Maxwell Constants Research Articles

Poincaré–Friedrichs type constants for operators involving grad, curl, and div: Theory and numerical experiments

We give some theoretical as well as computational results on Laplace and Maxwell constants, i.e., on the smallest constants cn>0 arising in estimates of the form |u|L2(Ω)≤c0|gradu|L2(Ω),|E|L2(Ω)≤c1|curlE|L2(Ω),|H|L2(Ω)≤c2|divH|L2(Ω).Besides the classical de Rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general relativity as well using the general functional analytical concept of Hilbert complexes. We consider mixed boundary conditions and bounded Lipschitz domains of arbitrary topology. Our numerical aspects are presented by examples for the de Rham complex in 2D and 3D which not only confirm our theoretical findings but also indicate some interesting conjectures.

On the Maxwell and Friedrichs/Poincaré constants in ND

We prove that for bounded and convex domains in arbitrary dimensions, the Maxwell constants are bounded from below and above by Friedrichs’ and Poincare’s constants, respectively. Especially, the second positive Maxwell eigenvalues in ND are bounded from below by the square root of the second Neumann-Laplace eigenvalue.

On Constants in Maxwell Inequalities for Bounded and Convex Domains

It is shown that for a bounded and convex domain Ω ⊂ ℝ3 , the Maxwell constants are bounded from below and above by the Friedrichs and Poincare constants of Ω, respectively. Bibliography: 14 titles.

On the Maxwell constants in 3D

Dedicated to Martin Costabel on the occasion of his 65th birthdayUsing tools from functional analysis, we show that for bounded and convex domains in three dimensions, the Maxwell constants are bounded from below and above by Friedrichs' and Poincaré's constants. Copyright © 2014 John Wiley & Sons, Ltd.

On Maxwell's and Poincaré's constants

We prove that for bounded and convex domains in three dimensions, the Maxwell constants are bounded from below and above by Friedrichs' and Poincaré's constants. In other words, the second Maxwell eigenvalues lie between the square roots of the second Neumann-Laplace and the first Dirichlet-Laplace eigenvalue.

Optical and rheological properties of the dibutyl phthalate-decalin binary liquid system

The flow birefringence method, diffusometry, and viscosimetry were used to study optical and rheological properties of dibutyl phthalate-decalin liquid mixtures of various compositions. The molecular refraction was demonstrated to obey the additivity rule, while the Maxwell constant was observed to deviate slightly from this rule. The nonmonotonic dependence of the molecular mobility of the medium on its composition was revealed and examined. The effects observed were interpreted. According to the diffusiometry and viscosimetry data obtained, the mean size of the mobile units in the system attains its maximum value at a dibutyl phthalate mole fraction of 0.2.

Temperature‐influenced ordering of mesogens in comb‐like liquid‐crystalline poly(methylsiloxane) macromolecules with side groups consisting of “phenyl benzoate” moieties in solution

AbstractComb‐like poly(methylsiloxane) macromolecules P4, P6, and P11, with side groups consisting of “phenyl benzoate” moieties and different alkyl spacers (from 4 to 11 C‐atoms) between the backbone and the mesogenic side groups have been synthesized and investigated as well as low‐molar‐mass compounds M4, M6, and M11, as model molecules analogous to side chains of the macromolecule. The liquid‐crystalline phase transition temperatures have been determined. Mobility of the mesogens and the macromolecule as a whole in dilute solution was studied in external fields (electric and mechanical shear field). In dilute solution under the influence of a sinusoidal electric field, the polymer behaviour is typical of flexible‐chain polymer molecules where each mesogenic unit is oriented independently of the other ones. This is in contrast to the effect observed in a mechanical shear flow field. The Maxwell constant in positive for the solutions of compounds M4, M6, M11 and negative in polymers P4, P6, P11. This indicates the orientation of the macromolecule as a whole in the shear flow field, the mesogenic group main axis being oriented in the direction preferably normal to the macromolecular axis of primary polarizability. Temperature effects are also discussed.

Direct connection between anisotropic optical properties, polarizability and rheological behaviour of single-phase glass melts

Rheological properties of glass melts can be determined by the method of uniaxial deformation of cylindrical glass specimens in tensile and compressive experiments. The glass melts investigated show non-Newtonian flow behaviour at high deformation rates due to alteration of the isotropic structure to an anisotropic structure. If this anisotropic state is frozen-in by a controlled cooling process under constant load, the glass samples show a permanent frozen-in birefringence very similar to that of an optically uniaxial crystal which represents - at least partly - the anisotropic structure of the glass melt during the flow process. This permanent birefringence varies in a characteristic manner analogously to the load- and deformation-dependent rheological properties. The two properties are connected to each other very closely. Under non-Newtonian flow conditions, the specific birefringence or the Maxwell constant is much larger than under Newtonian flow conditions, indicating a principal structural change of the glass melts from a slightly isotropic structure with frozen-in strains to an essentially larger anisotropic one with oriented flow units or network fragments, respectively, depending on the degree of interconnection of the tetrahedra. Characteristic examples are given for float, metasilicate and metaphosphate glass melts.

Structural aspects of highly deformed glass melts

Under normal and extreme conditions of deformation it is possible to introduce anisotropies and orientations into isotropic glass melts which are essentially of structural origin. This was shown in some systematic investigations on structure sensitive properties of glass fibers, on flow birefringence of glass melts, on non-Newtonian flow behaviour and on the stress optical coefficient above Tg. Characteristic indicators will be used in order to get conclusions on the structure of various types of glass melts (silicates, borates, phosphates and chalkogenides) along their extended viscosity temperature range, where the Maxwell constant, also called specific birefringence, is playing a central role.

Structural isomerism and flow birefringence of octanol, nonanol and decanol

The flow birefringence over a temperature range 10–40°C has been measured for structural isomers of octanol, nonanol and decanol. The Maxwell constants and stress-optical coefficients are found to be strongly dependent upon the position of the OH group along the carbon skeleton, being smaller when this group is nearest the middle of the molecule. The results are compared with predictions of molecular orientation theories of flow birefringence.

Variation of flow birefringence with temperature in some liquid n-alkanes

Flow birefringence measurements have been carried out on nine liquid n-alkanes over the temperature range 15–50°C. Five of these liquids have been investigated previously but at one temperature; results on the remainder represent new determinations. Our results invalidate a proposal that the optical polarizability parameter Γ is negative. The experimentally-determined temperature variations of Maxwell constant are compared with those calculated from the Raman-Krishnan theory and reveal a molecular weight anomaly in the region of the member n-tetradecane (C14H30). A qualitative explanation is proposed in terms of the presence of rotational isomers and their deformation in flow.

Streaming Birefringence and Optical Relaxation Time of Vanadium Pentoxide Sols

Maxwell constant and optical relaxation time is studied for V2O5 sols having concentrations which range from 0.01 to 0.1 volume percent using mercury green, 5461 A, and mercury red light, 6000 to 6800 A. These quantities were measured in a device which sheared the sols between two coaxial discs, that is, in a direction which corresponds to the radial one in a Couette device. The Maxwell constant ranged from about 1.5×10−2 to about 4.5×10−6 cm2/dyne for freshly made sols and from about 8.0×10−2 to 2.9×10−5 cm2/dyne for the same sols when they were 29 days old. It is shown that the Maxwell constant for these sols was an exponential function of concentration and that the optical relaxation time and Maxwell constant could be related for them. Because the Maxwell constant is an exponential function of sol concentration, it is suggested that the expression for the rotary diffusion constant of an infinitely dilute sol of rod-shaped particles can be made applicable to more concentrated sols by multiplying it by the term (esC−1)/sC, where C is concentration and s is the slope of the curve relating sol concentration to Maxwell constant, to account for the change in particle interaction with change in sol concentration. The viscosity and spectral transmission of these sols were also checked. It was found that the viscosity ranged from 1.0 cp for the most dilute sol to about 1.25 cp for the sol of concentration 0.1 volume percent. These sols were found to have a transmission which peaked at about 6800 A and dropped rapidly on either side of this value.