Page 668 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 668
PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS
denote by
A0 = EG − F 2 (7)
(8)
A1 = −eG − gE + 2f F
(9)
M= G −F
−F E
where E, F , G and e, f , g are the coefficients of the first and the second (respectively) funda-
mental forms of the surface parametrized by X.
We have observed that, depending on the boundary conditions, other models can be ob-
tained. We derive a shallow water model changing the boundary conditions that we had im-
posed: instead of assuming that we know the velocities on the upper and lower boundaries of
the domain, we assume that we know the tractions on these upper and lower boundaries. We
yield:
∂Vi0 + 2 ∂Vi0 + 2 2
∂t ∂ξl k=1
Vl0 − Cl0 Ri0k + Hi0lk Vl0 Vk0
l=1
l=1
=−1 αi0 ∂π00 + βi0 ∂π00
ρ0 ∂ξ1 ∂ξ2
+ν 2 2 ∂2Vi0 Jl0m + 2 2 ∂Vk0 (L0kli + ψ(h)i0kl)
m=1 l=1 ∂ξm∂ξl k=1 l=1 ∂ξl
2 + Fi0(h) − Q0i3 ∂X · N
∂t
+ Vk0(Si0k + χ(h)0ik) + κˆ(h)i0
k=1
(i = 1, 2) (10)
(11)
∂h + √h div √ hA1 ∂X · N =0
∂t A0 A0V 0 + A0 ∂t
where αi0, βi0, Cl0, Hi0lk, Jl0m, L0kli, Qi03, Ri0k, Si0k depend only on the parametrization X and
Fi0(h), ψ(h)0ikl, χ(h)0ik, κ(h)i0 depend on the parametrization X and on the gap h. The exact
definition of these coefficients can be found in [5], where the complete derivation of both models
is presented.
Once V10, V20 and π00 (the approximation of the pressure on the lower bound) are calculated
we have the following approximation of the velocities and the pressure
ui0 = Wi0 = Vi0 i = 1, 2 (12)
(13)
u03 = ∂X · N (14)
∂t
p0 = 2µ ∂h + π00
h ∂t
These models can not be found in the literature, as far as we know. We reach the conclusion
that the magnitude of the pressure differences at the lateral boundary of the domain is key when
deciding which of the two models best describes the fluid behavior.
Boundary conditions tell us which of the two models should be used when simulating the
flow of a thin fluid layer between two surfaces: if the fluid pressure is dominant (that is, it is
of order O(ε−2)), and the fluid velocity is known on the upper and lower surfaces, we must use
666
denote by
A0 = EG − F 2 (7)
(8)
A1 = −eG − gE + 2f F
(9)
M= G −F
−F E
where E, F , G and e, f , g are the coefficients of the first and the second (respectively) funda-
mental forms of the surface parametrized by X.
We have observed that, depending on the boundary conditions, other models can be ob-
tained. We derive a shallow water model changing the boundary conditions that we had im-
posed: instead of assuming that we know the velocities on the upper and lower boundaries of
the domain, we assume that we know the tractions on these upper and lower boundaries. We
yield:
∂Vi0 + 2 ∂Vi0 + 2 2
∂t ∂ξl k=1
Vl0 − Cl0 Ri0k + Hi0lk Vl0 Vk0
l=1
l=1
=−1 αi0 ∂π00 + βi0 ∂π00
ρ0 ∂ξ1 ∂ξ2
+ν 2 2 ∂2Vi0 Jl0m + 2 2 ∂Vk0 (L0kli + ψ(h)i0kl)
m=1 l=1 ∂ξm∂ξl k=1 l=1 ∂ξl
2 + Fi0(h) − Q0i3 ∂X · N
∂t
+ Vk0(Si0k + χ(h)0ik) + κˆ(h)i0
k=1
(i = 1, 2) (10)
(11)
∂h + √h div √ hA1 ∂X · N =0
∂t A0 A0V 0 + A0 ∂t
where αi0, βi0, Cl0, Hi0lk, Jl0m, L0kli, Qi03, Ri0k, Si0k depend only on the parametrization X and
Fi0(h), ψ(h)0ikl, χ(h)0ik, κ(h)i0 depend on the parametrization X and on the gap h. The exact
definition of these coefficients can be found in [5], where the complete derivation of both models
is presented.
Once V10, V20 and π00 (the approximation of the pressure on the lower bound) are calculated
we have the following approximation of the velocities and the pressure
ui0 = Wi0 = Vi0 i = 1, 2 (12)
(13)
u03 = ∂X · N (14)
∂t
p0 = 2µ ∂h + π00
h ∂t
These models can not be found in the literature, as far as we know. We reach the conclusion
that the magnitude of the pressure differences at the lateral boundary of the domain is key when
deciding which of the two models best describes the fluid behavior.
Boundary conditions tell us which of the two models should be used when simulating the
flow of a thin fluid layer between two surfaces: if the fluid pressure is dominant (that is, it is
of order O(ε−2)), and the fluid velocity is known on the upper and lower surfaces, we must use
666
