GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY

This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.

Incompressible Limit of the Oldroyd-B Model with Density-Dependent Viscosity

This paper studies the existence and uniqueness of local strong solutions to an Oldroyd-B model with density-dependent viscosity in a bounded domain Ω ⊂ Rd, d = 2 or 3 via incompressible limit, in which the initial data is “well-prepared” and the velocity field enjoys the slip boundary conditions. The main idea is to derive the uniform energy estimates for nonlinear systems and corresponding incompressible limit.

GLOBAL CLASSICAL SOLUTION TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

In this paper, we consider the global existence of classical solution to the 3-D compressible Navier-Stokes equations with a density-dependent viscosity coefficient λ(ρ)provided that the initial energy is small in some sense. In our result, we give a relation between the initial energy and the viscosity coefficient μ, and it shows that the initial energy can be large if the coefficient of the viscosity μ is taken to be large, which implies that large viscosity μ means large solution.

STABILITY OF VISCOUS SHOCK WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

We study the large-time behavior toward viscous shock waves to the Cauchy problem of the one-dimensional compressible isentropic Navier-Stokes equations with density- dependent viscosity. The nonlinear stability of the viscous shock waves is shown for certain class of large initial perturbation with integral zero which can allow the initial density to have large oscillation. Our analysis relies upon the technique developed by Kanel~ and the continuation argument.

FREE BOUNDARY VALUE PROBLEM FOR THE CYLINDRICALLY SYMMETRIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

In this paper, we investigate the free boundary value problem （FBVP） for the cylindrically symmetric isentropic compressible Navier-Stokes equations （CNS） with density- dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, we prove that there exists a unique global strong solution which tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.

ZERO DISSIPATION LIMIT TO RAREFACTION WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH SELECTED DENSITY-DEPENDENT VISCOSITY

This paper is devoted to studying the zero dissipation limit problem for the one-dimensional compressible Navier-Stokes equations with selected density-dependent viscosity.In particular,we focus our attention on the viscosity taking the formμ(ρ)=ρ^(ϵ)(ϵ>0).For the selected density-dependent viscosity,it is proved that the solutions of the one-dimensional compressible Navier-Stokes equations with centered rarefaction wave initial data exist for all time,and converge to the centered rarefaction waves as the viscosity vanishes,uniformly away from the initial discontinuities.New and subtle analysis is developed to overcome difficulties due to the selected density-dependent viscosity to derive energy estimates,in addition to the scaling argument and elementary energy analysis.Moreover,our results extend the studies in[Xin Z P.Comm Pure Appl Math,1993,46(5):621-665].

LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

In this article, we prove the local existence and uniqueness of the classical solution to the Cauchy problem of the 3-D compressible Navier-Stokes equations with large initial data and vacuum, if the shear viscosity μ is a positive constant and the bulk viscosity λ（ρ） = ρ^β with β≥0. Note that the initial data can be arbitrarily large to contain vacuum states.

GLOBAL SOLUTIONS TO A 3D AXISYMMETRIC COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY

In this paper,we consider the 3D compressible isentropic Navier-Stokes equations when the shear viscosityμis a positive constant and the bulk viscosity is λ(ρ)=ρ^(β) with β>2,which is a situation that was first introduced by Vaigant and Kazhikhov in[1].The global axisymmetric classical solution with arbitrarily large initial data in a periodic domain Ω={(r,z)|r=√x^(2)+y^(2),(x,y,z)∈R^(3),r∈I⊂(0,+∞),-∞<z<+∞} is obtained.Here the initial density keeps a non-vacuum state ρ>0 when z→±∞.Our results also show that the solution will not develop the vacuum state in any finite time,provided that the initial density is uniformly away from the vacuum.

Non-formation of vacuum states for Navier-Stokes equations with density-dependent viscosity

We consider the Cauchy problem, free boundary problem and piston problem for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity. Using the reduction to absurdity method, we prove that the weak solutions to these systems do not exhibit vacuum states, provided that no vacuum states are present initially. The essential re- quirements on the solutions are that the mass and energy of the fluid are locally integrable at each time, and the Lloc1-norm of the velocity gradient is locally integrable in time.

Remarks on One-Dimensional Compressible Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

The Navier-Stokes system for one-dimensional compressible fluids with density-dependent viscosities when the initial density connects to vacuum continuously is considered in the present paper. When the viscosity coefficient u is proportional to pθ with 0 〈 θ 〈 1, the global existence and the uniqueness of weak solutions are proved which improves the previous results in [Vong, S. W., Yang, T., Zhu, C. J.： Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum II. J. Differential Equations, 192（2）, 475-501 （2003）]. Here p is the density. Moreover, a stabilization rate estimate for the density as t → ＋∞ for any θ 〉 0 is also given.

Global Strong Solutions of the Cauchy Problem for 1D Compressible Navier-Stokes Equations with Density-dependent Viscosity

We consider the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity μ（ρ） = Aρα, where α〉 0 and A 〉0. The global existence of strong solutions is obtained, which improves the previous results by enlarging the interval of α. Moreover, our result shows that no vacuum is developed in a finite time provided the initial data does not contain vacuum.

The Cartesian analytical solutions for the Ndimensional compressible Navier-Stokes equations with density-dependent viscosity

In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integration theory.Two exact solutions are obtained by solving the reduced systems.

The global classical solution to a 1D two-fluid model with density-dependent viscosity and vacuum

In this paper, we consider the initial-boundary problem for a 1D two-fluid model with densitydependent viscosity and vacuum. The pressure depends on two variables but the viscosity only depends on one of the densities. We prove the global existence and uniqueness of the classical solution in the one-dimensional space with large initial data and vacuum. We use a new Helmholtz free energy function and the material derivative of the velocity field to deal with the general pressure with two variables, without the equivalence condition. We also develop a new argument to handle the general viscosity.

A Remark on the Cauchy Problem of 1D Compressible Navier-Stokes Equations with Density-dependent Viscosity Coefcients

This paper is concerned with the Cauchy problems of one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefcients.By assumingρ0∈L1（R）,we will prove the existence of weak solutions to the Cauchy problems forθ〉0.This will improve results in Jiu and Xin’s paper（Kinet.Relat.Models,1（2）：313–330（2008））in whichθ〉12is required.In addition,We will study the large time asymptotic behavior of such weak solutions.

Influence of variable viscosity and double diffusion on the convective stability of a nanofluid flow in an inclined porous channel

The influence of variable viscosity and double diffusion on the convective stability of a nanofluid flow in an inclined porous channel is investigated.The DarcyBrinkman model is used to characterize the fluid flow dynamics in porous materials.The analytical solutions are obtained for the unidirectional and completely developed flow.Based on a normal mode analysis,the generalized eigenvalue problem under a perturbed state is solved.The eigenvalue problem is then solved by the spectral method.Finally,the critical Rayleigh number with the corresponding wavenumber is evaluated at the assigned values of the other flow-governing parameters.The results show that increasing the Darcy number,the Lewis number,the Dufour parameter,or the Soret parameter increases the stability of the system,whereas increasing the inclination angle of the channel destabilizes the flow.Besides,the flow is the most unstable when the channel is vertically oriented.

Numerical Simulation of Slurry Diffusion in Fractured Rocks Considering a Time-Varying Viscosity

To analyze the effects of a time-varying viscosity on the penetration length of grouting,in this study cement slur-ries with varying water-cement ratios have been investigated using the Bingham’sfluidflow equation and a dis-crete element method.Afluid-solid coupling numerical model has been introduced accordingly,and its accuracy has been validated through comparison of theoretical and numerical solutions.For different fracture forms(a single fracture,a branch fracture,and a fracture network),the influence of the time-varying viscosity on the slurry length range has been investigated,considering the change in the fracture aperture.The results show that under different fracture forms and the same grouting process conditions,the influence of the time-varying viscosity on the seepage length is 0.350 m.

Study of the Viscosity and Specific Gravity of the Ternary Used Frying Oil (UFO)-Bioethanol-Diesel System

Fossil fuels cover around 80% of global energy consumption. However, the problems linked to their use justify the choice of using biofuel. In order to reduce as much as possible, diesel rate, an increase in the number of additives may be considered. Thus, in this work, the study of the used frying oil (UFO), bioethanol and diesel ternary system was undertaken. It emerges from this study that the addition of bioethanol reduces the viscosity and the density of the ternary system and permits a 90% substitution rate for diesel between the UFO and bioethanol. Finally, the percentage of oil becomes 40% after adding alcohol compared to the binary diesel crude vegetable oil mixture where this rate is 30%.

Vacuum states on compressible Navier-Stokes equations with general density-dependent viscosity and general pressure law

In this paper,we study the one-dimensional motion of viscous gas with a general pres- sure law and a general density-dependent viscosity coefficient when the initial density connects to the vacuum state with a jump.We prove the global existence and the uniqueness of weak solutions to the compressible Navier-Stokes equations by using the line method.For this,some new a priori estimates are obtained to take care of the general viscosity coefficientμ(ρ)instead ofρ~θ.

Molecular composition of naphthenic acids in a Chinese heavy crude oil and their impacts on oil viscosity

Most heavy crude oils underwent biodegradation and generated a significant amount of naphthenic acids. Naphthenic acids are polar compounds with the carboxylic group and are considered as a major factor affecting the oil viscosity. However, the relationship between the molecular composition of naphthenic acids and oil viscosity is not well understood. This study examined a “clean” heavy oil with low contents of heteroatoms but had a high content of naphthenic acids. Naphthenic acids were fractionated by distillation and caustic extraction. The molecular composition was characterized by high-resolution Orbitrap mass spectrometry. It was found that the 2- and 3-ring naphthenic monoacids with 15–35 carbon atoms are dominant components of the acid fractions;the caustic extraction is capable of isolating naphthenic acids with less than 35 carbons, which is equivalent to the upper limit of the distillable components, but not those in the residue fraction;the total acid number of the heavy distillates is higher than that of the residue fraction;the viscosity of the distillation fraction increases exponentially with an increased boiling point of the distillates. Blending experiments indicates that there is a strong correlation between the oil viscosity and acids content, although the acid content is only a few percent of the total oil.

THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS

This paper is concerned with the asymptotic behavior of solutions to the initial boundary problem of the two-dimensional density-dependent Boussinesq equations.It is shown that the solutions of the Boussinesq equations converge to those of zero thermal diffusivity Boussinesq equations as the thermal diffusivity tends to zero,and the convergence rate is established.In addition,we prove that the boundary-layer thickness is of the valueδ(k)=k^(α)with anyα∈(0,1/4)for a small diffusivity coefficient k>0,and we also find a function to describe the properties of the boundary layer.