Microscale equation

This section furnishes a brief overview of the general formulation of the hydrodynamics of suspensions. Basic kinematical and dynamical microscale equations are presented, and their main attributes are described. Solutions of the many-body problem in low Reynolds-number flows are then briefly exposed. Finally, the microscale equations are embedded in a statistical framework, and relevant volume and surface averages are defined, which is a prerequisite to describing the macroscale properties of the suspension. 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

It is important to note that the fluid-solid drag coefficients discussed above are valid only for monodisperse particles (i.e. particles with equal diameters and material densities). Using direct numerical simulations (DNS) of the microscale equations for fluid-particle flows, several authors (Beetstra et al, 2007 Buhrer-Skinner et al, 2009 Holloway et al, 2010 Tenneti et al, 2010, 2012 Yin Sundaresan, 2009) have proposed improved drag coefficients to account for polydisperse particles. 

Here (17) is used. Substituting this into (18) yields the following microscale equation [MiSE], together with X"-periodic boundary condition for the microscale domain 2 ... 

A mathematical scheme that can treat a micro-inhomogeneous material uniformly at the microscale and the macroscale is referred to as Homogenization Analysis (HA) (see Sanchez-Palencia 1980 Bakhvalov Panasenko 1984). In the HA method, we introduce a perturbation scheme by using both a macroscale coordinate system and a microscale one, and derive a microscale equation, which represents the geometry and material properties in the micro-domain. Then, using the solution of the microscale equation, we determine the macroscale equation (Fig. 1.2). However, since the HA method is implemented within a framework of continuum mechanics, it also experiences difficulties when the material properties of micro-inhomogeneous materials need to be obtained. 

The differential equation (7.12) is referred to as the microscale equation, which can be solved under the periodic boundary condition (7.6). 

In conclusion, the homogenization analysis procedure can be stated as follows (1) The microscale equation (7.12) is first solved under the periodic boundary condition, which gives the characteristic function Ai(xi). (2) Using the characteristic function Ai(xi) we then calculate the averaged elastic modulus E. (3) The macroscale equation (7.14) can then be solved, giving the first perturbed term o(- °). Sinee Mi(x ,x1) is calculated by (7.11), the first order approximation of M (x) can be represented as... 

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... 

A pure smectitic clay consists of stacks of clay minerals such as montmorillonite or beidellite. One mineral is a platelet of about 100 x 100 x 1 nm, and several crystals stack parallelly as shown in Fig. 1.7. Keeping this fact in mind, we consider a mieroseale structure, i.e., a unit cell, with flow between two parallel platelets as shown in Fig. 8.2c. If the viscosity r) of the fluid is constant, the solutions of microscale equations (8.20) and (8.21) are given by... 

When the viscosity r] is distributed inhomogeneously, an analytical solution of the microscale equations (8.20) and (8.21) is not possible we therefore perform a finite element calculation. We first give a penalized weak form of (8.20) with the... 

The water in the neighborhood of the smectitic clay surface is structured due to the hydrogen bond, and the viscosity varies inversely with distance from the surface. In this case we can apply the finite element method to solve the microscale equations (8.20) and (8.21), as described previously. 

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. ... 

Under the -periodicity condition we can solve the microscale equations (8.49) and (8.53), and obtain the characteristic functions vf (x ),... 

In order to solve the multiscale HA seepage problem, we proceed as follows First we solve the microscale equations (8.49) and (8.53), and obtain the characteristic functions vf and p. We next determine the permeabilities Kfj and Kij in the meso-domain and macro-domain using (8.54) and (8.60). Then we can solve the macroscale equation (8.59), and obtain the pressure p. By substituting these into... 

As described previously, the viscosity of water in the interlayer space of smectite clay stacks is strictly influenced by the processes at the clay surface. If the viscosity is location-dependent, the seepage problem is usually not solvable through an analytical technique. In this case, we solve the microscale equations (8.49) and (8.53) using a finite element method, and determine the HA-permeability Kij and C-permeability Kfj. 

The result (9.25) is referred to as the microscale equation for the two-scale diffusion problem in porous media. The boundary condition (9.25) is the A" -periodic condition (9.17). 

In certain situations it is useful to have an approximation to the problem with convection and adsorption in the micro-domain by including a term c. If we take the unknown of (9.27), and substitute the characteristic function of (9.24), we obtain the following higher (i.e., second) order microscale equation ... 

For terms 0(s ) Second order characteristic function and microscale equation Since c is a function of only at and c is a function of only a and a and we have the following equation in the micro-domain ... 

Substituting (9.51) into (9.49), we obtain the following microscale equation in the... 

Obtain the second order characteristic function by solving the microscale equation (9.52) under the Z -periodic condition. 

We present here the HA results for diffusion in bentonite and compare these with the experimental results. For a pure smectitic bentonite, Kunipai F , we apply the two-scale HA By solving the microscale equation (9.26) using a finite element method... 

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