A Space-Time Adaptive Algorithm to Illustrate the Lack of Collision of a Rigid Disk Falling in an Incompressible Fluid

A Derivation of the Penalty Method (2.1)

We present the derivation of the penalty formulation (2.1) used to approximate the motion of the rigid disk inside the cavity. We start from the physical model presented in [12]. We study the motion of a rigid disk of radius R inside a bounded, convex cavity Ω ⊂ ℝ 2 filled with a fluid of constant viscosity and density. The fluid is governed by the incompressible Navier–Stokes equations while the dynamic of the rigid disk is ruled by the Newton law. Given a final time T ∈ ( 0 , + ∞ ] , we denote by ℬ ⁢ ( 𝑿 ⁢ ( t ) ) the disk centered at 𝑿 ⁢ ( t ) , where t ∈ [ 0 , T ] . The equations of motions reads [12]: find ( 𝑿 , 𝐮 , p , 𝑽 , ω ) the solutions of

together with

where 𝐠 is gravitational acceleration, μ ℱ and ρ ℱ denote the viscosity and the density of the fluid, ρ ℬ the density of the disk, m and J its mass and inertia. In the above system, ( 𝐮 , p ) are the velocity and pressure fields of the fluid and ( 𝑽 , ω ) the translational and rotational speeds of ℬ ⁢ ( 𝑿 ⁢ ( t ) ) . Here above we note

Note that since 𝐠 is applied at the center of mass, its angular momentum satisfies in particular

which is also true for any constant vector 𝑽 inside the disk, i.e.,

To derive equations (2.1), the first step consists to establish a variational formulation that can easily be approximated with Galerkin methods, that is to say involving test functions that are defined on the whole cavity Ω. We follow the formulation introduced in [15]. Let us consider the problem at a given time t, and to lighten the presentation, we drop the dependence of 𝑿 upon the time. To extend the velocity field inside the disk, one may consider the space of rigid motion inside ℬ ⁢ ( 𝑿 ) given by

that has the equivalent definition (see for instance [27])

We now multiply the momentum equation of (A.1) by a test function 𝐯 ∈ H ℬ ⁢ ( 𝑿 ) and using the fact that for free divergence field

one has after integration by part

Using the Newton’s laws (A.2) and the fact that 𝐯 = 𝑽 v + ω v ⁢ ( 𝐱 - 𝑿 ) ⊥ in ℬ ⁢ ( 𝑿 ) , one can write that

Moreover, one has that

and

Thus since 𝐮 , 𝐯 ∈ H ℬ ⁢ ( 𝑿 ) , we can prove that

which leads to

Finally, since D ⁢ ( 𝐮 ) | ℬ ⁢ ( 𝑿 ) = D ⁢ ( 𝐯 ) | ℬ ⁢ ( 𝑿 ) = 0 (in particular div ⁡ 𝐮 = div ⁡ 𝐯 = 0 inside the disk), it is possible to write the previous equation on the whole domain Ω, which leads to the variational formulation

where we note

The constrains D ⁢ ( 𝐮 ) B ⁢ ( 𝑿 ) = D ⁢ ( 𝐯 ) B ⁢ ( 𝑿 ) = 0 can be approximated by a penalization of the viscous term. Thus the variational formulation can be written with test functions taken in H 0 1 ⁢ ( Ω ) by considering (we keep the same notations for the unknowns to lighten the presentation)

where

Written in the strong form, the approximated variational formulation gives

with div ⁡ 𝐮 = 0 . To obtain (2.1), we finally have to consider equations to compute the translational and rotational velocities inside the disk. In our approach, we assume that we are in a symmetric situation and that the disk does not spin. Thus it remains to have an equation to update only the position of the disk that is obtained by approximating the velocity inside the disk by taking its mean value

Observe that the last equation reduces to 𝑿 ˙ = 𝐮 in ℬ ⁢ ( t ) if 𝐮 is constant in ℬ ⁢ ( t ) , that is expected when ε → 0 . Thus it is justified to use (A.3) to approximate the motion of the disk.

B Justification of (2.8)

We now prove an a posteriori error estimate for the time discretization of a simplified problem. We simplify the first equation of (2.1) and consider the following problem:

where ρ 𝑿 and μ 𝑿 are still defined by (2.2). Here 𝐮 𝑿 is zero on the boundary ∂ ⁡ Ω and 𝑿 ⁢ ( 0 ) = 𝑿 0 .

We consider the forward Euler method. Given a integer N > 0 and a partition 0 = t 0 < t 1 < … < t N = T of [ 0 , T ] , let τ n + 1 = t n + 1 - t n , τ = max n = 0 , 1 , 2 , … , N - 1 ⁡ τ n + 1 . For every n = 0 , 1 , 2 , … , N - 1 , given 𝑿 n , we compute first ( 𝐮 𝑿 n , p 𝑿 n ) solution of

then 𝑿 n + 1 solution of

We assume that there exists a unique

solution of (B.1), that d ( ℬ ( 𝑿 ( t ) ; ∂ Ω ) > 0 for all t ∈ [ 0 , T ] , and that 𝑿 n , 𝐮 𝑿 n , p 𝑿 n converge to 𝑿 ⁢ ( t n ) , 𝐮 𝑿 ⁢ ( t n ) , p 𝑿 ⁢ ( t n ) when τ goes to zero, and prove an a posteriori error estimate.

Observe that problem (B.1) reads: starting from 𝑿 ⁢ ( 0 ) = 𝑿 0 , find 𝑿 ⁢ ( t ) , for t > 0 , such that

where, for any 𝒀 ∈ ℝ 2 such that ℬ ⁢ ( 𝒀 ) is compactly supported in Ω, 𝑭 ⁢ ( 𝒀 ) is given by

and ( 𝐮 𝒀 , p 𝒀 ) ∈ ( H 0 1 ⁢ ( Ω ) ) 2 × L 0 2 ⁢ ( Ω ) is the weak solution of

where μ 𝒀 , ρ 𝒀 are defined as in (2.2). In order to prove an a posteriori error estimate, we need to prove that 𝑭 is Lipschitz.

The key point to prove this lemma consists in mapping ℬ ⁢ ( 𝒀 ) into ℬ ⁢ ( 𝒁 ) as in [11]. For details we refer to [6, Appendix B]. The fact that estimate (B.3) blows up when d ⁢ ( ℬ ⁢ ( 𝑿 ) ; ∂ ⁡ Ω ) approaches zero seems reasonable; it is due to the mapping from ℬ ⁢ ( 𝒀 ) into ℬ ⁢ ( 𝒁 ) which becomes singular. However, we do not know if the power one (on d) is sharp.

Our main result is then the following.