Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies - Fluid-Structure Interaction and Biomedical Applications - page 305

We complete ( 4.86 )-( 4.88 ) with initial conditions

G.0/ D G 0 ; /D 0 ; ! /D ! 0 ; (4.89)

where initial data satisfy the initial no-contact assumption:

. 0 ;! 0 / 2 R d R d G 0 2fX D .X 1 ;:::;X d / 2 R d s.t. X d >1g: (4.90)

Because of a symmetry arguments that we skip for conciseness, discussing the
possibility of contact in solutions to ( 4.86 )-( 4.89 ) reduces to determine whether
in solutions of the form G.t/ D .0;:::;h.t/ C 1/, .t/ D .0;:::; h.t//, !.t/ D 0,
the distance function h might vanish in finite time or not. For such solutions ( 4.86 )-
( 4.89 ) reduces to

m 1 h DhF.h/C mg e d ;

(4.91)

where, denoting B h WD B..0;:::;0;.1C h//;1/ for arbitrary h>0,wehave:

Z

F.h/D

@B h T

. u h ;p h /nd e d ;

with . u h ;p h / solution to

u h rp h D 0; in R d

C n B h ;

(4.92)

r u h D 0; in R d

C n B h ;

(4.93)

u h D e d ; on @B h ; (4.94)

u h D 0; on @ R d and at infinity: (4.95)

C

An analytical expression for the drag force F.h/is provided in [ 7 ] and asymptotic
expansions on the basis of lubrication approximations are also given in [ 10 ]. We
present here an alternative approach for extracting the size of this drag when h<<1
(see [ 27 , 41 ]):

Proposition 4.9. Let h>0 . Then setting

n u 2 C c . R d

C n B h / s.t. r u D 0 and u j @B h D e d o ;

Y h WD

there holds:

F.h/D inf ( Z

) :

2

C n B h jr u j

I u 2 Y h (4.96)

R d