Mathematical Modeling of Nanomaterials - COMSOL.com

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Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt Figure 5 Figure 6 In Fig. 7 we show the density of the sample in dependence on the temperature and for different proportions of the hardening agent. In Fig. 8 we show the loss of mass in the sample in dependence on the temperature and for different proportions of the hardening agent. Figure 7 Figure 8 1.2 Magnetic nanoparticles for in vivo applications: A Numerical modeling study 1.2.1 Background In vivo applications of biocompatible magnetic nanoparticles in a carrier liquid controlled by an external magnetic field from outside the body has recently been proposed for specific drug delivery such as in locoregional cancer therapies or the occlusion aneurysms. They can also be used as guided contrast agents in myocardial imaging after myocardial infarction. However, the choice of the optimal clinical setting still remains a challenge for every of the mentioned applications. A numerical heterogeneous multiscale model can be used for the optimal a priori determination of the free parameters and might help to overcome this problem. In this study, we are going to present an approach to the implementation of such a
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.
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