Synchronization transitions induced by partial time delay in a excitatory–inhibitory coupled neuronal network

Information transmission delays are an inherent factor of neuronal systems as a consequence of the finite propagation speeds and time lapses occurring by both dendritic and synaptic processes. In real neuronal systems, some delay between two neurons is too small and can be ignored, which results in partial time delay. In this paper, we focus on investigating influences of partial time delay on synchronization transitions in a excitatory–inhibitory (E–I) coupled neuronal networks. Here, we suppose time delay between two neurons equals to \(\tau \) with probability \(p_{\mathrm{delay}}\) and investigate effect of partial time delay on synchronization transitions of the neuronal networks by controlling \(\tau \) and \(p_{\mathrm{delay}}\) under three cases. In these three cases, excitatory synapses are always considered to delayed with probability \(p_{\mathrm{delay}}\), while inhibitory synapses are considered to be without delays (case I), delayed with probability \(p_{\mathrm{delay}}\) (case II), and always delayed (case III), respectively. It is revealed that, in the first two cases, partial time delay has little influences on synchronization of the neuronal network for small \(p_{\mathrm{delay}}\), while it could induce synchronization transitions at \(\tau \) around integer multiples of the period of individual neuron T when \(p_{\mathrm{delay}}\) is large enough, while in the case III, partial time delay could induce synchronization transitions at \(\tau \) being around odd integer multiples of T / 2 for small \(p_{\mathrm{delay}}\) and at \(\tau \) being around integer multiples of T for large \(p_{\mathrm{delay}}\). Most interesting observation is that partial time delay could induce frequent synchronization transitions at \(\tau \) being around integer multiples of T / 2 for intermediate \(p_{\mathrm{delay}}\). Moreover, effect of rewiring probability on synchronization transitions induced by partial time delay has been discussed. It is found that synchronization transitions induced by partial time delay are robust to rewiring probability for large \(p_{\mathrm{delay}}\) under the three cases.     

Flow field in the wake of a bluff body driven through a steady recirculating flow

The wake produced by a bluff body driven through a steady recirculating flow is studied experimentally in a water facility using particle image velocimetry. The bluff body has a rectangular cross section of height, \(H\), and width, \(D\), such that the aspect ratio, AR = H/D, is equal to 3. The motion of the bluff body is uniform and rectilinear, and corresponds to a Reynolds number based on width, Re _D  = 9,600. The recirculating flow is confined within a hemicylindrical enclosure and is generated by planar jets emanating from slots of width, \(h\), such that \(Re_h=500\). Under these conditions, experiments are performed in a closed-loop facility that enables complete optical access to the near-wake. Velocity fields are obtained up to a distance of \(13D\) downstream of the moving body. Data include a selection of phase-averaged velocity fields representative of the wake for a baseline case (no recirculation) and an interaction case (with recirculation). Results indicate that the transient downwash flow typically observed in wakes behind finite bodies of small aspect ratio is significantly perturbed by the recirculating flow. The wake is displaced from the ground plane and exhibits a shorter recirculation zone downstream of the body. In summary, it was found that the interaction between a bluff body wake and a recirculating flow pattern alters profoundly the dynamics of the wake, which has implications on scalar transport in the wake.     

Measurements of Transitional and Turbulent Flow in a Randomly Packed Bed of Spheres with Particle Image Velocimetry

Particle image velocimetry (PIV) has been used to investigate transitional and turbulent flow in a randomly packed bed of mono-sized transparent spheres at particle Reynolds number, \(20<{{ Re}}_{\mathrm{p}}< 3220\). The refractive index of the liquid is matched with the spheres to provide optical access to the flow within the bed without distortions. Integrated pressure drop data yield that Darcy law is valid at \({{ Re}}_{\mathrm{p}} \approx 80\). The PIV measurements show that the velocity fluctuations increase and that the time-averaged velocity distribution start to change at lower \({{ Re}}_{\mathrm{p}}\). The probability for relatively low and high velocities decreases with \({{ Re}}_{\mathrm{p}}\) and recirculation zones that appear in inertia dominated flows are suppressed by the turbulent flow at higher \({{ Re}}_{\mathrm{p}}\). Hence there is a maximum of recirculation at about \({{ Re}}_{\mathrm{p}} \approx 400\). Finally, statistical analysis of the spatial distribution of time-averaged velocities shows that the velocity distribution is clearly and weakly self-similar with respect to \({{ Re}}_{\mathrm{p}}\) for turbulent and laminar flow, respectively.     

On the Justification of Viscoelastic Elliptic Membrane Shell Equations

We consider a family of linearly viscoelastic shells with thickness \(2\varepsilon\), clamped along their entire lateral face, all having the same middle surface \(S=\boldsymbol{\theta}(\bar{\omega})\subset \mathbb{R}^{3}\), where \(\omega\subset\mathbb{R}^{2}\) is a bounded and connected open set with a Lipschitz-continuous boundary \(\gamma\). We make an essential geometrical assumption on the middle surface \(S\), which is satisfied if \(\gamma\) and \(\boldsymbol{\theta}\) are smooth enough and \(S\) is uniformly elliptic. We show that, if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), the solution of the scaled variational problem in curvilinear coordinates, \(\boldsymbol{u}( \varepsilon)\), defined over the fixed domain \(\varOmega=\omega\times (-1,1)\) for each \(t\in[0,T]\), converges to a limit \(\boldsymbol{u}\) with \(u_{\alpha}(\varepsilon)\rightarrow u_{\alpha}\) in \(W^{1,2}(0,T,H ^{1}(\varOmega))\) and \(u_{3}(\varepsilon)\rightarrow u_{3}\) in \(W^{1,2}(0,T,L^{2}(\varOmega))\) as \(\varepsilon\to0\). Moreover, we prove that this limit is independent of the transverse variable. Furthermore, the average \(\bar{\boldsymbol{u}}= \frac{1}{2}\int_{-1}^{1} \boldsymbol{u}dx_{3}\), which belongs to the space \(W^{1,2}(0,T, V_{M}( \omega))\), where

$$V_{M}(\omega)=H^{1}_{0}(\omega)\times H^{1}_{0}(\omega)\times L ^{2}(\omega), $$

satisfies what we have identified as (scaled) two-dimensional equations of a viscoelastic membrane elliptic shell, which includes a long-term memory that takes into account previous deformations. We finally provide convergence results which justify those equations.

     

Periodic and Solitary Travelling Waves in Infinite Lattices without Ambrosetti–Rabinowitz Condition

In this paper, we consider FPU lattices with particles of unit mass. The dynamics of the system is described by the infinite system of second order differential equations

$$\begin{aligned} \ddot{q}_n= U^{\prime }(q_{n+1}-q_n)-U^{\prime }(q_n-q_{n-1}),\quad n\in \mathbb {Z}, \end{aligned}$$

where \(q_n\) denotes the displacement of the \(n\)-th lattice site and \(U\) is the potential of interaction between two adjacent particles. We investigate the existence of two kinds travelling wave solutions: periodic and solitary ones under some growth conditions on \(U\) which is different from the widely used Ambrosetti–Rabinowitz condition.

     

Long-time Dynamics of Resonant Weakly Nonlinear CGL Equations

Consider a weakly nonlinear CGL equation on the torus \(\mathbb {T}^d\):

$$\begin{aligned} u_t+i\Delta u=\epsilon [\mu (-1)^{m-1}\Delta ^{m} u+b|u|^{2p}u+ ic|u|^{2q}u]. \end{aligned}$$

(*)

Here \(u=u(t,x)\), \(x\in \mathbb {T}^d\), \(0<\epsilon <<1\), \(\mu \geqslant 0\), \(b,c\in \mathbb {R}\) and \(m,p,q\in \mathbb {N}\). Define \(I(u)=(I_{\mathbf {k}},\mathbf {k}\in \mathbb {Z}^d)\), where \(I_{\mathbf {k}}=v_{\mathbf {k}}\bar{v}_{\mathbf {k}}/2\) and \(v_{\mathbf {k}}\), \(\mathbf {k}\in \mathbb {Z}^d\), are the Fourier coefficients of the function \(u\) we give. Assume that the equation \((*)\) is well posed on time intervals of order \(\epsilon ^{-1}\) and its solutions have there a-priori bounds, independent of the small parameter. Let \(u(t,x)\) solve the equation \((*)\). If \(\epsilon \) is small enough, then for \(t\lesssim {\epsilon ^{-1}}\), the quantity \(I(u(t,x))\) can be well described by solutions of an effective equation:

$$\begin{aligned} u_t=\epsilon [\mu (-1)^{m-1}\Delta ^m u+ F(u)], \end{aligned}$$

where the term \(F(u)\) can be constructed through a kind of resonant averaging of the nonlinearity \(b|u|^{2p}+ ic|u|^{2q}u\).

     

Glimpses of Flow Development and Degradation During Type B Drag Reduction by Aqueous Solutions of Polyacrylamide B1120

Flow development and degradation during Type B turbulent drag reduction by 0.10 to 10 wppm solutions of a partially-hydrolysed polyacrylamide B1120 of MW \(=\) 18x10⁶ was studied in a smooth pipe of ID \(=\) 4.60 mm and L/D \(=\) 210 at Reynolds numbers from 10000 to 80000 and wall shear stresses Tw from 8 to 600 Pa. B1120 solutions exhibited facets of a Type B ladder, including segments roughly parallel to, but displaced upward from, the P-K line; those that attained asymptotic maximum drag reduction at low Re√ f but departed downwards into the polymeric regime at a higher retro-onset Re√ f; and segments at MDR for all Re√ f. Axial flow enhancement profiles of S\(^{\prime }\) vs L/D reflected a superposition of flow development and polymer degradation effects, the former increasing and the latter diminishing S\(^{\prime }\) with increasing distance downstream. Solutions that induced normalized flow enhancements S\(^{\prime }\)/S\(^{\prime }_{\mathrm {m}} <\) 0.4 developed akin to solvent, with L_e,p/D \(=\) L_e,n/D \(<\) 42.3, while those at maximum drag reduction showed entrance lengths L_e,m/D \(\sim \) 117, roughly 3 times the solvent L_e,n/D. Degradation kinetics were inferred by first detecting a falloff point (Re√f^∧, S\(^{{\prime }\wedge }\)), of maximum observed flow enhancement, for each polymer solution. A plot of S\(^{{\prime }\wedge }\)vs C revealed S\(^{{\prime }\wedge }\)linear in C at low C, with lower bound [S\(^{\prime }\)] \(=\) 5.0 wppm^??1, and S\(^{{\prime }\wedge }\) independent of C at high C, with upper bound S\(^{\prime }_{\mathrm {m}} =\) 15.9. The ratio S\(^{\prime }\)/S\(^{{\prime }\wedge }\) in any pipe section was interpreted to be the undegraded fraction of original polymer therein. Semi-log plots of (S\(^{\prime }\)/S\(^{{\prime }\wedge }\)) at a section vs transit time from pipe entrance thereto revealed first order kinetics, from which apparent degradation rate constants k_deg s^??1 and entrance severities ?ln(S\(^{\prime }\)/S\(^{{\prime }\wedge }\))₀ were extracted. At constant C, k_deg increased linearly with increasing wall shear stress Tw, and at constant Tw, k_deg was independent of C, providing a B1120 degradation modulus (k_deg/Tw) \(=\) (0.012 \(\pm \) 0.001) (Pa s)^??1 for 8 \(<\) Tw Pa \(<\) 600, 0.30 \(<\) C wppm \(<\) 10. Entrance severities were negligible below a threshold Twe \(\sim \) 30 Pa and increased linearly with increasing Tw for Tw \(>\) Twe. The foregoing methods were applied to Type A drag reduction by 0.10 to 10 wppm solutions of a polyethyleneoxide PEO P309, MW \(=\) 11x10⁶, in a smooth pipe of ID \(=\) 7.77 mm and L/D \(=\) 220 at Re from 4000 to 115000. P309 solutions that induced S\(^{\prime }\)/S\(^{\prime }_{\mathrm {m}} <\) 0.4 developed akin to solvent, with L_e,p/D \(=\) L_e,n/D \(<\) 23, while those at MDR had entrance lengths L_e,m/D \(\sim \) 93, roughly 4 times the solvent L_e,n/D. P309 solutions described a Type A fan distorted by polymer degradation. A typical trajectory departed the P-K line at an onset point Re√ f* followed by ascending and descending polymeric regime segments separated by a falloff point Re√f^∧, of maximum flow enhancement; for all P309 solutions, onset Re√ f* = 550 \(\pm \) 100 and falloff Re√f^∧ = 2550 \(\pm \) 250, the interval between them delineating Type A drag reduction unaffected by degradation. A plot of falloff S\(^{{\prime }\wedge }\) vs C for PEO P309 solutions bore a striking resemblance to the analogous S\(^{{\prime }\wedge }\) vs C plot for solutions of PAMH B1120, indicating that the initial Type A drag reduction by P309 after onset at Re√ f* had evolved to Type B drag reduction by falloff at Re√f^∧. Presuming that Type B behaviour persisted past falloff permitted inference of P309 degradation kinetics; k_deg was found to increase linearly with increasing Tw at constant C and was independent of C at constant Tw, providing a P309 degradation modulus (k_deg/Tw) \(=\) (0.011 \(\pm \) 0.002) (Pa s)^??1 for 4 \(<\) Tw Pa \(<\) 400, 0.10 \(<\) C wppm < 5.0. Comparisons between the present degradation kinetics and previous literature showed (k_deg/Tw) data from laboratory pipes of D \(\sim \) 0.01 m to lie on a simple extension of (k_deg/Tw) data from pipelines of D \(\sim \) 0.1 m and 1.0 m, along a power-law relation (k_deg/Tw) \(=\) 10^??5.4.D^??1.6. Intrinsic slips derived from PAMH B1120 and PEO P309-at-falloff experiments were compared with previous examples from Type B drag reduction by polymers with vinylic and glycosidic backbones, showing: (i) For a given polymer, [S\(^{\prime }\)] was independent of Re√ f and pipe ID, implying insensitivity to both micro- and macro-scales of turbulence; and (ii) [S\(^{\prime }\)] increased linearly with increasing polymer chain contour length Lc, the proportionality constant \(\beta =\) 0.053 \(\pm \) 0.036 enabling estimation of flow enhancement S\(^{\prime } =\) C.Lc.β for all Type B drag reduction by polymers.     

Topological Pressure of Generic Points for Amenable Group Actions

Let (X, G) be a G-action topological dynamical system (t.d.s. for short), where G is a countably infinite discrete amenable group. In this paper, we study the topological pressure of the sets of generic points. We show that when the system satisfies the almost specification property, for any G-invariant measure \(\mu \) and any continuous map \(\varphi \),

$$\begin{aligned} P\left( X_{\mu },\varphi ,\{F_n\}\right) = h_{\mu }(X)+\int \varphi d\mu , \end{aligned}$$

where \(\{F_n\}\) is a Følner sequence, \(X_{\mu }\) is the set of generic points of \(\mu \) with respect to (w.r.t. for short) \(\{F_n\}\), \(P(X_{\mu },\varphi ,\{F_n\})\) is the topological pressure of \(X_{\mu }\) for \(\varphi \) w.r.t. \(\{F_n\}\) and \(h_{\mu }(X)\) is the measure-theoretic entropy.

     

Further Dense Properties of the Space of Circle Diffeomorphisms with a Liouville Rotation Number

In continuation of Matsumoto’s paper (Nonlinearity 25:1495–1511, 2012) we show that various subspaces are \(C^{\infty }\)-dense in the space of orientation-preserving \(C^{\infty }\)-diffeomorphisms of the circle with rotation number \(\alpha \), where \(\alpha \in {\mathbb {S}}^1\) is any prescribed Liouville number. In particular, for every odometer \({\mathcal {O}}\) of product type we prove the denseness of the subspace of diffeomorphisms which are orbit-equivalent to \({\mathcal {O}}\).     

Generic Hamiltonian Dynamics

In this paper we contribute to the generic theory of Hamiltonians by proving that there is a \(C^2\)-residual \({\mathcal {R}}\) in the set of \(C^2\) Hamiltonians on a closed symplectic manifold \(M\), such that, for any \(H\in {\mathcal {R}}\), there is a full measure subset of energies \(e\) in \(H(M)\) such that the Hamiltonian level \((H,e)\) is topologically mixing; moreover these level sets are homoclinic classes.     

Models of Elastic Shells in Contact with a Rigid Foundation: An Asymptotic Approach

We consider a family of linearly elastic shells with thickness \(2\varepsilon\) (where \(\varepsilon\) is a small parameter). The shells are clamped along a portion of their lateral face, all having the same middle surface \(S\), and may enter in contact with a rigid foundation along the bottom face.We are interested in studying the limit behavior of both the three-dimensional problems, given in curvilinear coordinates, and their solutions (displacements \(\boldsymbol{u}^{\varepsilon}\) of covariant components \(u_{i}^{\varepsilon}\)) when \(\varepsilon\) tends to zero. To do that, we use asymptotic analysis methods. On one hand, we find that if the applied body force density is \(O(1)\) with respect to \(\varepsilon\) and surface tractions density is \(O(\varepsilon)\), a suitable approximation of the variational formulation of the contact problem is a two-dimensional variational inequality which can be identified as the variational formulation of the obstacle problem for an elastic membrane. On the other hand, if the applied body force density is \(O(\varepsilon^{2})\) and surface tractions density is \(O(\varepsilon^{3})\), the corresponding approximation is a different two-dimensional inequality which can be identified as the variational formulation of the obstacle problem for an elastic flexural shell. We finally discuss the existence and uniqueness of solution for the limit two-dimensional variational problems found.     

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (\({\textit{Re}}\ll 1\)) and high Péclet (\({\textit{Pe}}\gg 1\)) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length \(l_s\). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of \(Nu \sim {\textit{Pe}}^{1/3}\) and \(Nu \sim {\textit{Pe}}^{1/2}\) for no-slip (\(l_s \rightarrow 0\)) and shear-free (\(l_s \rightarrow \infty \)) boundaries, respectively. Boundary layer analysis also shows that the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) holds for a shear-free cylinder surface in the asymptotic limit of \({\textit{Re}}\gg 1\) so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite \({\textit{Re}}\), results from our two-dimensional simulations confirm the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) for a shear-free boundary over the range \(0.1 \le {\textit{Re}}\le 10^3\) and \(0.1\le {\textit{Pr}}\le 10\). A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with \(l_s\), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number \(Nu_{\theta }\) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high \({\textit{Re}}\) and Pr. At a Reynolds number of \(10^3\), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in \(Nu_{\theta }\) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variation with \(l_s\) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on \(l_s\) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low \(l_s\), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and \({\textit{Pr}}\) with Sherwood and Schmidt numbers, respectively.     

The Unstable Set of a Periodic Orbit for Delayed Positive Feedback

In the paper [Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, JDDE 23 (2011), no. 4, 727–790], we have constructed large-amplitude periodic orbits for an equation with delayed monotone positive feedback. We have shown that the unstable sets of the large-amplitude periodic orbits constitute the global attractor besides spindle-like structures. In this paper we focus on a large-amplitude periodic orbit \({\mathcal {O}}_{p}\) with two Floquet multipliers outside the unit circle, and we intend to characterize the geometric structure of its unstable set \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \). We prove that \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) is a three-dimensional \(C^{1}\)-submanifold of the phase space and admits a smooth global graph representation. Within \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \), there exist heteroclinic connections from \({\mathcal {O}}_{p}\) to three different periodic orbits. These connecting sets are two-dimensional \(C^{1}\)-submanifolds of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) and homeomorphic to the two-dimensional open annulus. They form \(C^{1}\)-smooth separatrices in the sense that they divide the points of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) into three subsets according to their \(\omega \)-limit sets.     

Blow-Up Rate Estimates and Liouville Type Theorems for a Semilinear Heat Equation with Weighted Source

We study the Liouville-type theorem for the semilinear parabolic equation \(u_t-\Delta u =|x|^a u^p\) with \(p>1\) and \(a\in {\mathbb R}\). Relying on the recent result of Quittner (Math Ann, doi: 10.1007/s00208-015-1219-7, 2015), we establish the optimal Liouville-type theorem in dimension \(N=2\), in the class of nonnegative bounded solutions. We also provide a partial result in dimension \(N\ge 3\). As applications of Liouville-type theorems, we derive the blow-up rate estimates for the corresponding Cauchy problem.     

Inertial Sensitivity of Porous Microstructures

Fluid flows through porous media are subject to different regimes, ranging from linear creeping flows to unsteady, chaotic turbulence. These different flow regimes at the pore scale have repercussions at larger scales, with the macroscale drag force experienced by a fluid moving through the medium becoming a nonlinear function of the average velocity beyond the creeping flow regime. Accurate prediction of the transition between different flow regimes is an important challenge with repercussions onto many engineering applications. Here, we are interested in the first deviation from Darcy’s law, when inertia effects become sizeable. Our goal is to define a Reynolds number, \(Re_{\mathrm{C}}\), so that the inertial deviation occurs when \(Re_{\mathrm{C}}\sim 1\) for any microstructure. The difficulty in doing so is to reduce the multiple length scales characterizing the geometry of the porous structure to a single length scale, \(\ell \). We analyze the problem using the method of volume averaging and identify a length scale in the form \(\ell =C_\lambda \sqrt{\nicefrac {K_\lambda }{\epsilon _\beta }}\), with \(C_\lambda \) a parameter that indicates the sensitivity of the microstructure to inertia. The main advantage of this definition is that an explicit formula for \(C_\lambda \) is given; \(C_\lambda \) is computed from a creeping flow simulation in the porous medium; and \(Re_{\mathrm{C}}\) can be used to predict the transition to a non-Darcian regime more accurately than by using Reynolds numbers based on alternative length scales. The theory is validated numerically with data from flow simulations for a variety of microstructures.     

Macro and micro mechanics behavior of granite after heat treatment by cluster model in particle flow code

In this paper, a cluster model in particle flow code was used to simulate granite specimens after heat treatment under uniaxial compression. The results demonstrated that micro-cracks are randomly distributed in the specimen when the temperature is below 300?C, and have partial coalescence when the temperature is up to 450?C, then form macro-cracks when the temperature is above 600?C. There is more inter-granular cracking than intra-granular cracking, and their ratio increases with increasing temperature.The micro-cracks are almost constant when the temperature decreases from 900?C to room temperature, except for quartz α–β phase transition temperature(573?C). The fracture evolution process is obviously affected by these cracks, especially at 600–900?C. Elevated temperature leads to easily developed displacement between the grains, and the capacity to store strain energy becomes weaker, corresponding to the plasticity of granite after heat treatment.     

Steady States of Fokker–Planck Equations: III. Degenerate Diffusion

This is the third paper in a series concerning the study of steady states of a Fokker–Planck equation in a general domain in \({\mathbb R}^n\) with \(L^{p}_{loc}\) drift term and \(W^{1,p}_{loc}\) diffusion term for any \(p>n\). In this paper, we give some existence results of stationary measures of the Fokker–Planck equation under Lyapunov conditions which allow the degeneracy of diffusion.     

Bifurcations of a creeping air–water flow in a conical container

This numerical study describes the eddy emergence and transformations in a slow steady axisymmetric air–water flow, driven by a rotating top disk in a vertical conical container. As water height \(H_{\mathrm{w}}\) and cone half-angle \(\beta \) vary, numerous flow metamorphoses occur. They are investigated for \(\beta =30^{\circ }, 45^{\circ }\), and \(60^{\circ }\). For small \(H_{\mathrm{w}}\), the air flow is multi-cellular with clockwise meridional circulation near the disk. The air flow becomes one cellular as \(H_{\mathrm{w}}\) exceeds a threshold depending on \(\beta \). For all \(\beta \), the water flow has an unbounded number of eddies whose size and strength diminish as the cone apex is approached. As the water level becomes close to the disk, the outmost water eddy with clockwise meridional circulation expands, reaches the interface, and induces a thin layer with anticlockwise circulation in the air. Then this layer expands and occupies the entire air domain. The physical reasons for the flow transformations are provided. The results are of fundamental interest and can be relevant for aerial bioreactors.     

On the Fiber Stretch in Shearing Deformations of Fibrous Soft Materials

The behavior of the fiber stretch in simple shear of soft materials fiber-reinforced with a single family of oriented parallel fibers is examined. The analysis is purely kinematical and the results are valid for both compressible and incompressible materials. It is shown that for a given amount of shear, for all fiber orientation angles in the range \(0 < \theta < \pi /4\), the fiber stretch increases with increasing \(\theta\) whereas in the range \(\pi /4 < \theta < \pi /2\), this is no longer the case and there is a particular fiber orientation for which the fiber stretch is a maximum. For a particular amount of shear corresponding to a special angle of shear (a “magic” angle of \(35.26^{\circ}\)), the fiber-orientation angle at which the fiber stretch is a maximum is its geometric complement namely a magic angle of \(54.74^{\circ}\). The results are also valid for torsion of a circular cylinder reinforced with a single family of helically wound fibers.