Finite Channel Research Articles

Suppression of Blow-Up in Multispecies Patlak–Keller–Segel–Navier–Stokes System via the Poiseuille Flow in a Finite Channel

Suppression of Blow-Up in Multispecies Patlak–Keller–Segel–Navier–Stokes System via the Poiseuille Flow in a Finite Channel

Scale dependence and non-isochoric effects on the thermohydrodynamics of rarefied Poiseuille flow

The plane Poiseuille flow of a rarefied gas in a finite length channel, driven by an axial pressure gradient, is analysed numerically to probe (i) the role of ‘dilatation’ ( $\varDelta ={\boldsymbol \nabla }\boldsymbol {\cdot }{\boldsymbol u}\neq 0$ ) on its thermohydrodynamics as well as to clarify (ii) the possible equivalence with its well-studied ‘dilatation-free’ or ‘isochoric’ ( ${\rm D}\rho /{\rm D}t=0$ ) counterpart driven by a constant acceleration. Focussing on the mass flow rate ${\mathcal {M}}({Kn})$ , which is an invariant quantity for both pressure-driven and acceleration-driven Poiseuille flows, it is shown that while ${\mathcal {M}}\sim \log {{Kn}}$ at ${Kn}\gg 1$ in the acceleration-driven case, the mass flow saturates to a constant value ${\mathcal {M}}\sim {{Kn}}^0$ at ${Kn}\gg 1$ in the pressure-driven case due to the finite length ( $L_x<\infty$ ) of the channel. The latter result agrees with prior theory and recent experiments, and holds irrespective of the magnitude of the axial pressure gradient ( $G_p$ ). The pressure-dilatation cooling ( $\varPhi _p=-p\varDelta <0$ ) is shown to be responsible for the absence of the bimodal shape of the temperature profile in the pressure-driven Poiseuille flow. The dilatation-driven reduction of the shear viscosity and the odd signs of two normal stress differences ( ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ ) in the pressure-driven flow in comparison with those in its acceleration-driven counterpart are explained from the Burnett-order constitutive relations for the stress tensor. While both ${\mathcal {N}}_1$ and ${\mathcal {N}}_2$ appear at the Burnett order $O({{Kn}}^2)$ in the acceleration-driven flow, the leading term in ${\mathcal N}_1$ scales as $(\mu/p)\varDelta$ due to the non-zero dilatation in the pressure-driven Poiseuille flow which confirms that the two flows are not equivalent even at the Navier–Stokes–Fourier order $O({{Kn}})$ . The heat-flow rate ( ${{\mathcal {Q}}_q}_x=\int q_x(x,y) \,{{\rm d} y}$ ) of the tangential heat flux is found to be negative (i.e. directed against the axial pressure gradient), in contrast to its positive asymptotic value (at ${Kn}\gg 1$ ) in the acceleration-driven flow. Similar to the scale-dependence of the mass flow rate, ${{\mathcal {Q}}_q}_x({{Kn}}, L_x)$ is found to saturate to a constant value at ${Kn}\gg 1$ in finite length channels. The double-well shape of the $q_x(y)$ -profile in the near-continuum limit agrees well with predictions from a generalized Fourier law. On the whole, the dilatation-driven signatures (such as the pressure-dilatation work and the ‘normal’ shear-rate differences) are shown to be the progenitor for the observed differences between the two flows with regard to (i) the hydrodynamic fields, (ii) the rheology and (iii) the flow-induced heat transfer.

Asymptotic Stability of Two-Dimensional Magnetohydrodynamic System Near the Couette Flow in a Finite Channel

In this paper, we consider the asymptotic stability of the incompressible two-dimensional(2D) magnetohydrodynamic(MHD) system near the Couette flow at high Reynolds number and high magnetic Reynolds number in a finite channel Ω=T×[-1,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega =\\mathbb {T}\ imes [-1,1]$$\\end{document}. We extend the results of the Navier–Stokes equations (for the previous results see[10]) to the MHD system. We prove that if the initial velocity Vin\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_{in}$$\\end{document} and the initial magnetic field Bin\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{in}$$\\end{document} satisfy ‖Vin-(y,0),Bin-(1,0)‖Hx,y2≤ϵmin{ν,μ}12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert \\left( V_{in}-(y,0), B_{in}-(1,0)\\right) \\Vert _{H_{x,y}^{2}}\\le \\epsilon \ ext {min}\\{\ u ,\\mu \\}^\\frac{1}{2}$$\\end{document} for some small ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon$$\\end{document} independent of ν,μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u ,\\mu$$\\end{document}, then the solution of the system remains within O(min{ν,μ}12)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(\ ext {min}\\{\ u ,\\mu \\}^\\frac{1}{2})$$\\end{document} of Couette flow, and close to Couette flow as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\rightarrow \\infty$$\\end{document}; the magnetic field remains within O(min{ν,μ}12)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(\ ext {min}\\{\ u ,\\mu \\}^\\frac{1}{2})$$\\end{document} of the (1, 0), and close to (1, 0) as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\rightarrow \\infty$$\\end{document}.

Semi-Analytical Modeling of Transient Stream Drawdown and Depletion in Response to Aquifer Pumping.

Analytical and semi-analytical models for stream depletion with transient stream stage drawdown induced by groundwater pumping are developed to address a deficiency in existing models, namely, the use of a fixed stream stage condition at the stream-aquifer interface. Field data are presented to demonstrate that stream stage drawdown does indeed occur in response to groundwater pumping near aquifer-connected streams. A model that predicts stream depletion with transient stream drawdown is developed based on stream channel mass conservation and finite stream channel storage. The resulting models are shown to reduce to existing fixed-stage models in the limit as stream channel storage becomes infinitely large, and to the confined aquifer flow with a no-flow boundary at the streambed in the limit as stream storage becomes vanishingly small. The model is applied to field measurements of aquifer and stream drawdown, giving estimates of aquifer hydraulic parameters, streambed conductance, and a measure of stream channel storage. The results of the modeling and data analysis presented herein have implications for sustainable groundwater management.

Adaptive Channel Equalization for Digital Communication with Tunicate Swarm Algorithm

In wireless communication systems, the information is transferred as digital symbols that get affected by noise interference while transmitting through the wireless channel. In today’s world, the increased demand for rapid data transmission rates leads to more inter-symbol interference in the received signal. To diminish the effects of inter-symbol interference, adaptive channel equalization is utilized in digital communication. In this paper, the inter-symbol interference effect in the finite impulse response channel is reduced in terms of an adaptive equalizer method. The weights/coefficients of the equalizer are optimized through a tunicate swarm algorithm. Channel equalization involves optimizing the channel coefficients to mitigate the impact of inter-symbol interference. This equalization process can be viewed as an iterative optimization task, aimed at lessening the mean square error among the transmitted signal and the equalizer’s output. The implementation of the proposed method is executed using MATLAB software. We assess its effectiveness through a comparative analysis, considering metrics such as mean square error, learning rate, bit error rate, convergence rate, and computational time. Furthermore, we conduct a comparative evaluation with other optimization approaches, including the Bat Algorithm, Slime Mould Algorithm, and Harris Hawks Optimization Algorithm. This comparison demonstrates the superiority of the proposed algorithm over traditional optimization techniques.

DISEASED BILE FLOW UNDER THE EFFECTS OF HEAT TRANSFER AND CHEMICAL REACTIONS

The movement of bile inside ducts in a diseased state is investigated using a mathematical model that takes into account heat transport and chemical interactions. Bile is considered as a viscous fluid, geometry of ducts is assumed as finite length asymmetric channel and the flow is induced by peristaltic wave along the length of channel walls. Under the presumption of a long wave length and a low Reynolds number, the formulas for axial velocity, pressure gradient, volume flow rate, stream function, pressure increase, shear stress, and heat transfer coefficient are produced. In the end, a plot and discussion are made of how different emergent factors affect the relevant physical quantities.

Heat and concentration analysis of two-layered muco-ciliary third grade fluid flow in human airways

Understanding the mechanisms involved in the movement and clearance of mucus within the human airways is important from a physiological perspective. A mathematical model of muco-ciliary two-layered fluid flow in a finite two-dimensional channel is proposed. Due to the presence of both the airway ciliary layer (ACL) and the periciliary liquid layer (PCL) in airways epithelial cells, the two-layered model is adopted. The third grade fluid models is used to characterize the viscoelastic nature of mucus secreted in the airways. The mathematical modeling of two-layer flow problem is simplified by using long wavelength and small Reynolds number approximation. A formulated set of partial differential equations (PDEs) is solved for series-form solutions of velocity, temperature, concentration, and pressure gradient by using the Adomian decomposition method (ADM). The analysis delineated that with an increase in pressure gradient at airways entrance ξ the velocity and temperature increases while concentration decreases. Deborah numbers impact ACL and PCL velocities differently, with De(A) influencing flow direction and De(P) causing fluid to flow faster in the forward direction. It also emphasizes the significance of the rate of thermal diffusion, viscous dissipation, and relaxation time. The results showed that the suggested two-layered muco-ciliary model provides a better description of mucus transport in the human airways. The pressure gradient at the airways entrance has a significant impact on mucus transport through the airways.

Nonlinear inviscid damping for a class of monotone shear flows in a finite channel

Nonlinear inviscid damping for a class of monotone shear flows in a finite channel

Layer separation of the 3D incompressible Navier–Stokes equation in a bounded domain

We provide an unconditional L 2 upper bound for the boundary layer separation of Leray–Hopf solutions in a smooth bounded domain. By layer separation, we mean the discrepancy between a (turbulent) low-viscosity Leray–Hopf solution u ν and a fixed (laminar) regular Euler solution u ¯ with similar initial conditions and body force. We show an asymptotic upper bound C | | u ¯ | | L ∞ 3 T on the layer separation, anomalous dissipation, and the work done by friction. This extends the previous result when the Euler solution is a regular shear in a finite channel. The key estimate is to control the boundary vorticity in a way that does not degenerate in the vanishing viscosity limit.

Diffusive wave model in a finite length channel with a concentrated lateral inflow subject to different types of boundary conditions

The diffusive wave model is one of the simplified forms of Saint-Venant equations, and it is often used instead of the full model. In this paper, we present an analytical solution for the linearized diffusive wave model represented by a simultaneous system of two first-order partial differential equations focused on spatial variation of a lateral inflow in a finite channel. A concentrated lateral inflow from a small-width tributary is considered through the Dirac delta function. We use the Laplace transform method to solve these equations analytically. Two types of upstream boundaries are considered here in the form of a flow-discharge hydrograph and a flow-depth hydrograph, while keeping a flow-depth hydrograph as the downstream boundary. Using unit-step responses of the lateral inflow, the effect of different boundaries on the flow-depth responses and the flow-discharge responses is studied for different values of the Peclet number (Pe). The flow depth is observed to be more sensitive to the downstream boundary and other parameters used in this work. Consideration of the flow depth as the upstream boundary reflects the effect of all the parameters on the unit-step responses presented. These responses are compared with the available semi-infinite channel responses, which are found to be an inappropriate substitute for the finite channel responses for Pe<5 which implies that the downstream boundary cannot be ignored for these cases. However, for the case Pe>5, although the semi-infinite channel responses are found to satisfactorily estimate the discharge along the entire channel, they can approximate the flow depth at the locations closer to the upstream boundary only.

Transition Threshold for the 3D Couette Flow in a Finite Channel

In this paper, we study nonlinear stability of the 3D plane Couette flow ( y , 0 , 0 ) (y,0,0) at high Reynolds number R e {Re} in a finite channel T × [ − 1 , 1 ] × T \mathbb {T}\times [-1,1]\times \mathbb {T} . It is well known that the plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox. One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. This work shows that if the initial velocity v 0 v_0 satisfies ‖ v 0 − ( y , 0 , 0 ) ‖ H 2 ≤ c 0 R e − 1 \|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1} for some c 0 > 0 c_0>0 independent of R e Re , then the solution of the 3D Navier-Stokes equations is global in time and does not transit away from the Couette flow in the L ∞ L^\infty sense, and rapidly converges to a streak solution for t ≫ R e 1 3 t\gg Re^{\frac 13} due to the mixing-enhanced dissipation effect. This result confirms the threshold result obtained by Chapman via an asymptotic analysis(JFM 2002). The most key ingredient of the proof is the resolvent estimates for the full linearized 3D Navier-Stokes system around the flow ( V ( y , z ) , 0 , 0 ) (V(y,z), 0,0) , where V ( y , z ) V(y,z) is a small perturbation(but independent of R e Re ) of y y .

Impact of a concentrated lateral inflow and stage–discharge relation imposed at the downstream end of a finite channel for the diffusive wave model

Impact of a concentrated lateral inflow and stage–discharge relation imposed at the downstream end of a finite channel for the diffusive wave model

The Fokker–Planck–Boltzmann equation in the finite channel

In this paper, we establish the existence of small-amplitude unique solutions near the Maxwellian for the Fokker–Planck–Boltzmann equation in a finite channel with specular reflection boundary conditions. The solution space we consider is denoted as Lk̄1LT∞Lx1,v2, introduced in Duan et al. [Commun. Pure Appl. Math. 74(5), 932–1020 (2021)]. In addition, we investigate the long-time behavior of solutions for both hard and soft potentials.

Performance Analysis for Reconfigurable Intelligent Surface Assisted MIMO Systems

This paper investigates the maximal achievable rate for a given maximal error probability, and blocklength for the reconfigurable intelligent surface (RIS) assisted multiple-input and multiple-output (MIMO) system. The result consists of a finite blocklength and finite alphabet constraints channel coding achievability and converse bounds based on the Berry-Esseen theorem, the Mellin transform and the closed-form expression of the mutual information and the unconditional variance. The numerical evaluation shows a fast speed of convergence to the maximal achievable rate as the blocklength increases and also proves that the channel variance is a sound measurement of the backoff from the maximal achievable rate due to finite blocklength.

The Boltzmann equation for plane Couette flow in a finite channel

The dynamics of a rarefied gas in a finite channel with the same temperatures and opposite velocities is a fundamental problem in kinetic theory. The relative motion of the planar boundaries can induce a non-equilibrium state which is referred to as the Couette flow. In this paper, we demonstrate that the unsteady Couette flow for the Boltzmann equation in 3D finite channel time asymptotically converges to the 1D steady state constructed in Duan et al. (2022), we also prove the exponential time decay rate as a byproduct. The validity of the analysis is established for all hard potentials.

Quantized feedback control of continuous time Lur'e systems with nested quantization over finite data rate channels

AbstractThis paper studies a control problem of continuous time Lur'e systems involving input quantization (for the output of the controller) and output quantization (for the output of the plant). When the output of one quantization operator becomes directly the input of the other one, the nested quantization is produced, which is difficult for the design of the systems. Based on spherical polar coordinate quantizer, a quantization method with finite data rate is proposed for handling the nested quantization. In addition, both the quantization errors and Lur'e‐type nonlinearity are converted to sector bound uncertainties that are modeled by norm bounded uncertain matrices, which facilitates the design of the systems. Owing to the quantization error the resulting closed‐loop system is described by a discontinuous right‐hand side differential equation and the notion of Krasovskii solution is adopted for the equation. By Krasovskii solution the update rule of the quantization region is designed to ensure the convergence to the origin of the system with the nested quantization and avoid the quantizer saturation. Further, since the quantizers cannot quantize the controller output and the system output directly, we develop an optimalization problem for each quantizer to obtain the estimates of the outputs of the plant and the controller. Finally, a method is presented to achieve the parameters of the controller.

Knot Formation on DNA Pushed Inside Chiral Nanochannels.

We performed coarse-grained molecular dynamics simulations of DNA polymers pushed inside infinite open chiral and achiral channels. We investigated the behavior of the polymer metrics in terms of span, monomer distributions and changes of topological state of the polymer in the channels. We also compared the regime of pushing a polymer inside the infinite channel to the case of polymer compression in finite channels of knot factories investigated in earlier works. We observed that the compression in the open channels affects the polymer metrics to different extents in chiral and achiral channels. We also observed that the chiral channels give rise to the formation of equichiral knots with the same handedness as the handedness of the chiral channels.

Quantized Feedback Control of Discrete-Time MIMO Linear Systems With Input and Output Quantization Over Finite Data Rate Channels

This paper considers a control problem of discrete-time MIMO linear systems involving input quantization (for the output of the controller) and output quantization (for the output of the plant). Based on spherical polar coordinate quantizer, a quantization method with finite data rate is proposed for handling the nested quantization and achieving the convergence to the origin of the system with input and output quantization. Further, an optimalization problem is developed for each quantizer, of which the optimal solution set contains a virtual augmented vector. The quantizers quantize their respective virtual augmented vectors to obtain the estimates of the outputs of the plant and the controller. We present the analytical expression of the optimal solution set for each quantizer. Finally, a method is presented to achieve the parameters of the quantizers and the controller.

Performance Analysis of Multiple-Antenna Ambient Backscatter Systems at Finite Blocklengths

This paper analyzes the maximal achievable rate for a given blocklength and maximal error probability over a multiple-antenna ambient backscatter channel. The result consists of a finite blocklength channel coding achievability bound and a converse bound for the legacy system with finite alphabet constraints and multiple-input-multiple-output based on the Neyman-Pearson test, the Berry-Esseen theorem, and the Mellin transform. Then, we derive the closed-form expression of the mutual information and the information variance to reduce the complexity of the computation. By applying the low-complexity ML detection, the relation between the maximal error probability of the RF source signal and the average error probability of the tag symbol with respect to the blocklength is proposed. Finally, numerical evaluation of these bounds shows fast convergence to the maximal achievable rate as the blocklength increases and also proves that the information variance is an accurate measure of the backoff from the maximal achievable rate due to finite blocklength.

Adaptive fuzzy finite‐time formation asymptotic tracking control for nonholonomic multirobot systems with inputs quantization

SummaryThis paper investigates the finite‐time formation tracking control problem for nonholonomic multirobot systems with inputs quantization. The fuzzy logic systems (FLSs) and adaptive method are introduced to approximate unknown nonlinear functions and parameters. Multirobot needs to keep safe and effective communicating range, the barrier function and prescribed performance method are employed to ensure collision avoidance and connectivity maintenance. Then, in order to make each robot more precisely and quickly track referenced leader's trajectory and consider the finite channel bandwidth in practical applications, the travel range of robots can be maintained at more secure region and formation can operate normally within the given bandwidth. By combining the command filter and finite‐time theory, a novel decentralized finite‐time formation asymptotic tracking (FTFAT) control strategy is further proposed based on inputs quantization technology, it guarantees that all signals are bounded and the formation tracking errors asymptotically converge to zero in finite‐time. Finally, the practicability of FTFAT strategy is demonstrated absolutely for nonholonomic multirobot systems.