Page 30 - Computational Modeling in Biomedical Engineering and Medical Physics

16 Computational Modeling in Biomedical Engineering and Medical Physics

express the interaction rates. The particular forms of these “assumed” relations are found
empirically or based on a different theory. For example, the diffusion (conduction) laws such
as Fourier, Ohm, and Fick empirical laws.
In this respect, Onsager relations (Bejan, 1988) provide for a succinct mathematical
representation of these relations of reciprocity, which presents a unified analytically
consistent expression of irreversible-flow processes. Most notably, and expected after
all, material properties such as the electrical conductivity, mass diffusivity, thermal con-
ductivity, viscosity, and thermoelectric power have their “trace” as “coefficients” in
these relations, relating the fluxes with the gradients.
Couplings between the PDEs that present the physics may occur through the
material properties (e.g., the properties are functions of temperature, Chapters 7:
Magnetic stimulation, and Chapter 8: Hyperthermia and ablation (Thermotherapy
methods), heat source terms (e.g., Joule, dielectric heating, or body forces (e.g., mag-
netic forces in MDT, Chapter 6: Magnetic Drug Targeting).
The physical and mathematical models then sum up all laws of Physics needed to
describe these phenomena in a consistent body of equations that has then to be solved.
It should be recognized that, again, a qualitative analysis has to be performed in the
first place to infer the strengths of such couplings with the aim to reduce the complex-
ity of the problem to be solved without losing the physical reality of the solution and
the meaningfulness of the results. The evaluation of the time and space scales, penetration
depths is part of this analysis.

1.8 Time and space scales

The physical phenomena that define a specific problem may evolve at different paces.
Their time scales may be estimated a priori, analytically, in the order of magnitude
sense, considering the time-dependent PDEs of their associated mathematical models.
The underlying mechanisms of evolution range from unsteady diffusion (conduction),
to propagation or transport (advection, convection) processes, and an earlier stage
qualitative analysis aims at evaluating their orders of magnitude, which may differ con-
siderably. Constitutive, matter properties, system structure, and its relative motion con-
tribute in defining their size.
Diffusion time scales are governing in unsteady conduction heat transfer, mass trans-
fer, and low-frequency EMFs. For instance, the unsteady conduction (diffusion) heat
3
transfer problem in immobile media with an internal heat source, qw [W/m ], is
described by the first law of thermodynamics (the energy equation; Bejan, 1993)

@T
ρc 5 r krTÞ 1 qw; ð1:17Þ
ð
@t