697 research outputs found
On Modeling the Response of Synovial Fluid: Unsteady Flow of a Shear-Thinning, Chemically-Reacting Fluid Mixture
We study the flow of a shear-thinning, chemically-reacting fluid that could
be used to model the flow of the synovial fluid. The actual geometry where the
flow of the synovial fluid takes place is very complicated, and therefore the
governing equations are not amenable to simple mathematical analysis. In order
to understand the response of the model, we choose to study the flow in a
simple geometry. While the flow domain is not a geometry relevant to the flow
of the synovial fluid in the human body it yet provides a flow which can be
used to assess the efficacy of different models that have been proposed to
describe synovial fluids. We study the flow in the annular region between two
cylinders, one of which is undergoing unsteady oscillations about their common
axis, in order to understand the quintessential behavioral characteristics of
the synovial fluid. We use the three models suggested by Hron et al. [ J. Hron,
J. M\'{a}lek, P. Pust\v{e}jovsk\'{a}, K. R. Rajagopal, On concentration
dependent shear-thinning behavior in modeling of synovial fluid flow, Adv. in
Tribol. (In Press).] to study the problem, by appealing to a semi-inverse
method. The assumed structure for the velocity field automatically satisfies
the constraint of incompressibility, and the balance of linear momentum is
solved together with a convection-diffusion equation. The results are compared
to those associated with the Newtonian model. We also study the case in which
an external pressure gradient is applied along the axis of the cylindrical
annulus.Comment: 25 pages, 11 figures, accepted in Computers & Applications with
Mathematic
Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index
We consider a system of nonlinear partial differential equations describing
the motion of an incompressible chemically reacting generalized Newtonian fluid
in three space dimensions. The governing system consists of a steady
convection-diffusion equation for the concentration and a generalized steady
power-law-type fluid flow model for the velocity and the pressure, where the
viscosity depends on both the shear-rate and the concentration through a
concentration-dependent power-law index. The aim of the paper is to perform a
mathematical analysis of a finite element approximation of this model. We
formulate a regularization of the model by introducing an additional term in
the conservation-of-momentum equation and construct a finite element
approximation of the regularized system. We show the convergence of the finite
element method to a weak solution of the regularized model and prove that weak
solutions of the regularized problem converge to a weak solution of the
original problem.Comment: arXiv admin note: text overlap with arXiv:1703.0476
Finite element approximation of an incompressible chemically reacting non-Newtonian fluid
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.Comment: 40 page
On the existence of classical solution to the steady flows of generalized Newtonian fluid with concentration dependent power-law index
Steady flows of an incompressible homogeneous chemically reacting fluid are
described by a coupled system, consisting of the generalized Navier--Stokes
equations and convection - diffusion equation with diffusivity dependent on the
concentration and the shear rate. Cauchy stress behaves like power-law fluid
with the exponent depending on the concentration. We prove the existence of a
classical solution for the two dimensional periodic case whenever the power law
exponent is above one and less than infinity
Methodology for computational fluid dynamics code verification /validation
The issues of verification, calibration, and validation of computational fluid dynamics (CFD) codes has been receiving increasing levels of attention in the research literature and in engineering technology. Both CFD researchers and users of CFD codes are asking more critical and detailed questions concerning the accuracy, range of applicability, reliability and robustness of CFD codes and their predictions. This is a welcomed trend because it demonstrates that CFD is maturing from a research tool to the world of impacting engineering hardware and system design. In this environment, the broad issue of code quality assurance becomes paramount. However, the philosophy and methodology of building confidence in CFD code predictions has proven to be more difficult than many expected. A wide variety of physical modeling errors and discretization errors are discussed. Here, discretization errors refer to all errors caused by conversion of the original partial differential equations to algebraic equations, and their solution. Boundary conditions for both the partial differential equations and the discretized equations will be discussed. Contrasts are drawn between the assumptions and actual use of numerical method consistency and stability. Comments are also made concerning the existence and uniqueness of solutions for both the partial differential equations and the discrete equations. Various techniques are suggested for the detection and estimation of errors caused by physical modeling and discretization of the partial differential equations
Non-Newtonian fluids with discontinuous-in-time stress tensor
We consider the system of equations describing the flow of incompressible
fluids in bounded domain. In the considered setting, the Cauchy stress tensor
is a monotone mapping and has asymptotically -growth with the parameter
depending on the spatial and time variable. We do not assume any smoothness
of with respect to time variable and assume the log-H\"{o}lder continuity
with respect to spatial variable. Such a setting is a natural choice if the
material properties are instantaneously, e.g. by the switched electric field.
We establish the long time and the large data existence of weak solution
provided that
Computational analysis of non-isothermal flow of non-Newtonian fluids
The dynamics of complex fluids under various conditions is a model problem in bio-fluidics and in process industries. We investigate a class of such fluids and flows under conditions of heat and/or mass transfer. Experiments have shown that under certain flow conditions, some complex fluids (e.g. worm-like micellar solutions and some polymeric fluids) exhibit flow instabilities such as the emergence of regions of different shear rates (shear bands) within the flow field. It has also been observed that the reacting mixture in reaction injection molding of polymeric foams undergoes self-expansion with evolution of heat due to exothermic chemical reaction. These experimental observations form the foundation of this thesis. We explore the heat and mass transfer effects in various relevant flow problems of complex fluids. In each case, we construct adequate mathematical models capable of describing the experimentally observed flow phenomena. The mathematical models are inherently intractable to analytical treatment, being nonlinear coupled systems of time dependent partial differential equations. We therefore develop computational solutions for the model problems. Depending on geometrical or mathematical complexity, finite difference or finite volume methods will be adopted. We present the results from our numerical simulations via graphical illustrations and validate them (qualitatively) against' similar' results in the literature; the quotes being necessary in keeping in mind the novelties introduced in our investigations which are otherwise absent in the existing literature. In the case where experimental data is available, we validate our numerical simulations against such experimental results
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