Microfluidic devices have been developing rapidly in recent years, and have been increasing in several applications such as drug delivery and cell sorting. Microfluidic devices have advantages over traditional devices, such as low energy cost and separation efficiency [1]. In addition to parallel processing, low material usage, and energy efficiency [2]. Moreover, The control of flows [2] in microchannels (MCs) and heat transfer is considered a complex process [3] and is one of the most important research fields in microfluidics [4]. Microchannels have various implementations in several fields as the cooling technology for addressing heat dissipation issues obtained via high heat flux devices [5]. For example, MC heat sinks are considered cutting-edge cooling technology for electronic systems according to their height of heat generation [6]. The heat transfer mechanism differs from the micro size to the macro size.

The Knudsen number (Kn), which is defined as the ratio of the particle mean free path to the characteristic length of the flow, serves as an indicator of the rarefaction effect [7]. The Kn number provides a way to categorize fluid flows in the micro-scale into four distinct regimes, each with unique characteristics. These regimes are classified as: continuum flow (for Kn numbers below 0.001), slip flow (for Kn numbers between 0.001 and 0.1), transition flow (for Kn numbers between 0.1 and 10), and free-molecular flow (for Kn numbers greater than 10). This categorization is based on the relative importance of viscous and molecular forces in determining the behavior of the fluid. The Kn number serves as a useful tool for predicting the regime of fluid flow and designing systems that operate in specific flow regimes [8,9].

In microflows, the slip boundary plays a critical role and cannot be neglected when analyzing the fluid’s flow and thermal behaviors [10,11]. The lattice Boltzmann method (LBM) can be used in microfluidics as an important computational tool [12,13]. According to the high level of parallel ability and the simple algorithm formulation [14], particularly the explicit formulation. The LBM method shows excellent implementation [15] as well as can handle complex boundary conditions [6]. Furthermore, the thermal LBM has been acknowledged as a powerful technique for investigating MC flows that incorporate slip boundary conditions [16]. The LBM method includes two steps collision and streaming [13]. Various models of LBM have been developed to increase the performance of the thermal field simulation. One approach that has been proposed for modeling microscale flows is the double population method, which involves the use of two distinct distribution functions. Specifically, this approach utilizes a density-momentum distribution function for the fluid domain and an internal energy distribution function for the thermal field [11,17].

Numerous studies have investigated the behavior of heat transfer and fluid flow in MCs utilizing the thermal LBM while taking into account slip velocity and temperature jump. For instance, Niu et al. [18] used the thermal LBM to simulate microthermal flow with velocity slip and temperature jump. Similarly, Liu et al. [19] investigated gas flow in a long MC with the LBM and examined the effects of slip and transition flow. Their results showed that the rarefaction effect becomes increasingly significant as the Kn number increases. Hatam et al. [20] studied Cu-H₂O heat transfer in a MC heat sink and demonstrated that increasing the interactions ‘nanoparticles-solid phase’ leads to an increase in channel aspect ratio, resulting in an elevated Nusselt number. Finally, Ghadirzadeh et al. [21] utilized the LBM to simulate laminar nanofluid flows within an annular MC in a slip flow regime. These studies provide valuable insights into the behavior of fluids in MCs under various conditions and have advanced our understanding of the mechanisms governing heat transfer and fluid flow in these systems.

Knupp et al. [22] analyzed slip flow in laminar gaseous flow with heat transfer in circular MCs. The study found a decrease in Nu numbers as the Kn number increased. Ahangar et al. [23] used LBM to investigate slip and transient flow regimes in MCs. The study found that in the slip regime, the velocity increased with the Kn number at the center and at the exit of the MC in the transient regime. Alipour Lalami et al. [24] employed LBM with slip boundary and temperature jump to examine heat transfer of nanofluids at conjugate heat in a wide MC with a thick wall. The study found that higher Re values caused an increase in the average Nu number on superhydrophobic surfaces. Finally, Rehman et al. [25] investigated fluid flow in a rectangular channel with a partially heated region by a circular, heated cylinder placed between channels as an obstacle. The study considered four different cases where rectangular ribs were installed and found that the circular obstacle showed the highest drag force when rectangular ribs were present on both the lower and upper walls heated.

Ashraf et al. [26] investigated the behavior of an oscillating mixed convection flow over a horizontal circular cylinder. The study revealed that buoyancy forces play a significant role in accelerating the fluid flow and increasing the velocity of fluid particles. Qiu et al. [27] examined the heat dissipation model of a copper MC heat sink, considering the impact of the Nu number on heat transfer. Their results indicated that the Nu number is a crucial factor affecting the heat transfer performance. Ma et al. [28] simulated 3D laminar flow in a rectangular MC and found that low Re numbers can considerably influence the local Nu numbers in the thermal zone. Overall, these studies provide insights into the complex nature of heat transfer and fluid flow in MCs, highlighting the importance of considering various factors such as buoyancy forces, Nu number, and Re number for achieving optimal heat transfer performance.

In recent years, improving the heat transfer in microdevices has become a significant area of research, especially with the rapid advancement of Micro Electro Mechanical Systems (MEMS) [12,13]. The demand for more efficient designs and technologies to enhance heat transfer efficiency in heat transfer systems has also increased. Slip velocity conditions and temperature jump implementation are some of the methods that have been explored. The Lattice Boltzmann Method (LBM) has been recognized as a useful method to simulate microflow due to its mesoscale characteristics [14]. LBM has several attractive features, including superior representation of microscopic interactions. The most recent and useful LBM framework for dealing with thermal flows is the thermal lattice Boltzmann method with a double distribution function [17]. In the literature, various studies have been conducted on microscale heat transfer, such as Ashraf et al. [29], who examined the impact of variable surface temperature on periodic mixed convection using a thermally and electrically conducting cone in a porous medium. The study by Lori et al. [30] investigated the heat transfer and fluid flow in solid ribs and alternating vertical porous with microchannel heat sinks, showing that the average Nusselt number of the porous-rib microchannel was more significant than the solid case. Another study by Lobasov et al. [4] explored the influence of temperature jump and velocity slip on heat transfer in the microchannel and found that the heat flux density and average heat transfer coefficient increased as the slip length increased.

In the study by Ashraf et al. [29], the impact of variable surface temperature on periodic mixed convection was analyzed through the use of a thermally and electrically conducting cone positioned in a porous medium. The results indicated that there was an increase in temperature distribution as the Pr number values rose. Meanwhile, Lori et al. [30] explored heat transfer and fluid flow in solid ribs and porous material that alternated vertically with MC heat sinks. According to their findings, the average Nu number was greater in the porous-rib MC than in the solid case. Lastly, Lobasov et al. [4] focused on investigating the effects of temperature jump and velocity slip on heat transfer in a MC. Their research revealed that the heat flux density and average heat transfer coefficient increased with an increase in slip length. These three studies provide important insights into the diverse factors that can impact heat transfer in different systems.

Microdevices have witnessed significant advancements in recent times, with Micro Electro Mechanical Systems (MEMS) driving a lot of innovation [12,13]. However, there is a pressing need for more efficient and better-designed heat transfer systems to improve heat transfer in microdevices [31]. Two crucial factors that can impact the performance of such systems are the slip velocity condition and temperature jump implementation. A promising approach that has gained considerable attention in this context is the LBM, which offers superior representation of microscopic interactions [32]. Of all the LBM frameworks available, the thermal LBM with a double distribution function [17] is the most recent and useful for simulating microflows owing to its mesoscale characteristics [14]. LBM has demonstrated its superiority in modeling rarefied gas flow in microchannels [8]. These observations highlight the potential of LBM in enhancing heat transfer in microdevices and the need for further research in this area.

The present study aims to provide a numerical investigation of laminar fluid flow and forced convective heat transfer in a rectangular MC, both with and without obstacles placed on the walls. To conduct this study, we employed the thermal LBM with the BGK model, while implementing slip velocity and temperature jump on both the walls and the obstacles. The novelty of this study lies in its focus on the impact of obstacles present in the MC walls and how they can enhance fluid flow and heat transfer. Additionally, the study examines the application of thermal LBM with slip velocity and temperature jump on MC and obstacle walls. To this end, the MC was simulated with different obstacle positions, and the effects of the Kn number, the implementation of obstacles on fluid flow and heat transfer, and the Nu number in the MC were analyzed. The results of this study hold significant implications for the design of more efficient microfluidic devices with enhanced heat transfer.

Figs. 5–7 represent the MC’s velocity distribution for Kn=0.012−0.02andKn=0.05. In the absence of obstacles, the velocity decreases with the increase of Kn numbers. In case of obstacles, the velocity drops as the Kn number values grow, and the velocity values increase in the case of obstacle implementation. From the velocity profiles, the performance of obstacles plays a vital role in increasing velocity. Also, the significant effect of obstacles on velocity distribution occurs in the case of three obstacles with low Kn numbers. However, the results indicated the importance of the rarefication effect on velocity distribution, especially can notice their influence is more significant in the middle of MC than next to wall. Nonetheless, the findings demonstrate the significance of the rarefication effect on the distribution of velocity. It is noteworthy that this influence is more prominent in the center as opposed to near the wall, as indicated by previous research [13]. Our analysis suggests that obstacles had a substantial impact on the increase in velocity.

Fig. 8 shows the velocity streamlines in various cases with Kn=0.05. The velocity field increased rapidly in the developing region until it achieved a maximum in the center of the MC, as shown in the streamlined structures without obstacles in Fig. 4a. Also, the fluid near the wall maintains a minimum velocity. In the cases of obstacles, the velocity field rises slowly due to obstacles but with a high value compared to the first case. Furthermore, the developing region experienced a substantial deceleration in the third scenario. The velocity of the rarefied flow reached its peak value upon exiting the MC, which was higher than the velocity observed in the first scenario that had no obstacles. These findings demonstrate that the presence of obstacles had a significant impact on accelerating the velocity of the rarefied flow in the MC.

Fig. 9 illustrates the velocity contours for different cases with Kn=0.05 along the MC. In the absence of obstacles, in Fig. 9a, it can be seen that the velocity contours are parallel and increase symmetrically from the top and bottom walls of the MC to reach their maximum values at the MC center. In the other cases, Figs. 9b and 9c, the velocity contours are distorted around the obstacles, and they increase slowly from the developing region with higher values than in the previous case. It can be noticed that in the case with nine obstacles, Fig. 9c shows that the fluid flows slowly through the MC but accelerates as the exits. Analyzing these findings is crucial in evaluating the effectiveness of the obstacles installed in the MC with velocity slip and temperature jump. Furthermore, the study of the alterations caused by the obstacles in the velocity contours of the microchannel aids in enhancing the flow acceleration.

Figs. 10–12 depict the distribution of temperature along MC for three scenarios with Kn values ranging from 0.012 to 0.05. The absence of obstacles in the first case exhibited a decrease in temperature with an increase in Kn number, as illustrated in Fig. 10. These outcomes indicate that the temperature of the fluid surges with the increase in dimensionless axial distance [28].

Even when adding obstacles, the temperature continues to drop. The temperature reduction is noticeable when nine obstacles are added, as seen in Figs. 12a and 12b, respectively. Moreover, the impact of rarefaction in the central area is more prominent compared to the region near the wall. This is attributed to the substantial temperature difference observed near the wall, which is caused by the rapid establishment of the hydrodynamic and thermal boundary layers in the entrance area, as depicted in Fig. 12. To sum it up, the MC with obstacles exhibits superior heat transfer enhancement in comparison to a smooth MC.

From the streamlined Fig. 13, the temperature decreases rapidly from the wall to the neighboring layers, reaching a low value in the central region of the MC with no obstacles case, Fig. 13a. In the second case, Fig. 13b, the temperature near the first obstacle decreases slowly compared with other obstacles. But in the case with nine obstacles, the fluid flows quickly, as confirmed by the temperature profile, Fig. 13c. However, when compared to the scenario without obstacles, the inclusion of obstacles results in fluctuations in the temperature of the fluid at the wall and its adjacent layers. Furthermore, it leads to an increase in temperature in the central area of the MC.

To illustrate the heat transfer improvement mechanism by obstacles, the temperature contour was added to the study. Fig. 14 depicts the temperature contours for Kn=0.05 along the MC. It can be seen that the temperature contours are parallel also. Their value increased in the central region with the absence of obstacles. The flow structure is noticeably different when obstacles are present. The figures show the heat transfer division between the obstacles in the other cases, Figs. 14b and 14c. In addition, the temperature contours are distorted, and a vortex is formed for the three obstacles case in the vicinity of the first obstacle. This vortex disappears for the nine obstacles case, Fig. 14c. However, low values are attained at the MC exit. And an increase in the temperature values along the MC is perceived with increasing the Kn number. These results are essential for assessing the overall efficacy of the obstacles installed in regulating the temperature of the MC. Moreover, the utilization of TLBM with a double distribution function enables the investigation of the thermal performance of rarefied flow in rectangular MCs with obstacles at varying Kn numbers while considering the slip velocity and temperature jump boundary conditions.

The evolution of the Nu number according to various Kn numbers is shown in Fig. 15. The Nu number varies from 0.05 to 0.08 along the MC for the different studied cases. According to the obtained results, it is obvious that the Nu number reduces as the Kn number rises. The results show that even small Kn number values cause a great impact on the Nu number. The Nu number reached its maximum value at the MC’s inlet.

Furthermore, due to the temperature gradient at the entrance region, the highest values were reached in the second case compared to the third one. The Nu number’s higher values denote that heat transfer is more efficient. Hence, rarefaction effects can substantially impact the Nu number in rarefied flows in the slip regime. From x=0.6, the temperature gradient is almost constant for various Kn numbers in the second and third cases, so the rarefication effect is no more predominant. In addition, the Nu number continues to reduce until the end of the MC [13]. Furthermore, from the results, conduction increases and convection decreases in the developing region.

Fig. 16 due to the rapid increase in velocity at the entrance zone and the reduction in the velocity gradient, the results showed a decreasing skin friction coefficient with a rise in Kn numbers according to the slip effect similar to [12]. Particularly, in case of no obstacles. However, the rarefication effect was more significant in the second and third cases. The findings indicate that the third scenario, which incorporates nine obstacles in the microchannel, exhibits a superior reduction in the skin friction coefficient of the rarefied flow in the slip regime.

Figs. 17 to 19 illustrate the axial variation of velocity slip along the lower and upper wall for various Kn numbers. In the absence of obstacles, the velocity slip along the top and lower walls gradually decreases, Fig. 17. In the presence of obstacles, the velocity slip continues to decrease and is separated into zones based on the obstacle, the result of a dropping slip velocity related to velocity gradient reduction at the wall [12], Figs. 18 and 19. An increase in slip velocity occurs before the obstacle, followed by a decrease in slip velocity until it gets the minimal value at x=0.6 for the top wall and before x=0.4 for the bottom wall in the second and third cases, Figs. 18 and 19. The low value of velocity slip is related to the second case, Fig. 18.

Figs. 20 to 22 depict the temperature jump along the upper and lower walls of the MC in the function of Kn number. In the developing region, the temperature jump keeps increasing rapidly in the absence of obstacles, Fig. 20. In the case of obstacles, the jumping temperature drops, and the low value of temperature jump occurs next to the obstacle. It increases when passing the obstacle. However, in both cases, with and without obstacles, the rarefaction effect does not influence the temperature jump on the top and bottom wall of MC for all Kn number values. Our study did not find any significant impact of rarefaction effect on temperature jump in slip regime of rarefied flow compared to other research such as [12].

The BGK model-based TLBM was employed using Python computing code to investigate forced convective heat and fluid flow in a unit length of a rectangular MC under three different configurations. The first configuration was without obstacles, while the second and third configurations featured three and nine obstacles, respectively. The temperature jump and slip velocity were considered at the boundary and extended to the obstacle level. This numerical investigation included an analysis of the impacts of temperature jump and velocity slip on heat transfer, temperature, and velocity profile, as well as the rarefaction effect’s influence. The results showed the rarefaction effect’s significance on velocity and temperature distribution. The current TLBM with a DDF allowed the modeling of MCs with and without obstacles with slip velocity and temperature jump. The study found that the fluid moved slowly through the third microchannel configuration (with nine obstacles) and accelerated at the MC outlet, as indicated by the lower temperature profile. No vortex was captured, but the results highlighted the critical role played by obstacles in MCs. Additionally, the rarefaction effect played a significant role in reducing the Nu number and skin friction coefficient in MCs containing obstacles. Furthermore, the proposed configurations offer a solution for enhancing the microchannel geometry and utilizing microelectromechanical systems and microdevices as cooling techniques.