Page 18 - Computational Modeling in Biomedical Engineering and Medical Physics

4 Computational Modeling in Biomedical Engineering and Medical Physics

system, these energies become energy densities, which are intensive, local quantities,
independent on the amount of substance in the system.
In the limits of reversible transformations, without heat transfer, the infinitesimal
work transfer interaction in Eq. (1.2), δW rev , is the scalar product force displacement
(deformation)
δQ 5 0 X
δW rev 5 2 Y i dX i ; ð1:5Þ
i
where Y i are the components (projections) of a generalized force with respect to
the generalized coordinate, of X i components. The minus sign denotes that the
interaction is performed by the environment upon the system, and not con-
versely. The subscript ( ) rev denotes that the process is reversible, that is, it is
_
without heat transfer. Such transformations may be either adiabatic, Q 5 0, or dia-
terman, @T 5 0(n is the coordinate associated with the outward pointing normal
@n
to the surface).
Eqs. (1.3) and (1.5) then yield
@E
Y i 52 ; ð1:6Þ
@Xi
which shows off that the generalized force is actually the gradient, (a vector quantity)
of the energy (a scalar quantity). This is the theorem of generalized forces and it provides a
method to compute forces that produce work. If the control volume is a continuum
medium then Eq. (1.6) yields the specific, per volume basis (or body) force as the gra-
dient of an energy density.
For isothermal internal processes, where the internal work interactions are of
electrical and or magnetic nature, the theorem of generalized forces states that
the electrical and, or magnetic body forces are the derivatives (the gradients) of
the electrical and, or magnetic energy density, respectively. For instance, for
3
magnetic linear media, the magnetic energy density, e mag [J/m ], is as follows
(Mocanu, 1981):

BH
e mag 5 ; ð1:7Þ
2
where B [T] is the magnetic flux density and H [A/m] is magnetic field strength.
Eqs. (1.6) and (1.7) then yield the magnetic body force

f mag 5 re mag : ð1:8Þ
This approach is used for instance in the magnetic drug targeting analysis
(Chapter 6: Magnetic Drug Targeting) to evaluate the magnetization body forces.