© Hindawi Publishing Corp. ON UNSTEADY TWO-PHASE FLUID FLOW DUE TO ECCENTRIC ROTATION OF A DISK

We examine the unsteady flow of a two-phase fluid generated by the nontorsional oscillations of a disk when the disk and the fluid at infinity rotate noncoaxially with the same angular velocity. The solutions are obtained for both the fluid and the particle velocities in closed form. It is found that the solutions remain valid for all values of the frequency of oscillations of the disk including the resonant frequency, which is equal to the angular velocity of rotation. But, in absence of particles, only in the case of resonance no oscillatory solution is possible, which is similar to that of solid-body rotation as pointed out by Thornley (1968). It is also shown that, unlike the case of single-disk configuration, no unique solution exists in a double-disk configuration, a result which is the reverse to that of solid-body rotation. Finally, the results are presented graphically to determine the quantitative response of the particle on the flow. 2000 Mathematics Subject Classification: 76B10. 1. Introduction. The dynamics of rotating fluids is particularly important in the analysis of flow phenomena associated with the atmospheric, oceanic, geophysical, and astrophysical problems. Thornley [3] investigated the flow generated in a semi-infinite expanse of viscous fluid bounded by the infinite rigid disk in the presence of the particles in the fluid. However, if the fluid is clean, no physically meaningful resonant solutions are possible in the existing flow configuration, which is an event similar to that of Thornley [3]. Moreover, it is found that, contrary to the case of single-disk geometry, infinite number of solutions exist for the flow confined between two noncoaxially rotating parallel disks. Finally, the results are evaluated quantitatively with a view to examine the effect of particles on the flow. 2. Formulation of the problem. We consider the flow of a two-phase fluid due to an oscillating disk in the xy-plane rotating about the z-axis normal to the disk with an angular velocity Ω in Cartesian coordinate system. The particulate fluid at z =∞ rotates, with the same angular velocity, about an axis parallel to the z-axis passing through the point (x1 ,y 1). For this type of motion the velocity fields for the fluid and the particles may be taken in the form


Introduction.
The dynamics of rotating fluids is particularly important in the analysis of flow phenomena associated with the atmospheric, oceanic, geophysical, and astrophysical problems. Thornley [3] investigated the flow generated in a semi-infinite expanse of viscous fluid bounded by the infinite rigid disk in the presence of the particles in the fluid. However, if the fluid is clean, no physically meaningful resonant solutions are possible in the existing flow configuration, which is an event similar to that of Thornley [3]. Moreover, it is found that, contrary to the case of single-disk geometry, infinite number of solutions exist for the flow confined between two noncoaxially rotating parallel disks. Finally, the results are evaluated quantitatively with a view to examine the effect of particles on the flow.

Formulation of the problem.
We consider the flow of a two-phase fluid due to an oscillating disk in the xy-plane rotating about the z-axis normal to the disk with an angular velocity Ω in Cartesian coordinate system. The particulate fluid at z = ∞ rotates, with the same angular velocity, about an axis parallel to the z-axis passing through the point (x 1 ,y 1 ). For this type of motion the velocity fields for the fluid and the particles may be taken in the form represent, respectively, the fluid and the particle velocities. Following Saffman [2], the unsteady motion of a two-phase fluid with uniformly distributed particles, occupying the semi-infinite space z > 0, is governed by the equations 2) where k and τ are, respectively, the concentration and the relaxation time of the particles in the fluid.
Substituting (2.1) in (2.2), we get Similarly, from (2.5) and (2.6), we get On eliminating w 2 from (2.8) with the help of (2.9), we have (2.10) Equation (2.10) is to be solved with the boundary conditions along with the assumption that the solutions are bounded at infinity.

Solution of the problem.
In view of the boundary condition (2.11a), we suggest the solution of (2.10) as (3.1) Substituting (3.1) in (2.10) and utilizing the boundary conditions, the fluid velocity for the case σ (= n/Ω) < 1 can be obtained as where m j z = ξ j A j + iB j , Equating the real and imaginary parts of (3.2) by taking a = a 1 + ia 2 and b = b 1 + ib 2 , we get In particular, where k = 0, the fluid velocity corresponding to clean fluid motion for σ < 1 is given by The distinctive feature of the solutions (3.4) is that the flow essentially consists of three distinct boundary layers on the disk. The thickness of these layers are of orders Clearly, the thickness of the layers is modified by the presence of particles in the fluid. In fact, it decreases with increase in particle concentration (k). On the other hand, in the absence of particles (k = 0), the above three layers modify themselves to an Ekman layer of thickness of the order (2ν/Ω) 1/2 surrounded by two more Stokes-Ekman layers of thickness of the orders (2ν/(Ω − n)) 1/2 and (2ν/(Ω + n)) 1/2 . These three layers combine into a single Ekman layer of thickness of the order (2ν/Ω) 1/2 when n = 0.
The fluid velocity for the case σ = (n/Ω) > 1 is given by (3.8) When the natural frequency of rotation is equal to the forced frequency n, that is, for σ = 1, the system resonates and in this case the solution is given by It is worth noting that, when k ≠ 0, the results (3.9) provide a meaningful resonant solution satisfying all boundary conditions. But when k = 0, the last terms of (3.9) do not satisfy the boundary condition at infinity. Accordingly, in the case of clean viscous fluids, no oscillatory solution exists at a resonant frequency n = Ω. This phenomenon is similar to that pointed out by Thornley [3] in the case of solid-body rotation.
To determine the particle velocity satisfying (2.9), we assume that Substituting (3.1) and (3.11) in (2.9) and utilizing the boundary conditions, the particle velocity is given by (3.13) It follows that the particles at infinity are unable to follow the fluid motion due to the presence of Ω and n. But when Ω and n equal zero, the particle and the fluid move in unison and we have W 1 = W 2 .
We next turn our attention to the case of another disk introduced at z = d which rotates with the angular velocity Ω about an axis parallel to the z-axis and passing through the point (x 1 ,y 1 ) so that the boundary condition of F 0 (z) = ∞ is replaced by W = x 1 + iy 1 at z = d. We focus our attention on the solution of F 0 (z) only because the essential nature of F 1 (z) and F 2 (z), for the unsteady case in (3.1), is similar to F 0 (z). The solution for F 0 (z) satisfying (2.10), (2.11a), and the boundary condition at z = d is given by where The result (3.14) contains an arbitrary constant C which remains undetermined under the stated boundary conditions, giving a possibility of infinite number of solutions. Similarly, the solutions of F 1 and F 2 also contain arbitrary constants each is undetermined. Thus, instead of getting a unique solution as in the case of single-disk configuration, the double-disk configuration provides an infinite number of solutions.
Finally, the fluid velocity near z = 0, corresponding to the case σ < 1, is obtained from (3.4) as (3.16) The above results (3.16) indicate that the velocity vector near the disk is inclined at an angle tan −1 (−f /g) to the disk. For a special case in which a 1 = a 2 = b 1 = b 2 = y 1 = 1, x 1 = 0, and nt = π/2, we have which, when k → 0, gives so that the velocity vector is inclined at an angle tan −1 (1 − 2N), where Thus, the angle of inclination of the velocity vector near the disk not only depends on the particles but also on σ = n/Ω. However, when both k → 0 and n → 0, the velocity vector is inclined at an angle 45 • to the disk.

Conclusion.
The analysis given above clearly indicates that in a noncoaxial system of rotation the resonance occurs at a frequency equal to angular velocity of rotation of the disk which is not the case in a coaxial system of solid-body rotation where the resonance occurs at a frequency equal to twice the angular velocity of rotation of the disk as pointed out by Thornley [3].
Secondly, the difficulty in obtaining the resonant solution in the case of clean fluid is resolved automatically in presence of the particles in the fluid.
Finally, the infinite number of solutions existing for the flow in the geometry of two parallel disks given by Berker [1] reduce to a single unique solution for the case of a single disk.
The quantitative evaluation of the results for f 1 and g 1 for various values of flow parameters is presented in Figures 4.1, 4.2, and 4.3.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable:

Manuscript Due
December 1, 2008 First Round of Reviews March 1, 2009