scholarly journals An Incompressible Polymer Fluid Interacting with a Koiter Shell

2021 ◽  
Vol 31 (1) ◽  
Author(s):  
Dominic Breit ◽  
Prince Romeo Mensah
Keyword(s):  
Moving Boundary ◽  
Classical Model ◽  
Elastic Shell ◽  
Planck Equation ◽  
Navier Stokes ◽  
Flexible Shell ◽  
Navier Stokes Equation ◽  
Shell System ◽  
Forward Equation ◽  
Polymer Fluid

AbstractWe study a mutually coupled mesoscopic-macroscopic-shell system of equations modeling a dilute incompressible polymer fluid which is evolving and interacting with a flexible shell of Koiter type. The polymer constitutes a solvent-solute mixture where the solvent is modelled on the macroscopic scale by the incompressible Navier–Stokes equation and the solute is modelled on the mesoscopic scale by a Fokker–Planck equation (Kolmogorov forward equation) for the probability density function of the bead-spring polymer chain configuration. This mixture interacts with a nonlinear elastic shell which serves as a moving boundary of the physical spatial domain of the polymer fluid. We use the classical model by Koiter to describe the shell movement which yields a fully nonlinear fourth order hyperbolic equation. Our main result is the existence of a weak solution to the underlying system which exists until the Koiter energy degenerates or the flexible shell approaches a self-intersection.

GLOBAL WEAK SOLUTIONS FOR A VLASOV–FOKKER–PLANCK/NAVIER–STOKES SYSTEM OF EQUATIONS

2007 ◽  
Vol 17 (07) ◽  
pp. 1039-1063 ◽  
Author(s):  
A. MELLET ◽  
A. VASSEUR
Keyword(s):  
Weak Solutions ◽  
Stokes System ◽  
Fluid Velocity ◽  
Planck Equation ◽  
Coupled System ◽  
Navier Stokes ◽  
Global Weak Solutions ◽  
Navier Stokes Equation ◽  
Reflection Boundary ◽  
Fokker Planck

We establish the existence of a weak solutions for a coupled system of kinetic and fluid equations. More precisely, we consider a Vlasov–Fokker–Planck equation coupled to compressible Navier–Stokes equation via a drag force. The fluid is assumed to be barotropic with γ-pressure law (γ > 3/2). The existence of weak solutions is proved in a bounded domain of ℝ3 with homogeneous Dirichlet conditions on the fluid velocity field and Dirichlet or reflection boundary conditions on the kinetic distribution function.

scholarly journals Mathematical problems in modeling artificial heart

1995 ◽  
Vol 1 (3) ◽  
pp. 245-254 ◽  
Author(s):  
N. U. Ahmed
Keyword(s):  
Blood Flow ◽  
Boundary Conditions ◽  
Stokes Equation ◽  
Artificial Heart ◽  
Moving Boundary ◽  
Hyperbolic Type ◽  
The Other ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Heart Chamber

In this paper we discuss some problems arising in mathematical modeling of artificial hearts. The hydrodynamics of blood flow in an artificial heart chamber is governed by the Navier-Stokes equation, coupled with an equation of hyperbolic type subject to moving boundary conditions. The flow is induced by the motion of a diaphragm (membrane) inside the heart chamber attached to a part of the boundary and driven by a compressor (pusher plate). On one side of the diaphragm is the blood and on the other side is the compressor fluid. For a complete mathematical model it is necessary to write the equation of motion of the diaphragm and all the dynamic couplings that exist between its position, velocity and the blood flow in the heart chamber. This gives rise to a system of coupled nonlinear partial differential equations; the Navier-Stokes equation being of parabolic type and the equation for the membrane being of hyperbolic type. The system is completed by introducing all the necessary static and dynamic boundary conditions. The ultimate objective is to control the flow pattern so as to minimize hemolysis (damage to red blood cells) by optimal choice of geometry, and by optimal control of the membrane for a given geometry. The other clinical problems, such as compatibility of the material used in the construction of the heart chamber, and the membrane, are not considered in this paper. Also the dynamics of the valve is not considered here, though it is also an important element in the overall design of an artificial heart. We hope to model the valve dynamics in later paper.

A Two-Phase Model for Analysis of Blood Flow and Rheological Properties in the Elastic Microvessel

2016 ◽  
Author(s):  
Xiaohui Lin ◽  
Chibin Zhang ◽  
Changbao Wang ◽  
Wenquan Chu ◽  
Zhaomin Wang
Keyword(s):  
Two Phase Flow ◽  
Volume Fraction ◽  
Transport Model ◽  
Planck Equation ◽  
Navier Stokes ◽  
Phase Flow ◽  
Two Phase ◽  
Navier Stokes Equation ◽  
Experimental Values ◽  
The Impact

The blood in microvascular is seemed as a two-phase flow system composed of plasma and red blood cells (RBCs). Based on hydrodynamic continuity equation, Navier-Stokes equation, Fokker-Planck equation, generalized Reynolds equation and elasticity equation, a two-phase flow transport model of blood in elastic microvascular is proposed. The continuous medium assumption of RBCs is abandoned. The impact of the elastic deformation of the vessel wall, the interaction effect between RBCs, the Brownian motion effect of RBCs and the viscous resistance effect between RBCs and plasma on blood transport are considered. Model does not introduce any phenolmeno-logical parameter, compared with the previous phenolmeno-logical model, this model is more comprehensive in theory. The results show that, the plasma velocity distribution is cork-shaped, which is apparently different with the parabolic shape of the single-phase flow model. The reason of taper angle phenomenon and RBCs “Center focus” phenomenon are also analyzed. When the blood vessel radius is in the order of microns, blood apparent viscosity’s Fahraeus-Lindqvist effect and inverse Fahraeus-Lindqvist effect will occur, the maximum of wall shear stress will appear in the minimum of diameter, the variations of blood apparent viscosity with consider of RBCs volume fraction and shear rate calculated by the model are in good agreement with the experimental values.

Use Multiphysics Simulations and Resistive Pulse Sensing to Study the Effect of Metal and Non-Metal Nanoparticles in Different Salt Concentration

2017 ◽  
Author(s):  
Chun-Lin Chiang ◽  
Che-Yen Lee ◽  
Yu-shan Yeh
Keyword(s):  
Metal Nanoparticles ◽  
Semiconductor Industry ◽  
Planck Equation ◽  
Ion Concentration ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Sensing Technology ◽  
Resistive Pulse ◽  
Multiphysics Coupling ◽  
Resistive Pulse Sensing

Wafer fabrication is a critical part of the semiconductor process when the finest linewidth with the improvement of technology continues to decline. The nanoparticles contained in the slurry or ultrapure water used for cleaning have a large influence on the manufacturing process. Therefore, semiconductor industry is hoping to find a viable method for on-line detection of the nanoparticles size and concentration. Resistive pulse sensing technology is one of the methods that may cover this question. There were a lot of reports showing that nanoparticles properties of materials differ significantly from their properties at nano length scales. So, we want to clear the translocation dynamic and ion current changes in measurement of metal nanoparticles or non-metal nanoparticles in different concentration electrolytes through the nanopore when resistive pulse sensing technology has been used. In this study, we try to use a finite element method that contains three governing equations to do multiphysics coupling simulations. The Navier-Stokes equation describes the laminar motion, the Nernst-Planck equation describes the ion transport, and the Poisson equation describes the potential distribution in the flow channel. Then, the reliability of the simulation results was verified by resistive pulse sensing test. The existing results showed that the lower the ion concentration the greater the effect of resistive pulse sensing was. We investigated the effect of resistive pulse sensing on different materials by both simulations and experiments. The results are discussed in this article.

scholarly journals Stochastic approach to the Navier-Stokes equation

2021 ◽  
Author(s):  
Takuya Yabu
Keyword(s):  
Diffusion Equation ◽  
Flow Velocity ◽  
Stokes Equation ◽  
Fluid Pressure ◽  
Planck Equation ◽  
Stochastic Approach ◽  
Molar Concentration ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Ito's Lemma

Take a stochastic approach to the Navier-Stokes equation. The pressure and flow velocity are used as probabilities, the xyz coordinates are replaced with molar concentrations, and the Navier-Stokes equation is transformed into the Fokker-Planck equation using Ito's lemma(formula). This made it possible to obtain the probability of turbulence from the Navier-Stokes equation. The molar concentration in the micro space can be obtained by separately solving the diffusion equation. Using these results, the probability of turbulence and the quantities such as fluid pressure and flow velocity can be analytically obtained.

Numerical Computation of Hydrodynamically and Thermally Developing Liquid Flow in Microchannels With Electrokinetics Effects

2004 ◽  
Vol 126 (1) ◽  
pp. 70-75 ◽  
Author(s):  
X. Y. Chen ◽  
K. C. Toh ◽  
C. Yang ◽  
J. C. Chai
Keyword(s):  
Liquid Flow ◽  
Electrical Potential ◽  
Planck Equation ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Electrokinetic Effect ◽  
Proposed Model ◽  
Parallel Slit ◽  
Temperature Dependent Properties ◽  
Volume Method

Developing fluid flow and heat transfer with temperature dependent properties in microchannels with electrokinetic effects is investigated numerically. The electrokinetic effect on liquid flow in a parallel slit is modeled by the general Nernst-Planck equation describing anion and cation distributions, the Poisson equation determining the electrical potential profile, the continuity equation, and the modified Navier-Stokes equation governing the velocity field. A Finite-Volume Method is utilized to solve the proposed model.

scholarly journals Effects of Salt Tracer Volume and Concentration on Residence Time Distribution Curves for Characterization of Liquid Steel Behavior in Metallurgical Tundish

Metals ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 430
Author(s):  
Changyou Ding ◽  
Hong Lei ◽  
Hong Niu ◽  
Han Zhang ◽  
Bin Yang ◽  
...  
Keyword(s):  
Residence Time ◽  
Time Distribution ◽  
Residence Time Distribution ◽  
Mass Concentration ◽  
Distribution Curve ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Residence Time Distribution Curve ◽  
Tracer Solution ◽  
Time Distribution Curve

The residence time distribution (RTD) curve is widely applied to describe the fluid flow in a tundish, different tracer mass concentrations and different tracer volumes give different residence time distribution curves for the same flow field. Thus, it is necessary to have a deep insight into the effects of the mass concentration and the volume of tracer solution on the residence time distribution curve. In order to describe the interaction between the tracer and the fluid, solute buoyancy is considered in the Navier–Stokes equation. Numerical results show that, with the increase of the mass concentration and the volume of the tracer, the shape of the residence time distribution curve changes from single flat peak to single sharp peak and then to double peaks. This change comes from the stratified flow of the tracer. Furthermore, the velocity difference number is introduced to demonstrate the importance of the density difference between the tracer and the fluid.

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