This work presents the numerical solution of laminar incompressible viscous flow in a three dimensional branching channel with circular cross section for generalized Newtonian fluids. This model can be generalized by cross model in shear thinning meaning. The governing system of equations is based on the system of balance laws for mass and momentum. Numerical tests are performed on a three dimensional geometry, the branching channel with one entrance and two outlet parts. Numerical solution of the...

The aim of this paper is to describe the numerical results of numerical modelling of steady flows of laminar incompressible viscous and viscoelastic fluids. The mathematical models are Newtonian and Oldroyd-B models. Both models can be generalized by cross model in shear thinning meaning. Numerical tests are performed on three dimensional geometry, a branched channel with one entrance and two output parts. Numerical solution of the described models is based on cell-centered finite volume method...

This work deals with the numerical solution of generalized Newtonian and Oldroyd-B fluids flow. The governing system of equations is based on the system of balance laws for mass and momentum for incompressible laminar viscous and viscoelastic fluids. Two different definition of the stress tensor are considered. For viscous case Newtonian model is used. For the viscoelastic case Oldroyd-B model is tested. Both presented models can be generalized. In this case the viscosity is defined as a shear rate...

The aim of this work is to present numerical results of non-Newtonian fluid flow in a model of bypass. Different angle of a connection between narrowed channel and the bypass graft is considered. Several rheology viscosity models were used for the non-Newtonian fluid, namely the modified Cross model and the Carreau-Yasuda model. The results of non-Newtonian fluid flow are compared to the results of Newtonian fluid. The fundamental system of equations is the generalized system of Navier-Stokes equations...

The motion of a three-dimensional glacier is considered. Ice is modeled as an incompressible non-Newtonian fluid. At each time step, given the shape of the glacier, a nonlinear elliptic system has to be solved in order to obtain the two components of the horizontal velocity field. Then, the shape of the glacier is updated by solving a transport equation. Finite element techniques are used to compute the velocity field and to solve the transport equation. Numerical results are compared to experiments...

We study the boundary value problem $-{\mathrm{d}iv\left(\right(\left|\nabla u\right|}^{p_1\left(x\right)-2}+{\left|\nabla u\right|}^{p_2\left(x\right)-2}\left)\nabla u\right)=f(x,u)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in ${ℝ}^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u{|}^{m\left(x\right)-2}u+\left|u{|}^{q\left(x\right)-2}u\right)$, where $m\left(x\right)=max\{p_1\left(x\right),p_2\left(x\right)\}$ for any $x\in \overline{\Omega }$ or $m\left(x\right)<q\left(x\right)<\frac{N·m\left(x\right)}{(N-m(x\left)\right)}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda$ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ${\mathbb{Z}}_2$-symmetric version for even functionals...

A model is presented for a Hele-Shaw flow with variable temperature in one space dimension. The problem to be solved is a free boundary problem for a parabolic equation with a non-linear and non-local free boundary condition. Existence and uniqueness are proved.

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,...

In classical constitutive models such as the Navier-Stokes fluid model, and the Hookean or neo-Hookean solid models, the stress is given explicitly in terms of kinematical quantities. Models for viscoelastic and inelastic responses on the other hand are usually implicit relationships between the stress and the kinematical quantities. Another class of problems wherein it would be natural to develop implicit constitutive theories, though seldom resorted to, are models for bodies that are constrained....

We consider a class of incompressible fluids whose viscosities depend on the pressure and the shear rate. Suitable boundary conditions on the traction at the inflow/outflow part of boundary are given. As an advantage of this, the mean value of the pressure over the domain is no more a free parameter which would have to be prescribed otherwise. We prove the existence and uniqueness of weak solutions (the latter for small data) and discuss particular applications of the results.

The boundary layer equations for the non-Newtonian power law fluid are examined under the classical conditions of uniform flow past a semi infinite flat plate. We investigate the behavior of the similarity solution and employing the Crocco-like transformation we establish the power series representation of the solution near the plate.

We consider the so-called Ladyzhenskaya model of incompressible fluid, with an additional artificial smoothing term ɛΔ3. We establish the global existence, uniqueness, and regularity of solutions. Finally, we show that there exists an exponential attractor, whose dimension we estimate in terms of the relevant physical quantities, independently of ɛ > 0.

We study the generalized Oldroyd model with viscosity depending on the shear stress behaving like ${\mu \left(D\right)\sim \left|D\right|}^{p-2}$ (p > 6/5), regularized by a nonlinear stress diffusion. Using the Lipschitz truncation method we prove global existence of a weak solution to the corresponding system of partial differential equations.