On regularity of a weak solution to the Navier–Stokes equation with generalized impermeability boundary conditions

2007 ◽  
Vol 66 (8) ◽  
pp. 1753-1769 ◽  
Author(s):  
Jiří Neustupa ◽  
Patrick Penel
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Navier Stokes Equation

On the Navier-Stokes equation with boundary conditions based on vorticity

2004 ◽  
Vol 269-270 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Hamid Bellout ◽  
Jiří Neustupa ◽  
Patrick Penel
Keyword(s):  
Boundary Conditions ◽  
Stokes Equation ◽  
Navier Stokes ◽  
Navier Stokes Equation

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.

scholarly journals On Regularity of a Weak Solution to the Navier–Stokes Equations with the Generalized Navier Slip Boundary Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Jiří Neustupa ◽  
Patrick Penel
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Regularity Criterion ◽  
Navier Stokes ◽  
Deformation Tensor ◽  
Slip Boundary Conditions ◽  
Navier Stokes Equation ◽  
Navier Slip ◽  
Slip Boundary

The paper shows that the regularity up to the boundary of a weak solution v of the Navier–Stokes equation with generalized Navier’s slip boundary conditions follows from certain rate of integrability of at least one of the functions ζ1, (ζ2)+ (the positive part of ζ2), and ζ3, where ζ1≤ζ2≤ζ3 are the eigenvalues of the rate of deformation tensor D(v). A regularity criterion in terms of the principal invariants of tensor D(v) is also formulated.

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