These classical problems (straight capillary, single diameter globular morphologies, and 1D layered systems) are already present examples of pure statements of the test-problem for more then hundred years. Few generations of researchers tried to solve these problems, but did not obtain a complete solution. Further, the conjugate problem of heat transport in capillary systems has neither been stated fully nor correctly. The studies referred to below are the solutions for both scales and directly explain how these problems can be treated using the general VAT governing equations.

To help those who are working in the field, or are just interested, obtain familiarity with the simplest multi-scale problems and give them an understanding how to correctly state the VAT problem and what are the possibilities to treat it.

Few fundamental features of treatment of these morphologies in fluid mechanics are given in - Fluid Mechanics::Classical Problems

Capillary Morphology

The most simple and at the same time trustworthy morphology as we all used is

Fig. 1 Capillary morphology model of porous medium: a bundle of parallel pores embedded in solid

Fig. 2 Temperatures local and upper scale non-local calculated for a system of two parallel pores with different diameters within the straight parallel capillary morphology model of porous medium

Fig. 3 Lower scale temperature distribution in and around the separate pore

Fig. 4 Upper scale volume averaged temperature distribution

Fig. 5 Estimation of terms in the upper scale fluid phase heat transfer VAT equation

Fig. 6 Balance of the upper scale heat transfer VAT equation in fluid phase

ExactClosure-IMECE-2001-24261 ( 589k)

Globular Morphology

There can be a variety of location modes for globular particles exposed into the space. When a certain volume is fully occupied by an unconsolidated highly porous medium composed of uniform packed spheres an immediate interests lie in the prediction of phenomena pertaining to the case of large heat conductivity coefficients ks/kf ratios. The specific case and physical parameter values considered well describe that of a steel spherical-bead matrix with incompressible air entering the interstices by forced non-Darcy convection.
ENHNC-2002 ( 516k)

Few results from the rather substantial study by Travkin and Kushch exploiting the exact closure and solutions for globular morphologies can be seen in the following excerpt - Globular Morph

and in the following subsection in Heterogeneous Electrodynamics section at this web site -
Electrodynamics::Globular Morphology Two Scale Electrostatic Exact Solutions

Layered and Superlattice Morphology

This is a simple 1D morphology, which, nevertheless have a great number of applications in various engineering fields -

Fig. 7 Layered regular 1D medium (2 different component layers) lower scale flux flow with perfect interface conductance

Fig. 8 Layered irregular 1D medium (n different component layers ) lower scale flux flow with perfect interface conductance

Fig. 9 Layered regular 1D medium (2 different component layers) lower scale flux flow with the second layer globulars of 3 kinds of amorphous substances (2 gradient and 1 isotropic phases)

Fig. 10 The 3rd lower scale averaged EM field {q₁}_r21 reflected from the interface between the second and first layer

It is interesting to note that this the most reviewed and analyzed morphology when being considered as a two - scale problem is not going to support the known definitions and textbook formulae for the effective coefficients and bulk properties of the medium - see analysis in our - HEAT AND CHARGE CONDUCTIVITIES IN SUPERLATTICES - TWO-SCALE MEASURING AND MODELING ( 468k)

Exact Solutions for Layered and Superlattice Morphology

Doing further the research, for some of superlattice upper (bulk) scale problems recently had been found the exact solutions. The linear problems for electrostatics and thermal conductivity problems were solved in analytical forms for both scales. Few results presented in the section devoted to Heterogeneous Electrodynamics at this web site - Electrodynamics::When the 2x2 is not going to be 4 - What to do?