where f is a vector field of body loads, g is the gravity acceleration, µ
f
is the fluid viscosity, (u) =
1
2
(∇u + ∇u
t
) is the strain rate tensor, and (d) =
1
2
(∇d + ∇d
t
) is the infinitesimal strain tensor, ρ
f
, ρ
s
are
the density of the fluid and solid, respectively, λ, µ
s
are the first and second Lam´e constants of the solid, κ is
the heterogeneous tensor of permeabilities (satisfying |w|
2
. w ·κ(x)w a.e. in Ω
P
and for all w ∈ R
d
); m
P
is a source/sink term for the fluid pressure (which also includes pressures in the previous backward Euler
time step); and C
0
, α are the total storage capacity and Biot-Willis poroelastic coefficient. Here we have
used the total pressure ϕ := αp
P
− λ div d, as an additional unknown [27, 20].
We furthermore supply boundary conditions as follows
u = 0 on Γ
u
F
, (2.2a)
[2µ
f
(u) −p
F
I]n = 0 on Γ
σ
F
, (2.2b)
d = 0 and
κ
µ
f
∇p
P
· n = 0 on Γ
d
P
, (2.2c)
[2µ
s
(d) −ϕI]n = 0 and p
P
= p
0
on Γ
p
P
P
. (2.2d)
In order to close the system, we consider the classical transmission conditions on Σ accounting for the
continuity of normal fluxes, momentum balance, equilibrium of fluid normal stresses, and the so-called
Beavers-Joseph-Saffman condition for tangential fluid forces [7, 2], which in the present setting reduce to
u ·n = (
1
∆t
d −
κ
µ
f
∇p
P
) ·n on Σ, (2.3a)
(2µ
f
(u) −p
F
I)n = (2µ
s
(d) −ϕI)n on Σ, (2.3b)
−n ·(2µ
f
(u) −p
F
I)n = p
P
on Σ, (2.3c)
−n ·(2µ
f
(u) −p
F
I)t
j
=
γµ
f
√
κ
(u −
1
∆t
d) ·t
j
, 1 ≤ j ≤ d −1 on Σ, (2.3d)
where γ > 0 is the slip rate coefficient depending on the geometry of the domain, and we recall that the
normal n on the interface is understood as pointing from the fluid domain Ω
F
towards the porous structure
Ω
P
, while t
j
, 1 ≤ j ≤ d −1 are orthonormal tangent vectors on Σ, normal to n.
We proceed to test (2.1a)-(2.1e) against suitable smooth functions and to integrate over the corresponding
subdomain. The challenging model parameters are µ
f
, C
0
, λ, γ, α, and the magnitude of κ. Therefore, we
will concentrate on the specific case where ∆t = 1, and the model parameters (in particular κ) are spatially
constant. Following [24], after applying integration by parts wherever adequate and using the transmission
conditions (2.3a)-(2.3d), we arrive at the following remainder on the interface
hp
P
, (v − w) ·ni
Σ
+
γµ
f
√
κ
d−1
X
j=1
h(u −d) ·t
j
, (v − w) ·t
j
i
Σ
+ h(u − d) · n, q
P
i
Σ
,
which is well-defined and therefore no additional Lagrange multipliers are required to realize the coupling
conditions. Also, in view of the character of the resulting variational forms in combination with the specifi-
cation of boundary conditions (2.2), we define the Hilbert spaces
H
1
?
(Ω
F
) = {v ∈ H
1
(Ω
F
) : v|
Γ
u
F
= 0}, H
1
?
(Ω
P
) = {w ∈ H
1
(Ω
P
) : w|
Γ
d
P
= 0},
H
1
?
(Ω
P
) = {q
P
∈ H
1
(Ω
P
) : q
P
|
Γ
p
P
P
= 0},
and the product space H
H = H
1
?
(Ω
F
) ×H
1
?
(Ω
P
) ×L
2
(Ω
F
) ×L
2
(Ω
P
) ×H
1
?
(Ω
P
). (2.4)
Consequently, we have the following weak form for the Biot-Stokes coupling: Find (u, d, p
F
, ϕ, p
P
) ∈ W
such that
2µ
f
((u), (v))
Ω
F
+
γµ
f
√
κ
hT
t
(u −d), T
t
vi
Σ
5
剩余24页未读，继续阅读
易小侠
- 粉丝: 6547
- 资源: 9万+
我的内容管理
展开
最新资源
- 多模态联合稀疏表示在视频目标跟踪中的应用
- Kubernetes资源管控与Gardener开源软件实践解析
- MPI集群监控与负载平衡策略
- 自动化PHP安全漏洞检测：静态代码分析与数据流方法
- 青苔数据CEO程永：技术生态与阿里云开放创新
- 制造业转型： HyperX引领企业上云策略
- 赵维五分享：航空工业电子采购上云实战与运维策略
- 单片机控制的LED点阵显示屏设计及其实现
- 驻云科技李俊涛：AI驱动的云上服务新趋势与挑战
- 6LoWPAN物联网边界路由器：设计与实现
- 猩便利工程师仲小玉：Terraform云资源管理最佳实践与团队协作
- 类差分度改进的互信息特征选择提升文本分类性能
- VERITAS与阿里云合作的混合云转型与数据保护方案
- 云制造中的生产线仿真模型设计与虚拟化研究
- 汪洋在PostgresChina2018分享：高可用 PostgreSQL 工具与架构设计
- 2018 PostgresChina大会：阿里云时空引擎Ganos在PostgreSQL中的创新应用与多模型存储
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
