Mathematical Modeling of Nanomaterials - COMSOL.com

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt
multiscale model. We applied a hybrid scheme which is based on Maxwell and Navier-
Stokes equations to paramagnetic liquids with particles of the size of 10nm (ferrofluids). For
a considered area of the body, the Maxwell equations for the static magnetic case were
solved. The vector potential was then coupled to a liquid flow problem described by the
Navier-Stokes equations by a volume force acting on the magnetic liquid. Time-dependent
boundary conditions were used to describe the systolic blood flow regime. The system of
coupled partial differential equations was solved by the finite element method on adaptive
meshes. For a representative geometry which can be extracted by medical imaging, our
hybrid model allowed the study of the hydrodynamics of the magnetic liquid. Moreover, our
model is open for an embedding of a nanoscale model which represents the particle
dynamics. It is concluded that the proposed model is a prerequisite for the optimal
computational choice of the free parameters of magnetic liquids and the external magnetic
field for in vivo applications.
1.2.2 Model equations
Maxwell‘s Equations for the Static Case:
∇ × H = J ∇ ⋅B = 0
Using a constitutive relation: B = µ ( H + M)
These equations can be combined to:
−1
∇× ( µ ∇× A − M)
= J
Where A denotes a magnetic vector potential, B is the Magnetic Flux density, M the
magnetization vector, J is the current density vector and H is the magnetic field vector.
The magnetization (gamma) of the ferrofluid is given by
⎛
⎜
⎛ −
γ =
α arctan⎜
βµ
⎝ ⎝
1
∂A ( x,
y)
⎞ ⎛ ∂ ⎞⎞
−1
A(
x,
y)
⎟,
α arctan⎜
βµ ⎟⎟
∂x
⎠
⎝ ∂y
⎠⎠
Incompressible Navier-Stokes Equation with Magnetic Force Term
∂u
ρ − ∇ ⋅
∂t
T
( ∇u
+ ( ∇u)
) + ρ(
u ⋅∇)
u + ∇p
= F
m
+ F
g
1.2.3 FEMLAB Modeling
Here we show two simulations, Fig. 9 is based on the macroscale model already described.
Fig. 10 shows a particle scale simulation, which is in development. It is noticeable that above
a time t, the magnetic nanoparticle carriers in the flow are disturbed in the vicinity of a
magnetic field.