A Venkatlaxmi, B S Padmavathi, T Amaranath
The completeness of solutions of homogeneous as well as non-homogeneous unsteady Stokes equations are examined. A necessary and sufficient condition for a divergence-free vector to represent the velocity field of a possible unsteady Stokes flow in the absence of body forces is derived.
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In this paper we study the deterministic homogenization problems for unsteady Navier-Stokes type equations, on one hand in an open set Ω of R^{N}, on the other hand in porous media Ω^{ε}. In the second case, the equations are classical unsteady Navier-Stokes one, and the porous media are periodic
In this paper we study the homogenization of unsteady Stokes type equations in the periodic setting. The usual Laplace operator involved in the classical Stokes equations is here replaced by a linear elliptic differential operator of divergence form with periodically oscillating coefficients. Our mean tool is the well known two-scale convergence method.
A Runge-type approximation theorem for the 3D unsteady Stokes system
本文证明了三维非定常Stokes系统在具有连通补集的紧集上，局部光滑解可被速度至多指数增长、压强多项式增长的全局解以任意精度逼近，并通过寄生解说明增长条件的必要性。
Homogenization of the unsteady compressible Navier-Stokes equations for adiabatic exponent $γ>3$
本文研究了穿孔三维域中非定常可压缩 Navier-Stokes 方程的均匀化，通过两种限制算子获得压力估计，将绝热指数下界从 $γ>6$ 降至 $γ>3$，并推广至 Navier-Stokes-Fourier 系统及讨论了多维结果的最优性。
Consider the Navier-Stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity $-h(t)u_\infty$ with constant vector $u_\infty\in \mathbb R^3\setminus\{0\}$. Finn raised the question whether his steady slutions are attainable as limits for $t\to\infty$ of unsteady solutions starting from motionless state when $h(t)=1$ after some finite time and $h(0)=0$ (starting problem). This was affirmatively solved by Galdi, Heywood and Shibata for small $u_\infty$. We study some generalized situation in which unsteady solutions start from large m…
Motivated by extrusion problems, we consider a non-stationary incompress-ible 3D fluid flow with a non-constant (temperature dependent) viscosity, subjected to mixed boundary conditions with a given time dependent velocity on a part of the boundary and Tresca's friction law on the other part. We construct a sequence of approximate solutions by using a regularization of the free boundary condition due to friction combined with a particular penalty method, reminiscent of the " incompressibility limit " of compressible fluids, allowing to get better insights into the links between the fluid veloc…
We study the diffusion-reaction-advection model for mobile chemical species together with the dissolution and precipitation of immobile species in a porous medium at the micro-scale. This leads to a system of semilinear parabolic partial differential equations in the pore space coupled with a nonlinear ordinary differential equation at the grain boundary of the solid matrices. The fluid flow within the pore space is given by unsteady Stokes equation. The novelty of this work is to do the iterative limit analysis of the system by tackling the nonlinear terms, monotone multi-valued dissolution r…
A New approach for the unsteady Stokes equations with time fractional derivative in Bounded Domains
针对有界域上含时间Caputo分数阶导数的非定常Stokes方程，本文构建适配分数阶场景的新型函数空间与变分框架，实现广义Galerkin格式，拓展经典理论并为相关分数阶偏微分方程研究提供基础。
We prove that there exists a weak solution to a system governing an unsteady flow of a viscoelastic fluid in three dimensions, for arbitrarily large time interval and data. The fluid is described by the incompressible Navier-Stokes equations for the velocity $v$, coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus models for a tensor $\mathbb{B}$. By a proper choice of the constitutive relations for the Helmholtz free energy (which, however, is non-standard in the current literature, despite the fact that this choice is well motivated from the point of view of p…
We study a system of nonlinear partial differential equations describing the unsteady motions of incompressible chemically reacting non-Newtonian fluids. The system under consideration consists of the generalized Navier-Stokes equations with a power-law type stress-strain relation, where the power-law index depends on the concentration of a chemical, coupled to a convection-diffusion equation for the concentration. This system of equations arises in the rheology of the synovial fluid found in the cavities of synovial joints. We prove the existence of global weak solutions of the non-stationary…
In this paper, we study the well-posedness of classical solutions to the nonlinear unsteady Prandtl equations with Robin boundary condition in half space in weighted Sobolev spaces. We firstly investigate the monotonic shear flow with Robin boundary condition and the linearized Prandtl-type equations with Robin boundary condition in weighted Sobolev spaces. Due to the degeneracy of the Prandtl equations and the loss of regularity, we apply the Nash-Moser-Hormander iteration scheme to prove the existence of classical solutions to the nonlinear Prandtl equations with Robin boundary condition whe…
We show the $H^1$ stability of shear flows of Prandtl type: $U^ν= (U_s(y/\sqrtν),0)$, in the steady two-dimensional Navier-Stokes equations, under the natural assumptions that $U_s(Y) > 0$ for $Y > 0$, $U_s(0) = 0$, and $U_s'(0) > 0$. Our result is in sharp contrast with the unsteady ones, in which at most Gevrey stability can be obtained, even under global monotonicity and concavity hypotheses. It provides the first positive answer to the inviscid limit problem in Sobolev regularity for a non-trivial class of steady Navier-Stokes flows with no-slip boundary condition.
Following the previous part of our study on unsteady non-New\-to\-nian fluid flows with boundary conditions of friction type we consider in this paper the case of pseudo-plastic (shear thinning) fluids. The problem is described by a $p$-Laplacian non-stationary Stokes system with $p<2$ and we assume that the fluid is subjected to mixed boundary conditions, namely non-homogeneous Dirichlet boundary conditions on a part of the boundary and a slip fluid-solid interface law of friction type on another part of the boundary. Hence the fluid velocity should belong to a subspace of $L^p \bigl(0,T; (W^…
We compute the solutions of Prandtl's and Navier-Stokes equations for the two dimensional flow induced by a rectilinear vortex interacting with a boundary in the half plane. For this initial datum Prandtl's equation develops, in a finite time, a separation singularity. We investigate the different stages of unsteady separation for Navier-Stokes solution at different Reynolds numbers $Re=10^3-10^5$, and we show the presence of a large-scale interaction between the viscous boundary layer and the inviscid outer flow. We also see a subsequent stage, characterized by the presence of a small-scale i…
On the existence of strong solutions for unsteady motions of incompressible chemically reacting generalized Newtonian fluids
该研究针对带化学反应的不可压缩广义牛顿流体非定常运动的偏微分方程组，证明了二维、三维周期域上满足幂律指数条件时全局强解的存在性，并给出了解唯一的幂律指数上下界条件。
On the local well-posedness of strong solutions to the unsteady flows of shear-thinning non-Newtonian fluids with a concentration-dependent power-law index
本文研究具有浓度依赖幂律指数的剪切变稀非牛顿流体非定常流动模型，在三维周期域中证明了局部时间强解的存在性，并在附加条件下证明了解的唯一性，同时将结果推广到二维情形。
On unsteady internal flows of incompressible fluids characterized by implicit constitutive equations in the bulk and on the boundary
本文针对三维不可压缩流体非定常内部流动，建立体内应力与速度梯度、边界切向速度与法向牵引均为隐式关系时弱解存在性的构造性理论，并讨论唯一性。
We study well-posedness and asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) full von Karman shallow shell equations that accounts for both transversal and lateral displacements on a flexible part of the boundary. We also take into account rotational inertia of filaments of the shell. Out main result shows that the problem generates a semiflow in an appropriate phase space. The regularity provided by viscous dissipation in the fluid allows us to consider simultaneously both cases of presence inertia in …
A mathematical analysis of local and nonlocal phase-field models of tumor growth is presented that includes time-dependent Darcy-Forchheimer-Brinkman models of convective velocity fields and models of long-range cell interactions. A complete existence analysis is provided. In addition, a parameter-sensitivity analysis is described that quantifies the sensitivity of key quantities of interest to changes in parameter values. Two sensitivity analyses are examined; one employing statistical variances of model outputs and another employing the notion of active subspaces based on existing observatio…
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in $\R^n$ with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure $p$ can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolut…
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