Properties of the solution of the adjoint problem describing the motion of viscous fluids in a tube

For the Navier-Stokes equations, we study a solution invariant with respect to a oneparameter group and modeling a nonstationary motion of two viscous fluids in a cylindrical tube; the fluid layer near the tube wall can be viewed as a lubricant. The motion is due to a nonstationary pressure drop. We obtain a priori estimates for the velocities in the layers. We find a stationary state of the system and show that it is the limit state as t → ∞ provided that the pressure gradient in one of the fluids stabilizes with time. We solve the inverse problem of finding the pressure gradients and the velocity field from a known flow rate.     

Exact analytical solution of a generalized multiple moving boundary model of one-dimensional non-Darcy flow in heterogeneous multilayered low-permeability porous media with a threshold pressure gradient

A nonlinear generalized multiple moving boundary model of one-dimensional non-Darcy flow in heterogeneous multilayered low-permeability porous media with a threshold pressure gradient is constructed, in which the total rate of fluid injection into the porous media remains constant. The number of layers in the model can be arbitrary, and thus the generalized model will be very suitable for describing the one-dimensional non-Darcy flow characteristics in low-permeability reservoirs with strong heterogeneity. Through the similarity transformation method, the exact analytical solution of the multiple moving boundary model is obtained, and the formula for the subrate of fluid injection into every layer is provided. Moreover, it is strictly proved that the exact analytical solution can reduce to the solution of Darcy flow as the threshold pressure gradient in different layers simultaneously tends to zero. Through the exact analytical solution, the effects of the layer threshold pressure gradient, the layer permeability ratio, and the layer elastic storage ratio on the moving boundaries, the spatial pressure distributions, the transient pressure, and the layer subrate in low-permeability porous media are discussed. Through comparison of the exact analytical solutions, it is also demonstrated that incorporation of the multiple moving boundary conditions is very necessary in the modeling of non-Darcy flow in heterogeneous multilayered porous media with a threshold pressure gradient, especially when the threshold pressure gradient is large. In particular, an explicit formula is presented for estimating the relative error of the transient pressure introduced by ignoring the moving boundaries in the modeling. All in all, solid theoretical foundations are provided for non-Darcy flow problems in stratified reservoirs with a threshold pressure gradient. They can be very useful for strictly verifying numerical simulation results, and for giving some guidance for project design and optimization of layer production or injection during the development of heterogeneous low-permeability reservoirs and heavy oil reservoirs so as to enhance oil recovery.     

Two-Dimensional Regular Shock Reflection for the Pressure Gradient System of Conservation Laws

We establish the existence of a global solution to a regular reflection of a shock hitting a ramp for the pressure gradient system of equations. The set-up of the reflection is the same as that of Mach＇s experiment for the compressible Euler system, i.e., a straight shock hitting a ramp. We assume that the angle of the ramp is close to 90 degrees. The solution has a reflected bow shock wave, called the diffraction of the planar shock at the compressive corner, which is mathematically regarded as a free boundary in the self-similar variable plane. The pressure gradient system of three equations is a subsystem, and an approximation, of the full Euler system, and we offer a couple of derivations.     

Spreading of Linear Disturbances in Boundary Layers at Mach Number Five and the Effect of Wall Cooling

We determine inviscid eigensolutions in zero pressure gradient flat plate boundary layers at Mach number five and evaluate the eigensolutions with the method of steepest descent. The resulting wave packets show that with wall cooling the tail of the packets becomes slower. Although the boundary layers investigated are all convectively unstable we interpret the slow tail of the wave packets as a trend towards an absolute instability. From a comparison of the wave packets with turbulent spot transition, we conclude that for wall cooling the general transition properties are close to those of an absolutely unstable flow. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact solution of the Navier-Stokes equations for the oscillating flow in a duct of a cross-section of right-angled isosceles triangle

The Navier-Stokes equations have been solved in order to obtain an analytical solution of the fully developed laminar flow in a duct having a cross section of a right-angled, isosceles triangle. We obtained a solution for the case of oscillating pressure gradient flow. The pulsating flow is obtained by the superposition of the steady and oscillating pressure gradient solutions.     

On a generalized Stokes problem

We deal with a generalization of the Stokes system. Instead of the Laplace operator, we consider a general elliptic operator and a pressure gradient with small perturbations. We investigate the existence and uniqueness of a solution as well its regularity properties. Two types of regularity are provided. Aside from the classical Hilbert regularity, we also prove the Hölder regularity for coefficients in VMO space.     

Electroosmosis law via homogenization of electrolyte flow equations in porous media

By the homogenization approach we justify a two-scale model of ion transport in porous media for one-dimensional horizontal steady flows driven by a pressure gradient and an external horizontal electrical field. By up-scaling, the electroosmotic flow equations in horizontal nanoslits separated by thin solid layers are approximated by a homogenized system of macroscale equations in the form of the Poisson equation for induced vertical electrical field and Onsager's reciprocity relations between global fluxes (hydrodynamic and electric) and forces (horizontal pressure gradient and external electrical field). In addition, the two-scale approach provides macroscopic mobility coefficients in the Onsager relations.     

Viscous-Inviscid Interactions in a Boundary-Layer Flow Induced by a Vortex Array

In this paper, we investigate the asymptotic validity of boundary-layer theory. For a flow induced by a periodic row of point-vortices, we compare Prandtl’s boundary-layer solution to Navier-Stokes solutions with different Reynolds numbers. We show how Prandtl’s solution develops a finite-time separation singularity. On the other hand, the Navier-Stokes solutions are characterized by the presence of two distinct types of viscous-inviscid interactions that can be detected by the analysis of the enstrophy and of the pressure gradient on the wall. Moreover, we apply the complex singularity-tracking method to Prandtl and Navier-Stokes solutions and analyze the previous interactions from a different perspective.     

Effect of bulk viscosity in supersonic flow past spacecraft

In this paper, we consider the effect of bulk viscosity in various hydrodynamic problems. We numerically study this effect on the front structure of the one-dimensional stationary shock wave and on the flow past blunt body. We estimate the effect of the bulk viscosity coefficient (BVC) on the heat transfer and drag of a sphere in a supersonic flow, apparently for the first time, by the numerical solution of parabolized Navier–Stokes equations. The solution is obtained by an original fast convergent method of global iterations of the longitudinal pressure gradient. The directions of further investigations of bulk viscosity are suggested.     

Decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions

When two inclusions get closer and their conductivities degenerate to zero or infinity, the gradient of the solution to the conductivity equation blows up in general. In this paper, we show that the solution to the conductivity equation can be decomposed into two parts in an explicit form: one of them has a bounded gradient and the gradient of the other part blows up. Using the decomposition, we derive the best possible estimates for the blow-up of the gradient. We then consider the case when the inclusions have positive permittivities. We show quantitatively that in this case the size of the blow-up is reduced.     

考虑二次梯度项影响的非线性不稳定渗流问题的精确解

考虑了二次梯度项影响的非线性径向流动问题的无限大地层和有界地层渗流模型．在井底定流量和定压生产时,对无限大地层及有界地层(包括封闭和定压地层)六种情况,利用广义Weber变换和广义Hankel变换求得了实空间的解析解,分析了非线性压力解与线性压力解的差异,发现在晚时段其差异可达8%以上．因此在试井长时要考虑二次梯度项的影响．     

Exact solution for the pulsating finite gap Dean flow

Analytical solutions of the Navier–Stokes equations for the fully developed laminar flow in a cylindrical annulus, when an oscillating circumferential pressure gradient is imposed (finite gap oscillating Dean flow), are presented. The solution for the case of steady flow, which has been given by Goldstein, is obtained as a limit case of the oscillating flow when the frequency of the oscillating pressure gradient tends to zero. The pulsating flow solution is obtained by the superposition of the constant and oscillating pressure gradient solutions.     

Pointwise and L2 bounds in some nonstandard problems for the heat equation

We consider some initial-boundary value problems for the linear and nonlinear heat equation where the gradient of the solution is prescribed on the boundary. Assuming that a solution exists, we obtain bounds for the solution and its gradient by maximum principle arguments or by means of differential and integral inequalities.     

Global existence of solutions for flows of fluids with pressure and shear dependent viscosities

There is clear and incontrovertible evidence that the viscosity of many liquids depends on the pressure. While the density, as the pressure is increased by orders of magnitude, suffers small changes in its value, the viscosity changes dramatically. It can increase exponentially with pressure. In many fluids, there is also considerable evidence for the viscosity to depend on the rate of deformation through the symmetric part of the velocity gradient, and most fluids shear thin, i.e., viscosity decreases with an increase in the rate of shear. In this paper, we study the flow of fluids whose viscosity depends on both the pressure and the symmetric part of the velocity gradient. We find that the shear thinning nature of the fluid can be gainfully exploited to obtain global existence of solution, which would not be possible otherwise. Previous studies of fluids with pressure dependent viscosity require strong restrictions to all data, or assume forms that are clearly contrary to experiments, namely that the viscosity decreases with the pressure. We are able to establish existence of space periodic solutions that are global in time for both the two- and three-dimensional problem, without restricting ourselves to small data.     

Laminar mixing layer on the boundary of two flows in the presence of a longitudinal pressure gradient : PMM vol. 38, n 6, 1974, pp. 1133–1136

We consider, in a linear formulation, the problem concerning the laminar mixing layer on the boundary of two flows of an incompressible liquid with a small difference in their Bernoulli constants; we assume the presence of longitudinal pressure gradient. We determine the velocity distribution in the mixing layer, the magnitude of the displacement thickness and the momentum loss thickness. For the case in which there is no longitudinal pressure gradient we calculate the force effect of the one flow on the other.     

Optimality Conditions and Characterizations of the Solution Sets in Generalized Convex Problems and Variational Inequalities

We derive necessary and sufficient conditions for optimality of a problem with a pseudoconvex objective function, provided that a finite number of solutions are known. In particular, we see that the gradient of the objective function at every minimizer is a product of some positive function and the gradient of the objective function at another fixed minimizer. We apply this condition to provide several complete characterizations of the solution sets of set-constrained and inequality-constrained nonlinear programming problems with pseudoconvex and second-order pseudoconvex objective functions in terms of a known solution. Additionally, we characterize the solution sets of the Stampacchia and Minty variational inequalities with a pseudomonotone-star map, provided that some solution is known.     

Radially Symmetric Solutions of the <Emphasis Type="Italic">p</Emphasis>-Laplace Equation with Gradient Terms

We consider the Dirichlet problem for the p-Laplace equation with nonlinear gradient terms. In particular, these gradient terms cannot satisfy the Bernstein—Nagumo conditions. We obtain some sufficient conditions that guarantee the existence of a global bounded radially symmetric solution without any restrictions on the growth of the gradient term. Also we present some conditions on the function simulating the mass forces, which allow us to obtain a bounded radially symmetric solution under presence of an arbitrary nonlinear source.     

Well-posedness of the free boundary problem in compressible elastodynamics

We study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid. At the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain. We prove the local-in-time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non-collinear vectors at each point of the initial free boundary. If this non-collinearity condition fails, the local-in-time existence is proved under the classical Rayleigh–Taylor sign condition satisfied at the first moment. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh–Taylor sign condition leads to Rayleigh–Taylor instability.     

Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by the patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space. We derive error estimates for all unknowns under both $L^2$ norms and energy norms. Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.     

A generalized collapsing sandpile model

In this paper, we introduce a new model for the collapsing sandpile and we prove existence and uniqueness of a solution for the corresponding initial value problem. Moreover, we prove the convergence of the time-stepping approximation of the solution. We use subgradient flows for variational problems with time dependent gradient constraints. These gradient constraints are interpreted as the critical angles of the sandpile. In particular, our model produces an evolution in time of avalanches in a drying of a sandpile, rather than instantaneous collapse.