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 $E_{0}$ .     

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.     

Besov Space Regularity Conditions for Weak Solutions of the Navier–Stokes Equations

Consider a bounded domain ${{\Omega \subseteq \mathbb{R}^3}}$ with smooth boundary, some initial value ${{u_0 \in L^2_{\sigma}(\Omega )}}$ , and a weak solution u of the Navier–Stokes system in ${{[0,T) \times\Omega,\,0 < T \le \infty}}$ . Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space $$B^{q,s}(\Omega ):=\left\{v\in L^2_{\sigma}(\Omega ); \|v\|_{B^{q,s}(\Omega )} := \left(\int\limits^{\infty}_0 \left\|e^{-\tau A}v\right\|^s_q {\rm d} \tau\right)^{1/s}<\infty \right\}$$ with ${{2 < s < \infty,\,3 < q <\infty,\,\frac2{s}+\frac{3}{q} = 1}}$ ; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012), is a subspace of the well known Besov space ${{{\mathbb{B}}^{-2/s}_{q,s}(\Omega )}}$ , see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002). Our main results on the regularity of u exploits a variant of the space ${{B^{q,s}(\Omega )}}$ in which the integral in time has to be considered only on finite intervals (0, δ ) with ${{\delta \to 0}}$ . Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition ${{u\in L^{\infty}_{\text{loc}}([0,T);L^3_{\sigma}(\Omega ))}}$ . Finally, we obtain a large class of regular weak solutions u defined by a smallness condition ${{\|u_0\|_{B^{q,s}(\Omega )} \le K}}$ with some constant ${{K=K(\Omega, q)>0}}$ .     

Experimental Characterization of Porosity Structure and Transport Property Changes in Limestone Undergoing Different Dissolution Regimes

Limestone dissolution by $\hbox {CO}_2$ -rich brine induces critical changes of the pore network geometrical parameters such as the pore size distribution, the connectivity, and the tortuosity which govern the macroscopic transport properties (permeability and dispersivity) that are required to parameterize the models, simulating the injection and the fate of $\hbox {CO}_2$ . A set of four reactive core-flood experiments reproducing underground conditions ( $T = 100\,^{\circ }\hbox {C}$ and $P = 12$ MPa) has been conducted for different $\hbox {CO}_2$ partial pressures $(0.034 < P_{\mathrm{CO}_2}< 3.4\; \hbox {MPa})$ in order to study the different dissolution regimes. X-ray microtomographic images have been used to characterize the changes in the structural properties from pore scale to Darcy scale, while time-resolved pressure loss and chemical fluxes enabled the determination of the sample-scale change in porosity and permeability. The results show the growth of localized dissolution features associated with high permeability increase for the highest $P_{\mathrm{CO}_2}$ , whereas dissolution tends to be more homogeneously distributed for lower values of $P_{\mathrm{CO}_2}$ . For the latter, the higher the $P_{\mathrm{CO}_2}$ , the more the dissolution patterns display ramified structures and permeability increase. For the lowest value of $P_{\mathrm{CO}_2}$ , the preferential dissolution of the calcite cement associated with the low dissolution kinetics triggers the transport that may locally accumulate and form a microporous material that alters permeability and produces an anti-correlated porosity–permeability relationship. The combined analysis of the pore network geometry and the macroscopic measurements shows that $P_{\mathrm{CO}_2}$ regulates the tortuosity change during dissolution. Conversely, the increase of the exponent value of the observed power law permeability–porosity trend while $P_{\mathrm{CO}_2}$ increases, which appears to be strongly linked to the increase of the effective hydraulic diameter, depends on the initial rock structure.     

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.     

Drainage of Capillary-Trapped Oil by an Immiscible Gas: Impact of Transient and Steady-State Water Displacement on Three-Phase Oil Permeability

In a previous paper (Dehghanpour et al., Phys Rev E 83:065302, 2011a), we showed that relative permeability of mobilized oil, $k_\mathrm{ro}$ , measured during tertiary gravity drainage, is significantly higher than that of the same oil saturation in other tests where oil is initially a continuous phase. We also showed that tertiary $k_\mathrm{ro}$ strongly correlates to both water saturation, $S_\mathrm{w}$ , water flux (water relative permeability), $k_\mathrm{rw}$ , and the change in water saturation with time, $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ . To develop a model and understanding of the enhanced oil transport, identifying which of these parameters ( $S_\mathrm{w},\,k_{\mathrm{rw}}$ , or $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ ) plays the controlling role is necessary, but in the previous experiments these could not be deconvolved. To answer the remaining question, we conduct specific three-phase displacement experiments in which $k_{\mathrm{rw}}$ is controlled by applying a fixed water influx, and $S_\mathrm{w}$ develops naturally. We obtain $k_{\mathrm{ro}}$ by using the saturation data measured in time and space. The results suggest that steady-state water influx, in contrast to transient water displacement, does not enhance $k_{\mathrm{ro}}$ . Instead, reducing water influx rate results in excess oil flow. Furthermore, according to our pore scale hydraulic conductivity calculations, viscous coupling and fluid positioning do not sufficiently explain the observed correlation between $k_{\mathrm{ro}}$ and $S_{\mathrm{w}}$ . We conclude that tertiary $k_{\mathrm{ro}}$ is controlled by the oil mobilization rate, which in turn is linked to the rate of water saturation decrease with time, $\mathrm{d}S_\mathrm{w}/\mathrm{d}t$ . Finally, we develop a simple model which relates tertiary $k_{\mathrm{ro}}$ to transient two-phase gas/water relative permeability.     

Resistance Coefficients of Polymer Membrane with Concentration Polarization

The Kedem-Katchalsky equations, modified by means of symmetric transformations of Peusner thermodynamic networks, were applied to interpret the membrane transport in concentration polarization conditions. The results from the study demonstrate that the resistance coefficients counted for membrane transport of aqueous solutions of glucose through Nephrophan membrane in horizontal plane are nonlinearly dependent on mean concentration of glucose in the membrane ${(\bar{C})}$ . It was also shown that the threshold value of concentration ${(\bar{C}_{cr})}$ existed, and for ${\bar{C} > \bar{C}_{cr}}$ , the resistance coefficients depend, while for ${\bar{C} < \bar{C}_{cr}}$ , they do not depend on the membrane system configuration. Increase of mean glucose concentration in the membrane (in the range ${\bar{C} > \bar{C}_{cr})}$ causes decrease of difference between resistance coefficients of the membrane system in homogeneous conditions (solutions mechanically stirred) and in conditions with hydrodynamic instabilities (configuration B). Besides increase of mean glucose concentration in the membrane (in the range ${\bar{C} > \bar{C}_{cr})}$ causes increase of the difference between resistance coefficients for membrane system with concentration polarization without hydrodynamic instabilities (configuration A) and membrane system in homogeneous conditions.     

Regularization of Point Vortices Pairs for the Euler Equation in Dimension Two

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem $$\left\{\begin{array}{l@{\quad}l} -\varepsilon^2 \Delta u = \sum\limits_{i=1}^m \chi_{\Omega_i^{+}} \left(u - q - \frac{\kappa_i^{+}}{2\pi} {\rm ln} \frac{1}{\varepsilon}\right)_+^p\\ \quad - \sum_{j=1}^n \chi_{\Omega_j^{-}} \left(q - \frac{\kappa_j^{-}}{2\pi} {\rm \ln} \frac{1}{\varepsilon} - u\right)_+^p , \quad \quad x \in \Omega,\\ u = 0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \in \partial \Omega,\end{array}\right.$$ where p > 1, ${\Omega \subset \mathbb{R}^2}$ is a bounded domain, ${\Omega_i^{+}}$ and ${\Omega_j^{-}}$ are mutually disjoint subdomains of Ω and ${\chi_{\Omega_i^{+}} ({\rm resp}.\; \chi_{\Omega_j^{-}})}$ are characteristic functions of ${\Omega_i^{+}({\rm resp}. \;\Omega_j^{-}})$ , q is a harmonic function. We show that if Ω is a simply-connected smooth domain, then for any given C ¹-stable critical point of Kirchhoff–Routh function ${\mathcal{W}\;(x_1^{+},\ldots, x_m^{+}, x_1^{-}, \ldots, x_n^{-})}$ with ${\kappa^{+}_i > 0\,(i = 1,\ldots, m)}$ and ${\kappa^{-}_j > 0\,(j = 1,\ldots,n)}$ , there is a stationary classical solution approximating stationary m + n points vortex solution of incompressible Euler equations with total vorticity ${\sum_{i=1}^m \kappa^{+}_i -\sum_{j=1}^n \kappa_j^{-}}$ . The case that n = 0 can be dealt with in the same way as well by taking each ${\Omega_j^{-}}$ as an empty set and set ${\chi_{\Omega_j^{-}} \equiv 0,\,\kappa^{-}_j=0}$ .     

On the effect of non-smooth Coulomb damping on flutter-type self-excitation in a non-gyroscopic circulatory 2-DoF-system

This article deals with self-excited vibrations, attractivity of stationary solutions, and the corresponding bifurcation behavior of two-dimensional differential inclusions of the type $\mathbf{M}\mathbf{q}'' + \mathbf{D}\mathbf{q}' + (\mathbf{K} + \bar{\mu}\mathbf{N})\mathbf{q} \in-\mathbf{R}\operatorname{Sign}(\mathbf{q}')$ . For the smooth case R=0, the equilibrium may become unstable due to non-conservative positional forces stemming from the circulatory matrix N. This type of instability is usually referred to as flutter instability and the loss of stability is related to a Hopf bifurcation of the steady state, which occurs for a critical parameter $\bar{\mu}= \bar{\mu}_{\mathrm{crit}}$ . For R≠0, the steady state is a set of equilibria, which turns out to be attractive for all values of the bifurcation parameter $\bar{\mu}$ . Depending on $\bar{\mu}$ , the basin of attraction of the equilibrium set can be infinite or finite. The transition from an infinite to a finite basin of attraction occurs at the stability threshold $\bar{\mu}_{\mathrm{crit}}$ of the underlying smooth problem. For the finite basin of attraction, its size is proportional to the Coulomb friction and inverse-proportional to $(\bar{\mu}- \bar{\mu}_{\mathrm{crit}})$ . By adding Coulomb damping the notion of steady state stability for the smooth problem is replaced by the question whether the basin of attraction of the steady state is infinite or finite. Simultaneously, the local Hopf-bifurcation is replaced by a global bifurcation. This implies that in the presence of Coulomb damping the occurrence of self-excited vibrations can only be investigated with regard to the perturbation level.     

Influence of \text{ CF }_{3}\text{ H } and \text{ CCl }_{4} additives on acetylene detonation

The influence of $\text{ CF }_{3}\text{ H }$ and $\text{ CCl }_{4}$ admixtures (known as detonation suppressors for combustible mixtures) on the development of acetylene detonation was experimentally investigated in a shock tube. The time-resolved images of detonation wave development and propagation were registered using a high-speed streak camera. Shock wave velocity and pressure profiles were measured by five calibrated piezoelectric gauges and the formation of condensed particles was detected by laser light extinction. The induction time of detonation development was determined as the moment of a pressure rise at the end plate of the shock tube. It was shown that $\text{ CF }_{3}\text{ H }$ additive had no influence on the induction time. For $\text{ CCl }_{4}$ , a significant promoting effect was observed. A simplified kinetic model was suggested and characteristic rates of diacetylene $\text{ C }_{4}\text{ H }_{2}$ formation were estimated as the limiting stage of acetylene polymerisation. An analysis of the obtained data indicated that the promoting species is atomic chlorine formed by $\text{ CCl }_{4}$ pyrolysis, which interacts with acetylene and produces $\text{ C }_{2}\text{ H }$ radical, initiating a chain mechanism of acetylene decomposition. The results of kinetic modelling agree well with the experimental data.     

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 ${\mu > \mu^*}$ .     

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 ${\mathbb{R}^4}$ .     

Investigation on the mechanism of convective mass transfer from a horizontal rotating cylinder without air jet flow

The local and mean convective mass transfer coefficients from the surface of a large-diameter horizontal circular rotating cylinder without air jet flow were investigated by measuring the concentration gradient. The results indicate that rotation performs different effects on the convective mass transfer at different regions. Based on the experimental data, the correlation equations of the mean convective mass transfer Sherwood number Sh and the critical Reynolds number Re _r,cri have been formulated as follows: $ Sh = 0.32[(8.5Re_{\text{r}}^{2} + Gr) \cdot Sc]^{1/3} $ and $ Re_{\text{r,cri}} = 0.44(Gr \cdot Sc)^{1/2} $ .     

Multistage γ-level \mathcal{H}_{\infty} control for input-saturated systems with disturbances

For input-saturated systems with disturbances, states in the domain of attraction cannot converge to the origin, but only to neighborhood around it. In order to design the smallest possible target invariant set and the largest possible domain of attraction, in this paper, we introduce a multistage γ-level $\mathcal{H}_{\infty}$ control for achieving a smaller target invariant set within a given $\mathcal{H}_{\infty}$ performance level and a larger domain of attraction than results obtained in previous studies. In particular, for the case in which the disturbances satisfy a matched condition, this paper introduces an $\mathcal{H}_{\infty}$ control with an extra control part to perfectly reject these disturbances despite the uncertainties; the introduction of the $\mathcal{H}_{\infty}$ control with an extra control part causes the target invariant set to shrink to the origin and the $\mathcal{H}_{\infty}$ performance level to become zero.     

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.     

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.     

A-Priori Direct Numerical Simulation Modelling of Co-variance Transport in Turbulent Stratified Flames

Three-dimensional Direct Numerical Simulations of statistically planar turbulent stratified flames at global equivalence ratios <???>?=?0.7 and <???>?=?1.0 have been carried out to analyse the statistical behaviour of the transport of co-variance of the fuel mass fraction Y _F and mixture fraction ξ (i.e. $\widetilde{Y_F^{\prime\prime} \xi ^{\prime\prime}}={\overline {\rho Y_F^{\prime\prime} \xi^{\prime\prime}} } \Big/ {\overline \rho })$ for Reynolds Averaged Navier Stokes simulations where $\overline q $ , $\tilde{q} ={\overline {\rho q} } \big/ {\overline \rho }$ and $q^{\prime\prime}= q-\tilde{q}$ are Reynolds averaged, Favre mean and Favre fluctuation of a general quantity q with ρ being the gas density and the overbar suggesting a Reynolds averaging operation. It has been found that existing algebraic expressions may not capture the statistical behaviour of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ with sufficient accuracy in low Damköhler number combustion and therefore, a transport equation for $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ may need to be solved. The statistical behaviours of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ and the unclosed terms of its transport equation (i.e. the terms originating from turbulent transport T ₁ , reaction rate T ₄ and molecular dissipation $\left( {-D_2 } \right))$ have been analysed in detail. The contribution of T ₁ remains important for all cases considered here. The term T ₄ acts as a major contributor in <???>?=?1.0 cases, but plays a relatively less important role in <???>?=?0.7 cases, whereas the term $\left( {-D_2 } \right)$ acts mostly as a leading order sink. Through an a-priori DNS analysis, the performances of the models for T ₁ , T ₄ and $\left( {-D_2 } \right)$ have been addressed in detail. A model has been identified for the turbulent transport term T ₁ which satisfactorily predicts the corresponding term obtained from DNS data. The models for T ₄ , which were originally proposed for high Damköhler number flames, have been modified for low Damköhler combustion. Predictions of the modified models are found to be in good agreement with T ₄ obtained from DNS data. It has been found that existing algebraic models for $D_2 =2\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} $ (where D is the mass diffusivity) are not sufficient for low Damköhler number combustion and therefore, a transport equation may need to be solved for the cross-scalar dissipation rate $\widetilde{\varepsilon }_{Y\xi } ={\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} } \big/ {\overline \rho }$ for the closure of the $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ transport equation.     

Wall-Driven Versus Pressure Gradient-Driven Flows in Porous Media

The heat transfer characteristics of two boundary layer flows past an isothermal plane surface adjacent to a saturated Darcy–Brinkman porous medium is compared to each other in this paper. The flows are driven either by a stretching of the adjacent plane boundary, or by an external pressure gradient. It is found that below a threshold value $\tilde{P}r_{*} $ of the modified Prandtl number $\tilde{P}r$ , the Nussselt number in case of the pressure gradient-driven flow is larger than in case of the wall- driven flow, while for $\tilde{P}r>\tilde{P}r_{*} $ the flow driven by the moving wall provides a more efficient heat transfer mechanism. The dependence of $\tilde{P}r_{*} $ on the Darcy number is also discussed in detail.     

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.