Finite element approximation of an incompressible chemically reacting non-Newtonian fluid
Finite element approximation of an incompressible chemically reacting non-Newtonian fluid
复制标题
不可压缩化学反应非牛顿流体的有限元近似
DOI：
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发表时间：
2017
期刊：
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通讯作者：
E. Suli
中科院分区：
文献类型：
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作者：
Seungchan Ko;Petra Pustejovsk'a;E. Suli
我们考虑一个非线性偏微分方程组，模拟不可压缩非牛顿流体的稳态运动，该流体具有化学反应。控制系统由浓度的定常对流扩散方程和广义定常Navier-Stokes方程组成，其中粘性系数是剪切速率的幂律型函数，方程之间的耦合是由幂律指数的浓度依赖性引起的。这个系统的非线性偏微分方程出现在数学模型中的滑液中发现的空腔的运动关节。我们构造了一个有限元近似的模型，并在两个空间维的情况下进行数值方法的数学分析。关键的技术工具包括离散对应的Bogovski\u{\i}算子，De Giorgi的正则性定理在两个维度上，和Acerbi-Fusco Lipschitz截断的Sobolev函数，在函数空间与可变的可积指数。
We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier-Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method in the case of two space dimensions. Key technical tools include discrete counterparts of the Bogovski\u{\i} operator, De Giorgi's regularity theorem in two dimensions, and the Acerbi-Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.