The Stokes flow induced by the motion of an elastic massless filament immersed in a twodimensional fluid is studied. Initially, the filament is deviated from its equilibrium state and the fluid is at rest. The filament will induce fluid motion while returning to its equilibrium state. Two different test cases are examined. In both cases, the motion of a fixedend massless filament induces the fluid motion inside a square domain. However, in the second test case, a deformable circular string is placed in the square domain and its interaction with the Stokes flow induced by the filament motion is studied. The interaction between the fluid and deformable body/bodies can become very complicated from the computational point of view. An immersed boundary method is used in the present study. In order to substantiate the accuracy of the numerical method employed, the simulated results associated with the Stokes flow induced by the motion of an extending star string are compared well with those obtained by the immersed interface method. The results show the ability and accuracy of the IBM method in solving the complicated fluidstructure and fluidmediated structurestructure interaction problems happening in a wide variety of engineering and biological systems.
There are abundant scientific and engineering applications where a flexible structure is immersed in a viscous incompressible fluid. The motion of the structure can induce the fluid flow motion and vice versa. The combination of the fluid and structure motions constitutes the socalled fluidstructure interaction (FSI) problem (see [
Peskin [
Our interest in IBM is motivated by the ability of this method in simulating a wide range of complex physical phenomena, which share the essential difficulties of the problems considered in the present work. The focus is mainly on the case of very low Reynolds number because many interesting applications occur in this regime. In the present work, the Stokes equations are solved in the presence of immersed boundaries on an unbounded domain. A number of researchers have used the IBM method to simulate the cellular and subcellular biological processes that occur at a very low Reynolds number [
In the present work, the immersed boundaryinduced Stokes flow is studied using IBM. Unlike most of the previous studies, the fluid in the present work is initially at rest. The motion of the immersed boundaries induces flow motion, which lies in the Stokes flow regime. The following three test cases are considered: (i) Stokes flow induced by motion of an extending star string (FSI); (ii) Stokes flow induced by motion of a filament fixed at one end (FSI); (iii) Stokes flow induced by motion of the same filament considered in the second test case, but now deforming a secondary elastic immersed boundary placed in the flow domain (fluidmediated structurestructure interaction). In order to benchmark the numerical simulation carried out in the present work, the results of the first test case are compared with those obtained by the immersed interface method [
The flow induced by the elastic boundary motion is assumed to be in the Stokes regime and hence it is governed by the Stokes equations. The Stokes flow refers to the highly viscous fluid flow, in the limit where the Reynolds number tends to zero and both the inertial and convection terms are omitted from the NavierStokes equations [
Equations (
It is worth mentioning that the density force
It may be noted that all the variables and forces described through (
The transformation between the Eulerian and Lagrangian variables can be realized by the Dirac delta function [
A staggered grid (marked nodal points) is used in the Lagrangian coordinate system (see Figure
Schematic representation of the Lagrangian coordinate system for a flexible filament located in the Eulerian grids.
Suppose that
It was mentioned in Section
We now consider the equations connecting the fluid lattice and the immersed boundary points. Since the positions of the immersed boundary points generally do not coincide with those of the lattice points, we have to interpolate the velocity field from the fluid lattice to these points and spread the Lagrangian force from the immersed boundary points to the nearby lattice points of the fluid. This can be done by introducing a sufficiently smooth approximation to the Dirac delta function:
The discretized form of the local fluid velocity at the structure position
The numerical algorithm used in the present work for simulating the fluidstructure interactions as well as the fluidmediated structurestructure interactions can be summarized as follows.
Compute Lagrangian forces
Communicate Lagrangian forces
Solve fluid equations from (
Find the local fluid velocity at the immersed boundary position
Update immersed boundary position using
In this test case, we study the motion of a nonequilibrium star string immersed in an incompressible fluid while relaxing to its circular equilibrium shape. The initial velocity and pressure of the fluid are set to zero, and the only driving force is the string tension. The fluid flow and the string motion are fully coupled. At equilibrium, the velocity is zero and the pressure is piecewise constant inside and outside the balloon. The initial shape of the distorted string is expressed in the cylindrical coordinates as
(a) Lagrangian forces associated with the elastic boundary at
Velocity and pressure fields at time
Figure
Interface of the star balloon at different times. The dotted circle represents the unstretched interface with
It can be seen from Figure
According to Figure
Plot of
The instantaneous maximum and minimum radii of the boundary are obtained from the following relations:
In this section, the Stokes flow induced by the motion of a bending filament is examined. An initial bending force is applied to the filament. Initially the fluid is at rest. After the first time step, the Lagrangian points representing the filament boundary move from their initial positions in the flow direction. This leads the points to become too close to (or too far away from) each other. To keep the inextensibility of the filament, a (relatively) large stiffness coefficient is applied. Under such conditions, only the angle between the nodes varies, but the distance between them remains unchanged. In addition, to circumvent the instability arising when proper tension and bending forces are not applied, very small time steps are considered. As shown in Figure
Initial geometry of the filament showing Lagrangian points.
Eulerian (a) and Lagrangian (b) forces at
By inserting the Eulerian forces to the flow field equations and solving them, the velocity and pressure fields can be obtained at different time steps. The values of various constants considered for the case of the bending filament are given in Table
Values of constants associated with the case of bending filament.
Domain  (−1, 1) 
Number of Eulerian grid points  64 

30 

0.001 

2 

0.0004 

1 
Initial radius  0.25 
Figure
Contours of
The pressure contours associated with the different times shown in Figure
Pressure contours at different time steps of 0.0, 0.008, 0.016, and 0.032 corresponding to the frames (a), (b), (c), and (d), respectively.
In this last test case, the motion of an initially bent filament in a square cavity causes the fluid motion, which in turn leads to the motion and deformation of an initially circular elastic boundary placed somewhere in the cavity. It may be noted that, for an immersed boundary to move in a fluid, either the fluid should have enough velocity to move the boundary or the initial deformation of the boundary itself would cause the fluid motion. In the present example, both of these mechanisms are present simultaneously.
The secondary immersed boundary considered here is an elastic circular string, which is initially in its equilibrium state without any deformation or internal forces. For this boundary, the tension forces are applied to avoid the Lagrangian points representing the boundary to move too far from (or too close to) each other. In addition, no bending force has been applied to this boundary in order to allow the angle between the points to change freely. As shown in Figure
Values of the constants associated with the filament and circular boundary.
Domain  (−1, 1) 
Number of Eulerian grid points 


30 

100 

2 

0.01 

0.001 

1 

0.0005 
Initial radius 1  0.25 
Initial radius 2  0.3 
Lagrangian coordinate systems defined in the Eulerian solution domain.
Initial configurations of the flexible boundaries immersed in the square flow domain.
For the circularimmersed boundary, the stiffness coefficient is set to be relatively small to allow the points to move easily far away from each other. At the outset of the calculations, this boundary exerts no force on the initially quiescent fluid domain. In fact, the fluid motion is induced by the motion of the filament, which is initially bent. The moving fluid can now exert forces on the circular boundary and move and deform it. The Lagrangian forces associated with the immersed boundaries and the consequent Eulerian forces imposed on the fluid domain by theses boundaries at time
Eulerian forces imposed by the immersed boundaries on the fluid (a) and Lagrangian forces associated with the immersed boundaries (b) at time
Figure
Contours of
In the present work, the Stokes flow induced by the motion of an elastic massless body immersed in a twodimensional fluid is studied using immersed boundary method. Initially, the immersed body is unstable and the fluid is at rest. The body can induce fluid motion while returning to its equilibrium (stable) state leading to a fluidstructure interaction problem. In the current study, two different test cases are examined. In both of the cases, the motion of a massless filament fixed at one end induces the fluid motion inside a square domain. However, in the second test case, a deformable circular string is placed in the square domain and its interaction with the Stokes flow induced by the filament motion is studied. In order to verify the accuracy of the numerical method employed, the simulated results associated with the Stokes flow induced by the motion of an extending star string are compared well with those obtained by the immersed interface method. The results show the ability and accuracy of the IBM method in solving the complicated FSI problems happening in a wide variety of engineering and biological systems.
The authors declare that they do not have conflict of interests regarding the publication of this paper.