Steady flow for incompressible fluids in domains with unbounded curved channels

We give an overview on the solution of the stationary Navier-Stokes equations for non newtonian incompressible fluids established by G. Dias and M.M. Santos (Steady flow for shear thickening fluids with arbitrary fluxes, J. Differential Equations 252 (2012), no. 6, 3873-3898), propose a definition for domains with unbounded curved channels which encompasses domains with an unbounded boundary, domains with nozzles, and domains with a boundary being a punctured surface, and argue on the existence of steady flowfor incompressible fluids with arbitrary fluxes in such domains.     

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

We provide a thermodynamic basis for the development of models that are usually referred to as ??phase-field models?? for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive ??phase-field models?? both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631?C651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier?CStokes?CFourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier?CStokes?CFourier fluid. As observed earlier in Heida and Málek (Int J Eng Sci 48(11):1313?C1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn?CHilliard?CNavier?CStokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn?CHilliard?CNavier?CStokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).     

Comparative study of some constitutive equations characterising non-Newtonian fluids

This paper compares, in a general way, the predictions of the constitutive equations given by Rivlin and Ericksen, Oldroyd, and Walters. Whether we consider the rotational problems in cylindrical co-ordinates or in spherical polar co-ordinates, the effect of the non-Newtonicity on the secondary flows is collected in a single parameterα which can be explicitly expressed in terms of the non-Newtonian parameters that occur in each of the above-mentioned constitutive equations. Thus, for a given value ofα, all the three fluids will have identical secondary flows. It is only through the study of appropriate normal stresses that a Rivlin-Ericksen fluid can be distinguished from the other two fluids which are indistinguishable as long as this non-Newtonian parameter has the same value.     

Restrictions placed on constitutive relations by angular momentum balance and Galilean invariance

In this note, we will show that for describing the response of a wide class of bodies, it is sufficient to invoke only the balance of angular momentum to obtain the restrictions on the constitutive functions that one obtains by appealing to frame indifference. While this result is known for hyperelastic materials (although it is not found in any standard text on the subject), we extend this result to classes of elasto-plastic and viscoelastic materials as well as for a class of implicit constitutive equations for viscous fluids. In particular, we show that for a class of bodies capable of instantaneous elastic response that is dictated by a stored energy function, the symmetry of the Cauchy stress alone is enough to obtain all the necessary restrictions. The result is related to Noether’s theorem; if we know that there is a conserved quantity (i.e., angular momentum), we can then show that the energy function must be invariant under a group of transformations. For a class of generalized Newtonian fluids (including the Navier Stokes fluid and the Bingham fluid), the symmetry of the stress and Galilean invariance of the response functions are all that are required to obtain restrictions that are usually obtained by enforcing frame indifference.     

On the Limit as the Density Ratio Tends to Zero for Two Perfect Incompressible Fluids Separated by a Surface of Discontinuity

We study the asymptotic limit as the density ratio ρ^?/ρ⁺ → 0, where ρ⁺ and ρ^? are the densities of two perfect incompressible 2-D/3-D fluids, separated by a surface of discontinuity along which the pressure jump is proportional to the mean curvature of the moving surface. Mathematically, the fluid motion is governed by the two-phase incompressible Euler equations with vortex sheet data. By rescaling, we assume the density ρ⁺ of the inner fluid is fixed, while the density ρ^? of the outer fluid is set to ε. We prove that solutions of the free-boundary Euler equations in vacuum are obtained in the limit as ε → 0.     

On boundary layer flow of a Sisko fluid over a stretching sheet

Abstract

In this paper, the steady boundary layer flow of a non-Newtonian fluid over a nonlinear stretching sheet is investigated. The Sisko fluid model, which is combination of power-law and Newtonian fluids in which the fluid may exhibit shear thinning/thickening behaviors, is considered. The boundary layer equations are derived for the two-dimensional flow of an incompressible Sisko fluid. Similarity transformations are used to reduce the governing nonlinear equations and then solved analytically using the homotopy analysis method. In addition, closed form exact analytical solutions are provided for n = 0 and n = 1. Effects of the pertinent parameters on the boundary layer flow are shown and solutions are contrasted with the power-law fluid solutions.     

Modeling bodies that can only undergo isochoric motions subject to mechanical stimuli but are compressible or expansible with respect to thermal stimuli

In this short note, we discuss a new constitutive approach to describing the response of bodies, both solid and fluid, that can only undergo isochoric motions in isothermal processes but which can undergo non-isochoric motions in arbitrary processes. Within this new framework, one finds that conditions that were perceived as constraints on the response of the body now arise naturally within the frame work of the constitutive definition of these bodies. For instance, a central approximation in fluid mechanics that is of great utility in the analysis of fluid flow problems in geophysics and astrophysics is that due to Oberbeck (Ann Phys Chem 1:271–292, 1879; Uber die bewengungsercheinungen der Atmosphare, Sitz Ber K Preuss Akad Miss, pp 383–395, 1129–1138, 1888) and Boussinesq (Theorie Analytique de la Chaleur. Gauthier-Villas, Paris, 1903) for describing the flow of fluids, which can undergo only isochoric motions in isothermal processes but which are otherwise capable of non-isochoric motions. A similar demand can be made concerning the response of solid bodies wherein one could carry out an approximation similar to that of the Oberbeck–Boussinesq equations, and such an approach might be of great value in the study of technologically relevant problems.     

On stationary flows with energy dependent nonlocal viscosities

A nonlocal constitutive law for an incompressible viscous flow in which the viscosity depends on the total dissipation energy of the fluid is obtained as the limit case of very large thermal conductivity when the viscosity varies with the temperature. A rigorous analysis is illustrated within the Hilbertian framework for unidirectional stationary flows of Newtonian and Bingham fluids with heating by viscous dissipation. An extension to quasi-Newtonian fluids of power law type and with temperature dependent viscosities is obtained in the context of the heat equation with an L¹-term. The nonlocal model proposed by Ladyzhenskaya in 1966 as a modification of Navier-Stokes equations can be, in particular, obtained with this procedure. Bibliography: 14 titles.Dedicated to O. A. Ladyzhenskaya on the occasion of her 80th birthday__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 295, 2003, pp. 99–117.     

ALMOST OPTIMAL LOCAL WELL-POSEDNESS OF THE MAXWELL-KLEIN-GORDON EQUATIONS IN 1 + 4 DIMENSIONS

Abstract

We study strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids in Ω ? R ³. Deriving higher a priori estimates independent of the lower bounds of the density, we prove the existence and uniqueness of local strong solutions to the initial value problem (for Ω =R ³) or the initial boundary value problem (for Ω ? ? R ³) even though the initial density vanishes in an open subset of Ω, i.e., an initial vacuum exists. As an immediate consequence of the a priori estimates, we obtain a continuation theorem for the local strong solutions.     

电场驱动直射流的动量方程

给出了电场驱动直射流的一维动量守恒方程．该方程是用应力分量表示的,适用于任何流体本构关系,只要流体是不可压缩的．结果显示,为了使方程封闭,需要沿轴向和径向两个方向的本构关系．然而,当附加应力张量的迹为0时,只需要沿轴向的一个本构关系就足够了．还发现,射流的第二主应力差的零阶近似总为0．与其他类型的动量方程做了比较．     

On the sharp singular limit for slightly compressible fluids

We consider the equations of motion to slightly compressible fluids and we prove that solutions converge, in the strong norm, to the solution of the equations of motion of incompressible fluids, as the Mach number goes to zero. From a physical point of view this means the following. Assume that we are dealing with a well-specified fluid, so slightly compressible that we assume it to be incompressible. Our result means that the distance between the (continuous) trajectories of the real and of the idealized solution is ‘small’ with respect to the natural metric, i.e. the metric that endows the data space.     

The Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid

The velocity field corresponding to the Rayleigh–Stokes problem for an edge, in an incompressible generalized Oldroyd-B fluid has been established by means of the double Fourier sine and Laplace transforms. The fractional calculus approach is used in the constitutive relationship of the fluid model. The obtained solution, written in terms of the generalized G-functions, is presented as a sum of the Newtonian solution and the corresponding non-Newtonian contribution. The solution for generalized Maxwell fluids, as well as those for ordinary Maxwell and Oldroyd-B fluids, performing the same motion, is obtained as a limiting case of the present solution. This solution can be also specialized to give the similar solution for generalized second grade fluids. However, for simplicity, a new and simpler exact solution is established for these fluids. For β → 1, this last solution reduces to a previous solution obtained by a different technique.      

Fluid Flow Equations with Variable Viscosity in Quaternionic Setting

For a large class of fluids the relation between shear stress and shear velocity is not longer a constant. The viscosity μ is now a function which depends on the position, the time and the shear-velocity. In our paper we will deal with a class of fluids with variable viscosity functions which correspond to fluid flow equations that permit a representation of the solution by the aid of a quaternionic operator calculus.     

Semi‐strong and strong solutions for variable density asymmetric fluids in unbounded domains

We establish the existence of local in time semi‐strong solutions and global in time strong solutions for the system of equations describing flows of viscous and incompressible asymmetric fluids with variable density in general three‐dimensional domains with boundary uniformly of class C³. Under suitable assumptions, uniqueness of local semi‐strong solutions is also proved. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence locale et unicité d'écoulements de fluides viscoélastiques dans des domaines non bornés

In this Note, we present a result of local existence and uniqueness, for any initial data, of the solutions to the equations of viscoelastic fluids of Jeffreys type (differential constitutive law). The system of equations is supposed to be verified in an unbounded domain Ω ⊂ ℝ^N (N = 2 or 3)), uniformly regular. The difficulty comes essentially from the loss of compactness in the case of unbounded domains. To overcome this difficulty we use a local compactness method, which allows us to construct a sequence of solutions on subdomains ;inn whi_n which union covers Ω After that, we pass to the limit to define a solution over the whole domain. Finally we show the uniqueness of this solution in its class of regularity, by using an energy estimate.     

The zero limits of angular and micro-rotational viscosities for the two-dimensional micropolar fluid equations with boundary effect

This paper deals with an initial-boundary value problem to the two-dimensional equations of incompressible micropolar fluids. We first prove that as the angular and micro-rotational viscosities go to zero (i.e., \({\gamma, \zeta \to 0}\) ), the solution converges to a global weak solution of the original equations with zero angular and micro-rotational viscosities. Convergence rates are also obtained. Then, we study the boundary effects and prove that a boundary-layer thickness is of the value \({\delta(\gamma) = \gamma^\alpha}\) with \({\alpha \in (0, 1/2)}\) , provided \({\lim_{\gamma \to 0} \zeta \gamma^{1/2} < \infty}\) .     

Two-layer flows of generalized immiscible second grade fluids in a rectangular channel

Unsteady one-dimensional flows of two incompressible and immiscible generalized second grade fluids in a rectangular channel are studied. A constant pressure gradient acts in the flow direction, while the channel walls have oscillating translational motions in their planes. The generalization considered in this paper consists into a mathematical model based on constitutive equations of second grade fluid with Caputo time-fractional derivative in which the history of the shear stress influences the velocity gradient. The velocity and shear stress fields in the Laplace transform domain are obtained. Numerical solutions for the real velocity and shear stress have been found by employing the Stehfest numerical algorithm for the inverse Laplace transform. The influence of the fractional parameters on the velocity and shear stress has been studied by numerical simulations and graphical illustrations. It is found that the memory effects are significant only for small values of the time t.     

High Accuracy Numerical Simulation of Non-Isothermal Viscoelastic Polymer Fluid Past a Cylinder北大核心CSCD

采用同位网格有限体积(coupled and linked equations algorithm revised,CLEAR)方法求解黏性和XPP (eXtended Pom-Pom)黏弹性流动的控制方程,基于延时修正方法构造了动量和本构方程对流项的高精度AVLsmart格式。首先,为了验证该文方法的有效性,对不同Reynolds数下不可压黏性流体圆柱绕流问题进行了模拟。随后,对等温及非等温不可压XPP黏弹性流体圆柱绕流问题进行了有效模拟,给出了速度矢量、应力分量、拉升量以及温度的分布规律,分析了We数对水平速度、法向应力及拉升量的影响。该文研究成果能为精确预测复杂型腔纤维增强黏弹性聚合物熔体动态充填过程提供理论基础。     

Navier–Stokes Equations on Weighted Graphs

Navier–Stokes equations arise in the study of incompressible fluid mechanics, star movement inside a galaxy, dynamics of airplane wings, etc. In the case of Newtonian incompressible fluids, we propose an adaptation of such equations to finite connected weighted graphs such that it produces an ordinary differential equation with solutions contained in a linear subspace, this subspace corresponding to the Newtonian conservation law. We discuss the particular case when the graph is the complete graph K _m , with constant weight, and provide a necessary and sufficient condition for it to have solutions.     

Existence of weak solutions for the generalized Navier–Stokes equations with damping

In this work we consider the generalized Navier–Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier–Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any ${q > \frac{2N}{N+2}}$ and any σ > 1, where q is the exponent of the diffusion term and σ is the exponent which characterizes the damping term.