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In parallel with this fixed system, we also introduce EQU local moving Cartesian coordinate systems with origins EQU at the centers of spheres coinciding with their centers of mass (in what follows, systems EQU ) so that sphere EQU does not move relative to system EQU.
Therefore, the radius vector EQU of any point of the space can be represented in the form EQU, where EQU is the radius vector of this point relative to the local system EQU.
The motion of the spheres is described by the equations where and EQU are, respectively, the translational and angular velocities of sphere EQU, EQU is its moment of inertia, EQU is the total force and EQU is the total torque (here and in what follows, all torques corresponding to particle EQU are considered relative to its center) acting on sphere EQU where EQU and EQU are the force and torque acting on sphere EQU due to the external force field and EQU and EQU are the force and torque exerted by the fluid on sphere EQU EQU is the surface of sphere EQU at the time EQU and EQU is the outward unit vector to this surface.
The fluid is described by the same equations as in the case of the absence of particles but defined only in the domain outside the spheres EQU, EQU, where EQU.
The problem is to determine the velocity and pressure fields of the fluid induced by moving particles in it as well as the forces and torques exerted by the fluid on the particles.
For the solution of this problem, we use the method of induced forces proposed in CITE and developed in CITE.
We extend the quantities EQU, EQU, and EQU defined in EQU and for EQU to the domains EQU occupied by the spheres as follows: We extend the external force field EQU given for EQU to the domains EQU so that in the entire space, it is described by the same analytic expression as in the domain EQU.
(For short, we call this extension an analytic extension.)
The nonzero extension of the fluid pressure defined by and the analytic extension of the external force field differ from the usually accepted zeroth extension of both the fluid pressure CITE and the external force field CITE to the domains EQU.
It is worth noting that the considered problem can be solved for various extensions of the quantities EQU and EQU to the domains EQU including the zeroth extension CITE and the extensions used in the present paper.
However, the complexity of calculations and the possibility of interpretation of certain results essentially depend on the specific method of extension.
If the fluid pressure is extended to the domains EQU according to relation , then the function EQU defined in the entire space has a discontinuity at the surfaces EQU.
For this reason, in what follows, we write the fluid pressure at the point EQU as EQU.
Note that for the analytic extension of the external force field EQU to the domains EQU, the first two terms in represent the pressure of the unbounded fluid without particles in the external force field EQU.
In the case where the fluid velocity satisfies the stick boundary conditions at the surfaces of the spheres CITE (this case is considered in the present paper), condition ensures the continuity of the function EQU given in the entire space at EQU.
In CITE, the external force field EQU is extended to the domains EQU by zero and the stress tensor EQU in these domains is defined as follows: We note that in the case where the fluid velocity EQU is extended to the domains EQU according to relation (in fact, this is used in CITE), the stress tensor EQU for EQU must have another analytic representation than .
Otherwise, if the stress tensor EQU in the domains EQU is defined by relation , where the quantity EQU is defined by relation in these domains, then for EQU, we have where the quantity EQU for EQU depends on the method of extension of the fluid pressure EQU defined for EQU to EQU.
However, for any extension of the fluid pressure EQU to the domains EQU, according to , the left-hand side of EQU contains the potential vector EQU, while the right-hand side of this equations contains the solenoidal vector - EQU because This means that the extension of the fluid velocity EQU to the domains EQU according to eliminates the possibility of extension (used in CITE) for the stress tensor EQU defined by relations in the entire space to these domains for the case of the time-dependent angular velocity EQU.
The quantities EQU and EQU extended to EQU according to and and EQU analytically extended to these domains are given in the entire space and satisfy the continuity equation and the equation given in the entire space.
Equation differs from EQU by the additional term (the induced force density) EQU that appears due to the extension of the linearized Navier–Stokes equation given for EQU to the domains EQU.
For the analytic extension of the external force field, the quantity EQU in EQU and is described by the same analytic expressions, while, for the zeroth extension of the external force field to EQU CITE, the analytic expressions describing the quantity EQU in the domains EQU and EQU are different.
Therefore, in the latter case, the procedure where the quantity EQU is formally considered to be equal to zero in EQU is not equivalent to the removal of the particles from the fluid because EQU given in the entire space and EQU with EQU describe the unbounded fluid in various force fields, namely, in EQU and in EQU, where EQU is the Heaviside function, respectively.
The induced force density EQU is presented in the form where EQU is the induced force density for particle EQU.
This quantity has the volume EQU and surface EQU components given, respectively, inside the volume EQU occupied by the particle EQU and on its surface EQU CITE Here, where EQU is the vector directed from the center of sphere EQU to a point on its surface characterized by the polar EQU and azimuth EQU angles in the local spherical coordinate system EQU, EQU is the solid angle, and EQU is a certain unknown density of the induced surface force distributed over the surface EQU.
Unlike CITE, the volume component EQU of the induced force does not contain the translational velocity EQU of particle EQU, which is caused by the different extensions of the fluid pressure to EQU used in the present paper and in CITE.
In addition, owing to extension for the fluid pressure to EQU, EQU is defined only by the solenoidal component of the analytically extended external force field.
Thus, in the particular case of conservative external forces, EQU is independent of external forces.
It is easy to verify that the representation of the induced force density in the form as well as relations and and the analytic extension of EQU to EQU insure the validity of EQU in the entire space.
In this case, Using relation , we can represent force and torque exerted by the fluid on sphere EQU as follows: where EQU and EQU are, respectively, the force and the torque exerted on the sphere EQU due to the induced force density distributed over its surface EQU and EQU are, respectively, the force and the torque acting on the fluid sphere occupying the volume EQU instead of spherical particle EQU due to the potential component EQU of the external force field analytically extended to this domain and is the inertial force that is necessary to be applied to a fluid sphere of volume EQU in order that it move with the acceleration EQU CITE, where EQU is the mass of the fluid displaced by particle EQU.
In the particular case of the time-independent homogeneous gravitational force field we have where EQU is the Archimedes force acting on particle EQU and EQU is the acceleration of gravity.
Therefore, in the considered case, EQU in relation is the gravity force of particle EQU corrected for the buoyancy force.
In view of relations and , the forces and torques exerted by the fluid on particles are completely determined by the surface induced force densities.
If it is necessary to determine only the forces EQU and the torques EQU in the linear approximation with respect to the velocities of the fluid and the spheres, it is convenient to use the procedure proposed in CITE, which does not require the explicit form of EQU.
On the basis of this procedure, for an arbitrary number of spheres in a fluid, one obtains the mobility tensors of particles that move and rotate in the fluid both in the stationary case CITE and with regard for the time dependence CITE.
In a more general case where the velocity field of the fluid should be determined, the problem becomes essentially more complicated because the explicit form of the induced surface force densities must be obtained.
In the nonstationary case, this problem is considered in CITE where the zeroth extension of the external forces to the domains occupied by spheres is used and the stress tensor in these domains is defined by relation .
Representation of the Required Quantities in Terms of Harmonics of Induced Surface Force Densities To determine the velocity and pressure fields of the fluid in the presence of EQU spheres in it, we note the same structure of EQU and .
Therefore, we can use solution of EQU with the substitution EQU for EQU.
We consider the problem linearized both with respect to the velocity of the fluid and the velocities of the spheres.
Within the framework of this approximation, we neglect the time dependence of the positions of centers of the spheres defined by EQU and their surfaces EQU.
This means that the spheres do not displace considerably for considered time intervals (for details, see CITE).
This enables us to represent EQU and EQU as follows: where EQU is the Stokes friction coefficient for a sphere of radius EQU uniformly moving in the fluid, EQU is the dimensionless parameter that characterizes the ratio of the radius EQU of sphere EQU to the length EQU, EQU is the spherical Bessel function of order EQU, and Note that the quantity EQU defined by relation differs from the Fourier transform EQU of the solenoidal component of the external force field because the integral in is taken over the finite space domain EQU, while for EQU, the Fourier integral is taken over the entire space.