Insight into Heterogeneity Effects in Methane Hydrate Dissociation via Pore-Scale Modeling

A large number (1253) of high-quality streaming potential coefficient (\(C_\mathrm{sp})\) measurements have been carried out on Berea, Boise, Fontainebleau, and Lochaline sandstones (the latter two including both detrital and authigenic overgrowth forms), as a function of pore fluid salinity (\(C_\mathrm{f})\) and rock microstructure. All samples were saturated with fully equilibrated aqueous solutions of NaCl (10\(^{-5}\) and 4.5 mol/dm\(^{3})\) upon which accurate measurements of their electrical conductivity and pH were taken. These \(C_\mathrm{sp}\) measurements represent about a fivefold increase in streaming potential data available in the literature, are consistent with the pre-existing 266 measurements, and have lower experimental uncertainties. The \(C_\mathrm{sp}\) measurements follow a pH-sensitive power law behaviour with respect to \(C_\mathrm{f}\) at medium salinities (\(C_\mathrm{sp} =-\,1.44\times 10^{-9} C_\mathrm{f}^{-\,1.127} \), units: V/Pa and mol/dm\(^{3})\) and show the effect of rock microstructure on the low salinity \(C_\mathrm{sp}\) clearly, producing a smaller decrease in \(C_\mathrm{sp}\) per decade reduction in \(C_\mathrm{f}\) for samples with (i) lower porosity, (ii) larger cementation exponents, (iii) smaller grain sizes (and hence pore and pore throat sizes), and (iv) larger surface conduction. The \(C_\mathrm{sp}\) measurements include 313 made at \(C_\mathrm{f} > 1\) mol/dm\(^{3}\), which confirm the limiting high salinity \(C_\mathrm{sp}\) behaviour noted by Vinogradov et al., which has been ascribed to the attainment of maximum charge density in the electrical double layer occurring when the Debye length approximates to the size of the hydrated metal ion. The zeta potential (\(\zeta \)) was calculated from each \(C_\mathrm{sp}\) measurement. It was found that \(\zeta \) is highly sensitive to pH but not sensitive to rock microstructure. It exhibits a pH-dependent logarithmic behaviour with respect to \(C_\mathrm{f}\) at low to medium salinities (\(\zeta =0.01133 \log _{10} \left( {C_\mathrm{f} } \right) +0.003505\), units: V and mol/dm\(^{3})\) and a limiting zeta potential (zeta potential offset) at high salinities of \({\zeta }_\mathrm{o} = -\,17.36\pm 5.11\) mV in the pH range 6–8, which is also pH dependent. The sensitivity of both \(C_\mathrm{sp}\) and \(\zeta \) to pH and of \(C_\mathrm{sp}\) to rock microstructure indicates that \(C_\mathrm{sp}\) and \(\zeta \) measurements can only be interpreted together with accurate and equilibrated measurements of pore fluid conductivity and pH and supporting microstructural and surface conduction measurements for each sample.     

Characterization for elastic constants of fused deposition modelling-fabricated materials based on the virtual fields method and digital image correlation

Fused deposition modelling (FDM), a widely used rapid prototyping process, is a promising technique in manufacturing engineering. In this work, a method for characterizing elastic constants of FDM-fabricated materials is proposed. First of all, according to the manufacturing process of FDM, orthotropic constitutive model is used to describe the mechanical behavior. Then the virtual fields method (VFM) is applied to characterize all the mechanical parameters \((Q_{11}\), \(Q_{22}\), \(Q_{12}\), \(Q_{66})\) using the full-field strain, which is measured by digital image correlation (DIC). Since the principal axis of the FDM-fabricated structure is sometimes unknown due to the complexity of the manufacturing process, a disk in diametrical compression is used as the load configuration so that the loading angle can be changed conveniently. To verify the feasibility of the proposed method, finite element method (FEM) simulation is conducted to obtain the strain field of the disk. The simulation results show that higher accuracy can be achieved when the loading angle is close to \(30^{\circ }\). Finally, a disk fabricated by FDM was used for the experiment. By rotating the disk, several tests with different loading angles were conducted. To determine the position of the principal axis in each test, two groups of parameters \((Q_{11}\), \(Q_{22}\), \(Q_{12}\), \(Q_{66})\) are calculated by two different groups of virtual fields. Then the corresponding loading angle can be determined by minimizing the deviation between two groups of the parameters. After that, the four constants \((Q_{11}\), \(Q_{22}\), \(Q_{12}\), \(Q_{66})\) were determined from the test with an angle of \(27^{\circ }\).     

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.

     

A Peridynamic Model of Fracture Mechanics with Bond-Breaking

Wave propagation in elastic dielectrics with flexoelectricity, micro-inertia and strain gradient elasticity is investigated in this paper. Dispersion phenomenon, which does not exist in classical elastic dielectric theory, is observed in the flexoelectric microstructured solids. Analytical solutions for the phase velocity \(C_{p}\), group velocity \(C_{g}\) and their ratio \(\gamma = C_{g} / C_{p}\) are calculated for the case of harmonic decomposition. The magnitudes of the phase velocity and group velocity changed with the increasing of the wave number, while they are constant in the classical elastic dielectric theory. It is shown that the flexoelectricity, micro-inertia and microstructural effects are significant to predict the real behavior of longitudinal wave propagating in flexoelectric microstructured solids. Microstructural effects are not sufficient for dealing with realistic dispersion curves in flexoelectric solids, the micro-inertia and flexoelectricity are needed to obtain a physically acceptable value of the phase and group velocities.     

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.     

Displacement and Dissolution Characteristics of CO$$_{}$$/Brine System in Unconsolidated Porous Media

This study investigated the dynamic displacement and dissolution of \(\hbox {CO}_{2}\) in porous media at 313 K and 6/8 MPa. Gaseous (\(\hbox {gCO}_{2}\)) at 6 MPa and supercritical \(\hbox {CO}_{2 }(\hbox {scCO}_{2}) \) at 8 MPa were injected downward into a glass bead pack at different flow rates, following upwards brine injection. The processes occurring during \(\hbox {CO}_{2}\) drainage and brine imbibition were visualized using magnetic resonance imaging. The drainage flow fronts were strongly influenced by the flow rates, resulting in different gas distributions. However, brine imbibition proceeded as a vertical compacted front due to the strong effect of gravity. Additionally, the effects of flow rate on distribution and saturation were analyzed. Then, the front movement of \(\hbox {CO}_{2}\) dissolution was visualized along different paths after imbibition. The determined \(\hbox {CO}_{2}\) concentrations implied that little \(\hbox {scCO}_{2}\) dissolved in brine after imbibition. The dissolution rate was from \(10^{-8}\) to \(10^{-9}\, \hbox {kg}\, \hbox {m}^{-3} \, \hbox {s}^{-1}\) and from \(10^{-6}\) to \(10^{-8}\, \hbox {kg}\, \hbox {m}^{-3} \, \hbox {s}^{-1}\) for \(\hbox {gCO}_{2}\) at 6 MPa and \(\hbox {scCO}_{2 }\) at 8 MPa, respectively. The total time for the \(\hbox {scCO}_{2}\) dissolution was short, indicating fast mass transfer between the \(\hbox {CO}_{2}\) and brine. Injection of \(\hbox {CO}_{2}\) under supercritical conditions resulted in a quick establishment of a steady state with high storage safety.     

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

     

Delayed detached eddy simulations of fighter aircraft at high angle of attack

The massively separated flows over a realistic aircraft configuration at 40?, 50?, and 60?angles of attack are studied using the delayed detached eddy simulation(DDES).The calculations are carried out at experimental conditions corresponding to a mean aerodynamic chord-based Reynolds number of 8.93 × 10~5 and Mach number of 0.088. The influence of the grid size is investigated using two grids, 20.0×10~6cells and 31.0 × 10~6 cells. At the selected conditions, the lift,drag, and pitching moment from DDES predictions agree with the experimental data better than that from the Reynoldsaveraged Navier–Stokes. The effect of angle of attack on the flow structure over the general aircraft is also studied, and it is found that the dominated frequency associated with the vortex shedding process decreases with increasing angle of attack.     

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (\({\textit{Re}}\ll 1\)) and high Péclet (\({\textit{Pe}}\gg 1\)) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length \(l_s\). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of \(Nu \sim {\textit{Pe}}^{1/3}\) and \(Nu \sim {\textit{Pe}}^{1/2}\) for no-slip (\(l_s \rightarrow 0\)) and shear-free (\(l_s \rightarrow \infty \)) boundaries, respectively. Boundary layer analysis also shows that the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) holds for a shear-free cylinder surface in the asymptotic limit of \({\textit{Re}}\gg 1\) so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite \({\textit{Re}}\), results from our two-dimensional simulations confirm the scaling \(Nu \sim {\textit{Pe}}^{1/2}\) for a shear-free boundary over the range \(0.1 \le {\textit{Re}}\le 10^3\) and \(0.1\le {\textit{Pr}}\le 10\). A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with \(l_s\), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number \(Nu_{\theta }\) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high \({\textit{Re}}\) and Pr. At a Reynolds number of \(10^3\), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in \(Nu_{\theta }\) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variation with \(l_s\) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on \(l_s\) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low \(l_s\), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and \({\textit{Pr}}\) with Sherwood and Schmidt numbers, respectively.     

A Simple Method for the Determination of Adsorption Kinetic Parameters Using Recycle Fixed-Bed Adsorber

This study focuses on the development of a novel analysis technique for determining the intraparticle diffusivity \((D_{\mathrm{S}})\) and fluid film mass transfer coefficient \((k_\mathrm{F})\) from a concentration history curve of a recycle fixed-bed reactor without using the linear driving force approximation and empirical equations for the estimation of \(k_\mathrm{F}\). The recycle fixed-bed method requires lesser amounts of working fluid for experiment purposes, which is an advantage over the usual fixed-bed method. Based on the characterization results of the concentration history curves, simulated by rigorous numerical calculations, a novel analysis technique was established. The \(D_{\mathrm{S}}\) value can be determined from the experimentally obtained time at which the concentration of the curve is minimal \((t_{C\mathrm{min}})\). The \(k_\mathrm{F}\) value can be determined from the \(D_{\mathrm{S}}\) value and Biot number (Bi), which can be estimated from the experimental ratio of the maximum concentration to the minimum concentration \((c_{\max }/c_{\min })\). The \(D_{\mathrm{S}}\) and \(k_\mathrm{F}\) values of phenol adsorption on an activated carbon material were determined experimentally using the proposed analysis method. This method enables the determination of reliable adsorption kinetic parameters through a simple and economical experiment. However, appropriate experimental data must be acquired under regulated experimental conditions, especially in the case of fluctuation of the concentration history curve.     

The Unstable Set of a Periodic Orbit for Delayed Positive Feedback

In the paper [Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, JDDE 23 (2011), no. 4, 727–790], we have constructed large-amplitude periodic orbits for an equation with delayed monotone positive feedback. We have shown that the unstable sets of the large-amplitude periodic orbits constitute the global attractor besides spindle-like structures. In this paper we focus on a large-amplitude periodic orbit \({\mathcal {O}}_{p}\) with two Floquet multipliers outside the unit circle, and we intend to characterize the geometric structure of its unstable set \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \). We prove that \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) is a three-dimensional \(C^{1}\)-submanifold of the phase space and admits a smooth global graph representation. Within \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \), there exist heteroclinic connections from \({\mathcal {O}}_{p}\) to three different periodic orbits. These connecting sets are two-dimensional \(C^{1}\)-submanifolds of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) and homeomorphic to the two-dimensional open annulus. They form \(C^{1}\)-smooth separatrices in the sense that they divide the points of \({\mathcal {W}}^{u}\left( {\mathcal {O}}_{p}\right) \) into three subsets according to their \(\omega \)-limit sets.     

Steady States of Fokker–Planck Equations: III. Degenerate Diffusion

This is the third paper in a series concerning the study of steady states of a Fokker–Planck equation in a general domain in \({\mathbb R}^n\) with \(L^{p}_{loc}\) drift term and \(W^{1,p}_{loc}\) diffusion term for any \(p>n\). In this paper, we give some existence results of stationary measures of the Fokker–Planck equation under Lyapunov conditions which allow the degeneracy of diffusion.     

Further Dense Properties of the Space of Circle Diffeomorphisms with a Liouville Rotation Number

In continuation of Matsumoto’s paper (Nonlinearity 25:1495–1511, 2012) we show that various subspaces are \(C^{\infty }\)-dense in the space of orientation-preserving \(C^{\infty }\)-diffeomorphisms of the circle with rotation number \(\alpha \), where \(\alpha \in {\mathbb {S}}^1\) is any prescribed Liouville number. In particular, for every odometer \({\mathcal {O}}\) of product type we prove the denseness of the subspace of diffeomorphisms which are orbit-equivalent to \({\mathcal {O}}\).     

Transitional and Turbulent Flow in a Bed of Spheres as Measured with Stereoscopic Particle Image Velocimetry

Stereoscopic particle image velocimetry has been used to investigate inertia dominated, transitional and turbulent flow in a randomly packed bed of monosized PMMA spheres. By using an index-matched fluid, the bed is optically transparent and measurements can be performed in an arbitrary position within the porous bed. The velocity field observations are carried out for particle Reynolds numbers, \({Re}_{{p}}\), between 20 and 3220, and the sampling is done at a frequency of 75 Hz. Results show that, in porous media, the dynamics of the flow can vary significantly from pore to pore. At \({Re}_{{p}}\) around 400 the spatially averaged time fluctuations of total velocity reach a maximum and the spatial variation of the time-averaged total velocity, \(u_\mathrm{tot}\) increases up to about the same \({Re}_{{p}}\) and then it decreases. Also in the studied planes, a considerable amount of the fluid moves in the perpendicular directions to the main flow direction and the time-averaged magnitude of the velocity in the main direction, \(u_{x}\), has an averaged minimum of 40% of the magnitude of \(u_\mathrm{tot}\) at \({Re}_{{p}}\) about 400. For \({Re}_{{p}} > 1600\), this ratio is nearly constant and \(u_{x}\) is on average a little bit less than 50% of \(u_\mathrm{tot}\). The importance of the results for longitudinal and transverse dispersion is discussed.     

Existence of $$C^{k}$$-Invariant Foliations for Lorenz-Type Maps

Under conditions similar to those in Shashkov and Shil’nikov (Differ Uravn 30(4):586–595, 732, 1994) we show that a \(C^{k+1}\) Lorenz-type map T has a \(C^{k}\) codimension one foliation which is invariant under the action of T. This allows us to associate T to a \(C^{k}\) one-dimensional transformation.     

Glimpses of Flow Development and Degradation During Type B Drag Reduction by Aqueous Solutions of Polyacrylamide B1120

Flow development and degradation during Type B turbulent drag reduction by 0.10 to 10 wppm solutions of a partially-hydrolysed polyacrylamide B1120 of MW \(=\) 18x10⁶ was studied in a smooth pipe of ID \(=\) 4.60 mm and L/D \(=\) 210 at Reynolds numbers from 10000 to 80000 and wall shear stresses Tw from 8 to 600 Pa. B1120 solutions exhibited facets of a Type B ladder, including segments roughly parallel to, but displaced upward from, the P-K line; those that attained asymptotic maximum drag reduction at low Re√ f but departed downwards into the polymeric regime at a higher retro-onset Re√ f; and segments at MDR for all Re√ f. Axial flow enhancement profiles of S\(^{\prime }\) vs L/D reflected a superposition of flow development and polymer degradation effects, the former increasing and the latter diminishing S\(^{\prime }\) with increasing distance downstream. Solutions that induced normalized flow enhancements S\(^{\prime }\)/S\(^{\prime }_{\mathrm {m}} <\) 0.4 developed akin to solvent, with L_e,p/D \(=\) L_e,n/D \(<\) 42.3, while those at maximum drag reduction showed entrance lengths L_e,m/D \(\sim \) 117, roughly 3 times the solvent L_e,n/D. Degradation kinetics were inferred by first detecting a falloff point (Re√f^∧, S\(^{{\prime }\wedge }\)), of maximum observed flow enhancement, for each polymer solution. A plot of S\(^{{\prime }\wedge }\)vs C revealed S\(^{{\prime }\wedge }\)linear in C at low C, with lower bound [S\(^{\prime }\)] \(=\) 5.0 wppm^??1, and S\(^{{\prime }\wedge }\) independent of C at high C, with upper bound S\(^{\prime }_{\mathrm {m}} =\) 15.9. The ratio S\(^{\prime }\)/S\(^{{\prime }\wedge }\) in any pipe section was interpreted to be the undegraded fraction of original polymer therein. Semi-log plots of (S\(^{\prime }\)/S\(^{{\prime }\wedge }\)) at a section vs transit time from pipe entrance thereto revealed first order kinetics, from which apparent degradation rate constants k_deg s^??1 and entrance severities ?ln(S\(^{\prime }\)/S\(^{{\prime }\wedge }\))₀ were extracted. At constant C, k_deg increased linearly with increasing wall shear stress Tw, and at constant Tw, k_deg was independent of C, providing a B1120 degradation modulus (k_deg/Tw) \(=\) (0.012 \(\pm \) 0.001) (Pa s)^??1 for 8 \(<\) Tw Pa \(<\) 600, 0.30 \(<\) C wppm \(<\) 10. Entrance severities were negligible below a threshold Twe \(\sim \) 30 Pa and increased linearly with increasing Tw for Tw \(>\) Twe. The foregoing methods were applied to Type A drag reduction by 0.10 to 10 wppm solutions of a polyethyleneoxide PEO P309, MW \(=\) 11x10⁶, in a smooth pipe of ID \(=\) 7.77 mm and L/D \(=\) 220 at Re from 4000 to 115000. P309 solutions that induced S\(^{\prime }\)/S\(^{\prime }_{\mathrm {m}} <\) 0.4 developed akin to solvent, with L_e,p/D \(=\) L_e,n/D \(<\) 23, while those at MDR had entrance lengths L_e,m/D \(\sim \) 93, roughly 4 times the solvent L_e,n/D. P309 solutions described a Type A fan distorted by polymer degradation. A typical trajectory departed the P-K line at an onset point Re√ f* followed by ascending and descending polymeric regime segments separated by a falloff point Re√f^∧, of maximum flow enhancement; for all P309 solutions, onset Re√ f* = 550 \(\pm \) 100 and falloff Re√f^∧ = 2550 \(\pm \) 250, the interval between them delineating Type A drag reduction unaffected by degradation. A plot of falloff S\(^{{\prime }\wedge }\) vs C for PEO P309 solutions bore a striking resemblance to the analogous S\(^{{\prime }\wedge }\) vs C plot for solutions of PAMH B1120, indicating that the initial Type A drag reduction by P309 after onset at Re√ f* had evolved to Type B drag reduction by falloff at Re√f^∧. Presuming that Type B behaviour persisted past falloff permitted inference of P309 degradation kinetics; k_deg was found to increase linearly with increasing Tw at constant C and was independent of C at constant Tw, providing a P309 degradation modulus (k_deg/Tw) \(=\) (0.011 \(\pm \) 0.002) (Pa s)^??1 for 4 \(<\) Tw Pa \(<\) 400, 0.10 \(<\) C wppm < 5.0. Comparisons between the present degradation kinetics and previous literature showed (k_deg/Tw) data from laboratory pipes of D \(\sim \) 0.01 m to lie on a simple extension of (k_deg/Tw) data from pipelines of D \(\sim \) 0.1 m and 1.0 m, along a power-law relation (k_deg/Tw) \(=\) 10^??5.4.D^??1.6. Intrinsic slips derived from PAMH B1120 and PEO P309-at-falloff experiments were compared with previous examples from Type B drag reduction by polymers with vinylic and glycosidic backbones, showing: (i) For a given polymer, [S\(^{\prime }\)] was independent of Re√ f and pipe ID, implying insensitivity to both micro- and macro-scales of turbulence; and (ii) [S\(^{\prime }\)] increased linearly with increasing polymer chain contour length Lc, the proportionality constant \(\beta =\) 0.053 \(\pm \) 0.036 enabling estimation of flow enhancement S\(^{\prime } =\) C.Lc.β for all Type B drag reduction by polymers.     

The Effect of Grain Size Distribution on Nonlinear Flow Behavior in Sandy Porous Media

The current study provides new experimental data on nonlinear flow behavior in various uniformly graded granular materials (20 samples) ranging from medium sands (\(d_{50 }>0.39\) mm) to gravel (\(d_{50}=6.3\) mm). Generally, theoretical equations relate the Forchheimer parameters a and b to the porosity, as well as the characteristic pore length, which is assumed to be the median grain size \((d_{50})\) of the porous medium. However, numerical and experimental studies show that flow resistance in porous media is largely determined by the geometry of the pore structure. In this study, the effect of the grain size distribution was analyzed using subangular-subrounded sands and approximately equal compaction grades. We have used a reference dataset of 11 uniformly graded filter sands. Mixtures of filter sands were used to obtain a slightly more well-graded composite sand (increased \(C_{u}\) values by a factor of 1.19 up to 2.32) with respect to its associated reference sand at equal median grain size \((d_{50})\) and porosity. For all composite sands, the observed flow resistance was higher than in the corresponding reference sand at equal \(d_{50}\), resulting in increased a coefficients by factors up to 1.68, as well as increased b coefficients by factors up to 1.44. A modified Ergun relationship with Ergun constants of 139.1 for A and 2.2 for B, as well as the use of \(d_{m}-\sigma \) as characteristic pore length predicted the coefficients a and b accurately.     

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.