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1.
Arkady I. Leonov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2008,59(2):333-359
The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity
effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially
symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of
weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids,
while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the
simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations
reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director
is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field
is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and
director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The
occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number
of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory
functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady
flows in simple shearing and simple elongation. 相似文献
2.
We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P
1, …, P
k
∈ Z[m] in one unknown m with P
1(0) = … = P
k
(0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with
1 \leqslant m \leqslant xe1 \leqslant m \leqslant x^\varepsilon, such that x + P
1(m), …, x + P
k
(m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P
j
= (j − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse
and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number
of points over finite fields in certain algebraic varieties. 相似文献
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