HELICAL SYMMETRIC SOLUTION OF 3D NAVIER-STOKES EQUATIONS ARISING FROM GEOMETRIC SHAPE OF THE BOUNDARY

Weifeng JIANG; Kaitai LI

doi:10.1016/S0252-9602(18)30801-4

Acta mathematica scientia, Series B >

2018 , Vol. 38 >Issue 3: 1057 - 1104

DOI: https://doi.org/10.1016/S0252-9602(18)30801-4

Articles

HELICAL SYMMETRIC SOLUTION OF 3D NAVIER-STOKES EQUATIONS ARISING FROM GEOMETRIC SHAPE OF THE BOUNDARY

Weifeng JIANG ,
Kaitai LI

Expand

1. School of Sciences, Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China;
2. School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

Received date: 2016-02-03

Revised date: 2018-03-21

Online published: 2018-06-25

Supported by

This work was supported by NSFC (91330116), State Major Basic Research and Development Project (2011CB 706505), NSFC (11371289), NSFC (11371288).

Fold

Abstract

In this article, we investigate three-dimensional solution with helical symmetry in a gap between two concentric rotating cylinders, inside is a helicoidal surface (screw propeller) while outside is a cylindrical surface. Establish the partial differential equations and its variational formulation satisfied by a helical solution in a helical coordinate system using tensor analysis method, we provide a computational method for the power and propulsion of the screw. The existence and uniqueness of weak helical solutions are proved.

Key words： Navier-Stokes equations; helicoidal surface; rotating helical flow

Cite this article

Weifeng JIANG , Kaitai LI . HELICAL SYMMETRIC SOLUTION OF 3D NAVIER-STOKES EQUATIONS ARISING FROM GEOMETRIC SHAPE OF THE BOUNDARY[J]. Acta mathematica scientia, Series B, 2018 , 38(3) : 1057 -1104 . DOI: 10.1016/S0252-9602(18)30801-4

References

[1] Agmon S. Lectures on elliptic boundary value problems. Princeton, N J:D Van Nostrand Co, Inc, 1965
[2] Constantin P, Foias C. Navier-Stokes Equations. Chicago:University of Chicago Press, 1988
[3] Ettinger B, Titi E S. Global existence and uniqueness of weak solutions of three-dimensional Euler equations with helical symmetry in the absence of vorticity stretching. SIAM J Math Anal, 2009, 41:269-296
[4] Ghidaglia J M. Régularité des solutions de certains problèmes aux limites linéaires liés aux équations d'Euler (French). Comm Partial Differential Equations, 1984, 9:1265-1298
[5] Heywood J G. Navier-Stokes Equations:On the existence, regularity and decay of solutions. Indiana Univ Math J, 1980, 29:639-681
[6] Hüttl T J, Wagner C, Friedrich R. Navier-Stokes solutions of laminar flows based on orthogonal helical co-ordinates. Int J Numer Meth Fluids, 1999, 29:749-763
[7] Ladyzhenskaya O A. The Mathematical Theory of Viscous Incompressible Flow. 2nd ed. New York:Gordon and Breach, 1969
[8] Li K, Huang A. Tensor Analysis and Its Applications. Science Press, Beijing, China and Alpha Science International Ltd, Oxford, UK, 2015
[9] Lions J L, Magenes E. Non-homogeneous Boundary Value Problems and Applications. Vol I. New York:Springer-Verlag, 1972
[10] Lopes Filho M C, Mazzucato A L, Niu D, Nussenzveig Lopes H J, Titi E S. Planar limits of ThreeDimensional incompressible flows with helical symmetry. J Dyn Diff Equations, 2014, 26:843-869
[11] Mahalov A, Titi E S, Leibovich S. Invariant helical subspaces for the Navier-Stokes equations. Arch Rational Mech Anal, 1990, 112:193-222
[12] Temam R. Navier-Stokes Equations. Theory and Numerical Analysis. 3rd ed. Amsterdam:North Holland Pub Co, 1984
[13] Wang S. On a new 3D model for incompressible Euler and Navier-Stokes equations. Acta Mathematica Scientia, 2010, 30B(6):2089-2102

Options

Abstract

Outlines

模态框（Modal）标题

Abstract

Cite this article

References