One-dimensional Navier-Stokes Equations Research Articles

Permeability evolution of defective wellbore cement under geologic carbon sequestration conditions by means of varying-radius pipeline modeling

X-ray imaging in combination with image registration was employed to unveil the microstructure alteration of defective wellbore cement cores when exposed to stagnant or flowing CO2-saturated brine under geologic carbon sequestration conditions. A varying-radius pipeline (VRP) modeling approach is proposed for estimating the permeability alteration from computed tomography (CT) images of artificial channels in the cement cores. The VRP model simplifies the three-dimensional channels into varying-radius pipelines, solves the problem of flow with one-dimensional Navier‒Stokes equations, and can readily be coupled with the Cahn–Hilliard equation to address multiphase flow problems. Furthermore, the accuracy of VRP modeling was verified by permeability testing and direct pore-scale modeling. The channel after CO2 exposure in the reactive-diffusion process was covered by substantial precipitation of calcite, and the measured channel permeability decreased by 28.67% from the original value 4.15 × 10−8 m2. In contrast, the channel after CO2 exposure in the reactive-flow process underwent significant dissolution, substantial tensile microfractures occurred at channel surface, and the measured channel permeability increased by 8.18% from the original 3.91 × 10−8 m2. The CT image–based microstructure investigation implied that when exposed to immobile CO2-saturated brine, even large fluid channels in wellbore cement may be shielded from further degradation; however, when encountering flowing CO2-saturated brine, cement with large fluid channels may fail. The proposed VRP model has the potential to assess field-scale CO2 leakage in wells.

Partially congested propagation fronts in one-dimensional Navier–Stokes equations

These notes are dedicated to the analysis of the one-dimensional free-congested Navier–Stokes equations. After a brief synthesis of the results obtained in Dalibard and Perrin (Commun Math Sci 18(7):1775–1813, 2020) related to the existence and the asymptotic stability of partially congested profiles associated to the soft congestion Navier-Stokes system, we present a first local well-posedness result for the one-dimensional free-congested Navier–Stokes equations.

Global existence, uniqueness and exponential stability of solutions for the one-dimensional Navier–Stokes equations with capillarity

In this paper, we investigate non-isothermal one-dimensional model of capillary compressible fluids as derived by Slemrod (1984) and Dunn and Serrin (1985). We establish the global existence, uniqueness and exponential stability of strong solutions in H+i(i=1,2,4) for the one-dimensional Navier–Stokes equations with capillarity, which implies the existence and exponential stability of the nonlinear C0-semigroups S(t) on H+i(i=1,2,4).

Dynamics of a Crankshaft Mechanism under the Pressure of a Viscous Gas

We study an initial–boundary value problem with free boundary for one-dimensional equations of viscous gas dynamics. The problem models the motion of a crankshaft mechanism under gas pressure. It is assumed that the gas fills a cylinder, which is modeled by the interval $$[0,1]$$ . A variable point $$a(t)\in[0,1]$$ models a piston moving inside the cylinder. The piston is assumed to be connected to a planar three-link crankshaft mechanism. We also assume that a velocity distribution on the boundary of the cylinder and a density distribution on gas inflow segments are given. The gas motion is described by the one-dimensional Navier–Stokes equations of viscous compressible fluid dynamics. It is required to determine the joint motion of the gas and crankshaft mechanism. We prove that this problem has a weak renormalized solution.

Knudsen number effects on the nonlinear acoustic spectral energy cascade.

We present a numerical investigation of the effects of gas rarefaction on the energy dynamics of resonating planar nonlinear acoustic waves. The problem setup is a gas-filled, adiabatic tube, excited from one end by a piston oscillating at the fundamental resonant frequency of the tube and closed at the other end; nonlinear wave steepening occurs until a limit cycle is reached, resulting in shock formation for sufficiently high densities. The Knudsen number, defined here as the ratio of the characteristic molecular collision timescale to the resonance period, is varied in the range Kn=10^{-1}-10^{-5}, from rarefied to dense regime, by changing the base density of the gas. The working fluid is Argon. A numerical solution of the Boltzmann equation, closed with the Bhatnagar-Gross-Krook model, is used to simulate cases for Kn≥0.01. The fully compressible one-dimensional Navier-Stokes equations are used for Kn<0.01 with adaptive mesh refinement to resolve the resonating weak shocks, reaching wave Mach numbers up to 1.01. Nonlinear wave steepening and shock formation are associated with spectral broadening of the acoustic energy in the wavenumber-frequency domain; the latter is defined based on the exact energy corollary for second-order nonlinear acoustics derived by Gupta and Scalo [Phys. Rev. E 98, 033117 (2018)2470-004510.1103/PhysRevE.98.033117], representing the Lyapunov function of the system. At the limit cycle, the acoustic energy spectra exhibit an equilibrium energy cascade with a -2 slope in the inertial range, also observed in freely decaying nonlinear acoustic waves by the same authors. In the present system, energy is introduced externally via a piston at low wavenumbers or frequencies and balanced by thermoviscous dissipation at high wavenumbers or frequencies, responsible for the base temperature increase in the system. The thermoviscous dissipation rate is shown to scale as Kn^{2} for fixed Reynolds number based on the maximum velocity amplitude, i.e., increasing with the degree of flow rarefaction; consistently, the smallest length scale of the steepened waves at the limit cycle, corresponding to the thickness of the shock (when present) also increases with Kn. For a given fixed piston velocity amplitude, the bandwidth of the inertial range of the spectral energy cascade decreases with increasing Knudsen numbers, resulting in a reduced resonant response of the system. By exploiting dimensionless scaling laws borrowed by Kolmogorov's theory of hydrodynamic turbulence, it is shown that an inertial range for spectral energy transfer can be expected for acoustic Reynolds numbers Re_{U_{max}}>100, based on the maximum acoustic velocity amplitude in the domain.

Modeling the hepatic arterial flow in living liver donor after left hepatectomy and postoperative boundary condition exploration.

Preoperative and postoperative hepatic perfusion is modeled with one-dimensional (1-D) Navier-Stokes equations. Flow rates obtained from ultrasound (US) data and impedance resulted from structured trees are the inflow and outflow boundary condition (BC), respectively. Structured trees terminate at the size of the arterioles, which can enlarge their size after hepatectomy. In clinical studies, the resistance to pulsatile arterial flow caused by the microvascular bed can be reflected by the resistive index (RI), a frequently used index in assessing arterial resistance. This study uses the RI in a novel manner to conveniently obtain the postoperative outflow impedance from the preoperative impedance. The major emphasis of this study is to devise a model to capture the postoperative hepatic hemodynamics after left hepatectomy. To study this, we build a hepatic network model and analyze its behavior under four different outflow impedance: (a) the same as preoperative impedance; (b) evaluated using the RI and preoperative impedance; (c) computed from structured tree BC with increased radius of terminal vessels; and (d) evaluated using structured tree with both increased radius of root vessel, ie, the outlets of the postoperative hepatic artery, and increased radius of terminal vessels. Our results show that both impedance from (b) and (d) give a physiologically reasonable postoperative hepatic pressure range, while the RI in (b) allows for a fast approximation of postoperative impedance. Since hemodynamics after hepatectomy are not fully understood, the methods used in this study to explore postoperative outflow BC are informative for future models exploring hemodynamic effects of partial hepatectomy.

Nonlinear stability of rarefaction waves for one-dimensional compressible Navier–Stokes equations for a reacting mixture

In this paper, we study the long-time behavior toward rarefaction waves for the Cauchy problem to a one-dimensional Navier–Stokes equations for a reacting mixture. It is shown that under the condition adiabatic exponent $$\gamma $$ is close to 1, the global stability is established. In this paper, the initial perturbation can be large.

Global classical solutions to 1D full compressible Navier–Stokes equations with the Robin boundary condition on temperature

In this paper, we consider the initial boundary value problem for the one-dimensional Navier–Stokes equations for viscous compressible and heat-conducting fluids in a bounded domain with the Robin boundary condition on temperature. There are few results until now about global existence of regular solutions to the full Navier–Stokes equations with the Robin boundary condition on temperature. By the analysis of the effect of boundary dissipation, we derive the global existence of classical solution to the corresponding initial boundary value problem with large initial data and vacuum. This result could be viewed as the first one on the global well-posedness of classical solutions to the full Navier–Stokes equations in a bounded domain with the Robin boundary condition on temperature.

Exponential Decay in Time of Density of One-dimensional Quantum Navier-Stokes Equations

The long-time behavior of the particle density of the compressible quantum Navier-Stokes equations in one space dimension is studied. It is shown that the particle density converges exponentially fast to the constant thermal equilibrium state as the time tends to infinity, the decay rate is also obtained. The results hold regardless of either the bigger of the scaled Planck constant or the viscosity constant. This improves the decay results of [5] by removing the crucial assumption that the scaled Planck constant is bigger than the viscosity constant. The proof is based on the entropy dissipation method and the Bresch-Desjardins type of entropy.

Spectral energy cascade and decay in nonlinear acoustic waves

We present a numerical and theoretical investigation of nonlinear spectral energy cascade of decaying finite-amplitude planar acoustic waves in a single-component ideal gas at standard temperature and pressure (STP). We analyze various one-dimensional canonical flow configurations: a propagating traveling wave (TW), a standing wave (SW), and randomly initialized Acoustic Wave Turbulence (AWT). We use shock-resolved mesh-adaptive direct numerical simulations (DNS) of the fully compressible one-dimensional Navier-Stokes equations to simulate the spectral energy cascade in nonlinear acoustic waves. We also derive a new set of nonlinear acoustics equations truncated to second order and the corresponding perturbation energy corollary yielding the expression for a new perturbation energy norm $E^{(2)}$. Its spatial average, <$E^{(2)}$> satisfies the definition of a Lyapunov function, correctly capturing the inviscid (or lossless) broadening of spectral energy in the initial stages of evolution -- analogous to the evolution of kinetic energy during the hydrodynamic break down of three-dimensional coherent vorticity -- resulting in the formation of smaller scales. Upon saturation of the spectral energy cascade i.e. fully broadened energy spectrum, the onset of viscous losses causes a monotonic decay of <$E^{(2)}$> in time.

A new insight into onset of inertial flow in porous media using network modeling with converging/diverging pores

The network modeling approach is applied to provide a new insight into the onset of non-Darcy flow through porous media. The analytical solutions of one-dimensional Navier-Stokes equation in sinusoidal and conical converging/diverging throats are used to calculate the pressure drop/flow rate responses in the capillaries of the network. The analysis of flow in a single pore revealed that there are two different regions for the flow coefficient ratio as a function of the aspect ratio. It is found that the critical Reynolds number strongly depends on the pore geometrical properties including throat length, average aspect ratio, and average coordination number of the porous media, and an estimation of such properties is required to achieve more reliable predictions. New criteria for the onset of non-Darcy flow are also proposed to overcome the lack of geometrical data. Although the average aspect ratio is the main parameter which controls the inertia effects, the effect of tortuosity on the onset of non-Darcy flow increases when the coordination number of media decreases. In addition, the higher non-Darcy coefficient does not essentially accelerate the onset of inertial flow. The results of this work can help to better understand how the onset of inertial flow may be controlled/changed by the pore architecture of porous media.

A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows

We consider the problem of finding an energy-based formulation of the Navier–Stokes equations for reactive flows. These equations occur in various applications, e. g., in combustion engines or chemical reactors. After modeling, discretization, and model reduction, important system properties as the energy conservation are usually lost which may lead to unphysical simulation results. In this paper, we introduce a port-Hamiltonian formulation of the one-dimensional Navier–Stokes equations for reactive flows. The port-Hamiltonian structure is directly associated with an energy balance, which ensures that a temporal change of the total energy is only due to energy flows through the boundary. Furthermore, the boundary ports may be used for control purposes.

Modeling digital pulse waveforms by solving one-dimensional Navier-stokes equations.

Mathematical modeling for composition distal arterial pulse wave in the blood vessels of the upper limbs was considered. Formation of distal arterial pulse wave is represented as a composition of forward and reflected pulse waves propagating along the arterial vessels. The formal analogy between pulse waves propagation along the human arterial system and the propagation of electrical oscillations in electrical transmission lines with distributed parameters was proposed. Dependencies of pulse wave propagation along the human arterial system were obtained by solving the one-dimensional Navier-Stokes equations for a few special cases.

Flow Field Analysis on the Stagnation Streamline of a Blunt Body

The hypersonic flow on the stagnation streamline of a blunt body is analyzed with quasi one-dimensional (1-D) Navier-Stokes equations approximated by adopting the local similarity to the two-dimensional (2-D)/axisymmetric Navier-Stokes equations. The governing equations are solved using the implicit finite volume method. The computational domain is confined from the stagnation point to the shock wave, and the shock fitting method is used to find the shock position. We propose a boundary condition at the shock, which employs the shock wave angle in the vicinity of the stagnation streamline using the shock shape correlation. As a result of numerical computation conducted for the hypersonic flow over a sphere, the proposed boundary condition is shown to improve the accuracy of the prediction of the shock standoff distance. The quasi 1-D Navier-Stokes code is efficient in computing time and is reliable for the flow analysis along the stagnation streamline and the prediction of heat flux at the stagnation point in the hypersonic blunt body flow.

Simulation of Generation of Peripheral Arterial Blood Pulsation

A mathematical model for the signal formation process of peripheral arterial blood pulsation in vessels of upper limbs is discussed. Formation of peripheral arterial blood pulsations is represented as a composition of forward and reflected pulse waves propagating through the arterial system. The study proposes a system of analogies between the propagation of pulse wave in the human arterial system and propagation of electrical oscillations in lines with distributed parameters. Dependences of pulse wave propagation in the arterial system were obtained by solving the one-dimensional Navier–Stokes equations for several special cases.

A Mathematical model of hemodynamic processes for distal pulse-wave formation

A mathematical model of the formation of distal arterial pulse wave signal in the blood vessels of the upper limbs was considered. The formation of distal arterial pulse wave is represented as a composition of forward and reverse pulse waves propagating along the human arterial system. The system of formal analogy between pulse waves propagation along the human arterial system and the propagation of electrical oscillations in electrical transmission lines with distributed parameters was proposed. Dependencies of pulse wave propagation along the human arterial system were obtained by solving the one-dimensional Navier-Stokes equations for a few special cases.

Benchmarks for time-domain simulation of sound propagation in soft-walled airways: steady configurations.

Time-domain computer simulation of sound production in airways is a widely used tool, both for research and synthetic speech production technology. Speed of computation is generally the rationale for one-dimensional approaches to sound propagation and radiation. Transmission line and wave-reflection (scattering) algorithms are used to produce formant frequencies and bandwidths for arbitrarily shaped airways. Some benchmark graphs and tables are provided for formant frequencies and bandwidth calculations based on specific mathematical terms in the one-dimensional Navier-Stokes equation. Some rules are provided here for temporal and spatial discretization in terms of desired accuracy and stability of the solution. Kinetic losses, which have been difficult to quantify in frequency-domain simulations, are quantified here on the basis of the measurements of Scherer, Torkaman, Kucinschi, and Afjeh [(2010). J. Acoust. Soc. Am. 128(2), 828-838].

Counterpropagation of waves with shock fronts in a nonlinear tissue-like medium

A numerical model for describing the counterpropagation of one-dimensional waves in a nonlinear medium with an arbitrary power law absorption and corresponding dispersion is developed. The model is based on general one-dimensional Navier-Stokes equations with absorption in the form of a time-domain convolution operator in the equation of state. The developed algorithm makes it possible to describe wave interactions in the presence of shock fronts in media like biological tissue. Numerical modeling is conducted by the finite difference method on a staggered grid; absorption and sound speed dispersion are taken into account using the causal memory function. The developed model is used for numerical calculations, which demonstrate the absorption and dispersion effects on nonlinear propagation of differently shaped pulses, as well as their reflection from impedance acoustic boundaries.

On the blowing up of solutions to one-dimensional quantum Navier-Stokes equations

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the one-dimensional quantum Navier-Stokes equations in a bounded domain is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differential operator, with the quantum Bohm potential, and a density-dependent viscosity. It is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time, if the viscosity constant is not bigger than the scaled Planck constant. The proof is inspired by an observable constructed by Gamba, Gualdani and Zhang, which has been used to study the blowing up of solutions to quantum hydrodynamic models.

NEWTONIAN FLOW IN CONVERGING-DIVERGING CAPILLARIES

The one-dimensional Navier–Stokes equations are used to derive analytical expressions for the relation between pressure and volumetric flow rate in capillaries of five different converging-diverging axisymmetric geometries for Newtonian fluids. The results are compared to previously derived expressions for the same geometries using the lubrication approximation. The results of the one-dimensional Navier–Stokes are identical to those obtained from the lubrication approximation within a nondimensional numerical factor. The derived flow expressions have also been validated by comparison to numerical solutions obtained from discretization with numerical integration. Moreover, they have been certified by testing the convergence of solutions as the converging-diverging geometries approach the limiting straight geometry.