On the Field-Induced Transport of Magnetic Nanoparticles in Incompressible Flow: Existence of Global Solutions
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Journal of Mathematical Fluid Mechanics
On the Field-Induced Transport of Magnetic Nanoparticles in Incompressible Flow: Existence of Global Solutions G. Gr¨ un
and P. Weiß
Communicated by Y. Giga
Abstract. We prove global-in-time existence of weak solutions to a pde-model for the motion of dilute superparamagnetic nanoparticles in fluids influenced by quasi-stationary magnetic fields. This model has recently been derived in Gr¨ un and Weiß(On the field-induced transport of magnetic nanoparticles in incompressible flow: modeling and numerics, Mathematical Models and Methods in the Applied Sciences, in press). It couples evolution equations for particle density and magnetization to the hydrodynamic and magnetostatic equations. Suggested by physical arguments, we consider no-flux-type boundary conditions for the magnetization equation which entails H(div, curl)-regularity for magnetization and magnetic field. By a subtle approximation procedure, we nevertheless succeed to give a meaning to the Kelvin force (m · ∇)h and to establish existence of solutions in the sense of distributions in two space dimensions. For the three-dimensional case, we suggest two regularizations of the system which each guarantee existence of solutions, too. Mathematics Subject Classification. 35D05, 35Q35, 65M60. Keywords. Ferrohydrodynamics, Magnetic nanoparticles, Magnetization, System of partial differential equations, Existence results in continuous and discrete settings.
1. Introduction Given two domains Ω ⊂⊂ Ω ⊂⊂ Rd , d ∈ {2, 3}, we are concerned with existence results for the model ρ0 ut + ρ0 (u · ∇)u + ∇p − div(2ηDu) μ0 curl(m × (α1 h + β2 ha )), = μ0 (m · ∇)(α1 h + β2 ha ) + 2
(1.1a)
ˆ =:h
div u = 0,
(1.1b)
ct + u · ∇c + div(cVpart ) = 0,
(1.1c)
Vpart
f2 (c) f2 (c) ∇g (c) + Kμ0 2 (∇ (α1 h + β2 ha − α3 m))T m, = −KD c c
(1.1d)
ˆ =:b
− ΔR = div(m),
(1.1e)
mt + div(m ⊗ (u + Vpart )) − σΔm =
1 1 curl u × m − (m − χ(c, h)h) 2 τrel
(1.1f)
in Ω × (0, T ) and −ΔR = 0
(1.1g)
in (Ω \Ω) × (0, T ). Supplemented with boundary conditions u=0 0123456789().: V,-vol
on ∂Ω × [0, T ],
(1.2a)
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G. Gr¨ un, P. Weiß
JMFM
c Vpart · ν = 0
on ∂Ω × (0, T ],
(1.2b)
(Vpart · ν)(m − (m · ν)ν) − σ curl m × ν = 0
on ∂Ω × (0, T ],
(1.2c)
(Vpart · ν)(m · ν) − σ div m = 0
on ∂Ω × (0, T ],
(1.2d)
transmission conditions [∇R + m] · ν = 0
on ∂Ω × (0, T ],
∇R · ν = ha · ν
on ∂Ω × (0, T ],
(1.2e) (1.2f)
and initial conditions u(·, 0) = u0 , 0
(1.3a)
c(·, 0) = c ,
(1.3b)
0
(1.3c)
m(·, 0) = m ,
this system has been proposed in [17] up to a change of boundary conditions to model the motion of dilute solutions of superparamagnetic nanoparticles influenced by external magnetic fields. Note that in points (x, t) ∈ ∂Ω × (0, T ] such that c(x, t) = 0, the boundary conditions (1.2b)–(1.2d) simplify to become Vpart · ν = 0, curl m × ν = 0, and div m = 0. These are the boundary conditions which have been used in the section on numerics in [17]. Here, (u, p) denote the hydrodynamic variables of the carrier fl
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