Biorthogonal stretching and shearing of an impermeable surface in a uniformly rotating fluid system

The flow induced by an impermeable flat surface executing orthogonal stretching and orthogonal shearing in a rotating fluid system is investigated. Both the stretching and shearing are linear in the coordinates. An exact similarity reduction of the Navier–Stokes equations gives rise to a pair of nonlinearly-coupled ordinary differential equations governed by three parameters. In this study we set one parameter and analyze the problem which leads to flow for an impermeable surface with shearing and stretching due to velocity u along the x-axis of equal strength a while the shearing and stretching due to velocity v along the y-axis of equal strength b. These solutions depend on two parameters—a Coriolis (rotation) parameter \(\sigma = \Omega /a\) and a stretching/shearing ratio \(\lambda =b/a\). A symmetry in solutions is found for \(\lambda = 1\). The exact solution for \(\sigma = 0\) and the asymptotic behavior of solutions for \(|\sigma | \rightarrow \infty\) are determined and compared with numerical results. Oscillatory solutions are found whose strength increases with increasing values of \(|\sigma |\). It is shown that these solutions tend to the well-known Ekman solution as \(|\sigma | \rightarrow \infty\).     

Infinitely Many Stability Switches in a Problem with Sublinear Oscillatory Boundary Conditions

We consider the elliptic equation \(-\Delta u +u =0\) with nonlinear boundary condition \(\frac{\partial u}{\partial n}= \lambda u + g(\lambda ,x,u), \) where \(\frac{g(\lambda ,x,s)}{s} \rightarrow 0, \hbox { as }|s|\rightarrow \infty \) and g is oscillatory. We provide sufficient conditions on g for the existence of unbounded sequences of stable solutions, unstable solutions, and turning points, even in the absence of resonant solutions.     

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.     

Spreading in a Shifting Environment Modeled by the Diffusive Logistic Equation with a Free Boundary

We investigate the influence of a shifting environment on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is homogeneous and favourable, this model was first studied in Du and Lin (SIAM J Math Anal 42:377–405, 2010), where a spreading–vanishing dichotomy was established for the long-time dynamics of the species, and when spreading happens, it was shown that the species invades the new territory at some uniquely determined asymptotic speed \(c_0>0\). Here we consider the situation that part of such an environment becomes unfavourable, and the unfavourable range of the environment moves into the favourable part with speed \(c>0\). We prove that when \(c\ge c_0\), the species always dies out in the long-run, but when \(0<c<c_0\), the long-time behavior of the species is determined by a trichotomy described by (a) vanishing, (b) borderline spreading, or (c) spreading. If the initial population is written in the form \(u_0(x)=\sigma \phi (x)\) with \(\phi \) fixed and \(\sigma >0\) a parameter, then there exists \(\sigma _0>0\) such that vanishing happens when \(\sigma \in (0,\sigma _0)\), borderline spreading happens when \(\sigma =\sigma _0\), and spreading happens when \(\sigma >\sigma _0\).     

New Derived from Anosov Diffeomorphisms with Pathological Center Foliation

In this paper we focused our study on derived from Anosov diffeomorphisms (DA diffeomorphisms ) of the torus \(\mathbb {T}^3,\) it is, an absolute partially hyperbolic diffeomorphism on \(\mathbb {T}^3\) homotopic to a linear Anosov automorphism of the \(\mathbb {T}^3.\) We can prove that if \(f: \mathbb {T}^3 \rightarrow \mathbb {T}^3 \) is a volume preserving DA diffeomorphism homotopic to a linear Anosov A,  such that the center Lyapunov exponent satisfies \(\lambda ^c_f(x) > \lambda ^c_A > 0,\) with x belongs to a positive volume set, then the center foliation of f is non absolutely continuous. We construct a new open class U of non Anosov and volume preserving DA diffeomorphisms, satisfying the property \(\lambda ^c_f(x) > \lambda ^c_A > 0\) for \(m-\)almost everywhere \(x \in \mathbb {T}^3.\) Particularly for every \(f \in U,\) the center foliation of f is non absolutely continuous.     

Threshold Behavior and Non-quasiconvergent Solutions with Localized Initial Data for Bistable Reaction–Diffusion Equations

We consider bounded solutions of the semilinear heat equation \(u_t=u_{xx}+f(u)\) on \(R\), where \(f\) is of the unbalanced bistable type. We examine the \(\omega \)-limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at \(x=\pm \infty \), the \(\omega \)-limit sets may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of \(f\).     

Initial and Boundary Blow-U Problem for $$$$-Lalacian Parabolic Equation with General Absortion

In this article, we investigate the initial and boundary blow-up problem for the \(p\)-Laplacian parabolic equation \(u_t-\Delta _p u=-b(x,t)f(u)\) over a smooth bounded domain \(\Omega \) of \(\mathbb {R}^N\) with \(N\ge 2\), where \(\Delta _pu=\mathrm{div}(|\nabla u|^{p-2}\nabla u)\) with \(p>1\), and \(f(u)\) is a function of regular variation at infinity. We study the existence and uniqueness of positive solutions, and their asymptotic behaviors near the parabolic boundary.     

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.     

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 \(b|u|^{2p}+ ic|u|^{2q}u\).

     

A Note on the Convergence of Singularly Perturbed Second Order Potential-Type Equations

In this paper we study the limit as \(\varepsilon \rightarrow 0\) of the singularly perturbed second order equation \(\varepsilon ^2 \ddot{u}_\varepsilon + \nabla _{\!x} V(t,u_\varepsilon (t))=0\), where V(t, x) is a potential. We assume that \(u_0(t)\) is one of its equilibrium points such that \(\nabla _{\!x}V(t,u_0(t))=0\) and \(\nabla _{\!x}^2V(t,u_0(t))>0\). We find that, under suitable initial data, the solutions \(u_\varepsilon \) converge uniformly to \(u_0\), by imposing mild hypotheses on V. A counterexample shows that they cannot be weakened.     

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 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.

     

Exact solution and invariant for fractional Cattaneo anomalous diffusion of cells in two-dimensional comb framework

This paper presents an investigation on anomalous diffusion of cells in a two-dimensional comb framework with effects of fractional Cattaneo flux. Formulated governing equation is an evolution equation with the coexisting characteristics of parabolic (diffusion) and hyperbolic (wave) for \(\alpha \) in (0, 1). Exact solution is obtained by the special fractional integral transformations, and a novel invariant is established, i.e., \(\left\langle {x^{2}\left( t \right) } \right\rangle \cdot \left\langle P \right\rangle = 0.5\) (the mean square displacement multiplied by the total number of cells along the x-axis = 0.5). Moreover, the characteristics of cells distribution, the total number and the mean square displacement of cells along the x-axis with different involved parameters, especially with the fractional parameter evolution, are shown graphically and analyzed in detail. For the cells distribution versus x, it turns from parabolic and hyperbolic with the decrease in t or the increase in \(\alpha \) or \(\xi \). It is monotonically decreasing for the cells distribution versus \(\alpha \) with different x, t and \(\xi \). For the distribution versus t with different \(\alpha \) and \(\xi \) or versus \(\alpha \) with different t, it is monotonically decreasing for the distribution of total number while monotonically increasing for the distribution of mean square displacement. It is remarkable that the anomalous subdiffusion happens along the x-axis for arbitrary parameters which is different from the classical Cattaneo diffusion.     

Validation of numerical solvers for liquid metal flow in a complex geometry in the presence of a strong magnetic field

Following the magnetohydrodynamic (MHD) code validation and verification proposal by Smolentsev et al. (Fusion Eng Des 100:65–72, 2015), we perform code to code and code to experiment comparisons between two computational solvers, FLUIDYN and HIMAG, which are presently considered as two of the prospective CFD tools for fusion blanket applications. In such applications, an electrically conducting breeder/coolant circulates in the blanket ducts in the presence of a strong plasma-confining magnetic field at high Hartmann numbers, \(\textit{Ha}\) (\(\textit{Ha}^2\) is the ratio between electromagnetic and viscous forces) and high interaction parameters, \(\textit{N}\) (\(\textit{N}\) is the ratio of electromagnetic to inertial forces). The main objective of this paper is to provide the scientific and engineering community with common references to assist fusion researchers in the selection of adequate computational means to be used for blanket design and analysis. As an initial validation case, the two codes are applied to the classic problem of a laminar fully developed MHD flows in a rectangular duct. Both codes demonstrate a very good agreement with the analytical solution for \(\textit{Ha}\) up to 15, 000. To address the capabilities of the two codes to properly resolve complex geometry flows, we consider a case of three-dimensional developing MHD flow in a geometry comprising of a series of interconnected electrically conducting rectangular ducts. The computed electric potential distributions for two flows (Case A) \(\textit{Ha}=515\), \(\textit{N}=3.2\) and (Case B) \(\textit{Ha}=2059\), \(\textit{N}=63.8\) are in very good agreement with the experimental data, while the comparisons for the MHD pressure drop are still unsatisfactory. To better interpret the observed differences, the obtained numerical data are analyzed against earlier theoretical and experimental studies for flows that involve changes in the relative orientation between the flow and the magnetic field.     

Chaotic behavior of logistic map in superior orbit and an improved chaos-based traffic control model

With a vital role of discrete chaos, standard logistic map has found a celebrated place in the dynamics of chaos theory and in various applications of science, such as a discrete traffic flow model, image encryption in cryptography, secure communication, and weather forecasting. Traditionally, this discrete chaos is controlled by one parameter \(\lambda \) using Picard orbit, a one-step feedback procedure. This article presents a one-step forward, applying Mann orbit (superior orbit) the chaotic properties such as period-doubling, period-3 window, and Lyapunov exponent of the standard logistic map is investigated. The results are illustrated analytically and experimentally followed by concluding remarks and a few counter examples. Due to the extra degree of freedom in parameter \(\lambda \), the map provides improved chaotic properties that increases the performance of dynamical phenomena. Moreover, this study describes an improved chaos-based discrete traffic control model. Surprisingly, added new parameter \(\alpha \) in Mann orbit works as control variable that increases the stability performance of the traffic model.     

The Effect of a Low-Frequency Structure on Passive Scalar Transport in the Flow Over a Surface-Mounted Rib

The unsteadiness of the flow over a surface-mounted rib involving passive scalar transport is studied using large-eddy simulation (LES) at a Reynolds number of 3000 (based on the rib height, \(h\), and the free-stream velocity, \(U_{0})\). The purpose of the present study is to gain further insight into the physical origin of the flow instability and its effect on passive scalar transport. Fourier spectral analysis of the velocity at different positions suggests that, in addition to the K-H instability in the shear layer (St ≈?0.42), two lower frequencies (St ≈?0.06 and 0.09) also exist. It is observed that the low-frequency instabilities accompany the shedding process of vortical structures. One low frequency, at \(\text {St}\approx 0.06\), is related to the pumping motion of the recirculation bubble, while the other, at \(\text {St}\approx 0.09\), is associated with the flapping mode of the shear layer. Through comparisons of velocity and temperature fields and the spectra of scalar fluctuations, it is found that the passive scalar is transported by the convection of vortical structures.     

Chaotic characteristics analysis of the sintering process system with unknown dynamic functions based on phase space reconstruction and chaotic invariables

We develop a local discontinuous Galerkin finite element method for the distributed-order time and Riesz space-fractional convection–diffusion and Schrödinger-type equations. The stability of the presented schemes is proved and optimal order of convergence \(\mathcal {O}(h^{N+1}+(\Delta t)^{1+\frac{\theta }{2}}+\theta ^{2})\) for the Riesz space-fractional diffusion and Schrödinger-type equations with distributed order in time, an order of convergence of \(\mathcal {O}(h^{N+\frac{1}{2}}+(\Delta t)^{1+\frac{\theta }{2}}\) \(+\theta ^{2})\) is provided for the Riesz space-fractional convection–diffusion equations with distributed order in time where h, \(\theta \) and \(\Delta t\) are space step size, the distributed-order variables and the step sizes in time, respectively. Finally, the performed numerical examples confirm the optimal convergence order and illustrate the effectiveness of the method.     

Spiky Steady States of a Chemotaxis System with Singular Sensitivity

This paper is concerned with the steady state problem of a chemotaxis model with singular sensitivity function in a one dimensional spatial domain. Using the chemotactic coefficient \(\chi \) as the bifurcation parameter, we perform local and global bifurcation analysis for the model. It is shown that positive monotone steady states exist as long as \(\chi \) is larger than the first bifurcation value \(\bar{\chi }_1.\) We further obtain asymptotic profiles of these steady states, as \(\chi \) becomes large. In particular, our results show that the cell density function forms a spike, which models the important physical phenomenon of cell aggregation.     

FEM solution to natural convection flow of a micropolar nanofluid in the presence of a magnetic field

The two-dimensional, laminar, unsteady natural convection flow in a square enclosure filled with aluminum oxide (\(\hbox {Al}_{2} \hbox {O}_{3}\))–water nanofluid under the influence of a magnetic field, is considered numerically. The nanofluid is considered as Newtonian and incompressible, the nanoparticles and water are assumed to be in thermal equilibrium. The mathematical modelling results in a coupled nonlinear system of partial differential equations. The equations are solved using finite element method (FEM) in space, whereas, the implicit backward difference scheme is used in time direction. The results are obtained for Rayleigh (Ra), Hartmann (Ha) numbers, and nanoparticles volume fractions (\(\phi\)), in the ranges of \(10^3 \le Ra \le 10^7\), \(0\le Ha \le 500\) and \(0 \le \phi \le 0.2\), respectively. The streamlines and microrotation contours are observed to show similar behaviors with altering magnitudes. For low Ra values, when \(Ha=0\), symmetric vortices near the walls and a central vortex in opposite direction are observed in vorticity. As Ra increases, the central vortex splits into two due to the circulation in the effect of the buoyant flow. Boundary layer formation is observed when Ha increases for almost all Rayleigh numbers in both streamlines and vorticity. The isotherms have horizontal profiles for high Ra values owing to convective dominance over conduction. As Ha is increased, the convection effect is reduced, and isotherms tend to have vertical profiles. This study presents the first FEM application for solving highly nonlinear PDEs defining micropolar nanofluid flow especially for large values of Rayleigh and Hartmann numbers.     

Oriented Shadowing Property and $$\Omega $$-Stability for Vector Fields

We call that a vector field has the oriented shadowing property if for any \(\varepsilon >0\) there is \(d>0\) such that each \(d\)-pseudo orbit is \(\varepsilon \)-oriented shadowed by some real orbit. In this paper, we show that the \(C^1\)-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with the \(\Omega \)-stability.