Mathematical modeling of bilayer flow
J. D. D. Zabsonr´e
∗, G. Narbona, E. D. Fern´andez, D. Bresch
Abstract
In this work we present a new two-dimensional bi-layer Shallow- Water model including viscosity and friction effect on the bottom and interface level. It is obtained from an asymptotic analysis of non- dimensional and incompressible Navier-Stokes equations with hydro- static approximation.
Keywords: Shallow Water equations, bi-layer models, viscosity, friction, capillarity, Finite Volume methods.
1 Derivation of the bilayer viscous Shallow Water model
In this section we give the derivation of the model proposed in this paper.
We consider the Navier-Stokes equations in a periodic domain Ω(t)∈R3. We assume a two layer environment of inmiscible fluids including three boundary regions. The global domain is Ω(t) = Ω1(t)∪Ω2(t)∪Γb∪Γ1,2(t)∪Γs(t), being:
Ω1(t) = {(x, z)∈R3/x∈ω, b(x)< z < η1,2(t, x)};
Ω2(t) = {(x, z)∈R3/x∈ω, η1,2(t, x)< z < η(t, x)};
Γb ={(x, z)∈R3/x∈ω, z =b(x)};
Γ1,2 ={(x, z)∈R3/x∈ω, z=η1,2(t, x)};
Γs ={(x, z)∈R3/x∈ω, z=η(t, x)};
see Figure 1. We consider ui = (vi, wi) the velocity for each layer, ρi the density, µi denote the dynamic viscosity and pi is the pressure, for i= 1,2.
With this notation, the Navier-Stokes equations for each layer i= 1,2 state
as: ½
ρi∂tui+ (ρiui∇)ui−div(σi) + 2ρi−→
Ω ×ui =−ρigez; div(ui) = 0.
∗Universit´e Polytechnique de Bobo-Dioulasso, 01 BP 1091 Bobo 01, [email protected]
h
h
2
1
b(x)
η1,2
η= b+h
= b+h1
1+h2
z
x
Figure 1: Domain.
In order to obtain a well-posed system we impose the following conditions on the boundaries:
• At the free surface:
σ2·ns=α2κ·ns, ∂tη+v2· ∇xη=w2.
• At the interface z =η1,2(t, x):
∂tη1,2+vi·∇xη1,2 =wi; (σi·n1,2)τ =−γ(u1−u2)τ i= 1,2.
(σ1·n1,2)n = (σ2·n1,2)n+ ((α1−α2)κ1,2·n1,2)n
• At the bottom z =b(x):
(σ1·nb)τ =α(u1)τ, u1·nb = 0;.
To obtain the model, first we shall write these equations under a non- dimensional form, secondly we shall develop the integration in each layer to obtain the Shallow Water system. Finally we shall perform the asymptotic analysis studding both first and second order approximations.
Final models
In this section we write the final equations for the two models obtained with dimension and dropping the cosines terms, having into account that
1
(BL1)
∂th2+ divx(h2v2) = 0;
∂t(h2v2) + divx(h2v2⊗v2) + 2Ω sinθh2(v2)⊥+ +1
2g∇xh22+gh2∇xη1,2 = ˜γ(v1−v2);
∂th1+ divx(h1v1) = 0;
∂t(h1v1) + divx(h1v1⊗v1) + 2Ω sinθh1(v1)⊥+1
2g∇xh21+ +gh1∇xb+rgh1∇xh2 =−r˜γ(v1 −v2)−αv˜ 1.
(1)
(BL2)
∂th2+ divx(h2v2) = 0;
∂t(h2v2) + divx(h2v2⊗v2) + 2Ω sinθh2(v2)⊥+1
2g∇xh22+ +gh2∇xη1,2 =δ−1γ˜
µ βαh˜ 1
6ν1v1+ (v1−v2)
¶
+ ˜α2h2∇x∆xh2+ +˜α2h2∇x∆xη1,2+ 2ν2divx(h2Dx(v2)) + 2ν2∇x(h2divxv2);
∂th1+ divx(h1v1) = 0;
∂t(h1v1) + divx(h1v1⊗v1) + 2Ω sinθh1(v1)⊥+1
2g∇xh21+ +gh1∇xb+rgh1∇xh2 =−δ−1γ r˜
µ βαh˜ 1
6ν1v1 + (v1−v2)
¶
−
−βα˜ µ
v1+δ−1rγh˜ 1 6ν1
(v1−v2)
¶
+ ˜α1h1∇x∆xh1+ +˜α1h1∇x∆xb+ 2ν1divx(h1Dx(v1)) + 2ν1∇x(h1divxv1),
(2)
being
β = µ
1 + α˜ 3ν1
h1
¶−1
, δ = 1 + ˜γ 3
µ rh1
ν1
+h2 ν2
¶ .
0 0.5 1 1.5 2 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
F.S.
Intf.
Bottom
Figure 2: Initial condition
0 0.5 1 1.5 2
0 0.5 1 1.5
(BL1) (BL2) Bottom
t=0.2
0 0.5 1 1.5 2
0 0.5 1 1.5
t=2
Figure 3: Longitudinal section of heights.
2 Numerical assessment
Test : Circular dam-break problem in a 2D domain.
We consider a circular dam-break problem in both, surface and interface with a no constant bottom. The domain is the square Ω = [0,2]×[0,2].
For the initial condition, see Figure 2. The heights of layers are shown in Figure 3 for the same times values. We can see that the difference between the solution of problems (BL1) and (BL2) is getting higher in time.
References
system for laminar Shallow-Water; Numerical validation. Disc. and Cont. Dynam. Syst. Series B, 1(1): 89-102, 2001.
[2] G. Narbona-Reina, J. D. D. Zabsonr´e, E. Fern´anadez-Nieto, D. Bresch, Derivation of a 2D bilayer shallow water equations with viscosity. Numerical validation, soumis.
[3] J.D.D Zabsonr´e, Mod`eles visqueux en s´edimentation et stratifica- tion: obtention formelle, stabilit´e th´eorique et sch´emas volumes finis bien ´equilibr´es, Th`ese `a l’Universit´e de Savoie, 2008.
[4] J.D.D Zabsonr´e, J. M. Castro-Diaz, E. Fern´andez-Nieto, C.
Par´es, Lax-Wendroff method for 2D non-conservative hyperbolic sys- tems over non-structured meshes, en pr´eparation, 2008.
[5] J.D.D Zabsonr´e, C. Lucas, E. Fern´anadez-Nieto, An energeti- cally consistent viscous sedimentation model, `a paraˆıtre dans M3AS.
[6] J.D.D Zabsonr´e, G. Narbona-Reina, Existence of a global weak solution for a 2D viscous bi-layer shallow water model, `a paraˆıtre dans Nonlinear Analysis: Real World Applications.
