Page 667 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 667
PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS
with r = x12 + x22 and u(x) = u(r, x3) to reduce the curl-curl operator to the vector Laplacian;
at the same time we consider an isometric isomorphism between D1,2(R3, R3) and H1(S3, R3)
to recover compactness.
Asymptotic study of a thin layer of viscous fluid between two surfaces
Raquel Taboada Vázquez, raqueltv@udc.es
Universidade da Coruña, Spain
Coauthor: José M. Rodríguez
In this work, we are interested in studying the behavior of an incompressible viscous fluid
moving between two closely spaced surfaces, also in motion.
We consider a three-dimensional thin domain, Ωtε, filled by a fluid, that varies with time
t ∈ [0, T ], given by
Ωεt = (x1ε, xε2, xε3) ∈ R3 : (1)
xi(ξ1, ξ2, t) ≤ xεi ≤ xi(ξ1, ξ2, t) + hε(ξ1, ξ2, t)Ni(ξ1, ξ2, t),
(i = 1, 2, 3), (ξ1, ξ2) ∈ D ⊂ R2
where X(ξ1, ξ2, t) is the lower bound surface parametrization, N (ξ1, ξ2, t) is the unit normal
vector and hε(ξ1, ξ2, t) is the gap between the two surfaces in motion assumed to be small with
regard to the dimension of the bound surfaces. We take into account that the fluid film between
the surfaces is thin by introducing a small non-dimensional parameter ε, and setting that
hε(ξ1, ξ2, t) = εh(ξ1, ξ2, t) (2)
We assume that the fluid motion is governed by Navier-Stokes equations and using the
asymptotic development technique, the following lubrication model in a thin domain with
curved mean surface has been obtained:
√1 div (√hε)3 M ∇p−2,ε ∂hε hεA1 ∂X · N
A0 A0 = 12µ + 12µ ∂t
∂t A0
− 6µ∇hε · (W 0 − V 0) + 6√µhε √ 0 + V 0)) (3)
div( A0(W
A0
It is a new generalized Reynolds equation where the pressure, pε, is approximated by p−2,ε =
ε−2p−2. The fluid velocities inside the domain are subsequently approximated from the pressure
using the equations
u10 h2(ξ32 − ξ3) ∂p−2 ∂p−2 + ξ3(W10 − V10) + V10
= 2µA0 G −F + ξ3(W20 − V20) + V20 (4)
∂ξ1 ∂ξ2 (5)
(6)
u02 = h2(ξ32 − ξ3) ∂p−2 − F ∂p−2
2µA0 E
∂ξ2 ∂ξ1
u03 = ∂X · N
∂t
where the velocity on the lower surface, V 0, and on the upper surface, W 0, are known. We
665
with r = x12 + x22 and u(x) = u(r, x3) to reduce the curl-curl operator to the vector Laplacian;
at the same time we consider an isometric isomorphism between D1,2(R3, R3) and H1(S3, R3)
to recover compactness.
Asymptotic study of a thin layer of viscous fluid between two surfaces
Raquel Taboada Vázquez, raqueltv@udc.es
Universidade da Coruña, Spain
Coauthor: José M. Rodríguez
In this work, we are interested in studying the behavior of an incompressible viscous fluid
moving between two closely spaced surfaces, also in motion.
We consider a three-dimensional thin domain, Ωtε, filled by a fluid, that varies with time
t ∈ [0, T ], given by
Ωεt = (x1ε, xε2, xε3) ∈ R3 : (1)
xi(ξ1, ξ2, t) ≤ xεi ≤ xi(ξ1, ξ2, t) + hε(ξ1, ξ2, t)Ni(ξ1, ξ2, t),
(i = 1, 2, 3), (ξ1, ξ2) ∈ D ⊂ R2
where X(ξ1, ξ2, t) is the lower bound surface parametrization, N (ξ1, ξ2, t) is the unit normal
vector and hε(ξ1, ξ2, t) is the gap between the two surfaces in motion assumed to be small with
regard to the dimension of the bound surfaces. We take into account that the fluid film between
the surfaces is thin by introducing a small non-dimensional parameter ε, and setting that
hε(ξ1, ξ2, t) = εh(ξ1, ξ2, t) (2)
We assume that the fluid motion is governed by Navier-Stokes equations and using the
asymptotic development technique, the following lubrication model in a thin domain with
curved mean surface has been obtained:
√1 div (√hε)3 M ∇p−2,ε ∂hε hεA1 ∂X · N
A0 A0 = 12µ + 12µ ∂t
∂t A0
− 6µ∇hε · (W 0 − V 0) + 6√µhε √ 0 + V 0)) (3)
div( A0(W
A0
It is a new generalized Reynolds equation where the pressure, pε, is approximated by p−2,ε =
ε−2p−2. The fluid velocities inside the domain are subsequently approximated from the pressure
using the equations
u10 h2(ξ32 − ξ3) ∂p−2 ∂p−2 + ξ3(W10 − V10) + V10
= 2µA0 G −F + ξ3(W20 − V20) + V20 (4)
∂ξ1 ∂ξ2 (5)
(6)
u02 = h2(ξ32 − ξ3) ∂p−2 − F ∂p−2
2µA0 E
∂ξ2 ∂ξ1
u03 = ∂X · N
∂t
where the velocity on the lower surface, V 0, and on the upper surface, W 0, are known. We
665
