Microscale equation Stokes’ flow

The Reynolds number in microreaction systems usually ranges from 0.2 to 10. In contrast to the turbulent flow patterns that occur on the macroscale, viscous effects govern the behavior of fluids on the microscale and the flow is always laminar, resulting in a parabolic flow profile. In microfluidic reaction systems, where the characteristic length is usually greater than 10 pm, a continuum description can be used to predict the flow characteristics. This allows commercially written Navier-Stokes solvers such as FEMLAB and FLUENT to model liquid flows in microreaction channels. However, modeling gas flows may require one to take account of boundary sUp conditions (if 10 < Kn < 10 , where Kn is the Knudsen number) and compressibility (if the Mach number Ma is greater than 0.3). Microfluidic reaction systems can be modeled on the basis of the Navier-Stokes equation, in conjunction with convection-diffusion equations for heat and mass transfer, and reaction-kinetic equations. 

By subsituting (8.18) and (8.19) into (8.14) and (8.16), we obtain the following incompressible flow equations in the micro-domain, referred to as the microscale equations for Stokes flow. 

Let us introduce a weak form for the microscale equations for Stokes flow... 

Given the importance of low-Re, viscous flow on microscale aerodynamics, it is possible to take advantage of the dominant heat transfer effects to enhance microrotorcraft flight. These heat effects can be characterized using the standard transport equations and a Navier-Stokes solver. In order to accurately apply the physical properties, it is important to include the effect of temperature on the viscosity (using, e.g., Sutherland s, Wilke s, or Keyes laws), thermal conductivity, and specific heat of the surrounding fluid (air). 

Many more CTD simulations at the microscale exist and the numerical reproduction of the behavior of pressure-driven flows in microchannels has been obtained often [2, 10]. Some discrepancies have been found between numerical solutions of the Navier-Stokes equations and experimental data obtained in viscosity-dominated shock tube investigations. It was noticed that even using fine grid cells of a few micrometers, the solution of the Navier-Stokes equations still does not match experimental data of shockwave reflection transition over a wedge [2]. It has been pointed out that the solution of shockwave motion at low Reynolds... 

It is clear from the above discussion that surface properties are extremely important in microscale systems and their importance grows as the characteristic channel dimension decreases. However, there is no straightforward way to take these effects into account, with the models developed to describe this phenomenon being problem specific. Generally they are based on a combination of classical solutions of the Navier-Stokes equations, coupled with ad hoc models of molecular slip flow. Therefore, in the simulation of microchannel flows, it is important to keep in mind that the use of the no-slip boundary condition may not be appropriate and that additional physics may need to be included in the modeling to capture the correct behavior. 

The Navier-Stokes (NS) equations can be used to describe problems of fluid flow. Since these equations are scale-independent, flow in the microscale structure of a porous medium can also be described by a NS field. If the velocity on a solid surface is assumed to be null, the velocity field of a porous medium problem with a small pore size rapidly decreases (see Sect. 5.3.2). We describe this flow field by omitting the convective term v Vv, which gives rise to the classical Stokes equation We recall that Darcy s theory is usually applied to describe seepage in a porous medium, where the scale of the solid skeleton does not enter the formulation as an explicit parameter. The scale effect of a solid phase is implicitly included in the permeability coefficient, which is specified through experiments. It should be noted that Kozeny-Carman s formula (5.88) involves a parameter of the solid particle however, it is not applicable to a geometrical structure at the local pore scale. 

If a homogenization analysis (HA) is applied to porous media flow, which is described by the Stokes equation, we can immediately obtain Darcy s formula and the seepage equation in a macroscale field while in the microscale field the distributions of velocity and pressure are specified (Sanchez-Palencia 1980). We can also apply HA for a problem with a locally varying viscosity. 

Eree surface flows and interfaces between two or more immiscible fluids or phases are observed in many natural and industrial processes at macro- and microscales. Different numerical techniques are developed to simulate these flows. However, due to the corrplexity of the problem, each technique is tailored to a particular category of flows. Einite element (EE), finite volume (EV) and finite difference (ED) methods are all potentially applicable to generalized Navier-Stokes equations. However, they have to be coupled with a technique to track moving fluid boundaries and interfaces. The difficulty in tackling interfacial flows is inherently related to the corrplexity of interface topology and the fact that the interface location is unknown. 

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