Gradient Terms Research Articles

Variable viscous flow resistance based on rotational inertia

Viscous flow resistance is dominated by viscous friction between fluid and wall. The flow resistance characteristic curve (i.e., the relationship curve between pressure drop and flow rates, represented as the Δp–Q curve) depends on some inherent characteristic variables, such as structural size, fluid viscosity, density, and temperature. Usually, to change the Δp–Q curve, these inherent characteristic variables must be changed. This paper proposes a new design of variable viscous flow resistance. The new design uses two disks to construct a slit flow channel, and rotate one of the disks to drive the fluid in the slit flow channel to form a rotational inertia effect. Therefore, by changing the rotating speed of the disk, the rotational inertia effect can be changed, thereby achieving the purpose of changing the Δp–Q curve. This paper derives a theoretical model for the pressure distribution of the rotating slit flow field and conducted experimental verification. It was found that the rotational inertia gradient and viscous gradient terms play major roles in governing the radial pressure gradient. The sum of the other two inertial gradient terms accounts for a maximum of about 1.58% of the total pressure gradient. There is a coupling relationship between circumferential velocity, radial velocity, and flow rates. An increase in Q can increase the rotational inertial gradient term by up to 24.9%. The rotating disk causes additional radial velocity and thus weakens the viscous gradient term by at least 16.41%.

A note on gradient estimates for p-Laplacian equations

The aim of this short paper is to show that some assumptions in Guarnotta et al. (Adv Nonlinear Anal 11:741–756, 2022) can be relaxed and even dropped when looking for weak solutions instead of strong ones. This improvement is a consequence of two results concerning gradient terms: an L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document} estimate, which exploits nonlinear potential theory, and a compactness result, based on the classical Riesz-Fréchet-Kolmogorov theorem.

Дифференцируемость функционала и необходимое условие для решения задачи оптимального управления с интегральным критерием качества по всей области для нелинейного уравнения Шрёдингера cо специальным градиентным слагаемым

This work is devoted to the study of the optimal control problem for the three-dimensional nonlinear Schrödinger equation with a special gradient term and with a complex potential. The controls are the real and imaginary parts of the complex potential and are selected from the class of measurable bounded functions depending on the time variable, and the performance criterion is an integral over throughout the region is of considerable scientific interest. We note that the existence and uniqueness of a solution to an optimal control problem with an integral performance criterion over the entire domain for a nonlinear Schrödinger equation with a special gradient term was previously studied by the author. And in this article, the differentiability of the previously considered problem is studied. A theorem on the uniqueness of the adjoint problem to the stated problem is given, which is proved by the Galerkin method. A theorem on the differentiability of the solution of the optimal control problem under consideration is given and proved. Along with these, a formula is found for the first variation of the quality criterion under consideration and with the help of which a necessary condition is established in the form of a variational inequality.

Pressure analysis in nonlinear waves by revisiting the breaking wave onset

Breaking waves are numerically simulated by solving the two dimensional nonlinear free surface boundary conditions in potential theory. Onset of breaking wave is determined for two configurations: A focused wave in intermediate depth and a dambreaking flow leading to a soliton. Two criteria are identified: first, a kinematic criterion follows from the ratio of the horizontal velocity at the crest to the phase velocity; second, a dynamic criterion is formulated in terms of the vertical pressure gradient. Thresholds of these criteria are proposed in the literature. For the two configurations studied here, it is shown that these thresholds are not universal but they are reasonable estimates of the onset of breaking wave. To facilitate this study, the Hessian matrix of the pressure is analyzed. It is shown that there exists a continuous curve that links the crest of the wave to the sea bottom. This line follows from the colinearity of the pressure gradient and one of the two eigenvectors of the Hessian matrix. The analysis of the pressure, its gradient and its second derivatives along this line gives more insights into the definition of the thresholds associated to the onset of breaking wave.

Dust density effects on electron density gradient driven lower hybrid waves in magnetized plasma

AbstractElectrostatic lower hybrid waves propagating transverse to the external magnetic field in magnetized dusty plasma are investigated, using the fluid equations and also including the effect of density gradient in plasma. Lower hybrid modes are considered important in laboratory, astrophysical as well as space plasmas in interpreting low frequency noises. Unlike other electrostatic waves, lower hybrid waves can resonantly interact with electrons as well as ions in plasma. The dust and gradient terms lead to novel growth of these waves in dusty plasma. A theoretical model is developed to investigate density gradient effects on lower hybrid waves in plasma comprised of electrons, ions and dust particles. The dependence of plasma components' density, strength of magnetic field, temperature and charge of electrons, ions and dust grains on the frequency and growth rate of electrostatic lower hybrid waves has been investigated and presented graphically. It has been observed that the number density of each plasma components directly affects the growth rate. The electrostatic waves are found to grow along the positive electron gradient and the dust effect depends on the gradient length chosen. Above a certain threshold value of gradient length, the waves are found to stabilize. A correlation of the obtained theoretical results with the experimental observations is also discussed.

A novel method applicable to large‐scale active magnetic shielding system which effectively compensates gradient magnetic fields originated from the surrounding buildings

AbstractWe propose the novel magnetic compensation method which is suitable to compensate gradient magnetic fields. The method is to be applied to a conventional large‐scale, tri‐axial active magnetic shielding system such as a spacecraft magnetic testing facility. The essences of this method are the adoption of compensation coils which have degrees of freedom in their coil orientations, and their installation position, which is chosen to have certain distances from the center of the shielding area. The method works with at least four compensation coils. In this research, we actually demonstrated the method with a scaled‐down experiment. Disturbance magnetic field (B) at the target zone was suppressed from 21.33 nT to −1.54 nT and magnetic field uniformity (ΔB) within the test zone was improved from 2.3 nT to 0.015 nT. In terms of magnetic field gradient, improvement from over 4 nT/m to below 0.5 nT/m was achieved.

A Covariant Non-Local Model of Bohm's Quantum Potential.

Assuming that the energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm's quantum potential and the Madelung equation are obtained, showing explicitly that some of the hypotheses that led to the formulation of quantum mechanics do admit a classical interpretation based on non-locality. Here, we generalize this approach imposing a finite speed of propagation of any perturbation, thus determining a covariant formulation of the Madelung equation.

CAIR: Combining integrated attention with iterative optimization learning for sparse-view CT reconstruction

Sparse-view CT is an efficient way for low dose scanning but degrades image quality. Inspired by the successful use of non-local attention in natural image denoising and compression artifact removal, we proposed a network combining integrated attention and iterative optimization learning for sparse-view CT reconstruction (CAIR). Specifically, we first unrolled the proximal gradient descent into a deep network and added an enhanced initializer between the gradient term and the approximation term. It can enhance the information flow between different layers, fully preserve the image details, and improve the network convergence speed. Secondly, the integrated attention module was introduced into the reconstruction process as a regularization term. It adaptively fuses the local and non-local features of the image which are used to reconstruct the complex texture and repetitive details of the image, respectively. Note that we innovatively designed a one-shot iteration strategy to simplify the network structure and reduce the reconstruction time while maintaining image quality. Experiments showed that the proposed method is very robust and outperforms state-of-the-art methods in terms of both quantitative and qualitative, greatly improving the preservation of structures and the removal of artifacts.

Improved seismic envelope full-waveform inversion

The envelope full-waveform-inversion method has the potential to greatly improve the estimation of long-wavelength components of subsurface seismic velocity, and simultaneously avoid or reduce inversion cycle skipping, by fitting the envelopes of observed and modeled seismic waveform data, rather than fitting the waveforms directly. However, some issues such as instability of the adjoint source and gradient term make it challenging to use in practice for many seismology applications. We examine the envelope inversion (EI) methods with different envelope function exponent p values from an adjoint source analysis perspective and reveal the adverse effects of the adjoint source weighting term on the data residuals, which degrade or mute important inversion gradient contributions, especially from reflected or scattered phases in the seismic data. We develop the theory and implementation for an improved EI (IEI) method by adding an upshift constant (zero-frequency direct current [DC] bias term) to the seismic waveform data before calculating the envelope functions, which can help solve the instability issue of the adjoint source of the EI with power value p = 1. Our approach is different than the common but simple addition of a “water-level” damping constant in the denominator of the data residual-weighting term, which can ignore important information contained in the data residuals. Our IEI method becomes a mixed envelope-waveform objective function, which introduces a new adjoint source weighting term that can preserve all of the information contained in the data residuals, by boosting the low-amplitude phases of the seismic waveform, while simultaneously preserving the correct envelope (energy shape) of the high-amplitude phases. We test the performance of all three methods in detail using the highly realistic SEAM4D model and data and find the advantages of our new IEI method compared with conventional EI and full-waveform inversion methods.

Existence and asymptotic behavior of strictly convex solutions for singular k-Hessian equations with nonlinear gradient terms

Abstract In this paper, we mainly consider the singular k-Hessian equations S k ⁢ ( λ ⁢ ( D 2 ⁢ u ) ) = h ⁢ ( x ) ⁢ f ⁢ ( - u ) + g ⁢ ( | D ⁢ u | ) in ⁢ Ω S_{k}(\lambda(D^{2}u))=h(x)f(-u)+g(|Du|)\quad\text{in }\Omega and S k ⁢ ( λ ⁢ ( D 2 ⁢ u ) ) = h ⁢ ( x ) ⁢ f ⁢ ( - u ) ⁢ ( 1 + g ⁢ ( | D ⁢ u | ) ) in ⁢ Ω S_{k}(\lambda(D^{2}u))=h(x)f(-u)(1+g(|Du|))\quad\text{in }\Omega with the Dirichlet boundary condition u = 0 {u=0} on ∂ ⁡ Ω {\partial\Omega} , where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 2 {N\geq 2} ) is a strictly convex, bounded smooth domain. Using the method of upper and lower solutions and the Karamata regular variation theory, we get new criteria of the existence and asymptotic behavior of strictly convex solutions under different conditions imposed on h, f and g. This problem is more difficult to solve than the k-Hessian problem without gradient terms, and requires additional new conditions in the proof process.

Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term

Abstract We prove Gibbons’ conjecture for the quasilinear elliptic equation - Δ p ⁢ u + a ⁢ ( u ) ⁢ | ∇ ⁡ u | q = f ⁢ ( u ) in ⁢ ℝ N , -\Delta_{p}u+a(u)|\nabla u|^{q}=f(u)\quad\text{in }\mathbb{R}^{N}, where N ≥ 2 {N\geq 2} , 2 ⁢ N + 2 N + 2 < p < 2 {\frac{2N+2}{N+2}<p<2} , q ≥ 1 {q\geq 1} and a and f are Lipschitz continuous functions which satisfy some relevant conditions. This conjecture states that every weak solution u ∈ C 1 ⁢ ( ℝ N ) {u\in C^{1}(\mathbb{R}^{N})} of the equation with | u | ≤ 1 {|u|\leq 1} and lim x N → ± ∞ ⁡ u ⁢ ( x ′ , x N ) = ± 1 {\lim_{x_{N}\to\pm\infty}u(x^{\prime},x_{N})=\pm 1} , uniformly in x ′ ∈ ℝ N - 1 {x^{\prime}\in\mathbb{R}^{N-1}} , must depend only on x N {x_{N}} and ∂ ⁡ u ∂ ⁡ x N > 0 {\frac{\partial u}{\partial x_{N}}>0} in ℝ N {\mathbb{R}^{N}} . In particular, our result holds for a being non-decreasing on [ - 1 , - 1 + δ ] {[-1,-1+\delta]} and on [ 1 - δ , 1 ] {[1-\delta,1]} and f ⁢ ( u ) = | u | r ⁢ u ⁢ | 1 - u 2 | s ⁢ ( 1 - u 2 ) {f(u)=|u|^{r}u|1-u^{2}|^{s}(1-u^{2})} , where r , s , δ ≥ 0 {r,s,\delta\geq 0} . The main tool we use is an adaptation of the sliding method to the corresponding quasilinear elliptic operator.

Horizontally Asymmetric Nanochannels of Graphene Oxide Membranes for Efficient Osmotic Energy Harvesting.

Reverse electrodialysis (RED) directly harvests renewable energy from salinity gradients, and the achievable potential power heavily relies on the ion exchange membranes. Graphene oxides (GOs) are considered a solid candidate for the RED membrane because the laminated GO nanochannels with charged functional groups provide an excellent ionic selectivity and conductivity. Yet, a high internal resistance and poor stability in aqueous solutions limit the RED performance. Here, we develop a RED membrane that concurrently achieves high ion permeability and stable operation based on epoxy-confined GO nanochannels with asymmetric structures. The membrane is fabricated by reacting epoxy-wrapped GO membranes with ethylene diamine via vapor diffusion, overcoming the swelling properties in aqueous solutions. More importantly, the resultant membrane exhibits asymmetric GO nanochannels in terms of both channel geometry and electrostatic surface charges, leading to the rectified ion transport behavior. The demonstrated GO membrane exhibits the RED performance up to 5.32 W·m-2 with >40% energy conversion efficiency across a 50-fold salinity gradient and 20.3 W·m-2 across a 500-fold salinity gradient. Planck-Nernst continuum models coupled to molecular dynamics simulations rationalize the improved RED performance in terms of the asymmetric ionic concentration gradient within the GO nanochannel and the ionic resistance. The multiscale model also provides the design guidelines for ionic diode-type membranes configuring the optimum surface charge density and ionic diffusivity for efficient osmotic energy harvesting. The synthesized asymmetric nanochannels and their RED performance demonstrate the nanoscale tailoring of the membrane properties, establishing the potentials for 2D material-based asymmetric membranes.

Hydraulic conductivity, microstructure and texture of compacted claystone/ bentonite mixtures saturated with different solutions

The French reference concept for the disposal of intermediate- and high-level nuclear waste in the Callovo-Oxfordian sedimentary rock formation (COX-claystone) considers the employment of crushed COX-claystone and its mixture with MX80-bentonite as potential backfill materials installed in drifts and shafts upon termination of the operational phase of the future repository. The fraction of MX80-bentonite in the mixture is limited to 30% in wet weight. Over time, the backfill will be saturated with solutions that originate in the surrounding rock formation and percolate through the concrete lining left in place. The main characteristic of the leachate will be its high pH-value. In addition to the swelling pressure, the sealing properties of the backfill are determined by the hydraulic conductivity. This laboratory experimental study aimed to understand the hydraulic conductivity evolution of potential backfill materials under realistic conditions in terms of compaction conditions, solution chemistry, temperature and hydraulic gradient, and to relate results determined at the macroscale to results of microstructural and textural analysis. Also, the impact of the solution chemistry was evaluated by these means. In the case of mixture-samples, no stabilization of hydraulic conductivity was detected, despite the duration of the experiments being about one year. There was no detectable impact of the solution chemistry in the course of experiments attributable to low reaction kinetics under imposed conditions. Subsequent microstructural analysis indicated material swelling triggering rearrangements in the pore space and lowering the fraction of hydraulic conductive voids. Interestingly, neither the saturation nor the solution chemistry had an impact on the texture of materials.

A weak form quadrature element formulation of geometrically exact strain gradient shells

A geometrically exact strain gradient shell model and its weak form quadrature element formulation are proposed. The governing equations, boundary conditions and corresponding variational form of the shell model are presented. The generalized differential quadrature analogue is employed for derivative approximations of displacements and rotations with C1 continuity conditions brought about by the second-order strain gradient terms. A novel scheme based on auxiliary vectors is developed to represent and update the derivatives of constrained rotations on element edges for the numerical approach. The present quadrature shell element is feasible to incorporate strain gradient terms and free from shear and membrane locking problems due to its adjustable high-order approximation property in nonlinear shell analysis. To demonstrate the potential of this formulation for nonlinear analysis of shells with size effects, five benchmark numerical examples are presented. The results verify the feasibility of this formulation and illustrate the strain gradient effects on elastic shells undergoing large displacements and rotations.

Non-Lifshitz invariants corrections to Dzyaloshinskii-Moriya interaction energy

We study the continuum limit of two-dimensional chiral magnets in which Dzyaloshinskii-Moriya interaction (DMI) is due to the interplay between a smooth magnetic texture and spin-orbit coupling. The resulting free-energy density of the system contains linear terms in the spatial gradient of the magnetic texture, which mark an instability of the system towards the formation of nontrivial magnetic orders such as skyrmions or chiral domain walls. We perform a microscopic analysis of DMI tensors responsible for this contribution to free energy based on a Berry phase formulation in the mixed space of momentum and position, and reveal that they exhibit non-Lifshitz invariants features. In particular, a perturbation theory shows in the case of Rashba spin-orbit interactions the presence of non-Lifshitz invariants to third order in the small spin-orbit interaction and fourth order in the small exchange coupling. The higher-order terms may even lead to an enhancement of DMI interaction at strong spin-orbit coupling due to divergences in the density of states at the bottom of the conduction band. Finally, we also study the DMI free energy generated from Rashba spin-orbit interaction in different symmetry groups.

Existence of positive radial solutions for nonlinear elliptic equations with gradient terms in an annulus

Existence of positive radial solutions for nonlinear elliptic equations with gradient terms in an annulus

A priori estimates for non-coercive Dirichlet problems with subquadratic gradient terms

We deal with some quasilinear elliptic problems posed in a bounded smooth convex domain Ω⊂RN (N≥3), namely{−Δu=λu+μ(x)|∇u|q+f(x),x∈Ω,u=0,x∈∂Ω, where the data satisfyμ∈L∞(Ω),μ⪈0;f∈Lp(Ω),p>N,f⪈0 and 1<q≤2. We provide sufficient conditions on f,μ (allowing μ to vanish on ∂Ω) that yield the sharp estimate λ‖u‖L∞(Ω)≤C for any bounded solution u with λ∈(0,λ1), which is the non-coercive regime. The estimate leads to remarkable consequences such as a multiplicity result and a precise asymptotic behavior of the bounded but blowing up solutions as λ→0+.

Effect of the down-slope on the structure and the pressure loss of an oil-water stream

The methodology and the results of an experimental campaign aimed at characterizing a two-phase flow of an oil-water-mixture in downward inclined pipes are here described. The goal was to assess the effects of the downslope on an oil-water stream in terms of flow patterns, pressure gradients and phase holdup inside a 40 mm I.D. pipe. The experiments ranged within (0.56−1.06ms) and (0.66−1.33ms) oil and water superficial velocities, respectively. The transition from annular to stratified-wavy flow pattern was analyzed and showed to occur at lower oil velocities with respect to the horizontal configuration. The frictional pressure gradients were measured, the results compared to mechanistic and empirical models showed to be in good agreement. The phase holdups were measured by quick-closing valves and compared to horizontal configuration results and literature models. Eventually, the drift-flux model was implemented confirming its applicability and a relationship for the of oil holdup was derived.

Concomitant magnetic-field compensation for 2D spiral-ring turbo spin-echo imaging at 0.55T and 1.5T.

To develop 2D turbo spin-echo (TSE) imaging using annular spiral rings (abbreviated "SPRING-RIO TSE") with compensation of concomitant gradient fields and B0 inhomogeneity at both 0.55T and 1.5T for fast T2 -weighted imaging. Strategies of gradient waveform modifications were implemented in SPRING-RIO TSE for compensation of self-squared concomitant gradient terms at the TE and across echo spacings, along with reconstruction-based corrections to simultaneously compensate for the residual concomitant gradient and B0 field induced phase accruals along the readout. The signal pathway disturbance caused by time-varying and spatially dependent concomitant fields was simulated, and echo-to-echo phase variations before and after sequence-based compensation were compared. Images from SPRING-RIO TSE with no compensation, with compensation, and Cartesian TSE were also compared via phantom and in vivo acquisitions. Simulation showed how concomitant fields affected the signal evolution with no compensation, and both simulation and phantom studies demonstrated the performance of the proposed sequence modifications, as well as the readout off-resonance corrections. Volunteer data showed that after full correction, the SPRING-RIO TSE sequence achieved high image quality with improved SNR efficiency (15%-20% increase), and reduced RF SAR (˜50% reduction), compared to the standard Cartesian TSE, presenting potential benefits, especially in regaining SNR at low-field (0.55T). Implementation of SPRING-RIO TSE with concomitant field compensation was tested at 0.55T and 1.5T. The compensation principles can be extended to correct for other trajectory types that are time-varying along the echo train and temporally asymmetric in TSE-based imaging.

Assessing habitat diversity and potential areas of similarity across protected areas globally

Biophysical characterization analyses of protected areas (PA) that provide information on their ecological values and potential areas with similar characteristics are needed to make informed PA network planning and management decisions. This study combines and further develops methodologies that use remote sensing and modelling to identify habitat functional types in PAs and map similar areas at the ecoregion level. The study also develops new terrestrial habitat diversity and irreplaceability indices at habitat and PA scale that allow the comparison and ranking of PAs in terms of biophysical gradients and singular environmental conditions. Six PAs were selected to highlight and discuss the results of the proposed methodology. Both individual and composite indices should be considered when trying to compare PAs to understand the overall complexity and ecological values of each PA. Results can inform planning and management of individual and protected area networks as well as identify new areas for conservation. The information provided by the model about similar habitats outside protected areas can also help assess their representativeness and support studies to strengthen ecological connectivity. Besides systematic comparisons, detailed assessments of protected areas can also be performed using medium and high-resolution input variables. This is especially relevant for protected areas in developing countries where undertaking fieldwork is very difficult and the budget devoted to conservation is limited.