Horizontal Layer Research Articles

Numerical Modeling of Hydraulic Fracturing Interference in Multi-Layer Shale Oil Wells

Multi-layer horizontal well development and hydraulic fracturing are key techniques for enhancing production from shale oil reservoirs. During well development, the fracturing performance and well-pad production are affected by depletion-induced stress changes. Previous studies generally focused on the stress and fracturing interference within the horizontal layers, and the infilled multi-layer development was not thoroughly investigated. This study introduces a modeling workflow based on finite element and displacement discontinuity methods that accounts for dynamic porous media flow, geomechanics, and hydraulic fracturing modeling. It quantitatively characterizes the in situ stress alteration in various layers caused by the historical production of parent wells and quantifies the hydraulic fracturing interference in infill wells. In situ stress changes and reorientation and the non-planar propagation of hydraulic fractures were simulated. Thus, the workflow characterizes infill-well fracturing interferences in shale oil reservoirs developed by multi-layer horizontal wells. Non-planar fracturing in infill wells is affected by the parent-well history production, infilling layers, and cluster number. They also affect principal stress reorientations and reversal of the fracturing paths. Interwell interference can be decreased by optimizing the infilling layer, infill-well fracturing timing, and cluster numbers. This study extends the numerical investigation of interwell fracturing interference to multi-layer development.

Shear-driven magnetic buoyancy in the solar tachocline: Dependence of the mean electromotive force on diffusivity and latitude

Abstract The details of the dynamo process that is responsible for driving the solar magnetic activity cycle are still not fully understood. In particular, whilst differential rotation provides a plausible mechanism for the regeneration of the toroidal (azimuthal) component of the large-scale magnetic field, there is ongoing debate regarding the process that is responsible for regenerating the Sun’s large-scale poloidal field. Our aim is to demonstrate that magnetic buoyancy, in the presence of rotation, is capable of producing the necessary regenerative effect. Building upon our previous work, we carry out numerical simulations of a local Cartesian model of the tachocline, consisting of a rotating, fully compressible, electrically conducting fluid with a forced shear flow. An initially weak, vertical magnetic field is sheared into a strong, horizontal magnetic layer that becomes subject to magnetic buoyancy instability. By increasing the Prandtl number we lessen the back reaction of the Lorentz force onto the shear flow, maintaining stronger shear and a more intense magnetic layer. This in turn leads to a more vigorous instability and a much stronger mean electromotive force, which has the potential to significantly influence the evolution of the mean magnetic field. These results are only weakly dependent upon the inclination of the rotation vector, i.e. the latitude of the local Cartesian model. Although further work is needed to confirm this, these results suggest that magnetic buoyancy in the tachocline is a viable poloidal field regeneration mechanism for the solar dynamo.

Stability analysis of Rayleigh–Bénard–Marangoni convection in fluids with cross-zero expansion coefficient

Cross-zero expansion coefficient Rayleigh–Bénard–Marangoni (CRBM) convection refers to the convective phenomenon where thermal convection with stratified positive and negative expansion coefficients in a liquid layer is coupled with the Marangoni convection. In the Bénard convection, fluids with a cross-zero expansion coefficient contain a neutral expansion layer where the expansion coefficient (α) is zero, and the local buoyancy-driven convection is coupled with the Marangoni convection, leading to unique flow instability phenomena. This paper uses linear stability theory to analyze the CRBM convection in a horizontal liquid layer under a vertical temperature gradient and performs numerical calculations for fluids under different Bond numbers (Bd) in both bottom-heated and bottom-cooled models, obtaining the critical destabilization conditions and modes. In the bottom-heated model, different combinations of buoyancy instability mechanism (BIM), tension instability mechanism, and coupled instability mechanism (CIM) appear depending on the dimensionless temperature for the neutral expansion layer (Tα0) and the Bd. In the bottom-cooled model, two mechanisms occur according to the variation of Tα0: BIM and CIM.

Failure mechanism of transversely isotropic schist under Brazilian test using real-time X-ray nano tomography scanning

The study tries to address Brazilian tests on a transversely isotropic rock to investigate failure conditions by using digital volume correlation (DVC). For this purpose, an in-situ Brazilian apparatus in a nano- X-ray computed tomography (CT) scanner is applied to investigate the 3D progressive failure mechanism of the schist. Two different anisotropy angles are tested. CT scans are conducted at every selected loading stage before peak load. DVC analyzes the constructed X-ray CT images to determine the effect of schistosity orientation on failure mechanism, including crack initiation and propagation. A calibration method is presented to verify DVC parameters, including the half size of the correlation window and the space between two sub-volumes. Using DVC, 3D deviatoric strain field, strain contours, and displacement increment are determined at all stages of loading. A CT value-based method (VG Studio) is also applied to validate the DVC results. It is found that the layer boundaries affect the failure pattern, with the agreement between the DVC results and VG Studio results being observed. For the specimen with horizontal layers, the crack initiates at the center part at the lower density layer and finishes out of the center. Also, for the specimen with horizontal layers, the crack initiates and propagates at the boundary of two layers out of the middle line of the specimen. These asymmetry failures are due to the heterogeneity of layers. The study also shows the type of failure using DVC by monitoring displacement increment vectors near the crack location.

Detection and tracking of ocean layers using an AUV with UKF based extremum seeking control in the Baltic Sea

Adaptive sampling and situational awareness are key features of modern autonomous underwater vehicles (AUVs) since data quality can be improved while operation time and cost can be reduced. An example for adaptive sampling in the marine environmental context is thermocline detection and tracking. The thermocline as horizontal ocean layer separates warm and cold water and is a key feature in many marine disciplines. For example, it influences the distribution and exchange of nutrients and is a habitat for many organisms. In this paper we use an unscented Kalman Filter (UKF) based extremum seeking control (ESC) to find and follow ocean layers such as the thermocline. Computer simulations and real-world tests show that the method is able to find and track non-trivial real-world ocean layers with sensors subject to hysteresis and delay effects.

Brinkman–Bénard convection in a box with temperature modulation

A bounded porous box saturated with Newtonian fluid and subjected to a sinusoidal temperature gradient has various practical applications, such as solar energy storage, groundwater remediation, food processing, and chemical reactors. We address the generalization of the classical Rayleigh–Bénard convection problem in a horizontal fluid layer in an infinitely large domain heated from below to a finite three-dimensional box. We also look into a more intricate form of the modulated Rayleigh–Bénard problem in which the temperature at the bottom boundary varies sinusoidally. The Rayleigh number quantifies the non-sinusoidal part of the temperature gradient, while the amplitude and frequency of modulation describe the sinusoidal one. The critical Rayleigh number is determined using linear and nonlinear stability analyses; for the latter, the energy method is used. There is a possibility of subcritical instabilities, as evidenced by the energy stability estimates being lower than the linear ones. Furthermore, eigenvalues are obtained as a function of aspect ratios, modulation amplitude, and frequency for varying Darcy numbers. Modulation amplitude more significantly triggers a change in flow patterns at the onset of convection compared to the effect of other parameters. Considering water-saturated porous media made up of different materials, we report the critical temperature difference between lower and upper surfaces required for the onset of convection. In addition, a comparison between such a temperature difference obtained from linear theory and the energy method is also provided in the same manner. It is observed that subharmonic instability occurs for all considered porous media packed densely or sparsely.

Out-of-plane behavior of unreinforced masonry walls with horizontal slip layers in RC frames

In some cases, the infilled walls may be subjected to in-plane（IP） earthquake and out-of-plane(OOP) earthquake action due to the uncertainty of earthquake action and infills’ position. As a new type of unreinforced masonry(URM) walls with horizontal slip layers, it can effectively reduce the in-plane damage of infilled walls of reinforced concrete frames under an earthquake. However，the infill wall out-of-plane damage has a greater security threat than in-plane damage. In this study, a numerical study about the difference of OOP behavior of URM walls with/without horizontal sliding layers are analyzed, mainly including failure mode, bearing capacity, displacement capacity and stiffness. The spacing of horizontal slip layers, length of annectent reinforcing bars and the friction efficient between slip layer and masonry unit are chosen to investigate the differences of OOP mechanical behaviors. The results show that the horizontal sliding layers reduce the integrity of URM walls in reinforced concrete frames, causing the constraint of surrounding frame on infilled wall and the arching effect inside infilled wall become weaker, and OOP bearing capacity to drop. It is found that i) the annectent reinforcing bars is conducive to reducing the out-of-plane failure of the wall; 2)the smaller the slip layer spacing of the wall is, the more serious the damage of the infill wall under the out-of-plane load is, and the smaller the out-of-plane bearing capacity and the out-of-plane stiffness of the wall are; iii) the influence of the friction coefficient between slip layers and masonry unit on OOP mechanical behaviors is small. Finally, Chinese building design code is also selected to estimate the OOP bearing capacity of URM walls with horizontal sliding layers. The value of λcr is recommended to be greater than 1.0, and the value of λP is greater than 3.3. The spacing of the slip layer should be greater than 1 / 5 of the wall height, and the length of the annectent reinforcing bars should not be less than 1 / 5 of the infilled wall length.

Emergent fault friction and supershear in a continuum model of geophysical rupture

Important physical observations in rupture dynamics such as static fault friction, short-slip, self-healing, and the supershear phenomenon in cracks are studied. A continuum model of rupture dynamics is developed using the field dislocation mechanics (FDM) theory. The energy density function in our model encodes accepted and simple physical facts related to rocks and granular materials under compression. We work within a 2-dimensional ansatz of FDM where the rupture front is allowed to move only in a horizontal fault layer sandwiched between elastic blocks. Damage via the degradation of elastic modulus is allowed to occur only in the fault layer, characterized by the amount of plastic slip. The theory dictates the evolution equation of the plastic shear strain to be a Hamilton–Jacobi (H-J) equation, resulting in the representation of a propagating rupture front. A Central-Upwind scheme is used to solve the H-J equation. The rupture propagation is fully coupled to elastodynamics in the whole domain, and our simulations recover static friction laws as emergent features of our continuum model, without putting in by hand any such discontinuous criteria in the formulation. Estimates of material parameters of cohesion and friction angle are deduced. Short-slip and slip-weakening (crack-like) behaviors are also reproduced as a function of the degree of damage behind the rupture front. The long-time behavior of a moving rupture front is probed, and it is deduced that equilibrium profiles under no shear stress are not traveling wave profiles under non-zero shear load in our model. However, it is shown that a traveling wave structure is likely attained in the limit of long times. Finally, a crack-like damage front is driven by an initial impact loading, and it is observed in our numerical simulations that an upper bound to the crack speed is the dilatational wave speed of the material unless the material is put under pre-stressed conditions, in which case supersonic motion can be obtained. Without pre-stress, intersonic (supershear) motion is recovered under appropriate conditions.

The Effective Vertical Anisotropy of Layered Aquifers.

Many sedimentary aquifers consist of small layers of coarser and finer material. When groundwater flow in these aquifers is modeled, the hydraulic conductivity may be simulated as homogeneous but anisotropic throughout the aquifer. In practice, the anisotropy factor, the ratio of the horizontal divided by the vertical hydraulic conductivity, is often set to 10. Here, numerical experiments are conducted to determine the effective anisotropy of an aquifer consisting of 400 horizontal layers of which the homogeneous and isotropic hydraulic conductivity varies over two orders of magnitude. Groundwater flow is simulated to a partially penetrating canal and a partially penetrating well. Numerical experiments are conducted for 1000 random realizations of the 400 layers, by varying the sequence of the layers, not their conductivity. It is demonstrated that the effective anisotropy of the homogeneous model is a model parameter that depends on the flow field. For example, the effective anisotropy for flow to a partially penetrating canal differs from the effective anisotropy for flow to a partially penetrating well in an aquifer consisting of the exact same 400 layers. The effective anisotropy also depends on the sequence of the layers. The effective anisotropy values of the 1000 realizations range from roughly 5 to 50 for the considered situations. A factor of 10 represents a median value (a reasonable value to start model calibration for the conductivity variations considered here). The median is similar to the equivalent anisotropy, defined as the arithmetic mean of the hydraulic conductivities divided by the harmonic mean.

Volumetric Reconstruction of Ionospheric Electric Currents From Tri‐Static Incoherent Scatter Radar Measurements

AbstractWe present a new technique for the upcoming tri‐static incoherent scatter radar system EISCAT 3D (E3D) to perform a volumetric reconstruction of the 3D ionospheric electric current density vector field, focusing on the feasibility of the E3D system. The input to our volumetric reconstruction technique are estimates of the 3D current density perpendicular to the main magnetic field, j⊥, and its covariance, to be obtained from E3D observations based on two main assumptions: (a) Ions fully magnetized above the E region, set to 200 km here. (b) Electrons fully magnetized above the base of our domain, set to 90 km. In this way, j⊥ estimates are obtained without assumptions about the neutral wind field, allowing it to be subsequently determined. The volumetric reconstruction of the full 3D current density is implemented as vertically coupled horizontal layers represented by Spherical Elementary Current Systems with a built‐in current continuity constraint. We demonstrate that our technique is able to retrieve the three dimensional nature of the currents in our idealized setup, taken from a simulation of an active auroral ionosphere using the Geospace Environment Model of Ion‐Neutral Interactions (GEMINI). The vertical current is typically less constrained than the horizontal, but we outline strategies for improvement by utilizing additional data sources in the inversion. The ability to reconstruct the neutral wind field perpendicular to the magnetic field in the E region is demonstrated to mostly be within ±50 m/s in a limited region above the radar system in our setup.

Instability of double-diffusive magnetoconvection in a non-Newtonian fluid layer with cross-diffusion effects

The study explores the initiation of two-dimensional double-diffusive convection in a horizontal layer of an electrically conducting non-Newtonian Navier–Stokes–Voigt fluid, subjected to a uniform vertical magnetic field and cross-diffusion effects. The numerical results are presented by obtaining the analytical solutions for both steady and oscillatory onset scenarios. The viscoelastic nature of the fluid either delays or hastens the onset of oscillatory convection depending on the strength of solute concentration. The analysis also uncovers contradictions in the linear instability characteristics with and without cross-diffusion terms, even when other input parameters are identical. Under specific conditions, three novel phenomena are observed that are not typically seen in double-diffusive cases: (i) an electrically conducting Navier–Stokes–Voigt fluid layer, initially linearly stable in the presence of a magnetic field, can become destabilized with the addition of a heavy solute to the fluid's bottom; (ii) a stable double-diffusive electrically conducting Navier–Stokes–Voigt fluid layer can be destabilized by the application of a magnetic field; and (iii) the requirement of three critical values of the thermal Rayleigh number to determine linear instability, as opposed to the usual single value owing to the existence of disconnected closed convex oscillatory neutral curves. The results are shown to align with previously published findings in the limiting cases.

Small-scale vortical motions in cool stellar atmospheres

Small-scale vortices in the solar atmosphere have received considerable attention in recent years. These events are considered potential conduits for the exchange of energy and mass between the solar atmospheric layers from the convective surface to the corona. Similar events may occur in the atmospheres of other stars and play a role in energy transfer within their atmospheres. Our aim is to study the presence and properties of small-scale swirls in numerical simulations of the atmospheres of cool main-sequence stars. Our particular focus is on understanding the variations in these properties for different stellar types and their sensitivity to the surface magnetic field. Furthermore, we aim to investigate the role of these events in the energy transport within the simulated atmospheres. We analyzed three-dimensional, radiative-magnetohydrodynamic, box-in-a-star, numerical simulations of four main-sequence stars of spectral types K8V, K2V, G2V, and F5V. These simulations include a surface small-scale dynamo responsible for amplifying an initially weak magnetic field. Thus, we can study models characterized by very weak, or, in near equipartition magnetic fields. To identify small-scale vortices in horizontal layers of the simulations, we employed the automated algorithm SWIRL. Small-scale swirls are abundant in the simulated atmospheres of all the investigated cool stars. The characteristics of these events appear to be influenced by the main properties of the stellar models and by the strength of the surface magnetic field. In addition, we identify signatures of torsional Alfv\'enic pulses associated with these swirls, which are responsible for a significant vertical Poynting flux in the middle photospheres of the simulated stellar models. Notably, this flux is particularly significant in the K8V model, suggesting that if sim 70<!PCT!> of it is dissipated in the low chromosphere, small-scale vortical motions may play a role in the enhanced basal Ca ii H and K fluxes observed in the range of $B-V$ color index $1.1 B - V 1.4$. Finally, we present a simple analytical model, along with an accompanying scaling relation, to explain a peculiar result of the statistical analysis that the rotational period of surface vortices increases with the effective temperature of the stellar model. Our study shows that small-scale vortical motions are not unique to the solar atmosphere and that their interplay with the stellar surface magnetic field may effect the observable chromospheric activity of main-sequence cool dwarf stars.

Weak time-scale separation at the onset of oscillatory magnetoconvection in rapidly rotating fluids

Convective instabilities are one of the integral parts of the dynamics of flows driven by thermal buoyancy. Naturally, physical phenomena exhibit a wide disparity in the length and timescales of the field variables in numerical simulations and experimental observations. Such variations are not represented in the traditional normal mode stability analysis attempting to understand the onset of convection. This study attempts to incorporate different time constants for different perturbation variables in the linear stability analysis with the help of a Taylor series expansion. The infinite horizontal layer model is chosen for simplicity. Apart from the classical Rayleigh-Bénard system, additional physical effects such as background rotation and magnetic field have been considered with plausible implications for geophysical flow applications. The time scale separation is implemented by considering a slight change in the frequency of temperature perturbation compared to that for other physical quantities. Both analytical and numerical methods have been utilised for the investigation. The threshold buoyancy force is reduced when the temperature perturbation has a smaller frequency than the frequencies of other variables. Besides that, the onset wavenumber and frequency of the oscillatory modes are modified due to weak scale separation from the onset characteristics of the reference case. In particular, enhanced frequency of temperature perturbations leads to smaller-scaled magnetically controlled convective rolls and larger-scaled viscously controlled instabilities at the onset. A robust dependence of the onset characteristics with the parameter quantifying the timescale separation is obtained. Additionally, two transitions in the convective onset modes with scale separation have been identified.

Linear and weakly nonlinear analyses of double‐diffusive convection in porous media with chemical reaction using LTNE model

AbstractThe onset of convection in a horizontal porous layer with chemical reaction and local thermal nonequilibrium is investigated. The nondimensional governing equations have been solved using the normal mode technique, which results in an eigenvalue problem. The analytical expressions for both stationary and oscillatory Rayleigh numbers are obtained. The effect of different parameters has been investigated and presented. The amplitude equation is derived using weakly nonlinear theory. Nusselt number is calculated using an amplitude equation to investigate heat transport. When modeling a fluid‐saturated porous medium, previous research on double‐diffusive convection has uniformly operated under the assumption of local thermal equilibrium (LTE) between the fluid and solid phases at all points within the medium. This standard practice assumes a minimal temperature gradient between the phases at any given location. However, in practical scenarios involving high‐speed flows or significant temperature differentials between the fluid and solid phases, the LTE assumption proves insufficient.

Effect of shape and thermal conductivity of microstructured capillary-porous coatings on heat transfer during evaporation and boiling in thin horizontal liquid layers

Investigation results on heat transfer during evaporation and boiling in thin horizontal liquid layers on 2D modulated capillary-porous coatings with a sinusoidal profile in a wide range of relative pressures and layer heights are presented in this paper. Coatings were made on a 3D laser printer using the technology of additive selective laser sintering (SLS). One coating was made of bronze with a modulation wavelength equal to the Laplace constant of the working fluid and two coatings were made of stainless steel and bronze with a doubled wavelength. The experimental data obtained on the coatings were compared with each other and with the values obtained by evaporation and boiling of n-dodecane on a smooth surface under the same conditions. It is shown that in the region of low relative pressures in liquid layers with a height less than the Laplace constant, heat transfer intensification achieved on a sample made of a material with low thermal conductivity (stainless steel) was five times higher as compared to a smooth surface. In the range of relative pressures, when nucleate boiling was observed in the layers, the temperature difference reached on the bronze coatings was approximately 3–4 times less than on a smooth surface. The temperature difference for the stainless steel coating changed little with an increase in the heat flux. In pre-crisis conditions, the temperature difference during boiling on the stainless steel coating was less than on the bronze coatings in the entire range of relative pressures and layer heights. The highest critical heat fluxes were obtained on a bronze coating with a modulation wavelength equal to the Laplace constant.

Electrothermosolutal convection in nanofluid

This article employs linear stability theory to investigate the thermosolutal instability phenomenon within a horizontal layer of nanofluid subjected to a vertical alternating current (AC) electric field while being heated from below. The model integrates the impacts of Brownian diffusion, thermophoresis, and electrophoresis into the nanofluid dynamics. Free-free boundary conditions are assumed at the upper and lower boundaries of the fluid layer. Employing the normal mode technique, the set of coupled differential equations is solved analytically for stress-free boundaries. The computed numerical values of the stationary thermal Rayleigh number are illustrated graphically. The conditions for nonexistence of overstability are also derived. The study focuses on stationary convection, examining the analytical and graphical implications of modified diffusivity ratio, AC electric Rayleigh number, solutal Lewis number, nanofluid Lewis number, solutal Rayleigh number, and nanoparticle Rayleigh number on the system dynamics. The analysis demonstrates that the nanofluid Lewis number and modified diffusivity ratio produce a stabilizing effect on the initiation of convection, notably delaying its onset. AC electric field and solutal Rayleigh number exhibit a destabilizing influence, notably accelerating the onset of convection.

Development characteristics and controlling mechanism of different microfracture combinations in shale reservoir: A case study of silurian longmaxi formation in Weiyuan area

Fractures in organic-rich shale are important reservoir spaces and seepage channels of shale gas, and they are closely related to the gas-bearing properties of shale. The development characteristics and laws of fractures are of great significance in the exploration and development of shale oil and gas. This study examines organic-rich shales of the Wufeng–Longmaxi Formation in the Weiyuan area of the Sichuan Basin. On the basis of two-dimensional large-area multi-scale combination electron microscopy characterization and digital core platform technology, the development degree and distribution of different fractures are quantitatively characterized. The results show the following. (1) The shale of the Wufeng and Longmaxi formations developed a variety of fractures with different occurrences, sizes, and origins. According to the number and combination relationship between fractures of different occurrences, the shale can be divided into four fracture combination types: horizontal bedding fractures; vein fractures; reticular fractures; and ring fractures. Of these, the horizontal bedding fracture group has the largest number of samples and a higher average fracture surface porosity. (2) The degree of fracture development in the shale is affected by many factors, such as the laminar type, mineral composition, mineral particle size, mineral distribution, and total organic carbon, and the controlling mechanisms of different fracture combination types differ. Factors such as horizontal stratification, high clay mineral content, and uneven mineral particle size are conducive to the development of horizontal bedding joints. (3) Differences in the sedimentary environment affect the variation laws of the vertical fracture combination types and density. The total organic carbon and organic quartz content of the Long111 layer with deeper sedimentary water is higher, and the vein fracture formation is more developed than in other small layers, while the clay mineral content of the Long112 and Long114 layers with shallower sedimentary water is higher and the horizontal layer is more developed; the fracture combination type is dominated by the horizontal bedding fracture combination. At the same time, the fractures at the junction of each layer of the Long11 sub-member are the most developed because sea level rise and fall make the mineral particle size heterogeneity most prominent at the junction of the small layer.

Weakly nonlinear analysis of rotating magnetoconvection with anisotropic thermal diffusivity effect

The influence of anisotropic thermal diffusive coefficient on the stability of the horizontal fluid planar layer rotating about its vertical axis and permeated by the horizontal homogeneous magnetic field is studied. The linear stability analysis is carried out using the normal mode method. The stationary cross/oblique and parallel modes are calculated for different ranges of control parameters arising in the system. The SA (Stratification Anisotropy) parameter, α (the ratio of horizontal and vertical thermal diffusivities), plays a key role in deciding the boundaries between these modes and their instability regions when there is a combination of high and low rotation with weak and strong magnetic fields. The obtained isotropic results coincide with those obtained by pioneers in the literature. The weakly nonlinear behavior of the stationary convective motion in the vicinity of primary instability threshold is studied using the two-dimensional Landau–Ginzburg (LG) equation with cubic nonlinearity. This equation derived using the multiple scale analysis is similar to the one obtained in the literature having different relaxation time, nonlinear coefficient, and coherence lengths. These coefficients are used to study the heat transfer rate. In the case of high rotation, Nusselt number gets decreased from atmospheric (α < 1) to oceanic (α > 1) SA types. The domain for secondary instability of Eckhaus is obtained using the spatiotemporal LG equation and it is observed that the Eckhaus instability region decreases with increasing α.

Numerical Modeling of Non-Boussinesq Natural Convection in a Square Cavity Containing a Porous Layer

Numerical simulations were conducted to study the fluid flow and heat transfer through a partially filled square enclosure with a horizontal porous layer at the bottom. The vertical walls of the enclosure were subjected to uniform heating and cooling at specified temperatures TH and TC, respectively. The problem formulation was based on the Darcy–Brinkman model, which took into account density variation using the Mach number law approximation and incorporated variable properties. Based on the developed code, a parametric study of fluid flow and heat transfer inside the cavity was carried out to examine the effects of temperature difference, medium permeability, and Darcy number. Additionally, the variation of the Nusselt number on the hot and cold walls was provided to demonstrate the overall nature of heat transfer within the enclosure. The results of this study highlight that fluid flow and heat transfer are influenced by the temperature difference between the vertical walls and the characteristic parameters of the porous layer.

Revealing the antipolar order in the antiferroelectric SmZA phase by means of circular alignment

Many ferroelectric nematic liquid crystals, like one of the archetype materials, DIO, do not have a direct paraelectric N to ferroelectric NF phase transition, but exhibit yet another phase between N and NF. This phase has recently been proposed to be antiferroelectric, with a layered structure of alternating polarization normal to the average director and is sometimes referred to as Smectic ZA (SmZA). We have examined the SmZA phase in circularly rubbed (CR) cells, known to discriminate between the polar NF and the non-polar N phase from the configuration of disclination lines formed. We find that the ground state of SmZA has the same disclination configuration as the non-polar N phase, demonstrating that the SmZA phase is also non-polar, i.e., it has no net ferroelectric polarization. At the same time, the SmZA texture generally has a grainy appearance, which we suggest is partly a result of the frustration related to layered order combined with the imposed twist in CR cells. We discuss possible orientations of the smectic layers, depending on the alignment conditions. While a horizontal SmZA layer structure is always compatible with surface-induced twist, a vertical layer structure would tend to break up in a twisted bookshelf structure to match non-parallel alignment directions at the two surfaces.