Existence and uniqueness of strong–weak solutions for chemically reacting generalized second grade fluids in 2 space dimensions

The present paper is devoted to the analysis of a nonlinear system modeling unsteady flows of an incompressible non‐Newtonian fluid mixed with a reactant. We are interested on generalized second grade fluids, which are chemically reacting and whose viscosity depends both on the shear‐rate and the concentration. We prove existence and uniqueness of strong–weak solution for a flow filling in the plane and subject to space periodic boundary conditions. This result is established under the fulfillment of some assumptions on the viscosity stress tensor and the flux vector of the diffusion–convection equation reflecting the chemical reaction. Copyright © 2016 John Wiley & Sons, Ltd.     

On Fully Developed Flows of Fluids with a Pressure Dependent Viscosity in a Pipe

Stokes recognized that the viscosity of a fluid can depend on the normal stress and that in certain flows such as flows in a pipe or in channels under normal conditions, this dependence can be neglected. However, there are many other flows, which have technological significance, where the dependence of the viscosity on the pressure cannot be neglected. Numerous experimental studies have unequivocally shown that the viscosity depends on the pressure, and that this dependence can be quite strong, depending on the flow conditions. However, there have been few analytical studies that address the flows of such fluids despite their relevance to technological applications such as elastohydrodynamics. Here, we study the flow of such fluids in a pipe under sufficiently high pressures wherein the viscosity depends on the pressure, and establish an explicit exact solution for the problem. Unlike the classical Navier-Stokes solution, we find the solutions can exhibit a structure that varies all the way from a plug-like flow to a sharp profile that is essentially two intersecting lines (like a rotated V). We also show that unlike in the case of a Navier-Stokes fluid, the pressure depends both on the radial and the axial coordinates of the pipe, logarithmically in the radial coordinate and exponentially in the axial coordinate. Exact solutions such as those established in this paper serve a dual purpose, not only do they offer solutions that are transparent and provide the solution to a specific but simple boundary value problems, but they can be used also to test complex numerical schemes used to study technologically significant problems.     

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.     

Global solutions of real compressible reactive gas with density-dependent viscosity and self-gravitation for higher-order kinetics

A mathematical model for viscous, real, compressible, reactive fluid flows is considered. The existence of global solutions for the free boundary problem with species diffusion in dynamic combustion is established when the viscosity λ depends on the density i.e., λ(ρ)=Aρα (), where A is a generic positive constant. Furthermore, the equations of state depend nonlinearly on density and temperature unlike the case of perfect gases or radiative flows. In addition, the shock wave, turbulence, vacuum, mass concentration or extremely hot spot will not be developed in any finite time if the initial data do not contain vacuum.     

Local Strong Solutions for the Compressible Non- Newtonian Models with Density-Dependent Viscosity and Vacuum

The one-dimensional compressible non-Newtonian models are considered in this paper. The extra-stress tensor in our models satisfies a kind of power law structure which was proposed by O. A. Ladyzhenskaya in 1970s. In particular, the viscosity coefficient in our models depends on the density. By using energy-estimate, the authors obtain the existence and uniqueness of local strong solutions for which the density is non-negative.     

Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity

In this paper, we consider an initial boundary value problem for the 3‐dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the L² norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed. Finally, we revisit the Navier‐Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic‐energy strong solution exists globally in time, which extends the recent result of Huang and Wang.     

Blow-up of viscous compressible reactive self-gravitating gas

In this paper,we prove some results concerning blow-up of viscous compressible reactive (selfgravitating) flows when the initial density is compactly supported and the other initial value satisfy proper conditions.It extends the work of Xin and Cho to the case of viscous compressible reactive self-gravitating flows equations.We control the lower bound of second moment by total energy and obtain the precise relationship between the size of the support of initial density and the existence time.     

A blow-up criterion for 3-D non-resistive compressible heat-conductive magnetohydrodynamic equations with initial vacuum

In this paper, we prove a blow-up criterion of strong solutions to the 3-D viscous and non-resistive magnetohydrodynamic equations for compressible heat-conducting flows with initial vacuum. This blow-up criterion depends only on the gradient of velocity and the temperature, which is similar to the one for compressible Navier-Stokes equations.     

Blowup of smooth solution for non‐isentropic magnetohydrodynamic equations without heat conductivity

In this paper, we establish a blow‐up criterion for the three‐dimentional viscous, compressible magnetohydrodynamic flows. It is shown that for the Cauchy problem and the initial‐boundary‐value problem with initial density allowed to vanish, the strong or smooth solution for the three‐dimentional magnetohydrodynamic flows exists globally if the density, temperature, and magnetic field is bounded from above. Copyright © 2016 John Wiley & Sons, Ltd.     

On the Regularity of Shear Thickening Viscous Fluids

The aim of this note is to improve the regularity results obtained by H. Beirao da Veiga in 2008 for a class of p-fluid flows in a cubic domain. The key idea is exploiting the better regularity of solutions in the tangential directions with respect to the normal one, by appealing to anisotropic Sobolev embeddings.     

A boundary value problem for the spherically symmetric motion of a pressureless gas with a temperature‐dependent viscosity

We consider an initial‐boundary value problem for the equations of spherically symmetric motion of a pressureless gas with temperature‐dependent viscosity µ(θ) and conductivity κ(θ). We prove that this problem admits a unique weak solution, assuming Belov's functional relation between µ(θ) and κ(θ) and we give the behaviour of the solution for large times. Copyright © 2009 John Wiley & Sons, Ltd.     

Free boundary value problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in this paper. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero also at an algebraic time-rate as the time tends to infinity.     

On Regularity of Incompressible Fluid with Shear Dependent Viscosity

The authors consider a non-Newtonian fluid governed by equations with p-structure in a cubic domain.A fluid is said to be shear thinning(or pseudo-plastic) if 1 < p < 2,and shear thickening(or dilatant) if p > 2.The case p > 2 is considered in this paper.To improve the regularity results obtained by Crispo,it is shown that the secondorder derivatives of the velocity and the first-order derivative of the pressure belong to suitable spaces,by appealing to anisotropic Sobolev embeddings.     

A blow-up criterion for two dimensional compressible viscous heat-conductive flows

We establish a blow-up criterion in terms of the upper bound of the density and temperature for the strong solution to 2D compressible viscous heat-conductive flows. The initial vacuum is allowed.     

Non‐linear stability for convection with quadratic temperature dependent viscosity

In this paper, we study the non‐linear stability of convection for a Newtonian fluid heated from below, where the viscosity of the fluid depends upon temperature. We are able to show that for Rayleigh numbers below a certain critical value, Ra_c, the rest state of the fluid and the steady temperature distribution remains non‐linearly stable, using the calculations of Diaz and Straughan (Continuum Mech. Thermodyn. 2004; 16 :347–352). The central contribution of this paper lies in a simpler proof of non‐linear stability, than the ones in the current literature, by use of a suitable maximum principle argument. Copyright © 2006 John Wiley & Sons, Ltd.     

Local strong solutions to a compressible non‐Newtonian fluid with density‐dependent viscosity

We consider the initial boundary problem for a compressible non‐Newtonian fluid with density‐dependent viscosity. The local existence of strong solution is established that is based on some compatibility condition. Moreover, it is also proved that the solutions are to blow up, and the maximum norm of velocity gradients controls the possible break down of the strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Melting heat and mass transfer in stagnation point micropolar fluid flow of temperature dependent fluid viscosity and thermal conductivity at constant vortex viscosity

Steady mixed convection micropolar fluid flow towards stagnation point formed on horizontal linearly stretchable melting surface is studied. The vortex viscosity of micropolar fluid along a melting surface is proposed as a constant function of temperature while dynamic viscosity and thermal conductivity are temperature dependent due to the influence of internal heat source on the fluid. Similarity transformations were used to convert the governing equation into non-linear ODE and solved numerically. A parametric study is conducted. An analysis of the results obtained shows that the flow-field is influenced appreciably by heat source, melting, velocity ratio, variable viscosity and thermal conductivity.     

Geometrical methods for analyzing the optimal management of tipping point dynamics

Natural resources are not infinitely resilient and should not be modeled as being such. Finitely resilient resources feature tipping points and history dependence. This paper provides a didactical discussion of mathematical methods that are needed to understand the optimal management of such resources: viscosity solutions of Hamilton–Jacobi–Bellman equations, the costate equation and the associated canonical equations, exact root counting, and geometrical methods to analyze the geometry of the invariant manifolds of the canonical equations. Recommendations for Resource Managers

• Management of natural resources has to take into account the possible breakdown of resilience and induced regime shifts.
• Depending on the characteristics of the resource and on its present and future economic importance, either for all initial states the same kind of management policy is optimal, or the type of the optimal management policy depends on the initial state.
• Modeling should reflect the finiteness of the data.

     

Three-dimensional numerical simulation of the inverse problem of thermal convection using the quasi-reversibility method

Inverse (time-reverse) simulation of three-dimensional thermoconvective flows is considered for a highly viscous incompressible fluid with temperature-dependent density and viscosity. The model of the fluid dynamics is described by the Stokes equations, the incompressibility and heat balance equations subject to the appropriate initial and boundary conditions. To solve the problem backward in time, the quasi-reversibility method is applied to the heat balance equation. The numerical solution is based on the introduction of a two-component vector potential for the velocity of the medium, on the application of the finite element method with a special tricubic spline basis for computing this potential, and on the application of the splitting method and the method of characteristics for computing the temperature. The numerical algorithm is designed to be executed on parallel computers. The proposed numerical algorithm is used to reconstruct the evolution of diapiric structures in the Earth’s upper mantle. The computational efficiency of the algorithm is analyzed on the basis of the appropriate functionals of residuals.     

Global classical solution to 1D compressible Navier–Stokes equations with no vacuum at infinity

C. Miao In this paper, we are concerned with the 1D Cauchy problem of the compressible Navier–Stokes equations with the viscosity μ(ρ) = 1+ρ^β(β≥0). The initial density can be arbitrarily large and keep a non‐vacuum state at far fields. We will establish the global existence of the classical solution for 0≤β < γ via a priori estimates when the initial density contains vacuum in interior interval or is away from the vacuum. We will show that the solution will not develop vacuum in any finite time if the initial density is away from the vacuum. To study the well‐posedness of the problem, it is crucial to obtain the upper bound of the density. Some new weighted estimates are applied to obtain our main results. Copyright © 2015 John Wiley & Sons, Ltd.