Page 32 - Computational Modeling in Biomedical Engineering and Medical Physics

18 Computational Modeling in Biomedical Engineering and Medical Physics

It is worth noting that any unsteady diffusion process governed by a parabolic PDE is
driven by such a diffusion time constant, which depends on the material properties and the
size of the physical system only, and the length of the path in the diffusion direction—the
largest of the diffusion paths if the process is multidirectional. For instance, in the magnetic
field analysis in the unsteady diffusion problem of electroconductive media, known also as
the eddy currents working conditions, the parabolic PDE to solve may be presented as

@H 1
5 ΔH; ð1:21Þ
@t μσ
where μ [H/m] is the magnetic permeability, σ [S/m] is the electrical conductivity,
and H [A/m] is the magnetic field strength. An analysis similar to Eq. (1.20) yields the
magnetic diffusion time constant (scale)

L 2 L 2
τ mag 5 5 ; ð1:22Þ
μσ α mg
21 2
which introduces the magnetic diffusivity, α mag 5 μσð Þ m =s , or the magnetic Reynolds
number.
Transport time scale characterizes, for instance, the unsteady diffusion advection heat flow of
a system without internal heat sources (Bejan, 1984) presented through the energy equation

@T
ρc 1 uUrÞT 5 r krTÞ; ð1:23Þ
ð
ð
@t
and the momentum equation (Navier Stokes)
@u
2
ρ 1 uUrÞu 52rp 1 ηr u; ð1:24Þ
ð
@t
for a Newtonian fluid in incompressible, laminar flow rUu 5 0Þ. Here p [Pa] is the
ð
pressure, u [m/s] is the velocity field, and η [Pa s] is the dynamic viscosity.
The momentum Eq. (1.24) may be scaled to indicate the order of magnitude balance

L P 0 1 η 1 1
; 1B 2 ; 5 ; or 1; 1B1; ; ð1:25Þ
U 0 τ transport ρU Re L ρ U 0 L Re L
0
where U 0 is a reference velocity [same in the heat transfer problem Eq. (1.23), if the
two problems are coupled], P 0 is a reference pressure, L is the size of the physical
domain [same in Eq. (1.20), if the two problems are coupled], and Re L is the nondi-
mensional Reynolds group based on the length scale L. Provided that U 0 is known
2
(e.g., out of the boundary conditions) one may chose P 0 5 ρU , and then define
0
τ transport 5 L=U 0 , which is a transport (velocity) time scale.