Critical Value Γc Research Articles

First passage percolation with recovery

First passage percolation with recovery is a process aimed at modeling the spread of epidemics. On a graph G place a red particle at a reference vertex o and colorless particles (seeds) at all other vertices. The red particle starts spreading a red first passage percolation of rate 1, while all seeds are dormant. As soon as a seed is reached by the process, it turns red and starts spreading red first passage percolation. All vertices are equipped with independent exponential clocks ringing at rate γ>0, when a clock rings the corresponding red vertex turns black. For t≥0, let Ht and Mt denote the size of the longest red path and of the largest red cluster present at time t. If G is the semi-line, then for all γ>0 almost surely lim suptHtloglogtlogt=1 and lim inftHt=0. In contrast, if G is an infinite Galton–Watson tree with offspring mean m>1 then, for all γ>0, almost surely lim inftHtlogtt≥m−1 and lim inftMtloglogtt≥m−1, while lim suptMtect≤1, for all c>m−1. Also, almost surely as t→∞, for all γ>0Ht is of order at most t. Furthermore, if we restrict our attention to bounded-degree graphs, then for any ɛ>0 there is a critical value γc>0 so that for all γ>γc, almost surely lim suptMtt≤ɛ.

Influence of Impurity Scattering on Surface Plasmons in Graphene in the Lindhard Approximation

We study the influence of impurity scattering on transverse magnetic (TM) and transverse electric (TE) surface plasmons (SPs) in graphene using the Lindhard approximation. We show how the behaviour and domains of TM SPs are affected by the impurity strength γ and determine the critical value γc below which no SPs exist. The quality factor of TM SPs, for single-band and two-band transitions, is proportional to the square of αλSP/γ, with α being the fine-structure constant and λSP being the plasmon wavelength. In addition, we show that impurity scattering suppresses TE SPs.

Evolution of the electrical resistivity at rest and during oscillatory shearing of co‐continuous morphology (PP/PMMA)/MWCNT systems

AbstractIn this work, a modified parallel‐disks configuration on a strain‐controlled ARES rheometer (TA Instrument) was used to study the evolution of the electrical resistivity at rest and during oscillatory shearing of a co‐continuous immiscible polymer blend morphology based on polypropylene and /polymethyl(methacrylate) (PP/PMMA) in which various amounts (0–3 wt%) of multiwall carbon nanotubes (MWCNT) were added. The co‐continuity of both PP and PMMA phases allowed the buildup of a conductive network due to the preferential localization of the conductive MWCNT at the interface between PP and PMMA. Under a stepwise increase of the oscillatory strain amplitude below a critical value (γc = 6.3%), a significant decrease in the electrical resistivity was observed for MWCNT concentrations above the percolation threshold (0.3 wt%) due to the conductive paths induced by both thermal (Brownian) motion and oscillatory shearing. However, for deformation amplitudes higher than γc, the resistivity increased due to the destruction of the MWCNT paths induced by the large deformation imposed on the PP/PMMA interface. These observations were also confirmed by the evolution of the storage modulus (G′) which remained constant for γc < 6.3% (linear viscoelastic regime), while the values decreased above γc due to the destruction of the system's morphology.

A comprehensive lattice-stability limit surface for graphene

The limits of reversible deformation in graphene under various loadings are examined using lattice-dynamical stability analysis. This information is then used to construct a comprehensive lattice-stability limit surface for graphene, which provides an analytical description of incipient lattice instabilities of all kinds, for arbitrary deformations, parametrized in terms of symmetry-invariants of strain/stress. Symmetry-invariants allow obtaining an accurate parametrization with a minimal number of coefficients. Based on this limit surface, we deduce a general continuum criterion for the onset of all kinds of lattice-stabilities in graphene: an instability appears when the magnitude of the deviatoric strain γ reaches a critical value γc which depends upon the mean normal strain E¯ and the directionality θ of the principal deviatoric stretch with respect to reference lattice orientation. We also distinguish between the distinct regions of the limit surface that correspond to fundamentally different mechanisms of lattice instabilities in graphene, such as structural versus material instabilities, and long-wave (elastic) versus short-wave instabilities. Utility of this limit surface is demonstrated in assessment of incipient failures in defect-free graphene via its implementation in a continuum finite elements analysis (FEA). The resulting scheme enables on-the-fly assessments of not only the macroscopic conditions (e.g., load and deflection) but also the microscopic conditions (e.g., local stress/strain, spatial location, temporal proximity, and nature of incipient lattice instability) at which an instability occurs in a defect-free graphene sheet subjected to an arbitrary loading condition.

Effect of cooling rates on clustering towards icosahedra in rapidly solidified Cu56Zr44 alloy

The rapid solidification processes of liquid Cu56Zr44 alloys at different cooling rates (γ) were simulated by a molecular dynamics (MD) method. In order to assess the influence of cooling rate on the clustering tendency and degree towards icosahedrons, a ten-indices' cluster-type index method was suggested to characterize the local atomic structures in the super-cooled liquid and the rapidly solidified solid. And their clustering and ordering degrees as well as the packing density of icosahedral clusters were also evaluated by an icosahedral clustering degree (fI), the chemical order parameter (ηαβ) and densification coefficients (D0, DI and DIS), respectively. Results show that the main local atomic configurations in Cu56Zr44 alloy system are Z12 clusters centered by Cu, and most of which are (12 0 12 0 0 0 0 0 0 0) standard icosahedra and (12 0 8 0 0 0 2 2 0 0) as well as (12 2 8 2 0 0 0 0 0 0) defective icosahedra. Below glass transition temperature (Tg), these icosahedral clusters will be coalesced to various icosahedral medium-range orders (IMROs) by IS linkages, namely, icosahedral bond, and their number N, size n, order parameter ηαβ as well as spatial distributions vary with γ. As the cooling rate exceeds the critical value (γc) at which a glassy transition can take place, a lower cooling rate, e.g., γ1=101 K/ns, is demonstrated to be favorable to uplift the number of icosahedra and enlarge the size of IMROs compared with the higher cooling rates, e.g., γ5=105 K/ns, and their packing density and clustering degree towards icosahedra in the rapidly solidified solid can also benefit from the slow cooling process.

Experiments on two immiscible fluids in spherical Couette flow

This paper deals with the experimental instabilities analysis of spherical Couette flow. We consider the flow of two immiscible fluids superimposed between concentric spheres when the outer sphere is fixed and the inner one rotates. The working fluids have rather different viscosities and thus different Reynolds numbers. The obtained results are compared with a reference case of filled gap using one fluid (Γmax = 20). Experiments are performed for different aspect ratio values, and Laser photometric technique is used for visualization. Our analysis is mainly focused on the type of instabilities and their relationship with the laminar-turbulent transition regime. We intend to explore the combined effects of the aspect ratio and the interaction between the two superposed fluids on the appearance of different instability evolutions. The evolution of the phase velocity for different aspect ratio of heavy fluid ΓHF = HHF/d is presented. The immiscible fluids are separated by a liquid–liquid interface (water–oil). In order to control instability occurrence, Taylor number variation is presented versus aspect ratio. Instability phenomena are found to be the same as for the nominal case for large heavy fluid aspect ratios. The first equatorial symmetry breaking of the flow is observed for a critical value Γc = 13 where the Taylor vortex flow is introduced with three stationary cells. For the same aspect ratio, the interaction of the immiscible fluids leads to the appearance of gravitational waves near the equatorial zone. A surface cell, starting before the appearance of Taylor vortices, is detected in the light fluid for low aspect ratios. This cell of Ekman type has not been observed before, to our best knowledge, in spherical Couette flow.

Emergence of a non-trivial fluctuating phase in the XY-rotors model on regular networks

We study an XY-rotor model on regular one-dimensional lattices by varying the number of neighbours. The parameter 2 ⩾ γ ⩾ 1 is defined. γ = 2 corresponds to mean field and γ = 1 to nearest-neighbours coupling. We find that for γ < 1.5 the system does not exhibit a phase transition, while for γ > 1.5 the mean-field second-order transition is recovered. For the critical value γ = γc = 1.5, the systems can be in a non-trivial fluctuating phase for which the magnetisation shows important fluctuations in a given temperature range, implying an infinite susceptibilty. For all values of γ the magnetisation is computed analytically in the low-temperatures range and the magnetised vs. non-magnetised states are recovered, confirming the critical value γc = 1.5.

Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer

Linear and nonlinear stability analyses were performed on a fluid layer with a concentration-based internal heat source. Clear bimodal behaviour in the neutral curve (with stationary and oscillatory modes) is observed in the region of the onset of oscillatory convection, which is a previously unobserved phenomenon in radiation-induced convection. The numerical results for the linear instability analysis suggest a critical value γc of γ, a measure for the strength of the internal heat source, for which oscillatory convection is inhibited when γ > γc. Linear instability analyses on the effect of varying the ratio of the salt concentrations at the upper and lower boundaries conclude that the ratio has a significant effect on the stability boundary. A nonlinear analysis using an energy approach confirms that the linear theory describes the stability boundary most accurately when γ is such that the linear theory predicts the onset of mostly stationary convection. Nevertheless, the agreement between the linear and nonlinear stability thresholds deteriorates for larger values of the solute Rayleigh number for any value of γ.

The development of forced convection heat transfer near a forward stagnation point with Newtonian heating

A mathematical model for the unsteady forced convection boundary-layer flow near a forward stagna- tion point is considered when there is Newtonian heating on the surface whereby the heat transfer is proportional to the local surface temperature. In a previous paper (Salleh et al. J Eng Math 69:101-110, 2011), a critical value γc, dependent on the Prandtl number σ , of the heat transfer coefficient γ was identified, with solutions for the corresponding steady problem possible only for γ γ c, the solution still continues to large time, now growing exponentially with time. This rate of growth is determined by an eigenvalue problem which we solve numerically for general values of γ and σ and asymptotically for large γ and both large and small σ .

Topologically non-trivial superconductivity in spin–orbit-coupled systems: bulk phases and quantum phase transitions

Topologically non-trivial superconductivity has been predicted to occur in superconductors with a sizable spin–orbit (SO) coupling in the presence of an external Zeeman splitting. Two such systems have been proposed: (a) s-wave superconductor pair potential is proximity induced on a semiconductor and (b) pair potential naturally arises from an intrinsic s-wave pairing interaction. As it is now well known, such systems in the form of a two-dimensional (2D) film or 1D nano-wires in a wire network can be used in topological quantum computation. When the external Zeeman splitting Γ crosses a critical value Γc, the system passes from a regular superconducting phase to a non-Abelian topological superconducting phase. In both cases (a) and (b) that we consider in this paper, the pair potential Δ is strictly s-wave in both the ordinary and the topological superconducting phases, which are separated by a topological quantum critical point at , where μ (≫Δ) is the chemical potential. On the other hand, since Γc≫Δ, the Zeeman splitting required for the topological phase (Γ>Γc) far exceeds the value (Γ∼Δ) above which an s-wave pair potential is expected to vanish (and the system to become non-superconducting) in the absence of SO coupling. We are thus led to the situation that the topological superconducting phase appears to set in a parameter regime at which the system is actually non-superconducting in the absence of SO coupling. In this paper, we address the question of how a pure s-wave pair potential can survive a strong Zeeman field to give rise to a topological superconducting phase. We show that the SO coupling is the crucial parameter for the quantum transition into and the robustness of the topologically non-trivial superconducting phase realized for Γ≫Δ.

Forced-convection heat transfer over a circular cylinder with Newtonian heating

A mathematical model for the forced convection boundary-layer flow over a circular cylinder is considered when there is Newtonian heating on the surface of the cylinder through which the heat transfer is proportional to the local surface temperature. The dimensionless version of the boundary-layer equations involve two parameters, the Prandtl number σ and γ measuring the strength of the surface heating. The solution near the stagnation point is considered first and this reveals that, to get a physically acceptable solution, γ must be less than some critical value γc, dependent on σ. Numerical solutions to the full boundary-layer problem are obtained which show that the surface temperature increases as the flow develops from the stagnation point.

Deformation modes of subducted lithosphere at the core‐mantle boundary: An experimental investigation

Ancient subducted lithosphere may be an important source of the seismically observed heterogeneity and anisotropy in the D″ layer. Proper interpretation of the seismic observations therefore requires an understanding of how subducted lithosphere deforms upon reaching the core‐mantle boundary (CMB). Here we study this process experimentally, using an analog model system in which a sheet of viscous sugar syrup (representing the subducted lithosphere) falls vertically onto an impermeable surface (the “CMB”) at the bottom of a reservoir of less viscous syrup (the “ambient mantle”). For sheet/ambient viscosity contrasts γ less than a critical value γc that depends on the ratio of the fall height to the sheet thickness, the sheet impinges on the CMB as a stagnation flow with bilateral symmetry. For γ &gt; γc, however, the sheet undergoes a periodic folding instability. We determine experimentally a universal scaling law for the folding amplitude δ as a function of the viscosities and densities of the two fluids and the thickness and vertical velocity of the sheet where it enters the folding region. To apply this law to the Earth, we assume that the sheet's excess viscosity and density relative to the ambient mantle are both controlled by the same temperature anomaly ΔT. Using estimated rheological parameters for diffusion creep in the lowermost mantle, we find that the folding criterion γ &gt; γc requires ΔT &gt; 300 K. The presence of folded lithosphere at the CMB beneath the Cocos plate, suggested recently on the basis of seismic data by Hutko et al. [2006], is unlikely to be the result of in situ folding because the temperature anomaly (400–700 K) required to explain the observed variations in the depth of the postperovskite phase transition is probably too large to be preserved in a deeply penetrating slab. However, the seismic observations might be explained in an alternative way by a “pile” of folded lithosphere that formed at 660 km depth and subsequently sank to the CMB.

On the effects of compressibility in marginally separated flows

AbstractIf the angle of attack α of a slender airfoil reaches a critical value αs flow separation is known to occur at the upper surface. Further increase of α initially leads to the formation of a short laminar separation bubble which has an extremely weak influence on the external flow field — a phenomenon known as marginal separation — but then rather rapidly causes a severe change of the flow behaviour (leading edge stall). According to the asymptotic theory of marginal separation holding in the limit of large Reynolds numbers t→∞ the flow in the neighbourhood of the separation bubble is governed by an integro–differential equation exhibiting a single controlling parameter Γ which relates the angle of attack to the Reynolds number, with a value Γs corresponding to αs. Numerical investigations have shown that solutions to this equation exist up to a critical value Γc&gt;Γs only and this has been taken as an indication that a substantial change of the flow field must take place if Γ exceeds Γc. In the case of unsteady three–dimensional perturbations of a two–dimensional steady marginally separated boundary layer in the limit Γ→Γc, the (generalized form of the) integro–differential equation reduces to a nonlinear diffusion equation of Fisher type. This asymptotic description serves as a basis for the investigations concerning compressibility effects, also in respect of the mechanism through which the hydromechanic motion inside the boundary layer could be transformed into aerodynamic sound. A modified interaction equation is obtained, which in addition to Γ, contains the Mach number at the point of separation as a second controlling parameter. The unsteady changes of the boundary layer displacement thickness in the limit ΔΓ=Γc−Γ→0, again governed by Fisher's equation, then are identified to act as a quadrupole source for sound radiation. The most surprising outcome is the fact that in the three–dimensional case, the second derivative with respect to the lateral direction contained in the Fisher equation changes its sign and becomes negative if the Mach number exceeds a certain value. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Aspect ratio influence on the stability of Taylor-Couette flow

The study deals with the effect of the fluid hight on the stability of the Taylor-Couette flow of a Newtonian fluid between two coaxial cylinders, the upper surface of the fluid system being free. The exploration of the flow regimes is carried out for different values of the Taylor number Ta and of the aspect ratio Γ. By means of polarographic technique, we have determined the temporal mean value of the gradient velocity at the inner wall of the outer cylinder maintained at rest. We also performed the spectral analysis of the fluctuations of the velocity gradient. In order to complete wall measurements, the LDA technique was used to measure particularly the axial velocity component and rotation speed of the azimuthal wave. In this wavy-mode regime, the analysis of the results associated with the fluctuations of the fundamental frequency shows a change in the circumferential wave number. We established that the appearance of the azimuthal wave is delayed when the aspect ratio decreases. In addition we found that below the critical value Γc, the azimuthal wave regime is no longer observed.

Stability of the flow of a viscoelastic fluid past a deformable surface in the low Reynolds number limit

The stability of the plane Couette flow of a viscoelastic fluid adjacent to a flexible surface is analyzed with the help of linear and weakly nonlinear stability theory in the limit of zero Reynolds number. The fluid is described by an Oldroyd-B model, which is parametrized by the viscosity η, the relaxation time λ, and the parameter β, which is the ratio of solvent-to-solution viscosity; β=0 for a Maxwell fluid and β=1 for a Newtonian fluid. The wall is modeled as an incompressible neo-Hookean solid of finite thickness and is grafted to a rigid plate at the bottom. The neo-Hookean constitutive model parametrized by the shear modulus G, augmented to include the viscous dissipation, is used for the solid medium. Previous studies for the Newtonian flow past a compliant wall predict an instability as the dimensionless shear rate Γ=(ηV∕GR) is increased beyond the critical value Γc. The present analysis investigates the effect of fluid elasticity, in terms of the Weissenberg number W=λG∕η, on the critical value of the imposed shear rate Γc for various parameters. The fluid elasticity is found to increase Γc, indicating the stabilizing influence of the polymer addition on the viscous instability. For dilute polymeric solutions with β⩾0.5, the flow is stable when the Weissenberg number is increased beyond a maximum value Wmax, and Wmax increases proportional to the ratio of solid-to-fluid thickness H. For concentrated polymer solutions and melts with β&amp;lt;0.5, the flow becomes unstable when the strain rate increases beyond a critical value for any large Weissenberg number. The weakly nonlinear analysis reveals that the bifurcation of the linear instability is subcritical when there is no dissipation in the solid. The nature of bifurcation, however, changes to supercritical when the viscous effects in the solid are taken into account and the relative solid viscosity ηr is large such that ηr∕H&amp;gt;1. The equilibrium amplitude and the threshold strain energy for the solid have been calculated, and the effect of parameters H, β, ηr, and interfacial tension on these quantities is analyzed.

Stability of the viscous flow of a polymeric fluid past a flexible surface

The instability in plane Couette flow of a viscoelastic fluid past a deformable surface is examined using the temporal linear stability theory in the zero Reynolds number limit. The polymeric fluid is described using the Oldroyd-B model and the flexible wall is modeled as a linear viscoelastic solid surface. The analysis shows that the wall flexibility tends to reduce the decay rate of the stable discrete modes for the polymeric flow past a rigid wall, and one of the discrete modes becomes unstable when the wall deformability parameter Γ=Vη∕(GR) exceeds a certain critical value Γc. Here, V is the top-plate velocity, η is the zero shear viscosity of the polymeric fluid, G is the shear modulus of the wall, and R is the width of the fluid layer. The analysis reveals the presence of two classes of modes, the first of which becomes unstable for perturbations with wavelength comparable to the channel width (finite wavelength modes), and the second becomes unstable for perturbations with wavelength small compared to the channel width (short wave modes). The latter class of modes are found to be absent for the highly concentrated polymer solutions with β≲0.23, where β is the ratio of solvent-to-solution viscosity. We have mapped out the regions in the parameter space (W¯-H) where the finite wavelength and short wave modes are unstable, where W¯=(λG∕η), and λ is the relaxation time of the viscoelastic fluid. Fluid elasticity is found to have a stabilizing influence on the unstable mode, such that when the shortwave instability is absent for β≲0.23, the flow becomes stable for any Weissenberg number W¯&amp;gt;W¯max. Here, W¯max increases proportional to H for H≫1. However, when the shortwave instability is present, the instability persists for W¯≫1. The behavior of both classes of modes with respect to the parameters, like W¯, H, β, and the ratio of solid-to-fluid viscosity ηr, is examined.

Shear-Induced Desorption in Polymer Brushes

Shear-Induced Desorption in Polymer Brushes

The single-particle pseudorelativistic Jansen–Hess operator with magnetic field

The pseudorelativistic no-pair Jansen–Hess operator is derived for the case where in addition to the Coulomb potential an external magnetic field B is permitted. With some restrictions on the vector potential, it is shown that this operator is positive provided the strength γ of the Coulomb potential is below a critical value (γc 0.35, depending on the magnetic field energy Ef ). Moreover, for γ< 0.32 and for B tending asymptotically to zero in a weak sense, the essential spectrum is given by [m, ∞) + Ef .

Shear-Induced Structures in Semidilute Solution of Ultrahigh Molecular Weight Polyethylene at Temperature Close to Equilibrium Dissolution Temperature

The shear-induced structure of a semidimute solution of ultrahigh molecular weight polyethylene (UHMWPE) with a paraffin wax as a solvent was investigated with shear small-angle light scattering (shear SALS) and shear microscopy. At a temperature of 124 °C, which is much higher than the nominal melting temperature (∼118 °C) and is approximately equivalent to or slightly lower than the equilibrium dissolution (crystal melting) temperature of the UHMWPE solution at quiescent state, the characteristic “butterfly” pattern was observed in shear SALS when the solution was subjected to shear flow at a shear rate larger than critical shear rate γc,butt. Lack of optical anisotropy in the sheared solution and disappearance of the butterfly pattern after cessation of the shear suggest that the pattern is attributed to shear-enhanced concentration fluctuations or phase separation. When γ further increased above a critical value γc,streak, a strong, sharp streaklike scattering pattern appeared perpendicular to the ...

The Breakdowns of BiCGStab

The effects of the three principal possible exact breakdowns which may occur using BiCGStab are discussed. BiCGStab is used to solve large sparse linear systems of equations, such as arise from the discretisation of PDEs. These PDEs often involve a parameter, say γ. We investigate here how the numerical error grows as breakdown is approached by letting γ tend to a critical value, say γc, at which the breakdown is numerically exact. We found empirically in our examples that loss of numerical accuracy due stabilisation breakdown and Lanczos breakdown was discontinuous with respect to variation of γ around γc. By contrast, the loss of numerical accuracy near a critical value γc for pivot breakdown is roughly proportional to |γ−γc|−1.