Couple stress fluid flow due to slow steady oscillations of a permeable sphere

Abstract The study of oscillating flow of a Couple Stress fluid past a permeable sphere is considered. Analytical solution for the flow field in terms of stream function is obtained using modified Bessel functions. The formula for Drag acting on the sphere due external flow is evaluated. Pressure field for the flow region past and inside the sphere is obtained. Effects of physical parameters like couple stress parameter, permeability, frequency and geometric parameters on the drag due to internal and external flows are represented graphically. It is observed that the drag for viscous fluid flow will be less than the case of couple-stress fluid flow and hence couple stress fluids offer resistance for flow.


Introduction
By the heavy technical demand of industries, many researchers are using Non-Newtonian uids extensively in the problems of extraction of petrol from porous wells, sedimentation, dilute polymers, suspensions and lubrications of journal bearings. The polar e ects namely couple stresses and non-symmetric tensors are well explained by a simple model of couple stress uids introduced by Stokes [1]. Stokes solved creeping ow of couple stress uid across a sphere [2]. The study of the ow of couple stress uid past axi-symmetric bodies was carried out [3]. Ramkisson [4] has derived an elegant and useful formula for drag on an axi-symmetric body in terms of a limit on the stream function. Uniform ow of a Couple stress uid past a permeable sphere was analyzed by Ramana Murthy and et al. [5]. Devakar et al. [6] studied analytical solutions of some fully developed ows of Couple Stress uids between two cylinders with slip boundary conditions. Couple stress uid ow past a porous spheroidal shell with solid core under Stokesian assumption was studied and analyzed by Iyengar and Radhika [7]. A study of a Couple acting on a couple-stress uid for rotary ows across a permeable sphere was carried out [8]. Vandana Mishra and Ram Gupta [9] studied the concept of analytically uniform ow of steady axi-symmetric creeping ow of an incompressible micro-polar uid around the permeable sphere. They considered non homogeneous boundary conditions for micro-rotation vector. Arbitrary oscilla-tory Stokes ow past a porous sphere for viscous uid was studied by Prakesh et al. [10]. The slow and stationary ow of a viscous uid was investigated by Leonov [11]. The concept of micro-polar uids was rst found by Eringen [12]. Gupta and Deo [13] examined Stokes ow of micro-polar uid past a porous sphere with hyper-stick condition on micro-rotation vector. Recently, Choudhuri et al. [14] developed a method to nd a solution to Stokes ow of a viscous and incompressible uid ow across a sphere coated by a thin uid of di erent viscosity. Ramkissoon [15] obtained a formula for drag coe cient of a micropolar uid ow past a sphere. Recently, Vainshtein and Shapiro [16] have examined the forces acting on a porous sphere oscillating in a viscous uid. Newtonian uid ow inside and outside sphere is governed by Darcy-Brinkman equations of porous medium. Jai Prakash and Raja Sekhar [17] analyzed the arbitrary oscillatory Stokes ow past a porous sphere using Brinkman model. Crittenden et al. [18] studied the in uence of oscillatory ow on axial dispersion in packed bed of spheres. They observed that the best reduction of axial dispersion coe cient (up to 50%) from the non-oscillation base value occurs when the column particle size is the smallest.
Many elastic properties of dilute polymers can be detected and measured conveniently by a suitable choice of oscillatory ows. The problems that are concerned with the e ects of free stream oscillations are of physical signi cance. The problems of unsteady ows are initiated by Lighthill [19] by giving analytical solution of functions in stream function due to heat transfer. Fatter [20] has discussed the problems of oscillating sphere in an elastic viscous uid. Latter many authors have studied the phenomena of oscillations of external ow over a non-zero mean velocity. Thomas and Walters [21] examined the ow due to the oscillatory motion of a sphere with convective terms present in a elastic viscous liquid using Laplace Transform technique. Lai and Fan [22] have considered the ow due to oscillating sphere in an elastico viscous uid by neglecting the nonlinear terms. They also studied the ow past a sphere accelerating with aperiodic and arbitrary motion in the visco-elastic uid using Fourier Transform technique and obtained expressions for drag experienced by the sphere. Entropy generation on non-Newtonian Eyring-Powell nano uid has been analysed through a permeable stretching sheet by Bhatti et al. [23]. Variable viscosity and inclined magnetic eld on the peristaltic motion of a non-Newtonian uid in an inclined asymmetric channel was studied by Khan et al. [24]. New analytical method for the study of natural convection ow of a non-Newtonian uid was studied by Rashidi et al. [25]. A steady ow of a sphere in a rotation motion in a micro-polar uid was ana-lyzed by the author [26]. Webster [27] has considered non-Newtonian and turbulent uid models. He developed anite di erence numerical technique to solve incompressible uid ow problems. Casanellas and Ortin [28] studied the laminar oscillatory ow of Maxwell and Oldroyd-B uids [28]. Jayalakshmamma et al. [29] studied numerically the steady ow of an incompressible micropolar uid past an impervious sphere. Mishra and Gupta [30] studied creeping ow of micro polar uid past composite sphere. Numerical and analytical study of ow past porous sphere embedded in micropolar uid by Ramalakshmi and Pankaj Shukla [31]. Ashmawy [32] developed a simple formula for drag acting on a sphere for couple stress uid. The problem of rotary oscillation of a rigid sphere in an incompressible couple stress uid is investigated by Shehadeh and Ashmawy [33]. Ashmawy [34] studied unsteady Stokes ow of a couple stress uid around a rotating sphere with slip condition on the boundary. Jaiswal and Gupta [35] have considered the ow over composite sphere: liquid core with permeable shell. Jaiwal [36] studied analytically, Stokes ow over Reiner-Rivlin liquid sphere embedded in a porous medium lled with micropolar uid using Brinkman's model. Nagaraju and Mahesh [37] studied the analytical investigation of two-dimensional heat transfer behavior of anaxisymmetric incompressible dissipative viscous uid ow in a circular pipe.
The oscillatory ow of incompressible couple stress uid ow past a permeable sphere is considered in the present study due to its practical importance. The velocity and pressure eld on the sphere are obtained. The drag experienced by the sphere is evaluated. E ects of couple stress parameter, permeability parameter, frequency parameter and geometric parameter on the drag due to internal and external ows are found numerically and are shown graphically.

Fundamental equations and formulation of the problem
Here we consider an oscillating ow of the form U∞e iωtk of incompressible couple stress uid, the direction of the oscillation being alongk. A spherical membrane of radius a with porous surface is introduced into the ow and held xed at the origin. Since the sphere is having a porous membrane, the couple stress uid ows across a xed permeable sphere and divides the entire region into ow region-I external to the sphere and region-II internal to the sphere.
The basic equations governing the ow of an incompressible couple stress uid as proposed by V.K. Stokes are Neglecting convective terms, on the basis of Stokes assumption that ow is very slow and Reynolds number Re is very small (Re 1), equation (2) reduces to Spherical coordinate system with origin at the center of the sphere and Z axis along the ow direction is considered. Velocity eld and pressure suitable for this oscillating ow are of the form, and P = P e iωt (4) where the scale factors for spherical coordinate system are h =1, h = R and h = Rsinθ and Ψ the stream function is taken to satisfy equation (1). By the choice of equation for velocity in (4), we note that (5) By taking curl to equation (3), the pressure is eliminated and we get, Using (5) in (6), the equation for stream function Ψ is obtained as is where We introduce the non-dimensional scheme and the non-dimensional parameters like σ the frequency parameter, "Re" Reynolds number and S the couple stress parameter as follows; In equation (9), the small letters on RHS indicate nondimensional quantities and the capital letters on LHS indicate dimensional quantities. By this non-dimensional scheme the equation (7) reduces to Now we nd the solution of the equation (10) for ψ under the following conditions: Condition (i) of (13) represents the uniform ow condition (after removing oscillation term e iωt ) far away from the sphere and nite velocity at the origin (centre of the sphere). Condition (ii) of (13) represents no slip tangential velocity on the surface of sphere.
Condition (iii) represents vanishing of couple stresses on the surface (this is called type A condition) or represents hyper-stick condition which means vanishing of micro-rotations (this condition is called type B condition). Here either type A or type B condition is taken. Both conditions are not valid simultaneously. Long chain uids satisfy type A condition and suspension like uids satisfy type B condition. Condition (iv) represents continuity condition for normal velocity which is equal to suction velocity V on the surface.

Solution for the problem undertaken
The solution for (10) Now from the condition (i) -(iv) we obtain the equations as given below: By solving the equations (16.a -d), in the following form, we get the constants where and The arbitrary constants a , a , b , b , c , c in (16) are expressed as a , a , b , b , c , c in (17). The arbitrary constants are six (6) in number, but the number of equations are ve (5). b , b , are expressed in terms of c , c . Hence 4 arbitrary constants are expressed in 3 equations as in (17). Hence one of the constants is arbitrary.
τ is arbitrary which is taken in the place of a . Hence τ need not take real values and need not start from zero value. Now τ is de ned as permeability parameter.

From equation (3) pressure is given by
The equations in non-dimensional form along radial and transverse directions are given by From these pressure can be obtained as For external and internal ows this reduces to, i.e. p i = iσ.τr cos θ

Bounds for permeability parameter τ
On the surface the ltration velocity V = gives τ = And Hence bounds for τ are as follows: It is to be observed that τ takes complex values. For the sake of calculations, for τ any real value between 0 and the maximum bound can be taken.

Drag on the sphere
The drag D on the sphere is given by the formula Stress tensor for couple stress uid is given by the constitutive equation The constitutive equation for Couple stress tensor M is given by The strain tensor E in (20)  where ϵ = η /η.+-Now the following quantities can be evaluated.
From this we get that the non -dimensional drag D ex due to external ow and D in . Drag due to external ow Dex is given by, By taking f = and f = V on r = , the drag simpli es to Similarly Drag due to internal ow= The non-dimensional drag D * is obtained by comparing the drag with Stokes drag.

Results and discussion
The geometric parameters λ and λ of equation (8) are computed by solving the quadratic equation Then, the constants in the stream function ψ in (14) and (15) for internal ow and external ow are obtained by using the equations (17). Filtration velocity V on the surface: Then the permeability parameter τ is xed by choosing a value within the bounds given in (18). It is to be noted that τ is not real. For choosing a value for τ we can x a real value which is less than the maximum bound in (18). Now ltration velocity V in (16.4) can be computed. This ltration velocity V is presented in gure2. With an increase of S there is an increase in the ltration velocity also. But for any value of S, ltration velocity is less than 50% of the velocity at in nity.

Radial velocity:
The stream function ψ in terms of radial function f(r) is shown in Figure 3 at di erent values of τ and S. As τ or S increases, f (radial velocity) increases. i.e as couple stresses increase, they increase the radial velocity. This means in the case of viscous ow, the radial velocity will be less than that in couple stress uid ow (since as S→ ∞, the ow reduces to viscous ow).
Drag: Drag on the sphere because of the ow of couple stress uid without the time factor e iσt is computed in Figure 4. Couple stress parameter S is not involved directly in the formula for drag. But it is found that with the increase in parameter S, there is a decrease in drag and tends to ). This indicates that the drag for viscous uid ow will be less than the case of couple-stress uids.
In Figure 5, drag is shown by including the time factor e iσt . Drag is drawn for a time period 2π/σ. We notice that as σ increases drag increases and as τ increases, magnitude of drag decreases. This can be expected. As frequency of oscillations increase, it is natural to expect high drag on the body.
Stream function: The stream function without time factor e iσt is shown in Figure 6. Three stream lines ψ=0.01, 0.05 and 0.12 are shown at di erent permeability parameter τ. The ow is as perceived by an observer travelling with the ow. It is to be noted that all stream lines are with positive sign only. The stream line ψ=0 passes through center of the sphere. As the value of ψ increases, the stream lines move away from the sphere and take uniform ow far away from the sphere. Three stream lines ψ=0.01, 0.05 and 0.12 are coming near to the axis of sphere as τ increases, which indicates that as τ increases more number of stream lines are owing through the sphere. It is observed that when τ is small, below the top pole near to it there is small circulation. When τ increases, this circulation disappears.
In Figure 7, the stream lines with time factor e iσt are shown. It is interesting to note that pattern of stream lines with time factor including and excluding di er completely. Now the stream lines take both negative and positive values. The ow is as per the observations of an observer xed in space. It is exciting to note that near the sphere there is another uid spherical region in which ow circulations take place. Within this uid sphere ψ takes negative values. Outside this uid sphere ψ is positive and ow is same as that of ow past an impermeable sphere. As τ increases,

Conclusions
The following observations are made in the study of the oscillating ow of couple stress uid past a permeable sphere. 1. As permeability parameter τ increases, ltration velocity increases. 2. As couple-stress parameter increases, there is an increase in ltration velocity. 3. The observer with the ow ( ow pattern excluding e iσt ) , observes a small circulation near the pole at small permeabilities. 4. The observer xed in space ( ow pattern including e iσt ) observes a circulation of uid within a uid sphere which passes through the permeable sphere. The center of circulation is below the pole of the sphere.