Microscale Stokes’ equation

Substituting (5) into (4>2 yields the following microscale Stokes equation ... 

Let us introduce a weak form of the microscale Stokes equation (8.49) as... 

In this work, microscale evaporation heat transfer and capillary phenomena for ultra thin liquid film area are presented. The interface shapes of curved liquid film in rectangular minichannel and in vicinity of liquid-vapor-solid contact line are determined by a numerical solution of simplified models as derived from Navier-Stokes equations. The local heat transfer is analyzed in term of conduction through liquid layer. The data of numerical calculation of local heat transfer in rectangular channel and for rivulet evaporation are presented. The experimental techniques are described which were used to measure the local heat transfer coefficients in rectangular minichannel and thermal contact angle for rivulet evaporation. A satisfactory agreement between the theory and experiments is obtained. 

The total number of independent variables appearing in Fq. (4.32) is thus quite large, and in fact too large for practical applications. However, as mentioned earlier, by coupling Eq. (4.32) with the Navier-Stokes equation to find the forces on the particles due to the fluid, the Ap-particle system is completely determined. Although not written out explicitly, the reader should keep in mind that the mesoscale models for the phase-space fluxes and the collision term depend on the complete set of independent variables. For example, the surface terms depend on all of the state variables A[p ( x ", ", j/p" j, V ", j/p" ). The only known way to determine these functions is to perform direct numerical simulations of the microscale fluid-particle system using all possible sets of initial conditions. Obviously, such an approach is intractable. We are thus led to reduce the number of independent variables and to introduce mesoscale models that attempt to capture the average effect of multi-particle interactions. 

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. 

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. 

When we solve the diffusion problem which is given by, e.g., (5.29), we encounter a problem that relates to the evaluation of the mean velocity v (note that dca/dt = dcoi/dt + v gradca,). It may be possible to solve a microscale problem based on the Navier-Stokes equation however, in classical soil mechanics we commonly use the seepage equation to determine v. Using the assumption of incompressibility of a fluid, we can derive the seepage equation from (5.18) and (5.37). 

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. 

The reason why the perturbation of v (x) starts with a " -term is to ensure reduction to the corresponding Stokes equation in the micro-domain as a microscale equation (to be discussed later). We assume that the first-order term of pressure is a function of only the macroscale coordinate system x. ... 

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. 

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). 

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. 

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