• Objective: Study fluid flow and heat-transfer behavior in microchannels under slip-flow conditions.

• Method: Use the Thermal Lattice Boltzmann Method (TLBM) with the D2Q9-BGK model.

• Key parameters: Investigate the effects of Knudsen number (0.01–0.10) and ramp height (0–10%).

• Findings: Increasing Kn and ramp height generally reduces mass flow, heat transfer, and COP, while ramp height increases friction.

• Best performance: Low/negative ramp heights provide better thermal-hydraulic performance, with results agreeing well with published studies.

Abstract:

This study investigates thermal and fluid behaviors in smooth microchannels under slip flow regime using an alternative numerical technique namely the thermal lattice Boltzmann method (TLBM). This method is based on D2Q9 model with lattice-BGK (Bhatnagar–Gross–Krook) approximations. In this procedure, an internal energy distribution function uses to calculate temperature, and a momentum distribution function to evaluate macroscopic quantities like density, pressure and velocity etc. With these macroscopic quantities, the important physical properties such as the average flow friction, mass flow rate, and the heat transfer rate are investigated and discussed for different governing parameters. The relative ramp heights (0–10) and Knudsen number (Kn) (0.01–0.10) are the most important parameters in this study. The average frictional resistance decrease with increasing Kn but increasing with ramps height, whereas the mass flow rate reduced both for ramps height and Kn. Moreover, the heat transfer rate decreased significantly with Kn and very slowly with ramps height. Another important properties, the combined effect of thermal and hydraulic properties called the coefficient of performance (COP) is studied to compare the efficiency of different microchannels. COP decreases with increasing ramp height as well as Kn. Optimal performance is observed with very low ramp heights. The microchannel with negative ramps perform better than positive ramps case. The COP of sawtooth microchannels is calculated to compare with the friction (pressure drop) and heat transfer of smooth microchannel. Finally, the obtained result is compared, and an excellent agreement is found with published work. 

• Objective: Analyze the thermal-hydraulic performance of upstream and downstream wedge-rib microchannels under slip-flow conditions.

• Method: Use the Thermal Lattice Boltzmann Method (TLBM) with Knudsen numbers (0.01–0.10) and rib heights up to 20% of channel height.

• Flow behavior: Upstream ribs create vortices behind the ribs, while downstream ribs create vortices in front of the ribs; vortex size changes with Kn.

• Main findings: Increasing Kn decreases friction and heat transfer, while increasing rib height generally increases friction and affects heat transfer.

• Best performance: Downstream wedge-rib microchannels perform better than upstream configurations, particularly at lower Kn and smaller rib heights.

Abstract:

The thermal lattice Boltzmann method (TLBM) is used to analyze the overall performance for upstream and downstream wedge ribs microchannels (MC) under slip flow conditions. The thermal–hydraulic enhancement criterion is investigated to evaluate the performance of the channel and compare it for various roughness MC. In order to improve the channel performance, two alternative artificial roughness geometry, upstream and downstream wedge ribs, are taken both on the top and bottom walls of the microchannel with aspect ratio (AR) 7, where AR = L/H; L and H are channel length and height respectively in micrometer (μm). This study focused on simulating temperature profiles, velocity vectors in terms of stream lines, pressure gradients, and friction factor in terms of Poiseuille number as well as heat transfer rate in terms of Nusselt number (Nu). The overall performance of the channel is calculated based on flow friction and heat transfer rate for different Knudsen numbers (Kn) ranging from 0.01 to 0.10 with upstream and downstream wedge ribs height up to 20% of channel height. The results have been compared with previously published work and are found a very good agreement. The analysis revels that, the vortices are formed behind each upstream wedge rib, whereas they are created in front of each downstream wedge rib. The size and shape of vortices are influenced by Kn. As Kn increases from 0.0 to 0.10, the fluid circulation area becomes smaller for upstream wedge ribs MC, while it is changing very slowly for downstream wedge ribs MC; hence, the pressure gradient is also responsible for changing Kn. The flow friction is linearly decreased with increasing Kn but significantly increased with ribs height. But compared to the smooth channel, the friction is significantly increased for upstream and downstream wedge ribs MC. The average rate of heat transfer in terms of Nu is also linearly decreased with increasing Kn, but Nu increased with ε for lower Kn and decreased for higher Kn. Therefore, compared to smooth MC, Nu increased and decreased for the same for upstream and downstream wedge ribs MC. Finally, the performance enhancement (η) is calculated, and it is found that η decreased with increasing Kn for upstream and downstream wedge ribs MC. The higher performances are indicated for lower Kn as well as lower ribs height. For all cases, the better performance is noted for downstream wedge ribs MC compared to upstream MC. 

• Objective: Improve the accuracy of Galerkin approximate solutions for higher-order boundary value problems (BVPs) using a residual correction procedure.

• Method: Construct an error differential equation from the residual and solve both the original and error equations using the Galerkin method.

• Basis functions: Use Bernstein and Bernoulli polynomials as basis functions.

• Validation: Apply the method to linear and nonlinear fourth- and sixth-order BVPs and compare the results with analytical and existing numerical solutions.

• Main finding: The proposed residual-correction method provides higher accuracy than several previously reported numerical approaches.

Abstract: 

This article uses residual correction procedure for improving the Galerkin ap proximate solutions to higher order boundary value problem (BVP). The residual function of a differential equation is found from the approximate solution of a BVP and setting it as nonhomogeneous term we get the error differential equation. We exploit Bernstein and Bernoulli polynomials as basis functions to solve the two differential equations, namely, original and its error equations, by Galerkin tech nique subject to the corresponding boundary conditions. Linear and nonlinear problems of fourth order BVPs are considered to verify the proposed method. The resulting numerical solutions are compared with the analytic solutions as well as the results of other approaches those have been reported in the literature. This me thod is also applied to sixth order BVPs. The comparison reveals that the current procedure is more accurate. 

• Objective & Method: Apply the Residual Galerkin Technique with Modified Legendre Polynomials to solve linear and nonlinear BVPs.

• Results: Tested on second-order BVPs, with excellent agreement between the approximate and analytical/numerical solutions.

• Main contribution: The method provides accurate solutions with less computational effort and can potentially be extended to higher-order nonlinear ODEs and PDEs.

Abstract:

In this paper, we have used the residual Galerkin technique to solve linear and nonlinear BVPs using Modified Legendre polynomials focused on the performance of the procedure. We have applied the formulation on secondorder BVP, compared approximate solutions with the analytical/numerical solutions available in the references, and found they are in excellent agreement. The results indicate that the residual correction procedure can obtain accurate numerical solutions to linear and nonlinear boundary value problems with less computational effort. This method may be applied for higher-order nonlinear BVP with ODEs and PDEs.