Non-uniqueness of solutions in asymptotically self-similar shock reflections

The present study addresses the self-similar problem of unsteady shock reflection on an inclined wedge. The start-up conditions are studied by modifying the wedge corner and allowing for a finite radius of curvature. It is found that the type of shock reflection observed far from the corner, namely regular or Mach reflection, depends intimately on the start-up condition, as the flow “remembers” how it was started. Substantial differences were found. For example, the type of shock reflection for an incident shock Mach number $M=6.6$ and an isentropic exponent $\gamma =1.2$ changes from regular to Mach reflection between $44^\circ $ and $45^\circ $ when a straight wedge tip is used, while the transition for an initially curved wedge occurs between $57^\circ $ and $58^\circ $ .     

On Formation of a Locally Self-Similar Collapse in the Incompressible Euler Equations

The paper addresses the question of the existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the L ^p -condition for velocity or vorticity and for a range of scaling exponents. In particular, in N dimensions if in self-similar variables ${u \in L^p}$ and ${u \sim \frac{1}{t^{\alpha/(1+\alpha)}}}$ , then the blow-up does not occur, provided ${\alpha > N/2}$ or ${-1 < \alpha \leq N\,/p}$ . This includes the L ³ case natural for the Navier–Stokes equations. For ${\alpha = N\,/2}$ we exclude profiles with asymptotic power bounds of the form ${ |y|^{-N-1+\delta} \lesssim |u(y)| \lesssim |y|^{1-\delta}}$ . Solutions homogeneous near infinity are eliminated, as well, except when homogeneity is scaling invariant.     

An Updated Portrait of Transition to Turbulence in Laminar Pipe Flows with Periodic Time Dependence (A Correlation Study)

Transition to turbulence in axially symmetrical laminar pipe flows with periodic time dependence classified as pure oscillating and pulsatile (pulsating) ones is the concern of the paper. The current state of art on the transitional characteristics of pulsatile and oscillating pipe flows is introduced with a particular attention to the utilized terminology and methodology. Transition from laminar to turbulent regime is usually described by the presence of the disturbed flow with small amplitude perturbations followed by the growth of turbulent bursts. The visual treatment of velocity waveforms is therefore a preferred inspection method. The observation of turbulent bursts first in the decelerating phase and covering the whole cycle of oscillation are used to define the critical states of the start and end of transition, respectively. A correlation study referring to the available experimental data of the literature particularly at the start of transition are presented in terms of the governing periodic flow parameters. In this respect critical oscillating and time averaged Reynolds numbers at the start of transition; Re _os,crit and Re _ta,crit are expressed as a major function of Womersley number, $\sqrt {\omega ^\prime } $ defined as dimensionless frequency of oscillation, f. The correlation study indicates that in oscillating flows, an increase in Re _os,crit with increasing magnitudes of $\sqrt {\omega ^\prime } $ is observed in the covered range of $1<\sqrt {\omega ^\prime } <72$ . The proposed equation (Eq. 7), ${\rm{Re}}_{os,crit} ={\rm{Re}}_{os,crit} \left( {\sqrt {\omega ^\prime } } \right)$ , can be utilized to estimate the critical magnitude of $\sqrt {\omega ^\prime }$ at the start of transition with an accuracy of ±12?% in the range of $\sqrt {\omega ^\prime } <41$ . However in pulsatile flows, the influence of $\sqrt {\omega ^\prime }$ on Re _ta,crit seems to be different in the ranges of $\sqrt {\omega ^\prime } <8$ and $\sqrt {\omega ^\prime } >8$ . Furthermore there is rather insufficient experimental data in pulsatile flows considering interactive influences of $\sqrt {\omega ^\prime } $ and velocity amplitude ratio, A ₁. For the purpose, the measurements conducted at the start of transition of a laminar sinusoidal pulsatile pipe flow test case covering the range of 0.21<?A ₁?<0.95 with $\sqrt {\omega ^\prime } <8$ are evaluated. In conformity with the literature, the start of transition corresponds to the observation of first turbulent bursts in the decelerating phase of oscillation. The measured data indicate that increase in $\sqrt {\omega ^\prime } $ is associated with an increase in Re _ta,crit up to $\sqrt {\omega ^\prime } =3.85$ while a decrease in Re _ta,crit is observed with an increase in $\sqrt {\omega ^\prime } $ for $\sqrt {{\omega }'} >3.85$ . Eventually updated portrait is pointing out the need for further measurements on i) the end of transition both in oscillating and pulsatile flows with the ranges of $\sqrt {\omega ^\prime } <8$ and $\sqrt {\omega ^\prime } >8$ , and ii) the interactive influences of $\sqrt {\omega ^\prime } $ and A ₁ on Re _ta,crit in pulsatile flows with the range of $\sqrt {\omega ^\prime } >8$ .     

Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

In this paper, we consider the Cauchy problem for a nonlinear parabolic system ${u^\epsilon_t - \Delta u^\epsilon + u^\epsilon \cdot \nabla u^\epsilon + \frac{1}{2}u^\epsilon\, {\rm div}\, u^\epsilon - \frac{1}{\epsilon}\nabla\, {\rm div}\, u^\epsilon = 0}$ in ${\mathbb {R}^3 \times (0,\infty)}$ with initial data in Lebesgue spaces ${L^2(\mathbb {R}^3)}$ or ${L^3(\mathbb {R}^3)}$ . We analyze the convergence of its solutions to a solution of the incompressible Navier?CStokes system as ${\epsilon \to 0}$ .     

Global Compensated Compactness Theorem for General Differential Operators of First Order

Let A ₁(x, D) and A ₂(x, D) be differential operators of the first order acting on l-vector functions ${u= (u_1, \ldots, u_l)}$ in a bounded domain ${\Omega \subset \mathbb{R}^{n}}$ with the smooth boundary ${\partial\Omega}$ . We assume that the H ¹-norm ${\|u\|_{H^{1}(\Omega)}}$ is equivalent to ${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_1u\|_{H^{\frac{1}{2}}(\partial\Omega)}}$ and ${\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_2u\|_{H^{\frac{1}{2}}(\partial\Omega)}}$ , where B _i  = B _i (x, ν) is the trace operator onto ${\partial\Omega}$ associated with A _i (x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to ${\partial\Omega}$ ). Furthermore, we impose on A ₁ and A ₂ a cancellation property such as ${A_1A_2^{\prime}=0}$ and ${A_2A_1^{\prime}=0}$ , where ${A^{\prime}_i}$ is the formal adjoint differential operator of A _i (i = 1, 2). Suppose that ${\{u_m\}_{m=1}^{\infty}}$ and ${\{v_m\}_{m=1}^{\infty}}$ converge to u and v weakly in ${L^2(\Omega)}$ , respectively. Assume also that ${\{A_{1}u_m\}_{m=1}^{\infty}}$ and ${\{A_{2}v_{m}\}_{m=1}^{\infty}}$ are bounded in ${L^{2}(\Omega)}$ . If either ${\{B_{1}u_m\}_{m=1}^{\infty}}$ or ${\{B_{2}v_m\}_{m=1}^{\infty}}$ is bounded in ${H^{\frac{1}{2}}(\partial\Omega)}$ , then it holds that ${\int_{\Omega}u_m\cdot v_m \,{\rm d}x \to \int_{\Omega}u\cdot v \,{\rm d}x}$ . We also discuss a corresponding result on compact Riemannian manifolds with boundary.     

Mixed Convection Boundary-Layer Flow Over a Vertical Surface Embedded in a Porous Material Subject to a Convective Boundary Condition

The mixed convection boundary-layer flow on one face of a semi-infinite vertical surface embedded in a fluid-saturated porous medium is considered when the other face is taken to be in contact with a hot or cooled fluid maintaining that surface at a constant temperature $T_\mathrm{{f}}$ . The governing system of partial differential equations is transformed into a system of ordinary differential equations through an appropriate similarity transformation. These equations are solved numerically in terms of a dimensionless mixed convection parameter $\epsilon $ and a surface heat transfer parameter $\gamma $ . The results indicate that dual solutions exist for opposing flow, $\epsilon <0$ , with the dependence of the critical values $\epsilon _\mathrm{{c}}$ on $\gamma $ being determined, whereas for the assisting flow $\epsilon >0$ , the solution is unique. Limiting asymptotic forms for both $\gamma $ small and large and $\epsilon $ large are also discussed.     

Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

We study the following nonlinear Stefan problem $$\left\{\begin{aligned}\!\!&u_t\,-\,d\Delta u = g(u) & &\quad{\rm for}\,x\,\in\,\Omega(t), t > 0, \\ & u = 0 \, {\rm and} u_t = \mu|\nabla_{x} u|^{2} &&\quad {\rm for}\,x\,\in\,\Gamma(t), t > 0, \\ &u(0, x) = u_{0}(x) &&\quad {\rm for}\,x\,\in\,\Omega_0,\end{aligned} \right.$$ where ${\Omega(t) \subset \mathbb{R}^{n}}$ ( ${n \geqq 2}$ ) is bounded by the free boundary ${\Gamma(t)}$ , with ${\Omega(0) = \Omega_0}$ , μ and d are given positive constants. The initial function u ₀ is positive in ${\Omega_0}$ and vanishes on ${\partial \Omega_0}$ . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary ${\Gamma(t)}$ is smooth outside the closed convex hull of ${\Omega_0}$ , and as ${t \to \infty}$ , either ${\Omega(t)}$ expands to the entire ${\mathbb{R}^n}$ , or it stays bounded. Moreover, in the former case, ${\Gamma(t)}$ converges to the unit sphere when normalized, and in the latter case, ${u \to 0}$ uniformly. When ${g(u) = au - bu^2}$ , we further prove that in the case ${\Omega(t)}$ expands to ${{\mathbb R}^n}$ , ${u \to a/b}$ as ${t \to \infty}$ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists ${\mu^* \geqq 0}$ such that ${\Omega(t)}$ expands to ${{\mathbb{R}}^n}$ exactly when ${\mu > \mu^*}$ .     

Tensor Products, Positive Linear Operators, and Delay-Differential Equations

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $$\begin{aligned} \dot{x}(t)=-\alpha (t)x(t)-\beta (t)x(t-1) \end{aligned}$$ with a single delay, where the delay coefficient is of one sign, say $\delta \beta (t)\ge 0$ with $\delta \in \{-1,1\}$ . Positivity properties are studied, with the result that if $(-1)^k=\delta $ then the $k$ -fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $\alpha (t)$ and $\beta (t)$ are periodic of the same period, and $\beta (t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$ -positivity of the exterior product is investigated when $\beta (t)$ satisfies a uniform sign condition.     

Elongational rheology of NIPAM-based hydrogels

Hydrogels of different composition based on the copolymerization of N-isopropyl acrylamide and surfmers of different chemical structure were tested in elongation using Hencky/real definitions for stress, strain, and strain rate, offering a more scientific insight into the effect of deformation on the properties. In a range between $\dot {\varepsilon }=10$ and 0.01 s $^{-1}$ , the material properties are independent of strain rate and show a very clear strain hardening with a “brittle” sudden fracture. The addition of surfmer increases the strain at break $\varepsilon _{\mathrm {H}}^{\max }$ and at the same time leads to a failure of hyperelastic models. The samples can be stretched up to Hencky strains $\varepsilon _{\mathrm {H}}^{\max }$ between 0.6 and 2.5, depending on the molecular structure, yielding linear Young’s moduli E $_{0}$ between 2,700 and 39,000 Pa. The strain-rate independence indicates an ideal rubberlike behavior and fracture in a brittle-like fashion. The resulting stress at break $\sigma _{\textrm max}$ can be correlated with $\varepsilon _{\mathrm {H}}^{\max } $ and $E_{0}$ as well as with the solid molar mass between the cross-linking points $M_{\mathrm {c}}^{\textrm {solids}} $ , derived from $E_{0}$ .     

Dynamical Parallelepipeds in Minimal Systems

For a topological dynamical system $(X,T)$ ( X , T ) and $d\in \mathbb N $ d ∈ N , the associated dynamical parallelepiped $\mathbf{Q}^{[d]}$ Q [ d ] was defined by Host–Kra–Maass. For a minimal distal system it was shown by them that the relation $\sim _{d-1}$ ～ d ? 1 defined on $\mathbf{Q}^{[d-1]}$ Q [ d ? 1 ] is an equivalence relation; the closing parallelepiped property holds, and for each $x\in X$ x ∈ X the collection of points in $\mathbf{Q}^{[d]}$ Q [ d ] with first coordinate $x$ x is a minimal subset under the face transformations. We give examples showing that the results do not extend to general minimal systems.     

Hadamard Variational Formula for the Green’s Function of the Boundary Value Problem on the Stokes Equations

For every ${\varepsilon > 0}$ , we consider the Green’s matrix ${G_{\varepsilon}(x, y)}$ of the Stokes equations describing the motion of incompressible fluids in a bounded domain ${\Omega_{\varepsilon} \subset \mathbb{R}^d}$ , which is a family of perturbation of domains from ${\Omega\equiv \Omega_0}$ with the smooth boundary ${\partial\Omega}$ . Assuming the volume preserving property, that is, ${\mbox{vol.}\Omega_{\varepsilon} = \mbox{vol.}\Omega}$ for all ${\varepsilon > 0}$ , we give an explicit representation formula for ${\delta G(x, y) \equiv \lim_{\varepsilon\to +0}\varepsilon^{-1}(G_{\varepsilon}(x, y) - G_0(x, y))}$ in terms of the boundary integral on ${\partial \Omega}$ of ${G_0(x, y)}$ . Our result may be regarded as a classical Hadamard variational formula for the Green’s functions of the elliptic boundary value problems.     

Generalized Resolvent Estimates of the Stokes Equations with First Order Boundary Condition in a General Domain

In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain ${\Omega}$ of the N-dimensional Eulidean space ${\mathbb{R}^N, N \geq 2}$ . This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter ${\lambda}$ varying in a sector ${\Sigma_{\sigma, \lambda_0} = \{\lambda \in \mathbb{C} \mid |\arg \lambda| < \pi-\sigma, \enskip |\lambda| \geq \lambda_0\}}$ , where ${0 < \sigma < \pi/2}$ and ${\lambda_0 \geq 1}$ . The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution ${p \in \hat{W}^1_{q, \Gamma}(\Omega)}$ to the variational problem: ${(\nabla p, \nabla \varphi) = (f, \nabla \varphi)}$ for any ${\varphi \in \hat W^1_{q', \Gamma}(\Omega)}$ . Here, ${1 < q < \infty, q' = q/(q-1), \hat W^1_{q, \Gamma}(\Omega)}$ is the closure of ${W^1_{q, \Gamma}(\Omega) = \{ p \in W^1_q(\Omega) \mid p|_\Gamma = 0\}}$ by the semi-norm ${\|\nabla \cdot \|_{L_q(\Omega)}}$ , and ${\Gamma}$ is the boundary of ${\Omega}$ . In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in ${(\lambda_0, \infty)}$ . Our assumption is satisfied for any ${q \in (1, \infty)}$ by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q =  2.     

On the Elastic Energy Density of Constrained Q-Tensor Models for Biaxial Nematics

Within the Landau–de Gennes theory, the order parameter describing a biaxial nematic liquid crystal assigns a symmetric traceless 3 × 3 matrix Q with three distinct eigenvalues to every point of the region Ω occupied by the system. In the constrained case of matrices Q with constant eigenvalues, the order parameter space is diffeomorphic to the eightfold quotient ${\mathbb{S}^3/\mathcal{H}}$ of the 3-sphere ${\mathbb{S}^3}$ , where ${\mathcal{H}}$ is the quaternion group, and a configuration of a biaxial nematic liquid crystal is described by a map from Ω to ${\mathbb{S}^3/\mathcal{H}}$ . We express the (simplest form of the) Landau–de Gennes elastic free-energy density as a density defined on maps ${q: \Omega \to \mathbb{S}^3}$ , whose functional dependence is restricted by the requirements that (1) it is well defined on the class of configuration maps from Ω to ${\mathbb{S}^3/\mathcal{H}}$ (residual symmetry) and (2) it is independent of arbitrary superposed rigid rotations (frame indifference). As an application of this representation, we then discuss some properties of the corresponding energy functional, including coercivity, lower semicontinuity and strong density of smooth maps. Other invariance properties are also considered. In the discussion, we take advantage of the identification of ${\mathbb{S}^3}$ with the Lie group of unit quaternions ${Sp(1) \cong SU(2)}$ and of the relations between quaternions and rotations in ${\mathbb{R}^3}$ and ${\mathbb{R}^4}$ .     

A New Insight into the Convective Boundary Condition

The steady mixed convection boundary layer flows over a vertical surface adjacent to a Darcy porous medium and subject respectively to (i) a prescribed constant wall temperature, (ii) a prescribed variable heat flux, $q_\mathrm{w} =q_0 x^{-1/2}$ q w = q 0 x ? 1 / 2 , and (iii) a convective boundary condition are compared to each other in this article. It is shown that, in the characteristic plane spanned by the dimensionless flow velocity at the wall ${f}^{\prime }(0)\equiv \lambda $ f ′ ( 0 ) ≡ λ and the dimensionless wall shear stress $f^{\prime \prime }(0)\equiv S$ f ′ ′ ( 0 ) ≡ S , every solution $(\lambda , S)$ ( λ , S ) of one of these three flow problems at the same time is also a solution of the other two ones. There also turns out that with respect to the governing mixed convection and surface heat transfer parameters $\varepsilon $ ε and $\gamma $ γ , every solution $(\lambda , S)$ ( λ , S ) of the flow problem (iii) is infinitely degenerate. Specifically, to the very same flow solution $(\lambda , S)$ ( λ , S ) there corresponds a whole continuous set of values of $\varepsilon $ ε and $\gamma $ γ which satisfy the equation $S=-\gamma (1+\varepsilon -\lambda )$ S = ? γ ( 1 + ε ? λ ) . For the temperature solutions, however, the infinite degeneracy of the velocity solutions becomes lifted. These and further outstanding features of the convective problem (iii) are discussed in the article in some detail.     

Semianalytical Solution for \text{ CO}_{2} Plume Shape and Pressure Evolution During \text{ CO}_{2} Injection in Deep Saline Formations

The injection of supercritical carbon dioxide ( $\text{ CO}_{2})$ in deep saline aquifers leads to the formation of a $\text{ CO}_{2}$ rich phase plume that tends to float over the resident brine. As pressure builds up, $\text{ CO}_{2}$ density will increase because of its high compressibility. Current analytical solutions do not account for $\text{ CO}_{2}$ compressibility and consider a volumetric injection rate that is uniformly distributed along the whole thickness of the aquifer, which is unrealistic. Furthermore, the slope of the $\text{ CO}_{2}$ pressure with respect to the logarithm of distance obtained from these solutions differs from that of numerical solutions. We develop a semianalytical solution for the $\text{ CO}_{2}$ plume geometry and fluid pressure evolution, accounting for $\text{ CO}_{2}$ compressibility and buoyancy effects in the injection well, so $\text{ CO}_{2}$ is not uniformly injected along the aquifer thickness. We formulate the problem in terms of a $\text{ CO}_{2}$ potential that facilitates solution in horizontal layers, with which we discretize the aquifer. Capillary pressure is considered at the interface between the $\text{ CO}_{2}$ rich phase and the aqueous phase. When a prescribed $\text{ CO}_{2}$ mass flow rate is injected, $\text{ CO}_{2}$ advances initially through the top portion of the aquifer. As $\text{ CO}_{2}$ is being injected, the $\text{ CO}_{2}$ plume advances not only laterally, but also vertically downwards. However, the $\text{ CO}_{2}$ plume does not necessarily occupy the whole thickness of the aquifer. We found that even in the cases in which the $\text{ CO}_{2}$ plume reaches the bottom of the aquifer, most of the injected $\text{ CO}_{2}$ enters the aquifer through the layers at the top. Both $\text{ CO}_{2}$ plume position and fluid pressure compare well with numerical simulations. This solution permits quick evaluations of the $\text{ CO}_{2}$ plume position and fluid pressure distribution when injecting supercritical $\text{ CO}_{2}$ in a deep saline aquifer.     

Predominant Gas Transport in Microporous Hydrotalcite–Silica Membrane

The prepared microporous hydrotalcite (HT)–silica membrane was found to exhibit the molecular sieving characteristic of pristine silica material and high $\mathrm{CO}_{2}$ adsorption capacity of HT. The combined properties made enhanced $\mathrm{CO}_{2}$ permeability and separability from $\mathrm{CH}_{4}$ possible. The gas transport in the membrane was predominantly surface adsorption. The porous membrane overcame the Knudsen limitation and yielded the highest separation selectivity of 120 at 40 % $\mathrm{CO}_{2}$ feed concentration, $30\,^{\circ }\mathrm{C}$ operating temperature, and 100 kPa pressure difference.     

Spatiotemporal complexity of a three-species ratio-dependent food chain model

In this paper, we investigate the complex dynamics of a ratio-dependent spatially extended food chain model. Through a detailed analytical study of the reaction–diffusion model, we obtain some conditions for global stability. On the basis of bifurcation analysis, we present the evolutionary process of pattern formation near the coexistence equilibrium point $(N^*,P^*,Z^*)$ via numerical simulation. And the sequence cold spots $\rightarrow $ stripe–spots mixtures $\rightarrow $ stripes $\rightarrow $ hot stripe–spots mixtures $\rightarrow $ hot spots $\rightarrow $ chaotic wave patterns controlled by parameters $a_1$ or $c_1$ in the model are presented. These results indicate that the reaction–diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics.     

Well-Posedness of Nematic Liquid Crystal Flow in {L^{3}_{\rm uloc}(\mathbb{R}^3)}

In this paper, we establish the local well-posedness for the Cauchy problem of a simplified version of hydrodynamic flow of nematic liquid crystals in ${\mathbb{R}^3}$ for any initial data (u ₀, d ₀) having small ${L^{3}_{\rm uloc}}$ -norm of ${(u_{0}, \nabla d_{0})}$ . Here ${L^{3}_{\rm uloc}(\mathbb{R}^3)}$ is the space of uniformly locally L ³-integrable functions. For any initial data (u ₀, d ₀) with small ${\|(u_0, \nabla d_0)\|_{L^{3}(\mathbb{R}^3)}}$ , we show that there exists a unique, global solution to the problem under consideration which is smooth for t > 0 and has monotone deceasing L ³-energy for ${t \geqq 0}$ .