UNSTEADY FLOW INDUCED BY VARIABLE SUCTION ON A POROUS DISK ROTATING ECCENTRICALLY WITH A FLUID AT INFINITY

We study the effect of variable suction or blowing on the flow of 
an incompressible viscous fluid due to noncoaxial rotations of a 
porous disk and a fluid at infinity. The inquiries are made about 
the components of fluid velocity and the shear stress at the 
disk. It is found that the effect of uniform suction or blowing 
on the flow is enhanced in the presence of variable suction or 
blowing.

1. Introduction.Thornley [5] has studied the unsteady flow developed in an incompressible viscous fluid due to nontorsional oscillations of an infinite rigid plate when both the fluid and the plate are in a state of solid body rotation.A similar problem of magnetohydrodynamic Ekman layer over an infinite rigid nonconducting plate was examined by Gupta [3].On the other hand, the flow due to noncoaxial rotations of a disk and a fluid at infinity was initiated by Berker [1].Subsequently, Erdogan [2] constructed solutions of the problem of steady flow due to eccentrically rotations of a porous disk and a fluid at infinity with the same angular velocity both for the cases of uniform suction and blowing at the disk.Of late, Kasiviswanathan and Rao [4] obtained an exact solution for the unsteady flow due to noncoaxial rotations of a porous disk oscillating in its own plane and a fluid at infinity.In the present paper, the flow due to noncoaxial rotations of a porous disk subjected to variable suction or blowing and a fluid at infinity has been investigated.Analytical solutions are obtained both for the components of fluid velocity and the components of shear stress at the disk.Quantitative evolution of the results are also made with a view to examine the effects of variable suction and blowing on the flow.It is found that the effects of uniform suction or blowing on the flow field enhances in presence of variable suction or blowing at the disk.

Formulation of the problem.
We consider the flow due to a porous disk lying in the xy-plane rotating about the z-axis perpendicular to the disk with uniform angular velocity Ω.The fluid at infinity rotates with the same angular velocity Ω about an axis parallel to the z-axis passing through the point (x, y).The unsteady motion is established in the fluid due to variable suction at the disk.For this motion, the velocity field has the form where V (t) > 0 represents the suction velocity which satisfies the equation of continuity.Introducing (2.1) in Navier-Stokes equations, we get ) We now suppose that the suction velocity normal to the disk oscillates in magnitude and not in direction about a nonzero mean given by where W 0 is a positive constant; ε > 0 is small and A is a real positive constant such that εA 1.From (2.4) and (2.5), we find that ∂p/∂z is small and hence can be neglected.This shows that p is independent of z.
Eliminating p from (2.2) and (2.3) by differentiating with respect to z and combining them, we get where U = f + ig.Since no unsteady motion other than suction is imposed on the disk, we must have the boundary conditions for U(z,t) as (2.7) In addition to these, we assume that the solutions are bounded at infinity.Again, from (2.5), we assumed that (2.8) Substituting (2.8) and (2.5) in (2.3), comparing harmonic terms and neglecting coefficient of ε 2 , we get (2.11) 3. Solution of the problem.We introduce ξ = Ω/2vz, S = W 0 /2 √ Ωv, and n = 1 + σ /Ω in (2.10) and (2.11) to obtain ) subject to conditions (2.11).
On solving (3.1) and (3.2) subject to (2.11), we get where In particular, when A = 0, the general results (3.3) and (3.4) reduce to These results coincide with the nonoscillating part of the results [4, (11), (12)] and describe the flow in absence of variable suction at the disk.Again, on putting x 1 = 0 and y 1 = l in (3.3) and (3.4), and replacing −f by g and g by f , we get which are exactly the same as those given in [2] when A = 0. Thus, the effect of variable suction at the disk introduces a transient part depending on ε, A, and σ superposed on the steady solution corresponding to uniform suction at the disk.The case of S = 0 corresponds to impermeable case and recovers the solution for steady Ekman layer on the disk.For the flow very near to the porous disk, we have, from (3.7) and (3.8), Consequently, the inclination of the fluid velocity vector to y-axis near z = 0 becomes where In the case σ t = π/2 and S ≠ 0, A ≠ 0, the inclination of the fluid velocity to y-axis near z = 0 will be tan  For the case of blowing, S < 0 and the components of fluid velocity in presence of variable blowing at the disk can be obtained easily from (3.7) on replacing S by −λ, where λ > 0. The results in the case of blowing are represented in Figures 3.7, 3.8, 3.9, 3.10, 3.11, and 3.12.Finally, the components of the shear stress at the disk z = 0 corresponding to the fluid velocity given by (3.7) and (3.8) can be obtained as which, when σ t = π/2, yields The components of shear stress at the disk in the presence of variable blowing can also be found similarly.

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: .12) which indicates a further reduction in the value of the inclination of the fluid velocity compared with its value in the case of uniform suction.The variations of f (ξ) and g(ξ) corresponding to (3.7) and (3.8) for various values of the suction parameter S, the magnitude of fluctuation of suction velocity A, and the frequency of fluctuation of suction velocity n are illustrated in Figures 3.1