A new thin layer model for viscous flow between two nearby non‐static surfaces

We propose a two‐dimensional flow model of a viscous fluid between two close moving surfaces. We show, using a formal asymptotic expansion of the solution, that its asymptotic behavior, when the distance between the two surfaces tends to zero, is the same as that of the the Navier‐Stokes equations.

[1]  R. Taboada-V'azquez,et al.  Asymptotic analysis of a thin fluid layer flow between two moving surfaces , 2021, Journal of Mathematical Analysis and Applications.

[2]  J. M. Rodriguez,et al.  Rigorous justification of the asymptotic model describing a curved‐pipe flow in a time‐dependent domain , 2018, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik.

[3]  F. J. Suárez-Grau,et al.  Homogenization of the Darcy–Lapwood–Brinkman Flow in a Thin Domain with Highly Oscillating Boundaries , 2018, Bulletin of the Malaysian Mathematical Sciences Society.

[4]  M. Anguiano,et al.  Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary , 2017, IMA Journal of Applied Mathematics.

[5]  G. Castiñeira,et al.  Asymptotic Analysis of a Viscous Flow in a Curved Pipe with Elastic Walls , 2016, 1602.06121.

[6]  G. Panasenko,et al.  Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. II. General case , 2015 .

[7]  G. Panasenko,et al.  Asymptotic analysis of the non-steady Navier–Stokes equations in a tube structure. I. The case without boundary-layer-in-time , 2015 .

[8]  F. J. Suárez-Grau.  Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary , 2015 .

[9]  J. M. Rodríguez,et al.  Derivation of a new asymptotic viscous shallow water model with dependence on depth , 2012, Appl. Math. Comput..

[10]  R. Stavre,et al.  Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube , 2012 .

[11]  J. M. Rodríguez,et al.  Bidimensional shallow water model with polynomial dependence on depth through vorticity , 2009 .

[12]  E. Miglio,et al.  ASYMPTOTIC DERIVATION OF THE SECTION-AVERAGED SHALLOW WATER EQUATIONS FOR NATURAL RIVER HYDRAULICS , 2009 .

[13]  Raquel Taboada-Vázquez,et al.  A new shallow water model with linear dependence on depth , 2008, Math. Comput. Model..

[14]  J. M. Rodríguez,et al.  A new shallow water model with polynomial dependence on depth , 2008 .

[15]  J. M. Rodríguez,et al.  From Euler and Navier-Stokes equations to shallow waters by asymptotic analysis , 2007, Adv. Eng. Softw..

[16]  Changbing Hu.  Asymptotic analysis of the primitive equations under the small depth assumption , 2005 .

[17]  E. Grenier.  On the derivation of homogeneous hydrostatic equations , 1999 .

[18]  Roger Temam,et al.  Asymptotic analysis of the Navier-Stokes equations in the domains , 1997 .

[19]  Philippe G. Ciarlet,et al.  Asymptotic analysis of linearly elastic shells. III. Justification of Koiter's shell equations , 1996 .

[20]  Philippe G. Ciarlet,et al.  Asymptotic analysis of linearly elastic shells: ‘Generalized membrane shells’ , 1996 .

[21]  S. Nazarov.  Asymptotic solution of the Navier-Stokes problem on the flow of a thin layer of fluid , 1990 .

[22]  M. Chipot.  On the Reynolds lubrication equation , 1988 .

[23]  G. Cimatti.  A rigorous justification of the Reynolds equation , 1987 .

[24]  M. Chipot,et al.  Existence and uniqueness of solutions to the compressible Reynolds lubrication equation , 1986 .

[25]  Guy Bayada,et al.  The transition between the Stokes equations and the Reynolds equation: A mathematical proof , 1986 .

[26]  G. Cimatti.  How the Reynolds equation is related to the Stokes equations , 1983 .

[27]  Philippe G. Ciarlet,et al.  A justification of the von Kármán equations , 1980 .

[28]  I W Dand,et al.  SHALLOW WATER HYDRODYNAMICS , 1971 .

[29]  K. O. Friedrichs,et al.  A boundary-layer theory for elastic plates , 1961 .

[30]  W. R. Dean,et al.  Note on the motion of fluid in a curved pipe , 1959 .

[31]  W. R. Dean.  LXXII. The stream-line motion of fluid in a curved pipe (Second paper) , 1928 .

[32]  W. R. Dean.  XVI. Note on the motion of fluid in a curved pipe , 1927 .

[33]  E. Marušić‐Paloka,et al.  Fluid flow through a helical pipe , 2007 .

[34]  F. Marche.  Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects , 2007 .

[35]  J. M. Rodríguez,et al.  From Navier–Stokes equations to shallow waters with viscosity by asymptotic analysis , 2005 .

[36]  Jean-Frédéric Gerbeau,et al.  Derivation of viscous Saint-Venant system for laminar shallow water , 2001 .

[37]  Francisco Guillén,et al.  Mathematical Justification of the Hydrostatic Approximation in the Primitive Equations of Geophysical Fluid Dynamics , 2001, SIAM J. Math. Anal..

[38]  E. Marušić‐Paloka.  The Effects of Flexion and Torsion on a Fluid Flow Through a Curved Pipe , 2001 .

[39]  G. Bayada,et al.  Asymptotic Navier–Stokes equations in a thin moving boundary domain , 1999 .

[40]  B. Miara,et al.  Asymptotic analysis of linearly elastic shells , 1996 .

[41]  P. G. Ciarlet,et al.  Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equations , 1996 .

[42]  P. G. Ciarlet,et al.  Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations , 1996 .

[43]  Ibrahim Aganović,et al.  A justification of the one‐dimensional linear model of elastic beam , 1986 .

[44]  A. Bermudez,et al.  Une justification des équations de la thermoélasticité des poutres à section variable par des méthodes asymptotiques , 1984 .

[45]  P. G. Ciarlet,et al.  JUSTIFICATION OF THE TWO-DIMENSIONAL LINEAR PLATE MODEL. , 1979 .

[46]  Philippe G. Ciarlet,et al.  A Justi cation of a Nolinear Model in Plate Theory , 1979 .

[47]  A. L. Gol'denveizer.  Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity , 1962 .