Inflow Problem Research Articles

The existence and stability of stationary solutions of the inflow problem for full compressible Navier-Stokes-Poisson system

In this paper, we consider an inflow problem for the non-isentropic Navier-Stokes-Poisson system in a half line (0, ∞). For the general gas including ideal polytropic gas, we first give some results for the existence of the stationary solution with the aid of center manifold theory on a 4 × 4 system of autonomous ordinary differential equations. We also show the time asymptotic stability of the stationary solutions with small strength under smallness assumptions on the initial perturbations by using an elementary energy method.

Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations

<p style='text-indent:20px;'>This paper is devoted to the study of the inflow problem governed by the radiative Euler equations in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact discontinuity solution. It is different from the case involved with the rarefaction wave for the inflow problem in our previous work [<xref ref-type="bibr" rid="b6">6</xref>], since the rarefaction wave is a nonlinear expansive wave, while the contact discontinuity wave is a linearly degenerate diffusive wave. So we need to take good advantage of properties of the viscous contact discontinuity wave instead. Moreover, series of tricky argument on the boundary is done carefully based on the construction and the properties of the viscous contact discontinuity wave for the radiative Euler equations. Our result shows that radiation contributes to the stabilization effect for the supersonic inflow problem.

In-Flow and Out-Flow Problem for the Stokes System

We investigate the existence and regularity of solutions to the stationary Stokes system and non-stationary Navier–Stokes equations in three dimensional bounded domains with in- and out-lets. We assume that on the in- and out-flow parts of the boundary the pressure is prescribed and the tangential component of the velocity field is zero, whereas on the lateral part of the boundary the fluid is at rest.

Nonlinear stability of the boundary layer and rarefaction wave for the inflow problem governed by the heat-conductive ideal gas without viscosity

Nonlinear stability of the boundary layer and rarefaction wave for the inflow problem governed by the heat-conductive ideal gas without viscosity

Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids

In this paper, we study the large time behaviors of boundary layer solution of the inflow problem on the half space for a class of isentropic compressible non-Newtonian fluids. We establish the existence and uniqueness of the boundary layer solution to the non-Newtonian fluids. Especially, it is shown that such a boundary layer solution have a maximal interval of existence. Then we prove that if the strength of the boundary layer solution and the initial perturbation are suitably small, the unique global solution in time to the non-Newtonian fluids exists and asymptotically tends toward the boundary layer solution. The proof is given by the elementary energy method.

Viscous Shock Wave to an Inflow Problem for Compressible Viscous Gas with Large Density Oscillations

This paper is concerned with the inflow problem for one-dimensional compressible Navier-Stokes equations. For such a problem, Huang, Matsumura, and Shi showed in [4] that there exists viscous shock wave solution to the inflow problem and both the boundary layer solution, the viscous shock wave, and their superposition are time-asymptotically nonlinear stable provided that both the initial perturbation and the boundary velocity are assumed to be sufficiently small. The main purpose of this paper is to show that similar stability results still hold for a class of large initial perturbation which can allow the initial density to have large oscillations. The proofs are given by an elementary energy method and our main idea is to use the smallness of the strength of the viscous shock wave and the boundary velocity to control the possible growth of the solutions induced by the nonlinearity of the compressible Navier-Stokes equations and the inflow boundary condition. The key point in our analysis is to deduce the desired uniform positive lower and upper bounds on the density.

Asymptotic Stability of Rarefaction Wave for the Inflow Problem Governed by the One-Dimensional Radiative Euler Equations

This paper is devoted to the study of the initial-boundary value problem on the half line for a one-dimensional radiative Euler equations, which is a system coupled by the classic compressible noni...

Asymptotic stability of superposition of stationary solutions and rarefaction waves for 1D Navier–Stokes/Allen–Cahn system

In this paper, we investigate the large time behavior of the solutions to the inflow problem for the one-dimensional Navier–Stokes/Allen–Cahn system in the half space. First, we assume that the space-asymptotic states (ρ+,u+,χ+) and the boundary data (ρb,ub,χb) satisfy some conditions so that the time-asymptotic state of solutions for the inflow problem is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Then, we show the existence of the stationary solution by the center manifold theorem. Finally, we prove that the nonlinear wave is asymptotically stable when the initial data is a small perturbation of the nonlinear wave. The proof is mainly based on the energy method by taking into account the effect of the concentration χ and the complexity of nonlinear wave.

Stability of composite wave for inflow problem on the planar magnetohydrodynamics

In this paper, we investigate the large time behavior of the solutions to an initial–boundary value problem for the planar magnetohydrodynamics in a half line R+≔(0,∞). Inspired by the relationship between magnetohydrodynamics and Navier–Stokes, we can prove that the composite wave consisting of the subsonic BL-solution, the contact wave, and the rarefaction wave for the inflow problem on the planar magnetohydrodynamics is time-asymptotically stable. Meanwhile, we obtain the global existence of solutions based on the basic energy method by taking into account the complexity of composite wave.

Stability of stationary solutions to the inflow problem for the two-fluid non-isentropic Navier–Stokes–Poisson system

In this paper, we study the asymptotic behavior of solution to the initial boundary value problem for the two-fluid non-isentropic Navier–Stokes–Poisson system in a half line (0,∞). We consider an inflow problem where the gas enter into the region through the boundary for general gases including ideal polytropic gas. On account of the quasineutral assumption and the absence of the electric field for the large time behavior, we show the existence of the stationary solutions with the aid of the center manifold theory, and then we give the rigorous proof of the stability theorem on the stationary solutions under small perturbations for the corresponding initial boundary value problem of the Navier–Stokes–Poisson system. Note that it is the first result for the stability of nonlinear waves on the inflow problem of the Navier–Stokes–Poisson system.

Stability of stationary solutions for inflow problem on the planar magnetohydrodynamics

In this paper, stability of stationary solutions for the inflow problem on the planar magnetohydrodynamics is being investigated. Motivated by the relationship between magnetohydrodynamics and Navier-Stokes, we can prove that the global solutions of the planar magnetohydrodynamics asymptotically converge to the corresponding stationary solutions of Navier-Stokes equations when time t → ∞ in both the subsonic and transonic cases. The proof is based on the basic energy method.

Fissure structure analysis to unravel groundwater inflow problem in gold mining site of Pongkor area, West Java, Indonesia

Hydrogeology of Pongkor area, West Java, Indonesia is characterised by fractured aquifer system. The aquifer consists of volcanic rocks of tertiary period, namely Cimapag formation. It is fractured with orientations axis of the fracture patterns are northwest - southeast and northeast - southwest. Pongkor area is drained by two main surface streams, called River Cikaniki and river Cimaja, normally flowing northward direction. In the study area, there is such a vein structure known as Vein Ciurug containing gold ore deposits. The vein is exploited by subsurface tunnelling system. During expansion of the mining activity, some ramps were needed to be constructed. When a ramp at the altitude of +484 m above sea level was expanded, before reaching the vein, the drilling penetrated a saturated zone, causing groundwater inflow with a relatively massive rate. Further, static groundwater flow flooded the opening ramp down with water level of +499 m above sea level, resulted in disruption of mining activities at the ramps under the elevation 500 m above sea level. This study is to analyse the source of groundwater inflow and set a recommendation to solve the problem.

Steady transport equation in Sobolev–Slobodetskii spaces

We show existence of a regular solution in Sobolev-Slobodetskii spaces to stationary transport equation with inflow boundary condition in a bounded domain $\Omega \subset \mathbb{R}^2$. Our result is subject to quite general constraint on the shape of the boundary around the points where the characteristics become tangent to the boundary which applies in particular to piecewise analytical domains. Our result gives a new insight on the issue of boundary singularity for the inflow problem for stationary transport equation, solution of which is crucial for investigation of stationary compressible Navier-Stokes equations with inflow/outflow.

Fissure structure analysis to unravel groundwater inflow problem in gold mining site of Pongkor area, West Java, Indonesia

Fissure structure analysis to unravel groundwater inflow problem in gold mining site of Pongkor area, West Java, Indonesia

Large-time behavior of solutions to the inflow problem of full compressible Navier–Stokes equations with large perturbation

In this paper, we investigate the one-dimensional motion of compressible viscous and heat-conductive fluid on the half line . The large-time behavior of the superposition of the transonic (or degenerate) boundary layer solution, 1-rarefaction wave, 2-viscous contact wave and 3-rarefaction wave is established for the inflow problem of full compressible Navier–Stokes system under a large initial perturbation for any adiabatic exponent , provide the wave strength is suitable small.

Stability of stationary solutions for inflow problem on the micropolar fluid model

In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the micropolar fluid model in a half-line \(\mathbb {R}_{+}:=(0,\infty ).\) We prove that the corresponding stationary solutions of the small amplitude to the inflow problem for the micropolar fluid model are time asymptotically stable under small \(H^{1}\) perturbations in both the subsonic and degenerate cases. The microrotation velocity brings us some additional troubles compared with Navier–Stokes equations in the absence of the microrotation velocity. The proof of asymptotic stability is based on the basic energy method.

Stability of Stationary Solutions to the Inflow Problem for Full Compressible Navier--Stokes Equations with a Large Initial Perturbation

The large-time behavior of stationary solutions to the inflow problem of full compressible Navier--Stokes equations in the half line is investigated. We first give some necessary and sufficient conditions for the existence of the stationary solution with the aid of center manifold theory. We also prove that if $L^2$-norm of initial perturbation is small in accordance with any given $L^2$-norm of its derivative, then the stationary solutions with small strength for the general gas including ideal polytropic gas are asymptotically stable. Last, we show that the subsonic stationary solution with small strength for the ideal polytropic gas is asymptotically stable under large perturbations in both the $H^1$-norm and the $L^\infty$-norm without the condition that the adiabatic exponent $\gamma$ be close to 1, provided that the strength of the stationary wave is suitably small.

Stability of the composite wave for the inflow problem on the micropolar fluid model

In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the micropolar fluid model in half line $\mathbb{R}_{+}:=(0, \infty).$ Inspired by the relationship between the micropolar fluid model and Navier-Stokes system, we can prove that the composite wave consisting of the subsonic BL-solution, the contact wave, and the rarefaction wave for the inflow problem on micropolar fluid model is time-asymptotically stable. Meanwhile, we obtain the global existence of solutions based on the basic energy method.

Application of artificial neural network coupled with genetic algorithm and simulated annealing to solve groundwater inflow problem to an advancing open pit mine

Summary In this study, hybrid models are designed to predict groundwater inflow to an advancing open pit mine and the hydraulic head (HH) in observation wells at different distances from the centre of the pit during its advance. Hybrid methods coupling artificial neural network (ANN) with genetic algorithm (GA) methods (ANN–GA), and simulated annealing (SA) methods (ANN–SA), were utilised. Ratios of depth of pit penetration in aquifer to aquifer thickness, pit bottom radius to its top radius, inverse of pit advance time and the HH in the observation wells to the distance of observation wells from the centre of the pit were used as inputs to the networks. To achieve the objective two hybrid models consisting of ANN–GA and ANN–SA with 4-5-3-1 arrangement were designed. In addition, by switching the last argument of the input layer with the argument of the output layer of two earlier models, two new models were developed to predict the HH in the observation wells for the period of the mining process. The accuracy and reliability of models are verified by field data, results of a numerical finite element model using SEEP/W, outputs of simple ANNs and some well-known analytical solutions. Predicted results obtained by the hybrid methods are closer to the field data compared to the outputs of analytical and simple ANN models. Results show that despite the use of fewer and simpler parameters by the hybrid models, the ANN–GA and to some extent the ANN–SA have the ability to compete with the numerical models.

The inflow problem for the systems of equations of a viscous heat-conducting gas in the noncylindrical domains expanding in time

For a complete system of equations of the one-dimensional nonstationary motion of some viscous heat-conducting gas, the global solvability is proved of the inflow problem in the noncylindrical domains expanding in time. The proof of the time-global existence and uniqueness theorem is connected with obtaining a priori estimates with the constants depending only on the data of the problem and the value of the time interval T but independent of the existence interval of the local solution.