scholarly journals Finite Element Approximation of Steady Flows of Incompressible Fluids with Implicit Power-Law-Like Rheology

2013 ◽  
Vol 51 (2) ◽  
pp. 984-1015 ◽  
Author(s):  
Lars Diening ◽  
Christian Kreuzer ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Power Law ◽  
Finite Element Approximation ◽  
Incompressible Fluids ◽  
Element Approximation ◽  
Steady Flows ◽  
Implicit Power

scholarly journals Finite element approximation of an incompressible chemically reacting non-Newtonian fluid

2018 ◽  
Vol 52 (2) ◽  
pp. 509-541 ◽  
Author(s):  
Seungchan Ko ◽  
Petra Pustějovská ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Newtonian Fluid ◽  
Power Law ◽  
Nonlinear Partial Differential Equations ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Partial Differential ◽  
Chemically Reacting

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 Bogovskiĭ 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.

On steady flows of incompressible fluids with implicit power-law-like rheology

2009 ◽  
Vol 2 (2) ◽  
Author(s):  
Miroslav Bulíček ◽  
Piotr Gwiazda ◽  
Josef Málek ◽  
Agnieszka Świerczewska-Gwiazda
Keyword(s):  
Power Law ◽  
Incompressible Fluids ◽  
Steady Flows ◽  
Implicit Power

scholarly journals Finite element analysis of the stationary power-law Stokes equations driven by friction boundary conditions

2015 ◽  
Vol 23 (1) ◽  
Author(s):  
Jules K. Djoko ◽  
Mohamed Mbehou
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Power Law ◽  
Stokes Equations ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Mixed Variational Inequality ◽  
Element Analysis ◽  
Slip Boundary ◽  
Nonlinear Slip Boundary Conditions

AbstractIn this work, we are concerned with the finite element approximation for the stationary power law Stokes equations driven by nonlinear slip boundary conditions of ‘friction type’. After the formulation of the problem as mixed variational inequality of second kind, it is shown by application of a variant of Babuska-Brezzi’s theory for mixed problems that convergence of the finite element approximation is achieved with classical assumptions on the regularity of the weak solution. Next, solution algorithm for the mixed variational problem is presented and analyzed in details. Finally, numerical simulations that validate the theoretical findings are exhibited.

An Angular Finite Element Method for the Solution of the Even-Parity Equation

2009 ◽  
Author(s):  
R. Becker ◽  
R. Koch ◽  
M. F. Modest ◽  
H.-J. Bauer
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Basis Functions ◽  
New Method ◽  
Discrete Ordinates ◽  
Test Case ◽  
Element Approximation ◽  
Promising Alternative ◽  
Element Discretization ◽  
Spatial Coordinates

The present article introduces a new method to solve the radiative transfer equation (RTE). First, a finite element discretization of the solid angle dependence is derived, wherein the coefficients of the finite element approximation are functions of the spatial coordinates. The angular basis functions are defined according to finite element principles on subdivisions of the octahedron. In a second step, these spatially dependent coefficients are discretized by spatial finite elements. This approach is very attractive, since it provides a concise derivation for approximations of the angular dependence with an arbitrary number of angular nodes. In addition, the usage of high-order angular basis functions is straightforward. In the current paper the governing equations are first derived independently of the actual angular approximation. Then, the design principles for the angular mesh are discussed and the parameterization of the piecewise angular basis functions is derived. In the following, the method is applied to two-dimensional test cases which are commonly used for the validation of approximation methods of the RTE. The results reveal that the proposed method is a promising alternative to the well-established practices like the Discrete Ordinates Method (DOM) and provides highly accurate approximations. A test case known to exhibit the ray effect in the DOM verifies the ability of the new method to avoid ray effects.

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