Boundary layer flow of the Newtonian fluid that is caused by the vibration of inner sphere while the outer sphere is at rest is calculated. Vishik-Lyusternik (Nayfeh refers to this method as the method of composite expansions) method is employed to construct an asymptotic expansion of the solution of the Navier-Stokes equations in the limit of high-frequency vibrations for Reynolds number of

In the present paper we consider boundary layer flow due to the vibrating sphere in a viscous fluid. Boundary layer flow for Newtonian and non-Newtonian fluid’s is studied by many researchers [

Mathematical modeling of the problem involves two parameters: the inverse of Reynolds number

Steady secondary flow was investigated numerically in [

Wang [

In the present paper, steady streaming between two spheres is studied. The results are obtained for

We consider a three-dimensional flow of a viscous incompressible fluid between two spheres with radii

We are interested in the asymptotic behavior of periodic solution of (

It is convenient to introduce parameter

In spherical polar coordinates

We seek a solution of (

Substituting (

Before we try to solve these equations, it is convenient to discuss boundary conditions at

Substituting (

Taking the derivative of (

Substitution of (

The only solution of this equation that satisfies the decay condition at infinity and the boundary condition

Using the same procedure as for the leading order equation we get the steady streaming at higher orders of

It is well-known that in oscillatory flows the observed averaged Lagrangian velocity differs from the Eulerian velocity by the term known as Stokes drift. The velocity observed in the experiments is the velocity of fluid particles, that is, the Lagrangian velocity. Our asymptotic expansion for the averaged Eulerian velocity has the form

Equation (

In (

Typical stream line pattern due to vibrating sphere is shown in Figure

Steady streaming between two spheres.

Steady streaming near inner sphere.

Steady streaming between two spheres.

Steady streaming between two spheres.

Radial component of the velocity for

Vishik-Lyusternik method is successfully applied to study the steady streaming between two spheres. In comparison with the method of matched asymptotic expansions, it has two advantages: (i) it does not require the procedure of matching the inner and outer expansions and (ii) the boundary layer part of the expansion satisfies the condition of decay at infinity (in boundary layers) in all the orders of the expansion, which is not the case in the method of matched asymptotic expansion.

Stokes drift is calculated not for the outer flow but for the entire flow domain including the boundary layers.

We also calculated the averaged lagrangian velocity which remains invariant with the change of reference frame and is observed in experiment.

The distance of stagnation points from the surface of sphere in each quadrant remains the same.

Qualitatively the graphs of the streamlines are very much similar to the experimental results which are done for

Functions

The author declares that there is no conflict of interests regarding the publication of this paper.

The author is thankful to King Fahd University of Petroleum and Minerals for providing adequate resource for carrying out this research.