Abstract
We prove global existence of weak solutions to regularized versions of balance equations representing the dynamics over a torus of complex fluids, with microstructure described by a vector field taking values in the unit ball. Regularization is offered by the presence of second-neighbor microstructural interactions and our choice of filtering the balance of macroscopic momentum by inverse Helmholtz operator with unit length scale.
Highlights
We prove global existence of weak solutions to regularized versions of balance equations representing the dynamics over a torus of complex fluids, with microstructure described by a vector field taking values in the unit ball
For sufficiently differentiable maps u : T2 × [0, T ] → R2 and ν : T2 × [0, T ] → S 2, with T2 a torus and S 2 the unit sphere, we have shown in reference [12] that the system ut + (u · ∇)u − ∆u + ∇π = −∇ · (∇ν ∇ν) − ∇ν ∆νt, ∇ · u = 0, ∆νt + ∆((u · ∇)ν) − ∆2ν = νt + (u · ∇)ν + |∇ν|2ν − ∆ν, reasonably describes the dynamics over T2 of oriented fluids, a representation in which we account for second-neighbor director interactions in a minimalistic way, the one giving us sufficient amount of regularity to allow existence of a certain class of weak solutions
In the balance of microstructural actions governing the evolution of ν, an hyper-stress behaving like ∇2ν accounts for second-neighbor interactions; it enters the equation through its double divergence, which generates the term ∆2ν
Summary
For sufficiently differentiable maps u : T2 × [0, T ] → R2 and ν : T2 × [0, T ] → S 2, with T2 a torus and S 2 the unit sphere, we have shown in reference [12] that the system ut + (u · ∇)u − ∆u + ∇π = −∇ · (∇ν ∇ν) − ∇ν ∆νt, ∇ · u = 0, ∆νt + ∆((u · ∇)ν) − ∆2ν = νt + (u · ∇)ν + |∇ν|2ν − ∆ν, reasonably describes the dynamics over T2 of oriented (i.e., polarized or spin) fluids, a representation in which we account for second-neighbor director interactions in a minimalistic way, the one giving us sufficient amount of regularity to allow existence of a certain class of weak solutions. We have explicitly underlined in reference [12] the terms neglected in the previous balance equations with respect to a complete representation of second-neighbor director interactions, and their contribution to the Ericksen stress. All these aspects will be matter of a forthcoming work
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