OON THE SEDIMENTATION OF A DROPLET IN STOKES FLOW
AMINA MECHERBET ∗† Abstract.
This paper is dedicated to the analysis of the transport-Stokes equation whichdescribes sedimentation of inertialess suspensions in a viscous flow at mesoscopic scaling.First we present a global existence and uniqueness result for L ∩ L ∞ initial densitieswith finite first moment. Secondly, we consider the case where the initial data is thecharacteristic function of an axisymmetric bounded domain and investigate the regularityof its surface. Using spherical parametrisation, a hyperbolic equation for the evolution ofthe radius of the droplet is derived and we present a local existence and uniqueness result.Finally, we investigate the case where the initial shape of the droplet is spherical andshow that the solution corresponds to the Hadamard and Rybczynski result. We presentnumerical simulations in the spherical case. Introduction
In this paper, we consider the sedimentation of a cloud of rigid particles in a viscousfluid. At the mesoscopic scaling, it has been showed [7, 10] that the equation describingthe dynamics is the transport-Stokes problem in the case where inertia of both fluid andparticles is neglected:(1) ∂ t ρ + div(( u + κg ) ρ ) = 0 , on R + × R , − ∆ u + ∇ p = 6 πr κρg , on R + × R ,, div u = 0 , on R + × R ,,ρ (0 , · ) = ρ , on R . Here, the function ρ stands for the density of the particles, ( u, p ) are the velocity andpressure of the fluid, g is the gravity vector, R = r N is the radius of the particles where N the(large) number of particles in the suspension and κg = R ( ¯ ρ − ρ ) g represents the fall speedof one particle sedimenting under gravitational force. Note in particular that the sourceterm in the Stokes equation corresponds to 6 πr κgρ = N πR ( ρ p − ρ f ) gρ = φ ( ρ p − ρ f ) gρ where φ is the solid volume fraction of the suspension in the case | supp ρ | = 1.At the microscopic scaling, the motion and shape evolution of a blob has been studied ∗ Sorbonne Universit´es, Laboratoire Jacques-Louis Lions (UMR 7598), F-75005, Paris, France † Universit´e de Paris, Institut de Math´ematiques de Jussieu-Paris Rive Gauche (UMR 7586), F-75205,Paris, FranceEmail address: [email protected] project has received funding from the European Research Council (ERC) under the EuropeanUnions Horizon 2020 research and innovation program Grant agreement No 637653, project BLOC Mathe-matical Study of Boundary Layers in Oceanic Motion. This work was supported by the SingFlows project,grant ANR-18- CE40-0027 of the French National Research Agency (ANR).. a r X i v : . [ m a t h . A P ] J u l AMINA MECHERBET ∗† in [9, 11, 12]. Experimental and numerical investigations lead to the conclusion that aspherical cloud of particles slowly evolves to a torus. Precisely, the particles at top of thecloud leak away from the cluster and form a vertical tail. The decrease of the numberof particles at the vertical axis of the cloud leads to the apparition of the toroidal form.Moreover, it has been observed that the unstable torus breaks into two secondary dropletswhich deform into tori themselves in a repeating cascade.At the macroscopic scaling, Hadamard [4] and Rybczynski [13] considered independently acoupled Stokes-Stokes model describing sedimentation of liquid spherical drop in a viscousfluid assuming a uniform surface tension on the sphere. Using the Stokes stream functionfor axisymmetric flow, authors show that the spherical shape of the drop is preserved.We are interested in investigating the mesoscopic model by considering the transport-Stokes equation (1) when the initial density of the cloud is the characteristic function ofa bounded domain B . First we present a global existence and uniqueness result for thetransport-Stokes equation for L ∩ L ∞ initial densities with finite first moment. Secondly,we derive a hyperbolic equation describing the evolution of the surface of axisymmetricdrop B and present a local existence and uniqueness result. We investigate then the casewhere the initial drop B is spherical and show that we recover the result of Hadamard andRybczynski on both the transport-Stokes equation and the hyperbolic equation. Finally wepropose a numerical scheme for solving the hyperbolic equation and present some numericalsimulations for the spherical case.1.1. Description of the main results.
Existence and uniqueness of (1) has been provedin [7] for regular initial data ρ . The first step of this study is to extend the result for lessregular data allowing to tackle blob distribution. Note that, as explained in [7], if ( ρ, u )are solutions to equation (1), then( ˜ ρ ( t, x ) , ˜ u ( t, x )) = ( ρ ( t, x + tκg ) , u ( t, x + tκg )) , is solution to ∂ t ρ + div( ρu ) = 0 , on R + × R , − ∆ u + ∇ p = 6 πr κρg , on R + × R , div u = 0 , on R + × R ,ρ (0 , · ) = ρ , on R . Since 6 πr κg = − πr κ | g | e , without loss of generality, we consider in this paper thefollowing transport-Stokes problem:(2) ∂ t ρ + div( ρu ) = 0 , on R + × R , − ∆ u + ∇ p = − ρe , on R + × R , div u = 0 , on R + × R ,ρ (0 , · ) = ρ , on R . where e is the third vector of the standard basis in R .The first result is a proof of existence and uniqueness of solutions for the transport-Stokesproblem. N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 3
Theorem 1.1.
Let ρ ∈ L ( R ) ∩ L ∞ ( R ) a measure with finite first moment. There exitsa unique couple ( ρ, u ) ∈ L ∞ (0 , T ; L ( R ) ∩ L ∞ ( R )) × L ∞ (0 , T ; W , ∞ ( R )) satisfying thetransport-Stokes equation (2) for all T ≥ . Moreover, for all s ∈ [0 , T ] there exists aunique characteristic flow X ( · , s, · ) ∈ L ∞ (0 , T, W , ∞ ( R )) (cid:26) ∂ t X ( t, s, x ) = u ( s, X ( t, s, x )) , ∀ t, s ∈ [0 , T ] ,X ( s, s, x ) = x, ∀ s ∈ [0 , T ] , For all s, t ∈ [0 , T ] the diffeomorphism X ( s, t, · ) is measure preserving and we have ρ ( t, · ) = X ( t, , · ) ρ . This result ensures the well posedness of the transport-Stokes equation when the initialdensity is the characteristic function of a bounded domain B ⊂ R . The proof relies onstability estimates using the first Wasserstein distance W . Moreover, the regularity of thecharacteristic flow ensures that if ρ = 1 B then we have for all time ρ ( t, · ) = 1 B t where B t is transported along the flow . Consequently, in the second part of this paper we focus oninvestigating the regularity of the surface of the drop B t . We consider the case of initialaxisymetric domains B (invariant under rotations around the vertical axis e ) describedusing a spherical parametrization and a radius function r depending only on θ ∈ [0 , π ](3) B = r ( θ ) cos( φ ) sin( θ )sin( φ ) sin( θ )cos( θ ) , ( θ, φ ) ∈ [0 , π ] × [0 , π ] . The motivation of considering such domains is that the Stokes equation preserves theinvariance and ensures that B t is axisymetric. We set then c ( t ) = (0 , , c ( t )) ∈ B t theposition at time t of a reference point such that c (0) = 0 and write B t = c ( t ) + ˜ B t where(4) ˜ B t = r ( t, θ ) cos( φ ) sin( θ )sin( φ ) sin( θ )cos( θ ) , ( θ, φ ) ∈ [0 , π ] × [0 , π ] . Remark 1.1.
The reference point c is not necessarily the center of mass of the droplet B t .The decomposition B t = c ( t ) + ˜ B t with ˜ B t defined in (4) is valid as long as c ( t ) ∈ B t . Using the weak formulation of the transport-Stokes equation we derive a hyperbolicequation for the evolution of the radius r .(5) (cid:26) ∂ t r + ∂ θ rA [ r ] = A [ r ] ,r (0 , · ) = r . The operators A and A are defined in (16) and (17) in Proposition 3.1. See also AppendixA for a summary of the formulas. These operators depend non linearly and non locally onthe unknown r , they also depend on the reference point c . We emphasize that there is acoupling between the evolution of the radius r and the motion of the reference center c .Precisely, the velocity of c can be seen as a parameter in the model. In particular, if we AMINA MECHERBET ∗† choose c to be transported along the flow we get c = c [ r ] = (0 , , c [ r ] ) with(6) ˙ c [ r ] ( t ) = − (cid:90) π r ( t, ¯ θ ) sin(¯ θ ) (cid:18) −
12 sin (¯ θ ) (cid:19) d ¯ θ,c [ r ] (0) = 0 , see Proposition 3.1. We present a local existence and uniqueness result of ( r, c ) for Lipschitzfunctions r such that | r | ∗ = inf (0 ,π ) r ( θ ) > . Theorem 1.2.
Let r ∈ C , [0 , π ] such that | r | ∗ > . There exists T > and a unique r ∈ C (0 , T ; C , (0 , π )) satisfying the hyperbolic equation (5) . Moreover, there exists a uniqueassociated reference point c = c [ r ] ∈ C (0 , T ) satisfying (6) . Remark 1.2.
The same result holds true if the motion of the center c is defined in anotherway. The only properties needed is a uniform bound on ˙ c and a stability estimate withrespect to r if c = c [ r ] , see (34) . We finish the second part by investigating the spherical case. We first prove that,analogously to the Hadamard-Rybczynski result, the spherical shape is preserved in thetransport-Stokes model, see Corollary 3.3. The proof relies on the property proven byHadamard-Rybczynski which states that the normal component of the velocity of thefluid is constant on the surface of the sphere. This constant velocity is denoted v ∗ andcorresponds to the velocity fall of the center of the droplet c ∗ and is given by formula (42).In particular we present direct computations showing the Hadamard-Rybczynski property,see Lemma 3.2.Regarding the hyperbolic equation, we set r = 1 and distinguish two cases. If we choose c = c ∗ , then a straightforward computation shows that A [1] = 0 and hence r = 1 is solutionof the hyperbolic equation and we recover the Hadamard-Rybczynski result. On the otherhand, if ˙ c (cid:54) = ˙ c ∗ , we show that the solution r corresponds to a spherical parametrizationof the Hadamard-Rybczynski sphere B ( c ∗ ,
1) as long as the reference center c belongsto B ( c ∗ , c is given by (6), explicitcomputations show that | c ( t ) − c ∗ ( t ) | ≤ t ≥ t →∞ | c ( t ) − c ∗ ( t ) | = 1, seeProposition 3.5. This ensures that c ( t ) ∈ B ( c ∗ ,
1) for all time and shows global existenceof the solution of equations (5), (6).We finish the paper by proposing a numerical scheme in order to investigate the sphericalcase r = 1. First, we present numerical simulations for the solution of (5) with ˙ c fixedas in (6) and recover numerically Hadamard-Rybczynski solution. Second we investigate atest case for which ˙ c (cid:54) = ˙ c ∗ and is such that the center c ( t ) leaves the sphere B ( c ∗ ,
1) after t = 0 .
5. Numerical computations show the validity of Proposition (3.4) until t = 0 . r after t = 0 .
5. The last test case illustrates thesteady state i.e. c = c ∗ for which r = 1 is solution for all time. We finish by a discussionon the approximation scheme and possible future investigations.This paper is divided into three main sections, the first one is dedicated to the existenceand uniqueness of the transport-equation (2). The second section concerns the derivation N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 5 and analysis of the hyperbolic equation. The last subsection of the second part is dedicatedto the discussion on the link between the spherical case and the Hadamard-Rybczynskiresult. Eventually, in the last section, we present numerical results for the spherical case.2.
Existence and uniqueness of the transport-Stokes equation
In order to prove Theorem 1.1 we recall some existence, uniqueness and stability esti-mates for Stokes and transport equations.2.1.
Reminder on the Steady Stokes and transport equations.
Equation (2) is asteady Stokes problem coupled with a transport equation. We recall here some propertiesconcerning the Stokes problem on R and the transport equations. Proposition 2.1.
Let η ∈ L ∞ ( R ) ∩ L ( R ) , The unique velocity field u solution to theStokes equation: (cid:26) − ∆ u + ∇ p = η , on R div( u ) = 0 , on R ,is given by the convolution of the source term η with the Oseen tensor Φ(7) Φ( x ) = 18 π (cid:18) I | x | + x ⊗ x | x | (cid:19) . Moreover, u ∈ W , ∞ ( R ) and there exists a positive constant independent of the data suchthat: (8) (cid:107) u (cid:107) ∞ + (cid:107)∇ u (cid:107) ∞ ≤ C (cid:107) η (cid:107) L ∩ L ∞ . A proof can be found in [7, Lemma 3.18] in the case η ∈ X β where X β is defined in [7,Definition 2.5]. The proof is mainly the same when considering η ∈ L ∩ L ∞ . We recallnow a stability estimate using the first Wasserstein distance W which is well defined formeasures with finite first moment. The following Proposition uses arguments similar to [6,Proposition 3] and [5, Theorem 3.1]. Proposition 2.2 (Steady-Stokes stability estimates) . Let η , η ∈ L ( R ) ∩ L ∞ ( R ) anddenote by u and u the associated Stokes solution. For all compact subset K ⊂ R onecan show that there exists a constant depending on K such that (cid:107) u − u (cid:107) L ( K ) + (cid:107)∇ u − ∇ u (cid:107) L ( K ) ≤ C ( K ) W ( η , η ) . Moreover, given a density ρ ∈ L ∩ L ∞ , there exists a positive constant independent of thedata such that: (9) (cid:90) R | u ( x ) − u ( x ) | ρ ( dx ) ≤ C (cid:107) ρ (cid:107) L ∩ L ∞ W ( η , η )Since similar computations will be used thereafter, we present the proof of the formerProposition. AMINA MECHERBET ∗† Proof.
According to [14, Theorem 1.5], there exists an optimal transport map T such that η := T η and we have: W ( η , η ) = (cid:90) R | T ( y ) − y | η ( dy ) . This yields: (cid:90) K | u ( x ) − u ( x ) | dx = (cid:90) K (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R Φ( x − y ) η ( dy ) − (cid:90) R Φ( x − T ( y )) η ( dy ) (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ C (cid:90) K (cid:90) R | T ( y ) − y | min( | x − y | , | x − T ( y ) | ) η ( dy ) dx ≤ (cid:90) R (cid:90) K (cid:18) | x − y | + 1 | x − T ( y ) | (cid:19) dx | T ( y ) − y | η ( dy ) ≤ C ( K ) W ( η , η ) . The proof of the last formula (9) is analogous to the estimate above where we replace C ( K )by (cid:107) ρ (cid:107) L ∩ L ∞ . (cid:3) Given a velocity field having the same regularity as above, we recall now an existence,uniqueness and stability estimates for the transport equations. The stability estimatepresented below is analogous to [6, Proposition 3] which is adapted from [8].
Proposition 2.3.
Let u ∈ L ∞ (0 , T ; W , ∞ ( R )) and ρ ∈ L ∩ L ∞ , for all T > thereexists a unique solution η ∈ L ∞ (0 , T ; L ∩ L ∞ ) to the transport equation (10) (cid:26) ∂ t ρ + div( ρu ) = 0 ,ρ (0 , · ) = ρ . Moreover, given two velocity fields u i , i = 1 , , if we denote by ρ i the solution to theassociated transport equation , we have for all t ≥ s ≥ : (11) W ( ρ ( t ) , ρ ( t )) ≤ (cid:18) W ( ρ ( s ) , ρ ( s )) + (cid:90) ts (cid:90) R | u ( τ, x ) − u ( τ, x ) | ρ ( τ, x ) dxdτ (cid:19) e Q ( t − s ) , where Q i := (cid:107) u i (cid:107) L ∞ (0 ,T ; W , ∞ ) .Proof. Classical transport theory ensures the existence and uniqueness. Precisely, thecharacteristic flow satisfying(12) (cid:26) ∂ t X ( t, s, x ) = u ( s, X ( t, s, x )) , ∀ t, s ∈ [0 , T ] ,X ( s, s, x ) = x, ∀ s ∈ [0 , T ] , is well defined in the sense of Carath´eodory since u is L ∞ in time and Lipschitz regardingthe space variable. Moreover, the following formula hods true(13) ρ ( t, · ) = X ( t, s, · ) ρ ( s, · ) . N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 7
Now, consider two velocity fields u i ∈ L ∞ (0 , T ; W , ∞ ) and denote by X i its associatedcharacteristic flow. For all x (cid:54) = y , i = 1 , | X i ( t, s, x ) − X i ( t, s, y ) | ≤ | x − y | + (cid:90) ts | u i ( τ, X i ( τ, s, x )) − u i ( τ, X i ( τ, s, y )) | dτ ≤ | x − y | + Q i (cid:90) ts | X i ( τ, s, x ) − X i ( τ, s, y ) | dτ , which yields, using Gronwall’s inequality, for all t ≥ s ≥ X i ( s, t, · )) ≤ e Q i ( t − s ) . We recall that at time s ≥
0, according to [14, Theorem 1.5], one can choose an optimalmapping T s such that ρ ( s ) = T s ρ ( s ) and W ( ρ ( s ) , ρ ( s )) := (cid:90) | T s ( y ) − y | ρ ( s, dy ) , on the other hand, thanks to the flows X i we can construct a mapping T t at time t ≥ s such that ρ ( t ) = T t ρ ( t ) defined by(14) T t := X ( t, s, · ) ◦ T s ◦ X ( s, t, · ) . According to the definition of the Wasserstein distance and formulas (13), (14) we have: W ( ρ ( t ) , ρ ( t )) ≤ (cid:90) | T t ( x ) − x ) | ρ ( t, dx )= (cid:90) | T t ( X ( t, s, y )) − X ( t, s, y ) | ρ ( s, dy )= (cid:90) | X ( t, s, T s ( y )) − X ( t, s, y ) | ρ ( s, dy ) ≤ Lip( X ( t, s, · )) W ( ρ ( s ) , ρ ( s )) + (cid:90) | X ( t, s, y ) − X ( t, s, y ) | ρ ( s, dy ) . Now we have: (cid:90) | X ( t, s, y ) − X ( t, s, y ) | ρ ( s, dy ) ≤ (cid:90) ts (cid:90) | u ( τ, X ( τ, s, y )) − u ( τ, X ( τ, s, y ) | ρ ( s, dy ) dτ ≤ Q (cid:90) ts (cid:90) | X ( τ, s, y ) − X ( τ, s, y )) | ρ ( s, dy ) dτ + (cid:90) ts (cid:90) | u ( τ, x ) − u ( τ, x ) | ρ ( τ, dx ) dτ. Gronwall’s inequality yields: (cid:90) R | X ( t, s, y ) − X ( t, s, y ) | ρ ( s, dy ) ≤ (cid:18)(cid:90) ts (cid:90) R | u ( τ, x ) − u ( τ, x ) | ρ ( τ, dx ) dτ (cid:19) e Q ( t − s ) . AMINA MECHERBET ∗† Finally we get W ( ρ ( t ) , ρ ( t )) ≤ Lip( X ( t, s, · )) W ( ρ ( s ) , ρ ( s ))+ (cid:18)(cid:90) ts (cid:90) R | u ( τ, x ) − u ( τ, x ) | ρ ( τ, dx ) dτ (cid:19) e Q ( t − s ) , with Lip( X ( s, t, · )) ≤ e Q ( t − s ) . (cid:3) proof of the existence and uniqueness result. Proof of Theorem 1.1.
Let T ≥ ρ ∈ L ∞ ∩ L a measure with finite first moment.We construct a sequence of solutions as follows : Given ρ N we define ( u N , ρ N +1 ) as thesolution to the system: ∂ t ρ N +1 + div( u N ρ N +1 ) = 0 , on [0 , T ] × R , − ∆ u N + ∇ p N = − ρ N e , on [0 , T ] × R , div u N = 0 , on [0 , T ] × R ,ρ N +1 (0 , · ) = ρ , on R , here u N is given by u N = − Φ ∗ ρ N e and p N its associated pressure. We choose ρ ( t, · ) = ρ as first step. Since ρ N is transported by an incompressible fluid we have for all time t ∈ [0 , T ]: (cid:107) ρ N ( t ) (cid:107) L ∩ L ∞ ≤ (cid:107) ρ (cid:107) L ∩ L ∞ . Formula (8) from Proposition 2.1 yields (cid:107) u N (cid:107) W , ∞ ≤ C (cid:107) ρ N (cid:107) L ∩ L ∞ . This shows that u N is uniformly bounded in W , ∞ and admits a weakly-* convergingsubsequence to a limit u .On the other hand, applying formula (11) from Proposition 2.3 together with formula (9)from Proposition 2.2, we have: W ( ρ N +1 , ρ N ) ≤ e Q N t (cid:90) t (cid:90) R | u N ( τ, x ) − u N +1 ( τ, x ) | ρ N ( t, dx ) dτ , ≤ Ce Q N t (cid:107) ρ N (cid:107) L ∩ L ∞ (cid:90) t W ( ρ N ( τ ) , ρ N − ( τ )) dτ, with Q N := sup τ ≤ t Lip( u N +1 ( τ, · )) ≤ sup τ ≤ t (cid:107) u N +1 ( τ, · ) (cid:107) W , ∞ ≤ C sup τ ≤ t (cid:107) ρ N (cid:107) L ∩ L ∞ ≤ C (cid:107) ρ (cid:107) L ∩ L ∞ . Hence(15) (cid:107) W ( ρ N +1 , ρ N ) (cid:107) L ∞ [0 ,T ] ≤ (cid:0) e C (cid:107) ρ (cid:107) T C (cid:107) ρ (cid:107) L ∩ L ∞ T (cid:1) N (cid:107) W ( ρ , ρ ) (cid:107) L ∞ [0 ,T ] . Note that, if we set X N the characteristic flow associated to u N , we have (cid:90) | x | ρ N +1 ( dx ) = (cid:90) | X N ( t, , x ) | ρ ( dx ) ≤ (cid:90) | x | ρ ( dx ) + T sup [0 ,T ] (cid:107) u N ( t, · ) (cid:107) ∞ (cid:107) ρ (cid:107) , N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 9 which ensures that the sequence ( ρ N ) N ∈ N is in the space of finite first moment measures. Ifwe take T small enough, formula (15) shows that ρ N is a Cauchy sequence in the (complete)space of L ∞ functions from [0 , T ] in the complete space of finite first moment measuresmetrized by the Wasserstein distance W , see [15, Theorem 6.16]. Hence there exists alimit ρ such that: (cid:107) W ( ρ N , ρ ) (cid:107) L ∞ [0 ,T ] → N →∞ . Recall that for all compact sets K we have for all M > N ≥ (cid:107) u N − u M (cid:107) L ∞ ( | ,T ] ,L ( K )) + (cid:107)∇ u N − ∇ u M (cid:107) L ∞ (0 ,T ; L ( K )) ≤ C ( K ) (cid:107) W ( ρ N , ρ M ) (cid:107) L ∞ [0 ,T ] . Hence, u N | K and ∇ u N | K are Cauchy sequences in L ∞ (0 , T ; L ( K )) and admit a limit in L ∞ (0 , T ; W , ∞ ( K )). Finally u ∈ L ∞ (0 , T ; W , ∞ ∩ W , ).Thanks to the convergence, in the space of measure-valued functions, of ρ N to ρ and thestrong convergence of u N towards u in L ∞ (0 , T ; W , ) one can show that ( u, ρ ) satisfiesweakly the system: ∂ t ρ + div( uρ ) = 0 , on [0 , T ] × R , − ∆ u + ∇ p = − ρe , on [0 , T ] × R , div u = 0 , on [0 , T ] × R ,ρ (0 , · ) = ρ , on R . Moreover, if we assume that there exists two fixed-points ( u i , ρ i ), i = 1 ,
2, then estimate(15) (cid:107) W ( ρ , ρ ) (cid:107) L ∞ [0 ,T ] ≤ CT (cid:107) ρ (cid:107) e C (cid:107) ρ (cid:107) T (cid:107) W ( ρ , ρ ) (cid:107) L ∞ [0 ,T ] , ensures uniqueness for T > ρ and u do not blow up in finite time and this is ensuredby the following estimates: (cid:107) ρ ( t ) (cid:107) L ∩ L ∞ ≤ (cid:107) ρ (cid:107) L ∞ ∩ L , (cid:107) u ( t ) (cid:107) L ∞ + (cid:107)∇ u ( t ) (cid:107) L ∞ ≤ C (cid:107) ρ ( t ) (cid:107) L ∩ L ∞ . (cid:3) Analysis of the surface of the drop
Derivation of the hyperbolic equation.
In this part we investigate the contourevolution in the case where the initial blob is axisymmetric. Using a spherical parametriza-tion we set B = r ( θ ) cos( φ ) sin( θ )sin( φ ) sin( θ )cos( θ ) , ( θ, φ ) ∈ [0 , π ] × [0 , π ] . and denote by B t the domain at time t . In order to use a spherical parametrization weset c ( t ) = (0 , , c ( t )) the position at time t of a reference point and write B t = c ( t ) + ˜ B t ∗† where ˜ B t = r ( t, θ ) cos( φ ) sin( θ )sin( φ ) sin( θ )cos( θ ) , ( θ, φ ) ∈ [0 , π ] × [0 , π ] . The velocity of the point c ( t ) can be choosen arbitrarily and in particular can be choosensuch that c ( t ) is transported along the flow meaning that ˙ c = u ( t, c ).Using the convolution formula for the velocity field u together with the weak formulationof (2) we get Proposition 3.1. r satisfies the following hyperbolic equation (cid:26) ∂ t r + ∂ θ rA [ r ] = A [ r ] ,r (0 , · ) = r . In the case where the reference point c = (0 , , c ) is transported along the flow i.e. u ( c ) = ˙ c we have c = c [ r ] = (0 , , c [ r ] ) and ˙ c [ r ] ( t ) = − (cid:90) π r ( t, ¯ θ ) sin(¯ θ ) (cid:18) −
12 sin (¯ θ ) (cid:19) d ¯ θ,c [ r ] (0) = 0 , The operators A [ r ] and A [ r ] are defined as follows (16) A [ r ]( t, θ ) := − πr ( t, θ ) (cid:90) π (cid:90) π r ( t, ¯ θ ) sin(¯ θ ) − ∂ θ r ( t, ¯ θ ) cos(¯ θ ) β [ r ]( t, θ, ¯ θ, φ ) r ( t, ¯ θ ) sin(¯ θ ) (cid:16) r ( t, θ ) cos( φ ) − r ( t, ¯ θ ) (cid:110) cos(¯ θ ) cos( θ ) cos( φ ) + sin(¯ θ ) sin( θ ) (cid:111)(cid:17) d ¯ θdφ + ˙ c sin( θ ) r ( t, θ )(17) A [ r ]( t, θ ) := − π (cid:90) π (cid:90) π r ( t, ¯ θ ) sin(¯ θ ) − ∂ θ r ( t, ¯ θ ) cos(¯ θ ) β [ r ]( t, θ, ¯ θ, φ ) r ( t, ¯ θ ) sin(¯ θ ) (cid:16) − r ( t, ¯ θ ) sin( θ ) cos(¯ θ ) cos( φ )+ r ( t, ¯ θ ) cos( θ ) sin(¯ θ ) (cid:17) d ¯ θdφ − ˙ c cos ( θ ) . (18) β [ r ]( θ, ¯ θ, φ ) = r ( θ ) + r (¯ θ ) − r ( θ ) r (¯ θ )(sin( θ ) sin(¯ θ ) cos( φ ) + cos( θ ) cos(¯ θ )) . Remark 3.1.
The volume of the drop is conserved in time (cid:90) π ∂ t r ( t, θ ) r ( t, θ ) sin( θ ) dθ = 0 . Proof of Proposition 3.1.
In what follows we drop the dependencies with respect to timesince the operators A and A depend on t only through r ( t, · ). N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 11
Using the change of variable x = c ( t ) + ˜ x ∈ B t , ˜ x ∈ ˜ B t the weak formulation of thetransport equation writes (cid:90) T (cid:90) ˜ B t ∂ t ψ + ∇ ψ · ( u ( c ( t ) + · ) − ˙ c ) d ˜ x = 0 , ∀ ψ ∈ C ∞ c ([0 , T ] × R ) , with ˙ c = u ( c ). Since the flow preserves the rotational invariance, we define the sphericalparametrization of ˜ B t as follows:˜ B t = { ze ( θ, φ ) , ( θ, φ ) ∈ [0 , π ] × [0 , π ] , ≤ z ≤ r ( t, θ ) } , where e ( θ, φ ) = cos( φ ) sin( θ )sin( φ ) sin( θ )cos( θ ) . . Passing to the spherical parametrization in the weak formulation and doing an integrationby parts we get for all ψ compactly supported in (0 , T ) × R (cid:90) T (cid:90) ˜ B t ∂ t ψ ( t, ˜ x ) d ˜ x = (cid:90) T (cid:90) [0 ,π ] × [0 , π ] (cid:90) r ( t,θ )0 ∂ t ψ ( t, ze ( θ, φ )) z sin( θ ) dzdθdφdt = − (cid:90) T (cid:90) [0 ,π ] × [0 , π ] ψ ( t, r ( t, θ ) e ( θ, φ )) ∂ t r ( t, θ ) r ( t, θ ) sin( θ ) dθdφdt, (19)for the second term a direct integration by parts yields (cid:90) T (cid:90) ˜ B t ∇ ψ ( t, ˜ x )( u ( c ( t ) + · ) − ˙ c ) d ˜ x = (cid:90) T (cid:90) ∂ ˜ B t ψ ( u ( c ( t ) + · ) − ˙ c ) · ndσdt = (cid:90) T (cid:90) [0 ,π ] × [0 , π ] ψ ( t, r ( t, θ ) e ( θ, φ ))( u ( c ( t ) + r ( t, θ ) e ( θ, φ )) − ˙ c ) · s ( θ, φ ) dθdφdt, (20)where s is the surface element on ∂ ˜ B t such that the unit normal vector satisfies n = s | s | and we have(21) s ( θ, φ ) = s [ r ]( θ, φ ) = ∂ θ ˜ y × ∂ φ ˜ y = r sin( θ ) e ( θ, φ ) − r (cid:48) ( θ ) r ( θ ) sin( θ ) ∂ θ e ( θ, φ ) . Gathering (19), (20) and (21) and droping the dependencies with respect to ( t, θ ) we get(22) − ∂ t r + ( u ( c + re ) − ˙ c ) · e − ∂ θ rr ( u ( c + re ) − ˙ c ) · ∂ θ e = 0 . Hence we set A [ r ] = 1 r ( u ( c + re ) − ˙ c ) · ∂ θ e, A [ r ] = ( u ( c + re ) − ˙ c ) · e. (23)We recall that for all x ∈ R : u ( x ) = 18 π (cid:90) B t (cid:18) − | x − y | e − ( x − y ) · e | x − y | ( x − y ) (cid:19) , ∗† which can be reformulated using an integration by parts as follows(24) u ( x ) = − π (cid:90) ∂B t (cid:18) ( x − y ) | x − y | n ( y ) − ( x − y ) · n ( y ) | x − y | e (cid:19) dσ ( y ) . Using again the spherical parametrization of ∂ ˜ B t , we set y = c + ˜ y , where ˜ y = r ( t, ¯ θ ) e (¯ θ, ¯ φ )and x = c + ˜ x = c + r ( t, θ ) e ( θ, φ ) ∈ ∂B t , ( θ, φ ) ∈ [0 , π ] × [0 , π ] . We recall that the velocity does not depend on the azimuth angle φ hence we can set φ = 0.We define the operator U [ r ] as U [ r ]( t, θ ) = u ( c ( t ) + r ( t, θ ) e ( θ, U [ r ]( t, θ )= u ( c + r ( t, θ ) e ( θ, − π (cid:90) [0 ,π ] × [0 , π ] (cid:32) (cid:0) r ( θ ) e ( θ, − r (¯ θ ) e (¯ θ, ¯ φ ) (cid:1) · e (cid:12)(cid:12) r ( θ ) e ( θ, − r (¯ θ ) e (¯ θ, ¯ φ ) (cid:12)(cid:12) s [ r ](¯ θ, ¯ φ ) − (cid:0) r ( t, θ ) e ( θ, − r (¯ θ ) e (¯ θ, ¯ φ ) (cid:1) · s [ r ](¯ θ, ¯ φ ) (cid:12)(cid:12) r ( θ ) e ( θ, − r (¯ θ ) e (¯ θ, ¯ φ ) (cid:12)(cid:12) e (cid:33) d ¯ θd ¯ φ. We recall that A and A are given by A [ r ] = 1 r ( U [ r ] − ˙ c ) · ∂ θ e, A [ r ] = ( U [ r ] − ˙ c ) · e. (26)We first compute the components of the vector U [ r ]. For sake of clarity we use the shortcut β = | x − y | = | ˜ x − ˜ y | = | r ( θ ) e ( θ, − r (¯ θ ) e (¯ θ, ¯ φ ) | , and we have:(27) β = r ( θ ) + r (¯ θ ) − r ( θ ) r (¯ θ ) (cid:16) cos ( ¯ φ ) sin( θ ) sin(¯ θ ) + cos( θ ) cos(¯ θ ) (cid:17) . This yields:(28) U [ r ] = − π (cid:90) π (cid:90) π r ( θ ) cos ( θ ) − r (¯ θ ) cos(¯ θ ) β r (¯ θ ) sin(¯ θ ) × (cid:16) r (¯ θ ) sin(¯ θ ) − r (cid:48) (¯ θ ) cos(¯ θ ) (cid:17) cos( ¯ φ ) d ¯ θd ¯ φ , (29) U [ r ] = − π (cid:90) π (cid:90) π r ( θ ) cos( θ ) − r (¯ θ ) cos(¯ θ ) β r (¯ θ ) sin(¯ θ ) × (cid:16) r (¯ θ ) sin(¯ θ ) − r (cid:48) (¯ θ ) cos(¯ θ ) (cid:17) sin( ¯ φ ) d ¯ θd ¯ φ , (30) U [ r ] = − π (cid:90) π (cid:90) π r (¯ θ ) sin(¯ θ ) − r (cid:48) (¯ θ ) cos(¯ θ ) β r (¯ θ ) sin(¯ θ ) × (cid:110) − r ( θ ) sin( θ ) cos( ¯ φ ) + r (¯ θ ) sin(¯ θ ) (cid:111) d ¯ θd ¯ φ N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 13
We can now compute U [ r ] · e = u · e ( θ,
0) and U [ r ] · ∂ θ e = u · ∂ θ e ( θ, U [ r ] · e = − π (cid:90) π (cid:90) π r (¯ θ ) sin(¯ θ ) − r (cid:48) (¯ θ ) cos(¯ θ ) β r (¯ θ ) sin(¯ θ ) (cid:16) − r (¯ θ ) sin( θ ) cos(¯ θ ) cos( ¯ φ )+ r (¯ θ ) cos( θ ) sin(¯ θ ) (cid:17) d ¯ θd ¯ φ , (32) U [ r ] · ∂ θ e = − π (cid:90) π (cid:90) π r (¯ θ ) sin(¯ θ ) − r (cid:48) (¯ θ ) cos(¯ θ ) β r sin(¯ θ ) (cid:16) r ( θ ) cos( ¯ φ ) − r (¯ θ ) (cid:110) cos(¯ θ ) cos( θ ) cos( ¯ φ ) + sin(¯ θ ) sin( θ ) (cid:111)(cid:17) d ¯ θd ¯ φ . Finally if we assume u ( c ) = ˙ c we get:˙ c = u ( c ) = − π (cid:90) ∂ ˜ B t (cid:18) − ˜ y [˜ y | s + e ˜ y · s | ˜ y | (cid:19) dσ (˜ y ) , recall that | ˜ y | = r ( θ ) and since e ⊥ ∂ θ e we get:˜ y · s = r ( θ ) e ( θ, φ ) · (cid:0) r ( θ ) sin( θ ) e ( θ, φ ) − r (cid:48) ( θ ) r ( θ ) sin( θ ) ∂ θ e ( θ, φ ) (cid:1) = r ( θ ) sin( θ ) . This yields:˙ c = − π (cid:90) (cid:90) − cos ( θ ) (cid:0) r ( θ ) sin( θ ) cos( φ ) sin( θ ) − r (cid:48) ( θ ) r ( θ ) sin( θ ) cos( φ ) cos( θ ) (cid:1) = 0 , ˙ c = − π (cid:90) (cid:90) − cos ( θ ) (cid:0) r ( θ ) sin( θ ) sin( φ ) sin( θ ) − r (cid:48) ( θ ) r ( θ ) sin( θ ) sin( φ ) cos( θ ) (cid:1) = 0 . ˙ c = − π (cid:90) π (cid:90) π (cid:16) − cos ( θ ) (cid:0) r ( θ ) sin( θ ) cos( θ ) + r (cid:48) ( θ ) r ( θ ) sin ( θ ) (cid:1) + r ( θ ) sin( θ ) (cid:17) dθdφ , = − (cid:90) π (cid:0) r ( θ ) sin ( θ ) − r (cid:48) ( θ ) r ( θ ) cos( θ ) sin ( θ ) (cid:1) dθ , = − (cid:90) π (cid:18) r ( θ ) sin ( θ ) + 12 r ( θ ) (cid:16) − sin ( θ ) + 2 cos ( θ ) sin( θ ) (cid:17)(cid:19) dθ , = − (cid:90) π r ( θ ) (cid:16) − sin ( θ ) + 2 sin( θ ) (cid:17) dθ , = − (cid:90) π r ( θ ) sin( θ ) (cid:16) −
12 sin ( θ ) (cid:17) dθ < . ∗† We conclude by replacing formulas (26) and (31), (32) in (22). For the volume conservation,direct computations using (21) yield (cid:90) π ∂ t r ( t, θ ) r ( t, θ ) sin( θ ) dθ = (cid:90) π A [ r ]( t, θ ) r ( t, θ ) sin( θ ) − ∂ θ r ( t, θ ) A [ r ]( t, θ ) r ( t, θ ) sin( θ ) dθ = (cid:90) θ r sin( θ )( u ( c + re ( θ, − ˙ c ) · e ( θ, − r∂ θ r sin( θ )( u ( c + re ( θ, − ˙ c ) · ∂ θ e ( θ, (cid:90) θ ( u ( c + re ( θ, − ˙ c ) · s ( θ, dθ = 12 π (cid:90) π (cid:90) θ u ( c + re ( θ, φ )) · s ( θ, φ ) dθdφ − ˙ c (cid:90) π ∂ θ (cid:18) r ( t, θ ) sin ( θ ) (cid:19) dθ = 12 π (cid:90) ∂ ˜ B u ( c + x ) · n ( x ) dσ ( x )= 12 π (cid:90) ∂B u · n = 0 . (cid:3) Proof of the local existence and uniqueness Theorem 1.2.
This section isdevoted to the proof of local existence and uniqueness of a solution for equation (5). Given r ∈ C (0 , π ), we recall the definition of the following quantity | r | ∗ = inf (0 ,π ) r ( θ ) . Proof.
The main idea is to apply a fixed-point argument. We recall that the operators A and A are defined using the velocity field u defined in (25). It is possible to formulateotherwise the velocity u using a spherical parametrization of the droplet B t = { c ( t ) + ze ( θ, φ ) , (¯ θ, ¯ φ ) ∈ (0 , π ) × (0 , π ) , ≤ z ≤ r (¯ θ ) } . this yields the following formula for u (33) U [ r ]( θ ) = (cid:90) (0 ,π ) × (0 , π ) (cid:90) r (¯ θ )0 Φ( r ( θ ) e ( θ, − ze (¯ θ, ¯ φ )) z sin(¯ θ ) dzd ¯ θd ¯ φ, with Φ the Oseen tensor, see (7). With this definition, the operator U [ r ] satisfies thefollowing estimates for r ∈ W , ∞ such that | r | ∗ > |U [ r ]( θ ) | ≤ C (cid:90) (0 ,π ) × (0 , π ) (cid:90) r (¯ θ )0 z dz | r ( θ ) e ( θ, − ze (¯ θ, ¯ φ ) | sin(¯ θ ) d ¯ θd ¯ φ, ≤ (cid:107) r (cid:107) / ∞ (cid:112) | r | ∗ (cid:90) π sin(¯ θ ) d ¯ θ | e (¯ θ, ¯ φ ) − e ( θ, | N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 15 where we used the fact that | r ( θ ) e ( θ, − ze (¯ θ, ¯ φ ) | = z + r ( θ ) − zr ( θ ) e ( θ, · e (¯ θ, ¯ φ )= ( z − r ( θ )) + zr ( θ ) | e (¯ θ, ¯ φ ) − e ( θ, | ≥ zr ( θ ) | e (¯ θ, ¯ φ ) − e ( θ, | , we conclude using Lemma B.1. For the derivative of U [ r ] we use the shortcuts e = e ( θ, e = e (¯ θ, ¯ φ ), r = r ( θ ), ¯ r = r (¯ θ ) and obtain after an integration on z | ∂ θ U [ r ]( θ ) | ≤ C ( | r ( θ ) | + | ∂ θ r ( θ ) | ) (cid:90) π (cid:90) π (cid:90) r (¯ θ )0 z dz | r ( θ ) e ( θ, − ze (¯ θ, ¯ φ ) | sin(¯ θ ) d ¯ θd ¯ φ, = C (cid:107) r (cid:107) , ∞ (cid:90) π (cid:90) π (cid:90) r ( θ )0 z dz ( z − re · ¯ e ) + r (1 − ¯ e · e ) sin(¯ θ ) d ¯ θd ¯ φ, = C (cid:107) r (cid:107) , ∞ (cid:90) π (cid:90) π (cid:16) r (¯ θ ) + re · ¯ e log | re − ¯ re | r + r (cid:18) e · ¯ e − √ − e · ¯ e (cid:19) (cid:20) arctan (cid:18) r (¯ θ ) − re · ¯ er √ − e · ¯ e (cid:19) + arctan (cid:18) ¯ e · e √ − e · ¯ e (cid:19)(cid:21) (cid:17) sin(¯ θ ) d ¯ θd ¯ φ ≤ C (cid:107) r (cid:107) , ∞ (cid:32) (cid:107) r (cid:107) ∞ | r | ∗ (cid:12)(cid:12)(cid:12)(cid:12) log (cid:107) r (cid:107) ∞ | r | ∗ (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) π (cid:90) π sin(¯ θ ) d ¯ θd ¯ φ | e (¯ θ, ¯ φ ) − e ( θ, | + (cid:90) π (cid:90) π sin(¯ θ ) d ¯ θd ¯ φ (cid:112) − e ( θ, · e (¯ θ, ¯ φ ) (cid:33) , where we used the fact that z log( z ) is uniformly bounded and that | re − ¯ r ¯ e | ≥ | r | ∗ | e − ¯ e | .We conclude using Lemma B.1.Let r , r ∈ C (0 , π ), | r | ∗ , | r | ∗ >
0, reproducing the same arguments as previously we havethe following stability estimate |U [ r ]( θ ) − U [ r ]( θ ) |≤ (cid:90) (0 ,π ) × (0 , π ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) r (¯ θ ) r (¯ θ ) z dz | r e − z ¯ e | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin(¯ θ ) d ¯ θd ¯ φ + (cid:107) r − r (cid:107) ∞ (cid:90) (0 ,π ) × (0 , π ) (cid:90) r (¯ θ )0 (cid:18) z dz | r e − z ¯ e | + z dz | r e − z ¯ e | (cid:19) dzd ¯ θd ¯ φ ≤ C (cid:107) r − r (cid:107) ∞ (cid:32) (cid:107) r (cid:107) / ∞ + (cid:107) r (cid:107) / ∞ (cid:112) | r | ∗ + ( (cid:107) r (cid:107) ∞ + (cid:107) r (cid:107) ∞ ) × (cid:34) (cid:107) r (cid:107) ∞ + (cid:107) r (cid:107) ∞ ) (cid:18) | r | ∗ + 1 | r | ∗ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) ( (cid:107) r (cid:107) ∞ + (cid:107) r (cid:107) ∞ ) | r | ∗ | r | ∗ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:35)(cid:33) . ∗† On the other hand since ˙ c [ r ] is defined in (6) we get | ˙ c [ r ] | ≤ C (cid:107) r (cid:107) ∞ , | ˙ c [ r ] − ˙ c [ r ] | ≤ C (cid:107) r − r (cid:107) ∞ ( (cid:107) r (cid:107) ∞ + (cid:107) r (cid:107) ∞ ) . (34)Since A [ r ], A [ r ] are defined in (26) using the estimates of U [ r ] and ˙ c we obtain (cid:107) A [ r ] (cid:107) , ∞ ≤ C | r | ∗ (cid:18) (cid:107) r (cid:107) , ∞ (cid:18) | r | ∗ (cid:19)(cid:19) ( (cid:107)U [ r ] (cid:107) , ∞ + (cid:107) r (cid:107) ∞ ) , (35) (cid:107) A [ r ] (cid:107) , ∞ ≤ C (cid:16) (cid:107)U [ r ] (cid:107) , ∞ + (cid:107) r (cid:107) ∞ (cid:17) , (36) (cid:107) A [ r ] − A [ r ] (cid:107) ∞ ≤ K (cid:18) | r | ∗ , | r | ∗ , (cid:107) r (cid:107) ∞ , (cid:107) r (cid:107) ∞ (cid:19) (cid:107) r − r (cid:107) ∞ (37)Now, given r , we introduce Θ[ r ] the characteristic flow of the transport equation (5) (cid:26) ˙Θ[ r ]( t, s, θ ) = A [ r ]( t, Θ[ r ]( t, s, θ )) , Θ[ r ]( t, t, θ ) = θ. Thanks to the regularity of A [ r ] the characteristic flow is well defined and in particularthe characteristic curves do not intersect and satisfyΘ[ r ]( t, s, · ) ◦ Θ[ r ]( s, t, · ) = id. In particular since A [ r ](0) = A [ r ]( π ) = 0 we have Θ[ r ]( t, s,
0) = 0 and Θ[ r ]( t, s, π ) = π for all t, s . Thanks to this properties, for a given r the unique solution of the transportequation(38) (cid:26) ∂ t ˜ r + ∂ θ ˜ rA [ r ] = A [ r ] , ˜ r (0 , · ) = r , satisfies ddt ˜ r ( t, Θ[ r ]( t, , θ )) = A [ r ]( t, Θ[ r ]( t, , θ )) , since the characteristic curves are well defined and do not intersect we have(39) ˜ r ( t, θ ) = r (Θ[ r ](0 , t, θ )) + (cid:90) t A [ r ]( s, Θ[ r ]( s, t, θ )) ds. Hence we define the mapping L : C (0 , T ; C , (0 , π )) → C (0 , T ; C , (0 , π )) which associatesto each r the solution ˜ r of equation 38 defined by (39). Thanks to estimates (35), (36) and(37) the operator L satisfies for all r, r , r such that (cid:107) r (cid:107) , ∞ ≤ (cid:107) r (cid:107) , ∞ λ and | r | ∗ ≥ β | r | ∗ with β < < λ (cid:107)L [ r ]( t, · ) (cid:107) , ∞ ≤ (cid:107) r (cid:107) , ∞ + T C ( λ, β, (cid:107) r (cid:107) , ∞ , | r | ∗ ) |L [ r ]( t, · ) | ∗ > | r | ∗ − T C ( λ, β, (cid:107) r (cid:107) , ∞ , | r | ∗ ) , (cid:107)L [ r ]( t, · ) − L [ r ]( t, · ) (cid:107) ∞ ≤ C ( λ, β, (cid:107) r (cid:107) , ∞ , | r | ∗ ) T (cid:107) r ( t, · ) − r ( t, · ) (cid:107) ∞ . If we define the sequence ( r n ) n ∈ N such that r = r and r n +1 = L [ r n ] i.e. (40) (cid:26) ∂ t r n +1 + ∂ θ r n +1 A [ r n ] = A [ r n ] r n +1 (0 , · ) = r , N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 17 previous estimates ensure that, for T small enough, r n converges (up to a subsequence) tosome ¯ r ∈ C (0 , T ; C (0 , π )) satisfying equation (5). Moreover, we have ¯ r ∈ C (0 , T ; C , (0 , π ))and | ¯ r | ∗ >
0. Uniqueness of the fixed-point is ensured thanks to the former stabilityestimates. Eventually, we recover the existence and uniqueness of c [ r ] thanks to (34). (cid:3) Comparison with Hadamard-Rybczynski analysis.
In this section we compareour analysis to the result of Hadamard [4] and Rybczynski [13] who investigate the motionof a liquid spherical drop B falling in a viscous fluid, see also [1, 2, 3]. The equationsconsidered are Stokes equations on both fluid and drop domain. Denoting by ¯ ρ , ¯ µ (resp. ρ , µ ) the density and viscosity of the drop (resp. density and viscosity of the fluid). Authorsshow that B t = v ∗ t + B i.e. the spherical form of the droplet is preserved and the velocityfall of the droplet v ∗ is given by(41) v ∗ = 29 R µ ( ¯ ρ − ρ ) µ + ¯ µ ¯ µ + µ g. In the case where we drop the coefficient R µ ( ¯ ρ − ρ ) and set µ = ¯ µ = 1, we get(42) v ∗ = − e . In particular, it is shown that the velocity of both the exterior fluid u and interior fluid ¯ u satisfy the following property Lemma 3.2 (Hadamard-Rybczynski) . Let u = − Φ ∗ B e . v ∗ = − e . We have ( u ( e ( θ, − v ∗ ) · e ( θ,
0) = 0 , for all θ ∈ [0 , π ] . We present below a proof relying on direct computationsProof.
Let θ ∈ [0 , π ] and e ( θ, ∈ ∂B (0 , u ( e ( θ, − π (cid:90) ∂B (0 , (cid:18) ( e ( θ, − y ) · e | e ( θ, − y | n ( y ) − e ( θ, · y − | e ( θ, − y | e (cid:19) dσ ( y ) . We set Q ( θ ) = cos( θ ) 0 sin( θ )0 1 0 − sin( θ ) 0 cos( θ ) the rotation matrix such that e ( θ,
0) = Q ( θ ) e with e = (0 , ,
1) and use the change of variable y = Q ( θ ) ω , ω ∈ ∂B (0 ,
1) such that n ( y ) = y = Q ( θ ) n ( ω ) = Q ( θ ) w and dσ ( y ) = dσ ( w ). We drop the dependencies withrespect to θ and write − πu ( e ) = Q (cid:18)(cid:90) ∂B (0 , ( Qe ) − ( Qω ) | e − ω | ωdσ ( ω ) (cid:19) − (cid:90) ∂B (0 , ( Qe ) · ( Qω ) − | e − ω | e dσ ( ω )= ( Qe ) Q (cid:18)(cid:90) ∂B (0 , w | e − ω | dσ ( ω ) (cid:19) − Q (cid:18)(cid:90) ∂B (0 , ( Qω ) | e − ω | ωdσ ( ω ) (cid:19) − (cid:18)(cid:90) ∂B (0 , ω | e − ω | dσ ( ω ) (cid:19) e + (cid:18)(cid:90) ∂B (0 , | e − ω | dσ ( ω ) (cid:19) e . ∗† We have ( Qω ) = − sin( θ ) ω + cos( θ ) ω , direct computations yield (cid:90) ∂B (0 , w | e − ω | dσ ( ω ) = 4 π e , (cid:90) ∂B (0 , | e − ω | dσ ( ω ) = 4 π (cid:90) ∂B (0 , w | e − ω | ωdσ ( ω ) = 1615 πe , (cid:90) ∂B (0 , w | e − ω | ωdσ ( ω ) = 1415 2 πe , where e = (1 , ,
0) hence we get − πu ( e ) = cos( θ ) 4 π Qe − Q (cid:18) − sin( θ ) 1615 πe + cos( θ ) 1415 2 πe (cid:19) − π e + 4 πe = − cos( θ ) Qe π
15 + sin( θ ) 16 π Qe + 8 π e , which yields the desired result. (cid:3) Comparison with the transport-Stokes equation.
The above Lemma yields directlythe following result.
Corollary 3.3.
The solution ( u, ρ ) of the transport-Stokes equation (2) in the case where ρ = 1 B is given by u ( t, x ) = u ( x − v ∗ t ) , ρ ( t, x ) = ρ ( x − v ∗ t ) , (43) u = − Φ ∗ ρ e , ρ = 1 B (0 , . (44) In other words, the drop B t remains spherical for all time.Proof. Indeed we have using (43), (44) ∂ t ρ + ∇ ρ · u = ( ∇ ρ · ( u − v ∗ )) | ( ·− v ∗ t ) = 0 , we conclude using Lemma 3.2 since ∇ ρ = ns where s is the surface measure on thesphere and n the unit normal on the sphere. (cid:3) Comparison with the hyperbolic equation.
We are interested now in showing thatthe solution of the hyperbolic equation (5) corresponds also to the Hadamard-Rybczynskisolution i.e. c + ∂ ˜ B t = ∂B ( v ∗ t, , with ∂ ˜ B t = { r ( t, θ ) e ( θ, φ ) , ( θ, φ ) ∈ [0 , π ] × [0 , π ] } . First, in the case where the referencepoint c corresponds to the center c ∗ ( t ) := v ∗ t given by Hadamard-Rybczynski, the result isstraightforward since the source term A [ r ] of the hyperbolic equations becomes accordingto formula (26) A [ r ]( θ ) = ( U [ r ] − ˙ c ∗ ) · e ( θ,
0) = ( U [ r ]( θ ) − v ∗ ) · e ( θ, , which vanishes for r = 1 since θ (cid:55)→ U [1]( θ ) corresponds to the velocity θ (cid:55)→ u ( e ( · , r = 1 is a solution to the hyperbolic equation N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 19 in the case c = c ∗ = v ∗ t .In the general case ˙ c (cid:54) = v ∗ , by symmetry, it is enough to show that | c ( t ) + r ( t, θ ) e ( θ, − c ∗ | = 1 for all θ ∈ [0 , π ] and t. Equivalently, we consider the function ¯ r satisfying the above formula and show that itsatisfies the hyperbolic equation. This is shown in the following Proposition. Proposition 3.4.
Let r = 1 and ( r, c ) the solution of (5) with ˙ c (cid:54) = v ∗ . Denote by T > the maximal time of existence of the solution such that | c − c ∗ | ≤ with c ∗ = v ∗ t = − e t .Then r is given by r ( t, θ ) = − ( c − c ∗ ) cos( θ ) + (cid:113) − ( c − c ∗ ) sin ( θ ) , ( t, θ ) ∈ [0 , T ] × [0 , π ] and satisfies | c ( t ) + r ( t, θ ) e ( θ, − v ∗ t | = 1 for all θ ∈ [0 , π ] and t ≤ T. In other words ∂B t := c + ∂ ˜ B t = ∂B ( c ∗ , on [0 , T ] . Proof.
First, note that | c ( t ) + r ( t, θ ) e ( θ, − c ∗ | = 1 corresponds to(45) r + 2( c − c ∗ ) r cos( θ ) + ( c − c ∗ ) − , Computing the solutions of the quadratic equation (45) we denote by ¯ r the solution whichsatisfies ¯ r (0 , · ) = 1 given by¯ r ( t, θ ) = − ( c − c ∗ ) cos( θ ) + (cid:113) − ( c − c ∗ ) sin ( θ ) , which is well defined provided that | c − c ∗ | ≤
1. We aim to prove that ¯ r satisfies thehyperbolic equation (5). We have ∂ t ¯ r ( t, θ ) = − ( ˙ c − ˙ c ∗ ) ¯ r cos( θ ) + ( c − c ∗ ) ¯ r + ( c − c ∗ ) cos( θ ) , ∂ θ ¯ r ( t, θ ) = ¯ r ( t, θ ) sin( θ )( c − c ∗ ) ¯ r + ( c − c ∗ ) cos( θ ) . Direct computations using formula (26) yield( ∂ t ¯ r + ∂ θ ¯ rA [¯ r ])(¯ r + ( c − c ∗ ) cos( θ )) = − ( ˙ c − ˙ c ∗ )¯ r cos( θ ) − ( ˙ c − ˙ c ∗ )( c − c ∗ ) + ( U [¯ r ] − ˙ c ) cos( θ ) sin( θ )( c ∗ c ) − ( U [¯ r ] − ˙ c ) ( c − c ∗ ) sin ( θ ) .A [¯ r ](¯ r + ( c − c ∗ ) cos( θ )) = ¯ r ( U [¯ r ] − ˙ c ) sin( θ )+( U [¯ r ] − ˙ c ) sin( θ ) cos( θ )( c − c ∗ ) + ( U [¯ r ] − ˙ c ) cos( θ )¯ r + ( U [¯ r ] − ˙ c ) cos( θ ) ( c − c ∗ ) . Taking the difference between the two above formulas we obtain(¯ r + ( c − c ∗ ) cos( θ ))( ∂ t ¯ r + ∂ θ ¯ rA [¯ r ] − A [¯ r ]) = ( ˙ c ∗ − U [¯ r ]) · (¯ re ( θ,
0) + c − c ∗ ) . It remains to proof that the right hand side in the above formula is equal to zero. Indeedthe term ¯ r + ( c − c ∗ ) cos( θ ) = (cid:112) − ( c − c ∗ ) sin ( θ ) in the above left hand side cannot ∗† be identically null for all t and θ ∈ [0 , π ] since we are in the case ˙ c (cid:54) = v ∗ . We recall theformula of U [ r ] given in (25) U [ r ]( t, θ ) = − π (cid:90) [0 ,π ] × [0 , π ] (cid:32) (cid:0) r ( θ ) e ( θ, − r (¯ θ ) e (¯ θ, ¯ φ ) (cid:1) · e (cid:12)(cid:12) r ( θ ) e ( θ, − r (¯ θ ) e (¯ θ, ¯ φ ) (cid:12)(cid:12) s [ r ](¯ θ, ¯ φ ) − (cid:0) r ( t, θ ) e ( θ, − r (¯ θ ) e (¯ θ, ¯ φ ) (cid:1) · s [ r ](¯ θ, ¯ φ ) (cid:12)(cid:12) r ( θ ) e ( θ, − r (¯ θ ) e (¯ θ, ¯ φ ) (cid:12)(cid:12) e (cid:33) d ¯ θd ¯ φ. We recall that ¯ r is such that | c + ¯ r ( t, θ ) e ( θ, − c ∗ | = 1. We claim that for all θ ∈ [0 , π ]there exists γ ∈ [0 , π ] such that c + ¯ r ( t, θ ) e ( θ, − c ∗ = e ( γ, , and the mapping γ (cid:55)→ θ is bijective. Indeed, let θ ∈ [0 , π ], we search for γ ∈ [0 , π ] satisfying c + ¯ r ( t, θ ) e ( θ, − c ∗ = e ( γ, , which yields cos( γ ) = ( c − c ∗ ) + ¯ r ( θ ) cos( θ ) , sin( γ ) = ¯ r ( θ ) sin( θ ) . Note that θ (cid:55)→ ¯ r ( θ ) cos( θ ) is monotone indeed ∂ θ [ θ (cid:55)→ ¯ r ( θ ) cos( θ )] = − r ( θ ) sin( θ ) (cid:112) − sin ( θ )( c − c ∗ ) ≤ , moreover,[ θ (cid:55)→ ( c − c ∗ ) + ¯ r ( θ ) cos( θ )] θ =0 = 1 , [ θ (cid:55)→ ( c − c ∗ ) + ¯ r ( θ ) cos( θ )] θ = π = − θ (cid:55)→ ( c − c ∗ ) + ¯ r ( θ ) cos( θ ) ∈ [ − ,
1] and bijective. This ensures that γ (cid:55)→ θ is bijective and in particular we have γ = 0 when θ = 0 and γ = π when θ = π , see Figure1 for an illustration.Consequently, we introduce the change of variable c + ¯ r (¯ θ ) e (¯ θ, ¯ φ ) − c ∗ = e (¯ γ, ¯ φ ) := ω ∈ ∂B (0 ,
1) and we set x (cid:48) = c + ¯ r ( θ ) e ( θ, − c ∗ = e ( γ, ∈ ∂B (0 , γ = arccos(( c − c ∗ ) + ¯ r ( θ ) cos( θ )) dγ = ¯ r ( θ )¯ r ( θ ) + ( c − c ∗ ) cos( θ ) dθ N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 21 c ∗ c r ( θ ) θγ c + r ( θ ) e ( θ, Figure 1.
Illustration of the bijective application [0 , π ]: θ (cid:55)→ γs [¯ r ]( θ, φ ) dθ = ¯ r ( θ ) + ( c − c ∗ ) cos( θ )¯ r ( θ ) s [¯ r ]( θ, φ ) dγ = ¯ r ( θ ) + ( c − c ∗ ) cos( θ )¯ r ( θ ) ¯ r ( θ ) sin( θ )(¯ re ( θ, φ ) − ∂ θ ¯ r ∂ θ e ( θ, φ )) dγ = ¯ r sin( θ ) ¯ r sin( θ ) cos( φ )¯ r sin( θ ) sin( φ )¯ r cos( θ ) + ( c − c ∗ ) dγ = sin( γ ) e ( γ, φ ) dγ = s [1]( γ, φ ) dγ where we used the fact that sin( γ ) = ¯ r ( θ ) sin( θ ) and cos( γ ) = ¯ r ( θ ) cos( θ ) + ( c − c ∗ ). Weget eventually U [¯ r ]( t, θ ) = − π (cid:90) ∂B (0 , (cid:32) ( e ( γ, − ω ) · e | e ( γ, − ω | n ( ω ) − ( e ( γ, − ω ) · n ( ω ) | e ( γ, − ω | e (cid:33) dσ ( ω )= U [1]( γ ) , using lemma 3.2 and the fact that U [1]( · ) corresponds to u ( e ( · , U [1]( γ ) · e ( γ,
0) = v ∗ · e ( γ,
0) which yields using the fact that c + r ( θ ) e ( θ, − c ∗ = e ( γ, U [¯ r ]( t, θ ) · ( c + r ( θ ) e ( θ, − c ∗ ) = v ∗ · ( c + r ( θ ) e ( θ, − c ∗ ) , which concludes the proof. (cid:3) Proposition 3.4 suggets that the existence time of the solution depends on the choice of˙ c . We complete the analysis by showing that the choice for which c is transported alongthe flow i.e. ˙ c is given by (6) is such that | c − c ∗ | ≤ ∗† Proposition 3.5.
Let r = 1 and ( r, c ) the solution of (5) and (6) . Then for all time t ≥ we have c ( t ) ≤ c ∗ ( t ) , | c ( t ) − c ∗ ( t ) | ≤ and lim t →∞ c ( t ) − c ∗ ( t ) = − Proof.
We recall the formula for r given by Proposition 3.4¯ r ( t, θ ) = − ( c − c ∗ ) cos( θ ) + (cid:113) − ( c − c ∗ ) sin ( θ ) , and we have r = 1 − ( c − c ∗ ) − c − c ∗ ) r cos( θ ) . This yields˙ c − ˙ c ∗ = − (cid:90) π r ( t, θ ) sin( θ ) (cid:18) −
12 sin ( θ ) (cid:19) dθ − v ∗ = − v ∗ − (cid:0) − ( c − c ∗ ) (cid:1) (cid:90) π sin( θ ) (cid:18) −
12 sin ( θ ) (cid:19) dθ + 12 ( c − c ∗ ) (cid:16) − ( c − c ∗ ) (cid:90) π cos ( θ ) sin( θ ) (cid:18) −
12 sin ( θ ) (cid:19) dθ + (cid:90) π cos( θ ) sin( θ ) (cid:113) − ( c − c ∗ ) sin ( θ ) dθ (cid:17) = − v ∗ − (cid:0) − ( c − c ∗ ) (cid:1) −
12 ( c − c ∗ ) θ (cid:48) = π − θ .We get ˙ c − ˙ c ∗ = −
115 + 115 ( c − c ∗ ) . solving the ODE ˙ x = − + x with x (0) = 0 we obtain c ( t ) − c ∗ ( t ) = 1 − e t e t + 1 , this shows that c ≤ c ∗ , | c − c ∗ | ≤ c − c ∗ → − t → ∞ . (cid:3) Numerical simulations
We present in this section numerical simulations in the spherical case i.e. r = 1.In what follows we set T >
0, we consider
N, M, L ∈ N ∗ and define(∆ t, ∆ θ, ∆ φ ) = (cid:18) TN , πM , πL (cid:19) , we set for i = 0 , · · · , M , j = 0 , · · · , L , n = 0 , · · · , Nθ i = ∆ θ i, φ j = ∆ φ j, t n = ∆ tn. N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 23 ( θ i ) ≤ i ≤ M is a subdivision of [0 , π ], ( t n ) ≤ n ≤ N a subdivision of [0 , T ] and ( φ ) ≤ j ≤ L a subdi-vision of [0 , π ]. We discretise the radius and the center by setting r ( t, θ ) ∼ ( r ni ) ≤ n ≤ N ≤ i ≤ M , r ni = r ( t n , θ i ) , c ( t ) ∼ ( c n ) ≤ n ≤ N . We use the following classical upwind finite difference scheme for the hyperbolic equation.Given ( r ni ) ≤ i ≤ N we define ( r n +1 i ) ≤ i ≤ N as(46) r n +1 i = r ni − ∆ t ∆ θ A i,n (cid:26) r ni − r ni − if A i,n ≥ ,r ni +1 − r ni if A i,n ≤ , + ∆ tA i,n , i = 2 , · · · , M − , where A i,n = A [ r n ]( t n , θ i ), A i,n = A [ r n ]( t n , θ i ) are computed by discretizing the integrals.For i = 1 , M we note that A [ r ]( t,
0) = A [ r ]( t, π ) = 0 for all function r and t ≥
0, hencewe set(47) r n +11 = r n + ∆ tA ,n , r n +1 M = r nM + ∆ tA M,n . For a fixed time
T >
0, the following conditions ensure a uniform bound of max ≤ n ≤ N max ≤ i ≤ M | r ni | max ≤ k ≤ N max ≤ i ≤ M | A i,k | ∆ t ∆ θ < , max ≤ k ≤ N max ≤ i ≤ M | A i,k | ≤ C. For the evolution of the center we set c ∼ ( c n ) ≤ n ≤ N with c = 0. We distinguish threetest cases according the choice of the velocity of the center c .4.1. First test case.
The first test case corresponds to the case where ˙ c is given by (6).We set (∆ t, M, L ) = (10 − , , ,
24] using the upwind finite difference scheme (46). Precisely we presentthe vertical section of the surface droplet parametrized with θ (cid:55)→ ( r ( θ ) sin( θ ) , r ( θ ) cos( θ )), θ ∈ [0 , π ].Table 1 gathers the following values for each t = 0 , . , · · · , • the distance | c n − c ∗ n | between the discretized centers c ∗ and c • The errors E n defined by E n = max i ( | r ni − ¯ r ( t n , θ i ) | ) , E n = 1 n (cid:88) i ( | r ni − ¯ r ( t n , θ i ) | ) , where ¯ r is the exact solution given by Proposition 3.4¯ r ( t, θ ) = − ( c − c ∗ ) cos( θ ) + (cid:113) − ( c − c ∗ ) sin ( θ ) , ( t, θ ) ∈ [0 , T ] × [0 , π ] . • the relative error for the volume conservation V n defined by discretizing the integralVol( t ) := 2 π (cid:90) π r ( θ ) sin( θ ) dθ = 4 π ,V n = (cid:12)(cid:12)(cid:12)(cid:12) V ol n − π (cid:12)(cid:12)(cid:12)(cid:12) π . ∗† Figure 2.
First test case. Droplet evolution for t = 0 , , · · · , t . . . . . | c − c ∗ | . .
166 0 .
322 0 .
462 0 .
58 0 .
68 0 .
76 0 .
82 0 .
865 0 .
898 0 . E n ( × . − ) 2 . − .
04 0 .
17 0 .
40 0 .
77 1 .
29 2 .
02 3 .
02 4 .
37 6 .
16 8 . E n ( × . − ) 8 . − .
011 0 .
043 0 .
095 0 .
17 0 .
25 0 .
36 0 .
54 0 .
85 1 .
33 2 . V n ( × . − ) 0 .
08 0 .
15 0 .
17 0 .
16 0 .
22 0 .
48 1 .
07 2 .
06 3 .
45 5 .
21 7 . Table 1.
First test case. Evolution of | c − c ∗ | , E n , E n and V n for theupwind finite difference scheme (46)Numerical computations are in agreement with Proposition 3.4 in the sense that r ni ∼ ¯ r ( t n , θ i ) i.e. the numerical result corresponds to the Hadamard-Rybczynski sphere whichcan also be noticed on Figure 2. N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 25 t . . . . . | c − c ∗ | . .
166 0 .
322 0 .
46 0 .
58 0 .
68 0 .
751 0 .
81 0 .
84 0 .
87 0 . E n ( × . − ) 2 . − .
04 0 . .
54 1 .
15 2 .
11 3 . .
44 8 .
04 11 .
47 15 . E n ( × . − ) 8 . − .
018 0 .
067 0 .
138 0 .
271 0 .
573 1 .
080 1 .
864 2 .
993 4 .
511 6 . V n ( × . − ) 0 .
08 0 .
26 0 .
98 2 .
59 5 .
16 8 .
55 12 . .
74 21 .
08 25 .
36 29 . Table 2.
First test case. Evolution of | c − c ∗ | , E n , E n and V n for thefinite volume scheme (48), (51)We provide in Table 2 the results obtained using the following finite volume scheme(48) r n +1 i = r ni − ∆ t ∆ θ (cid:16) F i + − F i − (cid:17) + ∆ tS i,n , i = 1 , · · · , M − , based on the conservative formula(49) ∂ t r + ∂ θ ( rA [ r ]) = A [ r ] + r∂ θ A [ r ] , with(50) S i,n = A i,n + r ni A i +1 ,n − A i − ,n θ . The flux is defined as follows(51) F i + = A i + (cid:26) r ni if A i + ≥ ,r ni +1 if A i + ≤ , A i + = A i,n + A i +1 ,n . Numerical computations show that A i +1 / ≤ i = 0 , · · · , M and all n = 0 , · · · , N .This means that the finite volume scheme can be rewritten as r n +1 i = r ni − ∆ t ∆ θ A i,n + A i +1 ,n r ni +1 − r ni ) + ∆ tA i,n , Figure 3 represents the error E n and the volume conservation V n for the two schemeswith M = 50 and M = 100 on the same time interval [0 ,
25] with (∆ t, L ) = (0 . , Second test case.
The second test case is chosen such that ˙ c = λ ˙ c ∗ with λ >
1. Wehave | c ( t ) − c ∗ ( t ) | = t ( λ − | v ∗ | = t ( λ −
1) 415 , if we set for instance λ = , the time ¯ t for which we have | c (¯ t ) − c ∗ (¯ t ) | = 1 is ¯ t = 0 . t, M, L ) = (0 . , , t = 0 . c . ∗† Figure 3.
Evolution of the error E n and the volume conservation V n for:(fd1, fd2) the finite difference scheme with ( M = 50, M = 100) respectively,(fv1, fv2) the finite volume scheme with ( M = 50, M = 100) respectively t . . . .
35 0 . .
45 0 .
49 0 . . | c − c ∗ | .
02 0 .
22 0 .
42 0 .
62 0 .
72 0 .
82 0 .
92 1 .
00 1 .
02 1 . E n ( × . − ) 0 .
02 0 .
22 0 .
48 0 .
83 1 .
08 1 . .
86 2 .
51 2 .
82 64 . E n ( × . − ) 0 .
01 0 .
11 0 .
23 0 .
38 0 .
48 0 .
59 0 .
69 0 .
72 0 .
92 2 . i r ni .
98 0 .
78 0 .
58 0 .
38 0 .
28 0 . . . − . − . V n ( × . − ) 0 .
03 0 .
436 0 .
87 1 .
38 1 .
68 2 .
014 2 .
42 2 .
84 - -
Table 3.
Second test case. Evolution of E n , E n , min i r ni and V n third test case. We investigate the case where c = c ∗ using the upwind finite differ-ence scheme (46). In this case we recall that ¯ r = 1 is a steady solution to the hyperbolicequation. We present in Table 4 the values of E n , E n , V n .4.4. Discussion on the approximation scheme.
In this last part we discuss the maindifficulties encountered regarding the numerical solving of the hyperbolic equation. Severalschemes have been tested in addition of the upwind finite difference scheme (46) andthe finite volume scheme (48),(51). First, a Lax-Friedrichs scheme for the conservative
N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 27 t . . . . . E n ( × . − ) 4 . − . .
22 0 .
43 0 .
78 1 .
24 1 .
82 2 .
53 3 .
38 4 .
41 5 . E n ( × . − ) 2 . − .
06 0 .
16 0 . .
49 0 .
73 1 .
02 1 .
36 1 .
76 2 .
22 2 . V n ( × . − ) 0 .
08 0 .
28 0 .
47 0 .
67 0 .
86 1 .
06 1 .
25 1 .
44 1 .
63 1 .
82 2 . Table 4.
Third test case. Evolution of E n , E n and V n formulation (49), (48), (50) defined using the following fluxes F i + = r i +1 A i +1 ,n + r i A i,n − ∆ θ t ( r i +1 − r i ) , yields less accurate estimate than previous schemes from the first iterations ( t ∈ [0 , ∂ t r ( t, θ ) + ∂ θ G ( r ( t θ ) , θ ) = A [ r ] + F ( r ( t, θ ) , θ ) , with ∂ r G ( r, θ ) = A [ r ], F ( r, θ ) = ∂ θ G ( r, θ ). An analogous Lax-Friedrichs scheme with adiscretization of the additional source term has been implemented but yields less accurateresults from the first iterations ( t ∈ [0 , . r ( θ ) = 1 (cid:113) − cos ( θ ) , r ( θ ) = 1 (cid:113) − sin ( θ ) , θ ∈ [0 , π ] , depending on the considered orientation of the ellipsoid. Acknowledgement
The author would like to thank Matthieu Hillairet for introducing the subject and sharinghis experience. The author expresses her gratitude to Anne-Laure Dalidard for the fruitfuldiscussions and for helping overcoming the difficulties. The author is also grateful toJacques Sainte-Marie for helping with the numerical part. ∗† Appendix A. Summary of formulas for the operators A [ r ] , A [ r ] and U [ r ] e ( θ,
0) = sin( θ )0cos( θ )) A [ r ]( θ ) = 1 r ( θ ) ( U [ r ]( θ ) − ˙ c ) · ∂ θ e ( θ, , A [ r ]( θ ) = ( U [ r ]( θ ) − ˙ c ) · e ( θ, U [ r ] ( θ ) = − π (cid:90) π (cid:90) π K (¯ θ, θ, φ ) (cid:110) r ( θ ) cos ( θ ) − r (¯ θ ) cos(¯ θ ) (cid:111) cos( ¯ φ ) d ¯ θd ¯ φ U [ r ] ( θ ) = − π (cid:90) π (cid:90) π K (¯ θ, θ, φ ) (cid:110) r ( θ ) cos ( θ ) − r (¯ θ ) cos(¯ θ ) (cid:111) sin( ¯ φ ) d ¯ θd ¯ φ U [ r ] ( θ ) = − π (cid:90) π (cid:90) π K (¯ θ, θ, φ ) (cid:110) − r ( θ ) sin( θ ) cos( ¯ φ ) + r (¯ θ ) sin(¯ θ ) (cid:111) d ¯ θd ¯ φ K (¯ θ, θ, φ ) = r (¯ θ ) sin(¯ θ ) − r (cid:48) (¯ θ ) cos(¯ θ ) β [ r ](¯ θ, θ, φ ) r (¯ θ ) sin(¯ θ ) β [ r ](¯ θ, θ, φ ) = r ( θ ) + r (¯ θ ) − r ( θ ) r (¯ θ ) (cid:16) cos ( ¯ φ ) sin( θ ) sin(¯ θ ) + cos( θ ) cos(¯ θ ) (cid:17) U [ r ]( θ ) · e ( θ,
0) = − π (cid:90) π (cid:90) π K (¯ θ, θ, φ ) r (¯ θ ) (cid:16) − sin( θ ) cos(¯ θ ) cos( ¯ φ )+ cos( θ ) sin(¯ θ ) (cid:17) d ¯ θd ¯ φ U [ r ]( θ ) · ∂ θ e ( θ,
0) = − π (cid:90) π (cid:90) π K (¯ θ, θ, φ ) (cid:16) r ( θ ) cos( ¯ φ ) − r (¯ θ ) (cid:110) cos(¯ θ ) cos( θ ) cos( ¯ φ ) + sin(¯ θ ) sin( θ ) (cid:111)(cid:17) d ¯ θd ¯ φ Appendix B. Technical lemma
Lemma B.1.
There exists a positive constant
C > . satisfying sup θ ∈ [0 ,π ] (cid:32)(cid:90) [0 ,π ] × [0 , π ] sin(¯ θ ) | e (¯ θ, ¯ φ ) − e ( θ, | d ¯ θd ¯ φ + (cid:90) [0 ,π ] × [0 , π ] sin(¯ θ ) d ¯ θd ¯ φ (cid:112) − e ( θ, · e (¯ θ, ¯ φ ) (cid:33) ≤ C. N THE SEDIMENTATION OF A DROPLET IN STOKES FLOW 29
Proof.
In fact we can show a stronger result. The idea is to note that (cid:90) [0 ,π ] × [0 , π ] sin(¯ θ ) d ¯ θd ¯ φ | e (¯ θ, ¯ φ ) − e ( θ, | d ¯ θd ¯ φ = (cid:90) ∂B (0 , dσ ( y ) | e ( θ, − y | . Let θ ∈ [0 , π ]. We set Q ( θ ) = cos( θ ) 0 sin( θ )0 1 0 − sin( θ ) 0 cos( θ ) the rotation matrix such that e ( θ,
0) = Q ( θ ) e with e = (0 , ,
1) and use the change of variable y = Q ( θ ) ω , ω ∈ ∂B (0 , | Q ( e − ω ) | = | ( e − ω ) | and dσ ( y ) = dσ ( w ). This yields (cid:90) ∂B (0 , dσ ( y ) | e ( θ, − y | = (cid:90) ∂B (0 , dσ ( y ) | e − y | = 4 π. We apply the same idea for the second integral using the fact that e ( θ, · e (¯ θ, ¯ φ ) = Q ( θ ) e · Q ( θ ) ω = e · ω . (cid:3) References [1]
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