Viscosity Limit Research Articles

Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks

Abstract In this article, we study the inviscid limit of the solution to the Cauchy problem of a one-dimensional viscous conservation law, where the second-order term is nonlinear. Under the assumption that the inviscid equation admits a piecewise smooth solution with two noninteracting entropy shocks, we prove that the solution of the viscous equation converges uniformly to the piecewise smooth inviscid solution away from the shocks, even the strength of shocks is not small.

Existence of solutions to k‐Wave models of nonlinear ultrasound propagation in biological tissue

AbstractWe investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k‐Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. We first consider a system that incorporates additional viscosity in the equation for momentum conservation. By constructing a Galerkin approximation procedure, we prove the local existence of its solutions. In view of inverse problems arising from imaging tasks, the theory allows for the variable background mass density, speed of sound, and the nonlinearity parameter in the systems. Second, under stronger conditions on the data, we take the vanishing viscosity limit of the problem, thereby rigorously establishing the existence of solutions for the limiting system as well.

Crosslinking assembly of Propenyl-Calix[4]-Crown-6 interpenetrating functional mesoporous carbon Spheres: High selectivity and stability to Remove Cs(I) from high acid solution

The calix[4]-crown-6 ligand shows high selectivity for Cs(I) over other alkali and alkaline earth metal cations, making it a promising candidate for radiocesium separation. However, its high viscosity and poor adhesion limit its practical use and stability. In this study, an alkenyl-functionalized calix[4]-crown-6 (p-C[4]C6) was developed. Using a mesoporous structure and post-assembly techniques, a composite (p-C[4]C6/MMCs) was created via crosslinking within mesoporous matrix channels. The p-C[4]C6/MMCs exhibited high stability and practicality. The p-C[4]C6/MMCs demonstrated high selectivity, good reusability, and significant adsorption capacity (47.76 mg/g) for Cs(I) in 3.0 M HNO3. Kinetic, isotherm, and thermodynamic studies suggest a spontaneous chemical adsorption mechanism. This research provides insights into the design of calix[4]-crown-6 based materials and offers a viable approach for Cs(I) separation from high-level liquid waste (HLW).

Study of molecular arrangement and density estimation of soybean oil biodiesel-diesel blends employing molecular dynamic simulation

Blends of diesel–biodiesel are a crucial alternative for petrodiesel substitution. However, biodiesel faces challenges such as high density and kinematic viscosity, which can affect the properties and performance of the blends. Based on this, the investigation of the impact of adding biodiesel into diesel fuel at percentages of 12%, 15%, 18%, and 21% using molecular dynamic simulations is presented. The density of the blends was experimentally measured and calculated with good agreement, which indicates a favorable description of the interactions by the OPLS-AA force field. Obtained results suggest that alkanes and isoalkanes showed higher fluctuations and were most sensitive to the variation of the percentage of biodiesel in diesel. The other classes of molecules did not show significant variations in interactions, suggesting that π-π interactions are unaffected by the biodiesel addition. The spatial distribution functions revealed that diesel molecules are organized homogeneously around methyl oleate and methyl linoleate, whereas aliphatic hydrocarbon and aromatic molecules are in distinct regions. The diffusion coefficients (Di) showed that the diesel molecules possess higher values, and the biodiesel addition increases the Di of the blend due to the methyl esters carbon chains, which present lower diffusion than diesel molecules. Finally, physical parameters and simulations revealed that the biodiesel addition into diesel fuel in the proportion up to 21% did not negatively impact the density and viscosity limits established by standards ASTM, EN, and ANP Resolution.

On the Support of Anomalous Dissipation Measures

By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier–Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler (Drivas and Eyink in Commun Math Phys 359(2):733–763, 2018. https://doi.org/10.1007/s00220-017-3078-4; Majda in Am Math Soc 43(281):93, 1983. https://doi.org/10.1090/memo/0281) and by recent constructions of dissipative incompressible Euler solutions (Brue and De Lellis in Commun Math Phys 400(3):1507–1533, 2023. https://doi.org/10.1007/s00220-022-04626-0 624; Brue et al. in Commun Pure App Anal, 2023), as well as passive scalars (Colombo et al. in Ann PDE 9(2):21–48, 2023. https://doi.org/10.1007/s40818-023-00162-9; Drivas et al. in Arch Ration Mech Anal 243(3):1151–1180, 2022. https://doi.org/10.1007/s00205-021-01736-2). For LtqLxr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q_tL^r_x$$\\end{document} suitable Leray–Hopf solutions of the d-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d-$$\\end{document}dimensional Navier–Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure Ps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {P}^{s}$$\\end{document}, which gives s=d-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s=d-2$$\\end{document} as soon as the solution lies in the Prodi–Serrin class. In the three-dimensional case, this matches with the Caffarelli–Kohn–Nirenberg partial regularity.

Vanishing shear viscosity limit for the compressible planar MHD system with boundary layer

This paper is devoted to the study of the vanishing shear viscosity limit and strong boundary layer problem for the compressible, viscous, and heat-conducting planar MHD equations. The main aim is to obtain a sharp convergence rate which is usually connected to the boundary layer thickness. However, The convergence rate would be possibly slowed down due to the presence of the strong boundary layer effect and the interactions among the magnetic field, temperature, and fluids through not only the velocity equations but also the strongly nonlinear terms in the temperature equation. Our main strategy is to construct some new functions via asymptotic matching method which can cancel some quantities decaying in a lower speed. It leads to a sharp L∞ convergence rate as the shear viscosity vanishes for global-in-time solution with arbitrarily large initial data.

Delta shock wave for ternary nonlinear chromatography equations as self-similar viscosity limit

In this paper, we investigate the existence and stability of the delta shock wave in ternary chromatography equations by the self-similar viscosity vanishing approach. Considering the appropriate initial values, we prove the existence of the self-similar solution for the corresponding Riemann problem of the ternary chromatography viscous equations. Furthermore, we rigorously demonstrate that the delta shock wave is the weak star limit of the self-similar solution as viscosity tends to disappear. The result implies that the structure of the delta shock wave is stable under the self-similar viscosity perturbation, which guarantees that the delta shock wave is a unique entropy solution. In addition, we present numerical simulations in agreement with the theoretical analysis.

On the Vanishing Viscosity Limit of Statistical Solutions of the Incompressible Navier–Stokes Equations

On the Vanishing Viscosity Limit of Statistical Solutions of the Incompressible Navier–Stokes Equations

Use of Nanobubbles to Improve Mass Transfer in Bioprocesses

Nanobubble technology has emerged as a transformative approach in bioprocessing, significantly enhancing mass-transfer efficiency for effective microbial activity. Characterized by their nanometric size and high internal pressure, nanobubbles possess distinct properties such as prolonged stability and minimal rise velocities, allowing them to remain suspended in liquid media for extended periods. These features are particularly beneficial in bioprocesses involving aerobic strains, where they help overcome common obstacles, such as increased culture viscosity and diffusion limitations, that traditionally impede efficient mass transfer. For instance, in an experimental setup, nanobubble aeration achieved 10% higher soluble chemical oxygen demand (sCOD) removal compared to traditional aeration methods. Additionally, nanobubble-aerated systems demonstrated a 55.03% increase in caproic acid concentration when supplemented with air nanobubble water, reaching up to 15.10 g/L. These results underscore the potential of nanobubble technology for optimizing bioprocess efficiency and sustainability. This review delineates the important role of the mass-transfer coefficient (kL) in evaluating these interactions and underscores the significance of nanobubbles in improving bioprocess efficiency. The integration of nanobubble technology in bioprocessing not only improves gas exchange and substrate utilization but also bolsters microbial growth and metabolic performance. The potential of nanobubble technology to improve the mass-transfer efficiency in biotechnological applications is supported by emerging research. However, to fully leverage these benefits, it is essential to conduct further empirical studies to specifically assess their impacts on bioprocess efficacy and scalability. Such research will provide the necessary data to validate the practical applications of nanobubbles and identify any limitations that need to be addressed in industrial settings.

Evaluation of warm mix effect in laboratory using a self-developed WMA additive Alube

In response to the problems of high mixing temperatures and energy consumption of polymer-modified asphalt mixtures in road construction, a surfactant-based warm mix additive ‘Alube’ was developed. The warm mix SBS-modified asphalt and its mixture were prepared. Their rheological performance and chemical structure were analysed by the rheological and microscopic tests. The active mixing power (P) of asphalt mixtures were examined by the variable speed mixing (VSM) test. The mixing flowability model was then established to evaluate the warm mix effect. The results indicated that the mixing flowability of asphalt mixtures obeyed the linear Bingham visco-plastic model, and the slope and intercept of the flow straight characterised the mixing viscosity and plastic limit. It is evaluated that 0.7% Alube1 and Evotherm reduced the mixing temperature of SBS-modified asphalt mixture by 14°C and 12°C, respectively. Moreover, Alube1 is suitable for a wide range of applications through warm mix asphalt technology.

Asymptotic analysis for the homogeneous model of wind‐driven oceanic circulation in the small viscosity limit

In this paper, we study the asymptotic expansion for the homogeneous model of wind‐driven oceanic circulation with the nonslip boundary condition when both of the beta‐plane parameter and the Reynolds number go to infinity. By asymptotic analysis under certain constraints on wind tensor, we derive a formal expansion including the geophysical boundary layer in the large beta‐plane parameter and high Reynolds number limit, from which one sees that the external force, wind tensor, has an important effect on the behavior of the boundary layers. Moreover, when the external force and initial data satisfy certain constraints, to exclude the appearance of strong boundary layers, we construct approximate solutions with steady boundary layers to the homogeneous model of wind‐driven oceanic circulation in the large beta‐plane parameter and high Reynolds number limit. By using an energy method, we obtain the ‐stability of small perturbation around this approximate solutions, which verifies the validity of the asymptotic expansion of weak boundary layers in the large beta‐plane parameter and high Reynolds number limit.

Enhancing heavy‐oil displacement efficiency through viscoelasticity of polymer solution by investigating the viscosity limit of crude oil: An experimental study

AbstractThe world is rich in ordinary heavy‐oil resources. Exploring the limit of the crude oil viscosity limits can further expand the application of polymer flooding technology in heavy‐oil reservoirs and improve the development efficiency of heavy‐oil reservoirs. On the basis of the polymer flooding that has been implemented in the Bohai offshore oilfield, this study characterized the in situ viscoelasticity of the polymer solution SNF3640C. We compared the polymer flooding and glycerol flooding performance in low‐viscous oil reservoirs (with a viscosity of 6.5 cp) and high‐viscous oil reservoirs (with a viscosity of 65 cp) by conducting microscopic dead‐end experiments and macroscopic core‐flooding experiments, and then analyzing the impact of polymer's viscoelasticity on the oil displacement efficiency of heavy oil. Finally, multiple crude oils with a wide range of viscosity were used to explore the crude oil viscosity limits under which the efficiency of heavy‐oil displacement could be improved through polymer viscoelasticity. The results demonstrate that: (1) the SNF3640C polymer solution can exhibit a certain degree of viscoelasticity in deep formation. At a distance of 140 m from the wellbore in the reservoir, a solution with a concentration of 1500 mg/L can still have a first normal stress difference of 2.5 Pa. Polymers can effectively improve the efficiency of heavy‐oil displacement through elastic action. (2) The viscosity limit that can improve the efficiency of heavy‐oil displacement under current experimental conditions does not exceed 5486 cp. The efficiency of polymer flooding for crude oil has decreased to 31.68%, and the contribution of elastic flooding is only 1.5%. The research method of using viscous glycerol as a comparison to demonstrate the contribution of polymer elasticity to oil displacement effectiveness effectively solves the problem of isolating the contribution by polymer solution's viscosity and elasticity, and clarifies the development direction of polymer flooding for developing heavy crude oil. Enhancing the elastic effect of polymer solution in porous media can increase the viscosity range of polymer flooding heavy oil.

Use of Sulfuric Acid-Impregnated Biochar Catalyst in Making of Biodiesel From Waste Cooking Oil Via Leaching Method

The biodiesel synthesis of waste cooking oil (WCO) over a impregnated biochar catalyst was systematically studied. This research aimed to prepare Biochar-based material that comes from coconut coir, activate it, and apply it as a catalyst to the esterification reaction of high-FFA waste cooking oil. Activation of the catalyst was done by impregnation H2SO4 solution in Biochar. The obtained catalyst was characterized by FTIR, XRF, XRD, surface area analyzer, and SEM-EDS. The esterification process was conducted by varying the catalyst weight (5, 7, and 10 wt%) and the reaction temperature (55 and 60 °C). The obtained liquid yields were characterized by GC-MS. The study found that the esterification process worked best with 10 wt% catalysts, a 1:76 mole ratio of oil to alcohol, and a reaction temperature of 60 °C. The waste cooking oil was successfully converted into biodiesel, reaching 84.50% of yield and 77.30% of purity (methyl ester content). Meanwhile, testing using national biodiesel standards with parameter limits of density, viscosity, iodine number, and acid number shows results that meet the requirements. Copyright © 2024 by Authors, Published by BCREC Publishing Group. This is an open access article under the CC BY-SA License (https://creativecommons.org/licenses/by-sa/4.0).

Vanishing viscosity limit for incompressible axisymmetric flow in the exterior of a cylinder

In this paper, we study the initial boundary value problem and vanishing viscosity limit for incompressible axisymmetric Navier–Stokes equations with swirls in the exterior of a cylinder under Navier-slip boundary condition. In the first part, we prove the existence of a unique global solution with the axisymmetric initial data u0ν∈Lσ2(Ω) and axisymmetric force f∈L2([0,T];L2(Ω)) . This result improves the initial regularity condition on the global well-posedness result obtained by K Abe and G Seregin (2020 Proc. R. Soc. Edinburgh A 150 1671–98) and extends their boundary condition. In the second part, we make the first attempt to investigate the inviscid limit of unforced viscous axisymmetric flows with swirls and prove that the viscous axisymmetric flows with swirls converge to inviscid axisymmetric flows without swirls under the condition ‖ru0θν‖L2(Ω)=O(ν) . Some new uniform estimates, independent of the viscosity, are obtained here. The second result can be thought as a follow-up work to the previous work by K Abe (2020 J. Math. Pures Appl. 137 1–32), where the inviscid limit for the same equations without swirls in an infinite cylinder was studied.

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.

On the vanishing viscosity limit for the viscous and resistive 3D magnetohydrodynamic system with axisymmetric data

The current paper aims to explore the three-dimensional axisymmetric viscous and resistive magnetohydrodynamic system. This article is concerned with two outcomes, in the first result we prove the global existence of a unique solution for this system under initial data lying in the Sobolev spaces Hs × Hs−2 with s>52, furthermore, we obtain uniform estimates with respect to the viscosity. The second outcome shows that the viscous and resistive solutions (vμ,bμ)μ>0 strongly converge to the non-viscous one (v, b). This convergence is performed in the space L∞(R+,Hs−2×Hs−2), also we obtain that the rate of convergence is (μt).

Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier–Stokes equations with temperature-dependent transport coefficients

Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier–Stokes equations with temperature-dependent transport coefficients

Vanishing capillarity–viscosity limit of the incompressible Navier–Stokes–Korteweg equations with slip boundary condition

In this paper, we investigate the vanishing capillarity–viscosity limit of the incompressible Navier–Stokes–Korteweg (NSK) equations in a three-dimensional horizontally periodic strip domain, in which the velocity of the fluid is supplemented with slip boundary condition and the gradient of density with Dirichlet boundary condition on the boundary. We prove that there exists an positive constant T0 independent on the capillarity and viscosity coefficients, such that the incompressible NSK equations have a unique strong solution on [0,T0] and the solution is uniformly bounded in H3. Based on the uniform estimates, we further give the convergence rate in H1 from the solutions of the incompressible NSK equations to the solution of the inhomogeneous incompressible Euler equations as the capillarity and viscosity coefficients go to zero simultaneously.

A KAM Approach to the Inviscid Limit for the 2D Navier–Stokes Equations

In this paper, we investigate the inviscid limit ν→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u \\rightarrow 0$$\\end{document} for time-quasi-periodic solutions of the incompressible Navier–Stokes equations on the two-dimensional torus T2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {T}}^2$$\\end{document}, with a small time-quasi-periodic external force. More precisely, we construct solutions of the forced Navier–Stokes equation, bifurcating from a given time quasi-periodic solution of the incompressible Euler equations and admitting vanishing viscosity limit to the latter, uniformly for all times and independently of the size of the external perturbation. Our proof is based on the construction of an approximate solution, up to an error of order O(ν2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\ u ^2)$$\\end{document} and on a fixed point argument starting with this new approximate solution. A fundamental step is to prove the invertibility of the linearized Navier–Stokes operator at a quasi-periodic solution of the Euler equation, with smallness conditions and estimates which are uniform with respect to the viscosity parameter. To the best of our knowledge, this is the first positive result for the inviscid limit problem that is global and uniform in time and it is the first KAM result in the framework of the singular limit problems.

Semi-Lagrangian finite element exterior calculus for incompressible flows

We develop a semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions on arbitrary simplicial meshes. We recast the equations as a nonlinear transport problem for a momentum 1-form and discretize in space using methods from finite element exterior calculus. Numerical experiments show that the linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing viscosity limit and is applicable to inviscid incompressible Euler flows. We obtain second-order convergence and conservation of energy is achieved through a Lagrange multiplier.