (14)), the velocity of the microfluid flow can be obtained. e mean value for w(x, y) is derived as Furthermore, fRe can be expressed as a function of the volumetric flow rate in the following equation: where D h is the hydraulic diameter. In a more detailed form, e velocity distribution can be obtained by the above calculations; however, the temperature and heat transfer data are also important parts of a microchannel fluid. e calculation of the temperature distribution is described below.
And, the present paper is the temperature jump problem describing highly conductive boundaries. e temperature is temperature jump conditions on the boundary for each cross section and varies linearly downstream. e given heat flux varies peripherally but is constant with respect to axial distance.
First of all, the calculated analytic solution of velocity distribution is substituted into the temperature distribution function to obtain the governing equation for the dimensionless temperature distribution of fluid in the microchannel: e eigenvalue α n is expressed as e boundary conditions can be expressed as Mathematical Problems in Engineering where θ 1 and θ 2 are local dimensionless temperatures and β � β θ /β w . Eigenvalue η m : e boundary conditions can be expressed as where θ w is set as 0, and the temperature boundary condition is substituted into the aforesaid temperature distribution governing equation. e temperature distribution function can thus be worked out: 2α n x cos α n y e α n (x−c) + e −α n (x+c) + n D n cos α n y e α n (x−c) + e −α n (x+c) , Temperature distribution equations (31) and (32) and coefficients C n , D n , E m , and F m are calculated using the modified point matching method. e boundary conditions are expressed as follows: Equations (30)-(32) are expanded to obtain the following three equations: 6 Mathematical Problems in Engineering where y i in equations (34)-(36), we choose 2N points along the boundary at x � c and obtain And, truncate C n and D n to 2N terms and E m and F m to 2M terms. Note that M can be calculated from the following equation: e linear system of 2M + 2N equations with 2M + 2N unknowns is solved for the coefficients of C n , D n , E m , and F m . e numerical calculations in the present study were based on a new numerical analysis scheme that was used to name modified point-matching methods. In order to estimate the series convergence, the present research tests the term of the coefficients N. When the term of the coefficient N is set as N � 40, the precision of the coefficient convergences to four decimal places. erefore, the present investigation uses N � 60 to calculate the results in order to ensure convergence.
Moreover, the heat flux in the microchannel with microgrooves is calculated as follows: (39) e mean Nusselt number is calculated as follows:

Results and Discussion
In the first section of the results and discussion, for validating the convergence of the series, the present study predicts the term of the coefficients. When the term of series N is set as N � 40, the precision of the coefficient convergences to five decimal places. erefore, the present investigation will apply N � 60 to compute the numerical data from Figures 4-8 for ensuring convergence. Table 1 shows some selected coefficients obtained by the modified pointmatching method under N � 40. e numerical solution in the present investigation can be exactly computed by the modified point-match methods. Convergence is fairly fast. For realizing coefficients A n , B m , C n , D n , E m , and F m , the values of the some chosen coefficients are listed in Table 1. Figure 9 shows the applied grids in the present investigation. Figure 10 discusses the velocity vector of the microchannel with microgrooves. Figure 10 represents velocity vector of the L-shaped region based on the contour map of the fluid velocity. e present investigation is based on the two-dimensional analysis of microchannel. e main flow direction of the velocity in the microchannel with microgrooves was normal in this paper (z-dimension). It is observed that the velocity vectors in microchannel with microgrooves decrease from the center of the domain (x � 0, 0 ≤ y≤ (a/2); y�0, 0 ≤ x≤ 1) to the boundary of the domain (y � (a/2), 0 ≤ x≤ c; x � c,(a/2) − b ≤ y≤ (a/2); y � (a/2) − b, c ≤ x ≤1).      Figures 5 and 6 show the local velocity distribution of the microchannel and microgroove and the local temperature distribution of the microchannel and microgroove, respectively (β w Kn � 0.025). It was observed that the local velocity was lower at the slip boundary and that the local velocity increased as the distance to the geometric center shortened. e local temperature distribution presented a larger value on the wall surface. is study assumed there was a linear temperature change in the z direction and that the temperature of each section in the z direction was constant. It was observed that the temperature dropped as the distance to the center shortened; on the contrary, the closer to the wall surface of microchannel and grooves, the higher the temperature became.

Mathematical Problems in Engineering
e node-density effect on the predicted solutions, in terms of fRe and Nu, is calculated under different grid size and is listed in Table 2. e parameters setting in Table 2 is under β w Kn � 0, c � 1.0, and b � 0. at is equal to the  special case for the laminar flow between two infinitely long plates [21], and fRe � 24 and Nu � 7.54. e relative percent errors for fRe and Nu are defined as error � analytical solution − present results analytical solution × 100%. (41) For the density of 40 × 40 grid size, the relative error of fRe values is less than 0.31%, and for the density of 80 × 80 grid size, the relative error of Nu numbers is less than 0.12%. Besides, the hardware of the computer applying in the present numerical analysis uses 4-core 3.4 GHZ CPU and 8G DDR3-1600 RAM. e computational time, seconds, are also listed in Table 2. Figure 7 shows the effect of rarefaction parameter β w Kn on the fRe under different dimensionless microgroove depths. First, for validating the fluid dynamic characteristic of the present study, one set the present parameters of    microchannel with the microgrooves as c � 1.0, b � 0, and β w Kn � 0. at is equal to the special case for the laminar flow between two infinitely long plates [21], and fRe � 24. Under the special case, according to the calculated results of the present numerical methods, fRe value under the special case is exactly 24.017. Figure 7 shows that the results of the present paper are in a good agreement with the velocity characteristic of laminar flow between two infinitely long plates. Moreover, dimensionless depth b of the microgroove was increased. As the contact area of the fluid and the shoulder wall surface was enlarged, the fRe increased. It was observed that the fRe decreased as rarefaction parameter β w Kn increased, because the fluid was rarefied and the friction was reduced when the rarefaction effect was enhanced. Figure 8 shows the effect of rarefaction parameter β w Kn on Nu under different dimensionless microgroove lengths. Next, for further validating the thermocharacteristic of the present investigation, the conditions of the present microchannel with microgrooves is set the same as the laminar fluid flow between two infinitely long plates [21], i.e., under β w Kn � 0, c � 1.0, and b � 0, the Nusselt number was similar to the laminar flow between two infinitely long plates. Based on the numerical analysis of the present modified pointmatching method, the Nusselt number under the special case is 7.542, i.e., under the present theoretical method prediction, the Nusselt number approximated the Nu of the laminar fluid flow between two infinitely long plates; the Nu equally is equal to 7.54. e data of the present study are in a good agreement with the thermocharacteristic of laminar flow between two infinitely long plates. e comparison is shown in Figure 8.
In addition, the average Nu decreased as rarefaction parameter β w Kn increased. If dimensionless depth b of the microgroove was increased, the heat transfer area would be enlarged and Nu would increase accordingly.

Conclusion
is study found that the slip velocity decreased gradually at the top of the microchannel under the shear force of the microgroove shoulder, whereas the slip velocity increased gradually at the shoulder of the microgroove far from the top of the microchannel. e variability decreased at the bottom of the microgroove. e discontinuous temperature rose at the top of the microchannel under the thermal effect of the microgroove shoulder, dropped at the shoulder of the microgroove far from the top of the microchannel, and rose at the bottom of the microgroove. Moreover, the contact area of the fluid and wall surface was enlarged when dimensionless depth b of the microgroove was increased, the fRe increased accordingly, the fluid was rarefied as β w Kn increased, and the fRe was reduced.
e fluid would be rarefied if β w Kn was increased, and the average Nu would be reduced as a result. If dimensionless depth b of the microgroove was increased, the average Nu would be increased as the heat transfer area was enlarged.

Data Availability
e data used to support the findings of the present investigation are not available because our research data come from computer simulation.

Conflicts of Interest
e author declares there are have no conflicts of interest.