Multi-component Reynolds-averaged Navier–Stokes simulations of Richtmyer–Meshkov instability and mixing induced by reshock at different times

Turbulent mixing generated by shock-driven acceleration of a perturbed interface is simulated using a new multi-component Reynolds-averaged Navier–Stokes (RANS) model closed with a two-equation $K$ – $\epsilon $ model. The model is implemented in a hydrodynamics code using a third-order weighted essentially non-oscillatory finite-difference method for the advection terms and a second-order central difference method for the gradients in the source and diffusion terms. In the present reshocked Richtmyer–Meshkov instability and mixing study, an incident shock with Mach number $M\!a_{\mathrm{s}}=1.20$ is generated in air and progresses into a sulfur hexafluoride test section. The time evolution of the predicted mixing layer widths corresponding to six shock tube test section lengths are compared with experimental measurements and three-dimensional multi-mode numerical simulations. The mixing layer widths are also compared with the analytical self-similar power-law solution of the simplified model equations prior to reshock. A set of model coefficients and initial conditions specific to these six experiments is established, for which the widths before and after reshock agree very well with experimental and numerical simulation data. A second set of general coefficients that accommodates a broader range of incident shock Mach numbers, Atwood numbers, and test section lengths is also established by incorporating additional experimental data for $M\!a_{\mathrm{s}}=1.24$ , $1.50$ , and $1.98$ with $At=0.67$ and $M\!a_{\mathrm{s}}=1.45$ with $At=-0.67$ and previous RANS modeling. Terms in the budgets of the turbulent kinetic energy and dissipation rate equations are examined to evaluate the relative importance of turbulence production, dissipation and diffusion mechanisms during mixing. Convergence results for the mixing layer widths, mean fields, and turbulent fields under grid refinement are presented for each of the $M\!a_{\mathrm{s}}=1.20$ cases.     

Towards the Development of an Evolution Equation for Flame Turbulence Interaction in Premixed Turbulent Combustion

Flame turbulence interaction is one of the leading order terms in the scalar dissipation \(\left (\widetilde {\varepsilon }_{c}\right )\) transport equation [35] and is thus an important phenomenon in premixed turbulent combustion. Swaminathan and Grout [36] and Chakraborty and Swaminathan [15, 16] have shown that the effect of strain rate on the transport of \(\widetilde {\varepsilon }_{c}\) is dominated by the interaction between the fluctuating scalar gradients and the fluctuating strain rate, denoted here by \(\overline {\rho }\widetilde {\Delta }_{c}= \overline {\rho {\alpha }\nabla c^{\prime \prime }S_{ij}^{\prime \prime }\nabla c^{\prime \prime }}\) ; this represents the flame turbulence interaction. In order to obtain an accurate representation of this phenomenon, a new evolution equation for \(\widetilde {\Delta }_{c}\) has been proposed. This equation gives a detailed insight into flame turbulence interaction and provides an alternative approach to model the important physics represented by \(\widetilde {\Delta }_{c}\) . The \(\widetilde {\Delta }_{c}\) evolution equation is derived in detail and an order of magnitude analysis is carried out to determine the leading order terms in the \(\widetilde {\Delta }_{c}\) evolution equation. The leading order terms are then studied using a Direct Numerical Simulation (DNS) of premixed turbulent flames in the corrugated flamelet regime. It is found that the behaviour of \(\widetilde {\Delta }_{c}\) is determined by the competition between the source terms (pressure gradient and the reaction rate), diffusion/dissipation processes, turbulent strain rate and the dilatation rate. Closures for the leading order terms in \(\widetilde {\Delta }_{c}\) evolution equation have been proposed and compared with the DNS data.     

A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler Equations: the Use of Kolmogorov’s Dissipation Range

Motivated by Kolmogorov’s theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber ${\Lambda(t)}$ that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we prove that Leray-Hopf solutions are regular provided ${\Lambda \in L^{5/2}}$ , which improves our previous ${\Lambda \in L^\infty}$ condition. We also show that ${\Lambda \in L^1}$ for all Leray-Hopf solutions. Finally, we prove that Leray-Hopf solutions are regular when the time-averaged spatial intermittency is small, i.e., close to Kolmogorov’s regime.     

Double-Diffusive Natural Convection in Anisotropic Porous Medium Bounded by Finite Thickness Walls: Validity of Local Thermal Equilibrium Assumption

Double-diffusive natural convection in fluid-saturated porous medium inside a vertical enclosure bounded by finite thickness walls with opposing temperature, concentration gradients on vertical walls as well as adiabatic and impermeable horizontal ones has been performed numerically. The Darcy model was used to predict fluid flow inside the porous material, while thermal fields are simulated based on two-energy equations for fluid and solid phases on the basis of a local thermal non-equilibrium model. Computations have been performed for different controlling parameters such as the buoyancy ratio $N$ , the Lewis number Le, the anisotropic permeability ratio $R_\mathrm{p}$ , the fluid-to-solid thermal conductivity ratio $R_\mathrm{c}$ , the interphase heat transfer coefficient $\mathcal{H}$ , the ratio of the wall thickness to its height $D$ , the wall-to-porous medium thermal diffusivity ratio $R_\mathrm{w}$ , and the solid-to-fluid heat capacity ratio $\gamma $ . Thus, the effects of the controlling parameters on heat and mass transfer characteristics are discussed in detail. Moreover, the validity domain of the local thermal equilibrium (LTE) assumption has been delimited for different set of the governing parameters. It has been shown that Le has a noticeable significant effect on fluid temperature profiles and that higher $N$ values lead to a significant enhancement in heat and mass transfer rates. Moreover, for higher $\mathcal{H}, R_\mathrm{c}$ , $R_\mathrm{p}, R_\mathrm{w}$ , or $D$ values and/or lower $\gamma $ values, the solid and fluid phases tend toward LTE.     

A Comment on the Flow and Heat Transfer Past a Permeable Stretching/Shrinking Surface in a Porous Medium: Brinkman Model

An analytical solution is presented for the boundary-layer flow and heat transfer over a permeable stretching/shrinking surface embedded in a porous medium using the Brinkman model. The problem is seen to be characterized by the Prandtl number $Pr$ , a mass flux parameter $s$ , with $s>0$ for suction, $s=0$ for an impermeable surface, and $s<0$ for blowing, a viscosity ratio parameter $M$ , the porous medium parameter $\Lambda $ and a wall velocity parameter $\lambda $ . The analytical solution identifies critical values which agree with those previously determined numerically (Bachok et al. Proceedings of the fifth International Conference on Applications of Porous Media, 2013) and shows that these critical values, and the consequent dual solutions, can arise only when there is suction through the wall, $s>0$ .     

The Onset of Convection in a Layered Porous Medium with Vertical Throughflow

An analytical investigation of the effect of vertical throughflow on the onset of convection in a composite porous medium consisting of two horizontal layers has been made. The cases of iso-flux and iso-temperature boundaries are both investigated. The critical Rayleigh number depends on a Péclet number $Q$ , a permeability ratio $K_{r}$ , a thermal conductivity ratio $k_{r}$ , and a depth ratio $\delta $ . For the case of small $Q$ an approximate solution is obtained, which shows that in general throughflow has a stabilizing effect whose magnitude may be increased or decreased by the heterogeneity. This solution is supplemented by an asymptotic solution valid for large $Q.$      

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.     

Some Aspects of Stability for Semigroup Actions and Control Systems

The present paper introduces both the notions of Lagrange and Poisson stabilities for semigroup actions. Let \(S\) be a semigroup acting on a topological space \(X\) with mapping \(\sigma :S\times X\rightarrow X\) , and let \(\mathcal {F}\) be a family of subsets of \(S\) . For \(x\in X\) the motion \(\sigma _{x}:S\rightarrow X\) is said to be forward Lagrange stable if the orbit \(Sx\) has compact closure in \(X\) . The point \(x\) is forward \(\mathcal {F}\) -Poisson stable if and only if it belongs to the limit set \(\omega \left( x,\mathcal {F}\right) \) . The concept of prolongational limit set is also introduced and used to describe nonwandering points. It is shown that a point \(x\) is \( \mathcal {F}\) -nonwandering if and only if \(x\) lies in its forward \(\mathcal {F} \) -prolongational limit set \(J\left( x,\mathcal {F}\right) \) . The paper contains applications to control systems.     

On the Characterization of the Navier–Stokes Flows with the Power-Like Energy Decay

We study the energy decay of the turbulent solutions to the Navier–Stokes equations in the whole three-dimensional space. We show as the main result that the solutions with the energy decreasing at the rate \({O(t^{-\alpha}), t \rightarrow \infty, \alpha \in [0, 5/2]}\) , are exactly characterized by their initial conditions belonging into the homogeneous Besov space \({\dot{B}^{-\alpha}_{2, \infty}}\) . Similarly, for a solution u and \({p \in [1, \infty]}\) the integral \({\int_{0}^{\infty} \|t^{\alpha/2} u(t)\|^p \frac{1}{t} dt}\) is finite if and only if the initial condition of u belongs to the homogeneous Besov space \({\dot{B}_{2, p}^{-\alpha}}\) . For the case \({\alpha \in (5/2, 9/2]}\) we present analogical results for some subclasses of turbulent solutions.     

Global Regularity for the Two-Dimensional Anisotropic Boussinesq Equations with Vertical Dissipation

This paper establishes the global in time existence of classical solutions to the two-dimensional anisotropic Boussinesq equations with vertical dissipation. When only vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, we bound the derivatives in terms of the ${L^\infty}$ -norm of the vertical velocity v and prove that ${\|v\|_{L^{r}}}$ with ${2\leqq r < \infty}$ does not grow faster than ${\sqrt{r \log r}}$ at any time as r increases. A delicate interpolation inequality connecting ${\|v\|_{L^\infty}}$ and ${\|v\|_{L^r}}$ then yields the desired global regularity.     

Toward a New Method of Porosimetry: Principles and Experiments

Current experimental methods used to determine pore size distributions (PSD) of porous media present several drawbacks such as toxicity of the employed fluids (e.g., mercury porosimetry). The theoretical basis of a new method to obtain the PSD by injecting yield stress fluids through porous media and measuring the flow rate $Q$ at several pressure gradients $\nabla P$ was proposed in the literature. On the basis of these theoretical considerations, an intuitive approach to obtain PSD from $Q(\nabla P)$ is presented in this work. It relies on considering the extra increment of $Q$ when $\nabla P$ is increased, as a consequence of the pores of smaller radius newly incorporated to the flow. This procedure is first tested and validated on numerically generated experiments. Then, it is applied to exploit data coming from laboratory experiments and the obtained PSD show good agreement with the PSD deduced from mercury porosimetry.     

A Fast Algorithm for Invasion Percolation

We present a computationally fast Invasion Percolation (IP) algorithm. IP is a numerical approach for generating realistic fluid distributions for quasi-static (i.e., slow) immiscible fluid invasion in porous media. The algorithm proposed here uses a binary-tree data structure to identify the site (pore) connected to the invasion cluster that is the next to be invaded. Gravity is included. Trapping is not explicitly treated in the numerical examples but can be added, for example, using a Hoshen–Kopelman algorithm. Computation time to percolation for a 3D system having $N$ total sites and $M$ invaded sites at percolation goes as $O(M \log M)$ for the proposed binary-tree algorithm and as $O(M N)$ for a standard implementation of IP that searches through all of the uninvaded sites at each step. The relation between $M$ and $N$ is $M = N^{D/E}$ , where $D$ is the fractal dimension of an infinite cluster and $E$ is Euclidean space dimension. In numerical practice, on finite-sized cubic lattices with invasion structures influenced by the injection boundary and boundary conditions lateral to the flow direction, we observe the scaling $M = N^{0.852}$ in 3D (valid through the second decimal place) instead of $M= N^{0.843}$ based on the infinite cluster fractal dimension $D=2.53$ .     

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.     

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.     

Stable disarrangement phases of elastic aggregates: a setting for the emergence of no-tension materials with non-linear response in compression

A recent field theory of elastic bodies undergoing non-smooth submacroscopic geometrical changes (disarrangements) provides a setting in which, for a given homogeneous macroscopic deformation \(F\) of the body, there are typically a number of different states \(G\) of smooth, submacroscopic deformation (disarrangement phases) available to the body. A tensorial consistency relation and the inequality \(\det G\le \det F\) that guarantees that \(F\) accommodates \(G\) determine the totality of disarrangement phases \(G\) corresponding to \(F\) , and it is natural to seek for a given \(F\) those disarrangement phases that minimize the Helmholtz free energy (stable disarrangement phases). We introduce these concepts in the particular context of continuous bodies comprised of many small elastic bodies (elastic aggregates) and in the context where disarrangements do not contribute to the Helmholtz free energy (purely dissipative disarrangements). In this setting, the Helmholtz free energy response \(G\longmapsto \varPsi (G)\) of the pieces of the aggregate determines the totality of disarrangement phases corresponding to \(F\) , which necessarily includes the phase \(G=F\) (compact phase) in which every piece of the aggregate undergoes the given macroscopic deformation \(F\) . When the response function \(\varPsi \) is isotropic and smooth, and when \(\varPsi \) possesses standard semiconvexity and growth properties, the body also admits phases of the form \(G=\zeta _{\min }R\) (loose phases) with \(R\) an arbitrary rotation, provided that \(\zeta _{\min }R \) satisfies the accommodation inequality \(\zeta _{\min }^{3}\le \det F\) . Loose phases, when available, achieve the global minimum \(\varPsi (\zeta _{\min }R)\) of the free energy and consequently are stable and stress-free. When \( \varPsi (G)\) has the specific form \(\varPsi _{\alpha \beta }(G)=(\alpha /2)(\det G)^{-2}+(\beta /2)tr(GG^{T})\) , with \(\alpha \) , \(\beta \) given elastic constants, we determine all of the disarrangement phases corresponding to \(F\) . These include not only the compact and loose phases, but also disarrangement phases \(G\) in which the stress \(D\varPsi (G)\) is uniaxial or planar. Our main result (“stability implies no-tension”) is the assertion that every stable disarrangement phase for \(\varPsi _{\alpha \beta }\) cannot support tensile tractions, and our treatment of elastic aggregates thus provides a natural setting for the emergence of no-tension materials whose response in compression is non-linear. Existing treatments of no-tension materials assume at the outset that the body cannot support tension and that the response in compression is linear.     

Tomographic PIV investigation of roughness-induced transition in a hypersonic boundary layer

The disagreement between free surface scalar experiments and the two-dimensional (2D) transport equation is discussed. An effective diffusivity coefficient, \(\kappa _{{\rm eff}}\) , is introduced and defined as the quotient between variance decay and mean gradient square. In all the experiments performed, \(\kappa _{{\rm eff}}\) is significantly larger than the scalar diffusivity, \(\kappa \) . Three mechanisms are identified as responsible for the differences between the quasi two-dimensional (Q2D) experiments and the 2D behaviour of a diffusive scalar. These are the vertical velocity gradients, the free surface divergence and the gravity currents induced by the scalar. These mechanisms, which affect the diffusive term in the 2D transport equation for large Péclet number ( \(Pe\gg 1\) ), are evaluated for steady and time-dependant laminar flows driven by electromagnetic body forces.     

A fast algorithm to calculate the critical coupling strength for synchronization in a chain of Kuramoto oscillators

Synchronization in a one-dimensional chain of Kuramoto oscillators with periodic boundary conditions is studied. An algorithm to rapidly calculate the critical coupling strength \(K_c\) for complete frequency synchronization is presented according to the mathematical constraint conditions and the periodic boundary conditions. By this new algorithm, we have checked the relation between \(\langle K_c\rangle \) and \(N\) , which is \(\langle K_c\rangle \sim \sqrt{N}\) , not only for small \(N\) , but also for large \(N\) . We also investigate the heavy-tailed distribution of \(K_c\) for random intrinsic frequencies, which is obtained by showing that the synchronization problem is equivalent to a discretization of Brownian motion. This theoretical result was checked by generating a large sample of \(K_c\) for large \(N\) from our algorithm to get the empirical density of \(K_c\) . Finally, we derive the permutation for the maximum coupling strength and its exact expression, which grows linearly with \(N\) and would provide the theoretical support for engineering applications.     

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.     

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 $E_{0}$ .     

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.