Case Of Free Boundary Conditions Research Articles

Point mobility of a cylindrical plate incorporating a tapered hole of power-law profile

The paper describes the results of experimental measurements of point mobility carried out on circular plates containing tapered holes of quadratic power-law profile with attached damping layers. The obtained results are compared to the developed numerical model, as a means of validation. The profiles of the tapered hole in the plates are designed to replicate near zero reflection of quasi-plane waves from a tapered hole in geometrical acoustics approximation, also known as acoustic black hole effect. The driving point mobility measurements are provided, showing a comparison of the results for a constant thickness circular plate, a constant thickness plate with a layer of damping film applied and a plate with a quadratic power-law profile machined into the center, which is tested with a thin layer of elastic damping material attached. The results indicate a substantial suppression of resonant peaks, agreeing with a numerical model, which is based on the analytical solution available for the vibration of a plate with a central quadratic power-law profile. The paper contains results for the case of free boundary conditions on all edges of the plates, with emphasis placed on the predictions of resonant frequencies and the amplitudes of vibration and loss factor.

Density profiles in open superdiffusive systems

We numerically solve a discretized model of Lévy random walks on a finite one-dimensional domain with a reflection coefficient r and in the presence of sources. At the domain boundaries, the steady-state density profile is nonanalytic. The meniscus exponent μ, introduced to characterize this singular behavior, uniquely identifies the whole profile. Numerical data suggest that μ = α/2 + r(α/2 - 1), where α is the Lévy exponent of the step-length distribution. As an application, we show that this model reproduces the temperature profiles obtained for a chain of oscillators displaying anomalous heat conduction. Remarkably, the case of free-boundary conditions in the chain corresponds to a Lévy walk with negative reflection coefficient.

Growth and morphology transitions in anisotropic disordered media

Depinning transitions of an interface between two magnetic domains are studied within the random-field Ising model in three dimensions. The fields are uniformly distributed with width $2\ensuremath{\Delta}$ in units of the exchange coupling. The morphology of the interface at the depinning transition varies with $\ensuremath{\Delta}$. Self-similar morphologies characteristic of percolation are observed at large disorder, and self-affine interfaces with roughness exponent 2/3 are formed at intermediate disorder. The multicritical point at the self-similar to self-affine transition is analyzed through finite-size scaling. The critical disorder ${\ensuremath{\Delta}}_{c}$ is substantially higher than previous estimates and lies near a local maximum in the depinning field ${H}_{c}(\ensuremath{\Delta})$. The scaling exponent for the correlation length near ${\ensuremath{\Delta}}_{c}$ is consistent with results for a Gaussian distribution of random fields, suggesting that the form of the distribution does not change the universality class. However there is another transition to faceted growth at small disorder that is not present for Gaussian distributions. The critical field for faceted growth depends on orientation and boundary conditions. There is a unique ``hard'' growth direction, (001), for periodic boundary conditions, whose ${H}_{c}$ is larger than that for all other orientations. For the more physical case of free boundary conditions, growth leads to facets with this hard orientation. This inhibits further growth and increases ${H}_{c}$ with respect to the value for periodic boundary conditions.

The shape of invasion percolation clusters in random and correlated media

The shape of two-dimensional invasion percolation clusters are studied numerically for bothnon-trapping (NTIP) and trapping (TIP) invasion percolation processes. Two differentanisotropy quantifiers, the anisotropy parameter and the asphericity, are used for probingthe degree of anisotropy of clusters. We observe that, in spite of the difference in scalingproperties of NTIP and TIP, there is no difference in the values of anisotropyquantifiers of these processes. Furthermore, we find that, in completely randommedia, the invasion percolation clusters are on average slightly less isotropic thanstandard percolation clusters. Introducing isotropic long-range correlations into themedia reduces the isotropy of the invasion percolation clusters. The effect is morepronounced for the case of persisting long-range correlations. The implicationof boundary conditions on the shape of clusters is another subject of interest.Compared to the case of free boundary conditions, IP clusters of conventionalrectangular geometry turn out to be more isotropic. Moreover, we see that inconventional rectangular geometry the NTIP clusters are more isotropic than the TIPclusters.

Backward Uniqueness for Thermoelastic Plates with Rotational Forces

This paper provides a backward uniqueness theorem for thermoelastic plate models which account for rotational forces, under all sets of canonical boundary conditions, including the most challenging case of so-called free boundary conditions. The proof is abstract and accomodates space variable coefficients in the model. This result is derived in two steps. First, in Section 3, a new backward uniqueness theorem for strongly continuous semigroups is given, which is of interest in itself. It is based on the assumption that the resolvent operator of the generator be bounded on suitable rays of the complex plane. Its proof uses the Phragmen-Lindelof Theorem. Next, the paper verifies a fortiori the required resolvent conditions, under all sets of canonical boundary conditions. The explicit proof (in Section 4) considers the most demanding case of free boundary conditions. An abstract version of this proof, and a corresponding backward uniqueness result, are then noted in Section 5, which gives the most general result of this paper. It covers thermoelastic wave equations as well.

Generic aspects of axonemal beating

We study the dynamics of an elastic rod-like filament in two dimensions, driven by internally generated forces. This situation is motivated by cilia and flagella which contain an axoneme. These hair-like appendages of many cells are used for swimming and to stir surrounding fluids. Our approach characterizes the general physical mechanisms that govern the behaviour of axonemes and the properties of the bending waves generated by these structures. Starting from the dynamic equations of a filament pair in the presence of internal forces we use a perturbative approach to systematically calculate filament shapes and the tension profile. We show that periodic filament motion can be generated by a self-organization of elastic filaments and internal active elements, such as molecular motors, via a dynamic instability termed Hopf bifurcation. Close to this instability, the behaviour of the system is shown to be independent of many microscopic details of the active system and only depends on phenomenological parameters such as the bending rigidity, the external viscosity and the filament length. Using a two-state model for molecular motors as an active system, we calculate the selected oscillation frequency at the bifurcation point and show that a large frequency range is accessible by varying the axonemal length between 1 and 50 µm. We discuss the effects of the boundary conditions and externally applied forces on the axonemal wave forms and calculate the swimming velocity for the case of free boundary conditions.

The influence of finite dimensions on the static ordering of the S*Cphase in an electric field

Abstract A ferroelectric liquid-crystalline sample of a finite length along the helical axis is studied in an external electric field applied perpendicular to the helical axis. By taking into account the linear coupling to the field, the equilibrium state is found for the case of free boundary conditions. Just below the critical field which induces the transition into the homogeneous S*C phase the domain-like structure appears as in the case of an infinite sample. The helical period in a finite sample is not a continuous function of the field, but it increases in finite jumps. We show that the finiteness of the sample has also an influence on the dielectric response at zero field.

Boundary correlation functions and interface formation in two-dimensional solids

Two-point correlation functions at the rectilinear boundary of a two-dimensional crystal half-space on a reference lattice are studied for the case of free boundary conditions. We show that the power of the decay law of the positional correlation function at the surface satisfies η surf = 2η bulk, within the approximation that dislocation pair correlations are taken into account. Dislocations at the boundary form a quasi-one-dimensional plasma for T ≠ 0 and this may lead to a dynamic roughening of the surface. Formation of dynamic interfaces in grain boundaries and near free boundaries is discussed in terms of 1-dimensional charged gas models with long-range interaction. Melting of boundary layers at T surf c< T bulk c is suggested to be possible only if topological effects of moving dislocations on the reference lattice are taken into account.

Qualitative study of the boundary correlation function in the two-dimensional XY-model

Abstract The two-point correlation function at the rectilinear boundary of the two-dimensional half-space is studied for the XY-model for the case of free boundary conditions. It is shown that defects of vortex-type at the boundary form a one-dimensional plasma for T ≠ 0, but that this does not lead to exponential decay of the boundary correlation function for T

On the convected linear stability of a viscoelastic Oldroyd B fluid heated from below

We investigate the linear stability problem of convection by the general Oldroyd B fluid and its Maxwell limit in the presence of rigid or free boundaries and fixed temperature or fixed flux. Comparison with recent results by Rosenblat [9] for the analytically accessible case of free boundary conditions shows a qualitative similarity in the shape of the neutral stability curves. But while Newtonian and Jeffreys (general Oldroyd B) fluids are sharply stabilized by the presence of rigid boundaries, the Maxwell fluid is largely unaffected at even moderately large Prandtl number. The reasons for this are discussed. Also, a discrepancy between the earlier works by Vest and Arpaci [3], and Sokolov and Tanner [4], which treat the case of a Maxwell fluid, is found to be due to algebraic error, and not multivaluedness of the stress-strain rate relation as earlier suggested by Eltayeb [6].