Page 53 - Computational Modeling in Biomedical Engineering and Medical Physics

Physical, mathematical, and numerical modeling 39

Scalar fields
The internal thermodynamic disequilibrium of a system may be accompanied by gra-
dients and fluxes of the scalar, state quantities (e.g., temperature, pressure, mass density,
concentration, potential), and state, vector fields (e.g., EMF, flow field, stress field).
Scalar fields are presented (visualized) using surfaces (lines in 2D models), which
are loci of constant value scalars. For instance the electrical potential field, V(P),is
visualized using equipotential surfaces of V(P) 5 const. The regional distribution of the
equipotential surfaces, or how fast the scalar varies along a specific direction, is pre-
sented by the directional derivatives. A particular directional derivative, oriented in the
direction of scalar growth, and orthogonal to the local equipotential surface, is the
gradient

@V
rV 5 n ; ðA1:1Þ
@n
where r 5 @ i 1 @ j 1 @ k (in Cartesian coordinates) is Hamilton’s vector operator
@x @x @x
(nabla), and n is the unit vector orthogonal to the local equipotential surface pointing
in the direction of V(P) increase.
The gradient is an invariant with respect to the system of coordinates. It may also
be introduced using Gauss-divergence theorem for the volume integral of a gradient
field (Purcell, 1984; Mocanu, 1981)

H
Σ VndA
rV 5 lim ; ð1:2Þ
Δv Σ -0 Δv Σ
where v Σ is the volume and Σ is its boundary.
Gradients to drive fluxes are postulated by the fundamental law of thermodynamics for
the systems outside the internal equilibrium (Bejan, 1988). Onsager relations present
the particular analytical forms of these relations, which turn to be laws that relate gra-
dients and fluxes (e.g., Fourier, Fick, Ohm, Peltier, Seebeck). These relations may
provide for the basis of analysis of the systems with internal multiple conjugated and
coupled gradients and fluxes.

Vector fields
The fundamental theorem of vector fields states that a vector field, F(r), is uniquely deter-
mined everywhere in the ROI of volume v Σ , bounded by the closed surface Σ, if and
only if its divergence, ρ(r), and curl (rotor), R(r), are known as follows (Mocanu,
1981):
r 3 F 5 Rr ðÞ; rUF 5 ρ r ðÞ; ðA1:3Þ