1. Introduction
Due to the rapid development of micro-electro-mechanical systems (MEMS), the flow mechanisms in microchannels have been a hot research topic worldwide [
1,
2,
3,
4,
5,
6]. At the same time, microfluidics are increasingly applied in aero engines, gas turbines, electrical devices, and so on. It has been observed that the conventional formulations applicable to macro-size structures may be invalid for micro-structures [
1,
2,
3,
4,
5,
6,
7,
8,
9]. Therefore, research investigating the flow behavior in microchannels is in great demand.
In contrast to macroscale flows, the majority of microscale flows are laminar, and the relatively small dimensions of microchannels may cause a high pressure drop. The aspect ratio of the microchannel plays an important role due to the relatively small size. Chen et al. [
10] studied the effect of the aspect ratio on laminar flow bifurcations in curved rectangular tubes driven by pressure gradients, and derived the ranges of stable flow solutions. In their further study [
11], they investigated the effects of the aspect ratio on multiple flow solutions in a two-sided parallel motion lid-driven cavity. They distinguished the regions of stable and unstable flows according to the different aspect ratios. Besides, unlike in conventional devices, in many micro-size channels, the length of the channel is always not sufficient to give rise to fully developed flows [
12]. Hence, it is of great significance to estimate the entrance length of microchannels, especially under relatively lower Reynolds numbers.
The entrance region can be defined as the area from the inlet of the channel to a location where the maximum local velocity has gained 99% of its fully developed value [
10]. Since the fluid needs a much longer distance to shape into a fully developed flow pattern, this criterion is applicable in engineering for estimation of the length of the entrance region. It is important to estimate the entrance length in microchannels because the transport properties are highly dependent on this region. Besides, considering the relatively short dimension of the channel length, the entrance effects of micro-size flow should be paid more attention.
Previous investigations of the entrance length have mainly focused on the Reynolds number and the geometric parameters of the channels such as the hydraulic diameter and aspect ratio. The entrance length in conventional channels has been studied by many scholars since the 1960s. Atkinson et al. [
13] and Chen et al. [
14] conducted a numerical investigation to estimate the effect of the Reynolds number on the entrance region in macroscale circular pipes and between two parallel plates. Atkinson et al. [
13] found that the dimensionless entrance region length was linearly related to the Reynolds number as in Equation (1); meanwhile, Chen et al. [
14] proposed the correlation shown in Equation (2), where
is the length of the entrance region and
is the hydraulic diameter of the channel. Coefficients
and
are listed in
Table 1. Schlichting et al. [
15] considered boundary theory and proposed that the dimensionless entrance region length in macro-size devices was directly proportional to the Reynolds number as shown in Equation (3). Muzychka and Yovanovich et al. [
16] simplified the mathematical models and then proposed new models for square channels which showed a good agreement with correlation proposed by Schlichting et al. [
15].
However, the behaviors of entrance length in macrochannels and microchannels are obviously different.
The effect of viscous force plays a bigger role in microscale flow than at the conventional scale [
17,
18], arousing controversy as to whether the classical fluid theory is applicable to microscale fluids. Hence, it is important to estimate the entrance length in microscale flow. However, unlike conventional macroscale channels, the applicable results of entrance lengths in microscale devices are limited.
Micro-PIV (particle image velocimetry) experiments carried out by Lee and Kim et al. [
19,
20] using deionized water flowing about
Re = 1 in a rectangular channel with 58 μm depth, 100 μm width, and 30 mm length showed that the entrance lengths of the microscale channels were much shorter than those of macroscale channels. Ahmad and Hassan et al. [
21] conducted experimental investigations to estimate the entrance lengths of rectangular microchannels with micro-PIV. They proposed new correlations of Reynolds numbers and entrance lengths. Hao et al. [
22] analyzed the development process of laminar flow in trapezoidal microchannels. The hydraulic diameter was 273 μm, and micro-PIV was adopted. They found that the correlation between entrance length (
) and the Reynolds number (Re) was
.
Renksizbulut and Niazmand et al. [
23] carried out simulations to study the laminar flow as well as heat transfer in the entrance region of the trapezoidal models. The aspect ratio ranged from 0.5 to 2 and the Reynolds numbers from 10 to 1000. The results showed that the previous studies that calculated the entrance lengths based on fully developed flow were inaccurate, and the entrance length was a function of Reynolds number and geometric parameters. They proposed new correlations to estimate the length of the entrance region. However, the inflow condition they used in their simulation was a uniform velocity profile, which is different from the realistic conditions in the inlet of a channel.
Galivis and Yarusevych et al. [
24] numerically studied the entrance region length of microchannels with hydraulic diameters between 100 and 500 µm. The Reynolds numbers ranged from 0.5 to 200 and aspect ratios from 1 to 5. The authors concluded that when the Reynolds number (
Re) was under 50, the dimensionless entrance lengths changed nonlinearly with Reynolds numbers but this correlation became linear for higher Reynolds numbers. Besides, the dimensionless entrance length increased with increasing channel aspect ratios under a certain
Re. However, as in the research of Renksizbulut and Niazmand et al. [
23], the velocity in the entrance was given as a uniform profile, which does not represent real inlet conditions.
Although some studies on the entrance length in microscale channels have been done, as
Table 2 presents, the previous studies were mainly qualitative, and the applicable results are still limited—especially for laminar flow at low Reynolds numbers. Even though some of them proposed correlations of entrance length [
23,
24], the inlet conditions were given as a uniform velocity in these studies, which does not simulate real inlet velocities. Therefore, further study is necessary to explore the factors influencing the entrance length in microscale channels. This paper conducted simulations to estimate the factors influencing the dimensionless entrance region length. In addition, we aimed to quantitatively research the entrance lengths in different channels. Compared with giving a uniform velocity profile at the entrance as in previous studies, we used a new inlet configuration to simulate a realistic inlet velocity profile in this research, making the results more reliable.