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

1 Introduction

By the heavy technical demand of industries, many researchers are using Non-Newtonian fluids extensively in the problems of extraction of petrol from porous wells, sedimentation, dilute polymers, suspensions and lubrications of journal bearings. The polar effects namely couple stresses and non-symmetric tensors are well explained by a simple model of couple stress fluids introduced by Stokes [1]. Stokes solved creeping flow of couple stress fluid across a sphere [2]. The study of the flow of couple stress fluid 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 flow of a Couple stress fluid past a permeable sphere was analyzed by Ramana Murthy and et al. [5]. Devakar et al. [6] studied analytical solutions of some fully developed flows of Couple Stress fluids between two cylinders with slip boundary conditions. Couple stress fluid flow 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 fluid for rotary flows across a permeable sphere was carried out [8]. Vandana Mishra and Ram Gupta [9] studied the concept of analytically uniform flow of steady axi-symmetric creeping flow of an incompressible micro-polar fluid around the permeable sphere. They considered non homogeneous boundary conditions for micro-rotation vector. Arbitrary oscillatory Stokes flow past a porous sphere for viscous fluid was studied by Prakesh et al. [10]. The slow and stationary flow of a viscous fluid was investigated by Leonov [11]. The concept of micro-polar fluids was first found by Eringen [12]. Gupta and Deo [13] examined Stokes flow of micro-polar fluid past a porous sphere with hyper-stick condition on micro-rotation vector. Recently, Choudhuri et al. [14] developed a method to find a solution to Stokes flow of a viscous and incompressible fluid flow across a sphere coated by a thin fluid of different viscosity. Ramkissoon [15] obtained a formula for drag coefficient of a micropolar fluid flow past a sphere. Recently, Vainshtein and Shapiro [16] have examined the forces acting on a porous sphere oscillating in a viscous fluid. Newtonian fluid flow inside and outside sphere is governed by Darcy–Brinkman equations of porous medium. Jai Prakash and Raja Sekhar [17] analyzed the arbitrary oscillatory Stokes flow past a porous sphere using Brinkman model. Crittenden et al. [18] studied the influence of oscillatory flow on axial dispersion in packed bed of spheres. They observed that the best reduction of axial dispersion coefficient (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 flows. The problems that are concerned with the effects of free stream oscillations are of physical significance. The problems of unsteady flows 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 fluid. Latter many authors have studied the phenomena of oscillations of external flow over a non- zero mean velocity. Thomas and Walters [21] examined the flow 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 flow due to oscillating sphere in an elastico viscous fluid by neglecting the nonlinear terms. They also studied the flow past a sphere accelerating with aperiodic and arbitrary motion in the visco-elastic fluid using Fourier Transform technique and obtained expressions for drag experienced by the sphere. Entropy generation on non-Newtonian Eyring-Powell nanofluid has been analysed through a permeable stretching sheet by Bhatti et al. [23]. Variable viscosity and inclined magnetic field on the peristaltic motion of a non-Newtonian fluid in an inclined asymmetric channel was studied by Khan et al. [24]. New analytical method for the study of natural convection flow of a non-Newtonian fluid was studied by Rashidi et al. [25]. A steady flow of a sphere in a rotation motion in a micro-polar fluid was analyzed by the author [26]. Webster [27] has considered non-Newtonian and turbulent fluid models. He developed a finite difference numerical technique to solve incompressible fluid flow problems. Casanellas and Ortin [28] studied the laminar oscillatory flow of Maxwell and Oldroyd-B fluids [28]. Jayalakshmamma et al. [29] studied numerically the steady flow of an incompressible micropolar fluid past an impervious sphere. Mishra and Gupta [30] studied creeping flow of micro polar fluid past composite sphere. Numerical and analytical study of flow past porous sphere embedded in micropolar fluid by Ramalakshmi and Pankaj Shukla [31]. Ashmawy [32] developed a simple formula for drag acting on a sphere for couple stress fluid. The problem of rotary oscillation of a rigid sphere in an incompressible couple stress fluid is investigated by Shehadeh and Ashmawy [33]. Ashmawy [34] studied unsteady Stokes flow of a couple stress fluid around a rotating sphere with slip condition on the boundary. Jaiswal and Gupta [35] have considered the flow over composite sphere: liquid core with permeable shell. Jaiwal [36] studied analytically, Stokes flow over Reiner-Rivlin liquid sphere embedded in a porous medium filled with micropolar fluid using Brinkman’s model. Nagaraju and Mahesh [37] studied the analytical investigation of two-dimensional heat transfer behavior of anaxisymmetric incompressible dissipative viscous fluid flow in a circular pipe.

The oscillatory flow of incompressible couple stress fluid flow past a permeable sphere is considered in the present study due to its practical importance. The velocity and pressure field on the sphere are obtained. The drag experienced by the sphere is evaluated. Effects of couple stress parameter, permeability parameter, frequency parameter and geometric parameter on the drag due to internal and external flows are found numerically and are shown graphically.

2 Fundamental equations and formulation of the problem

Here we consider an oscillating flow of the form U_∞e^iωtk̄ of incompressible couple stress fluid, the direction of the oscillation being along k̄. A spherical membrane of radius a with porous surface is introduced into the flow and held fixed at the origin. Since the sphere is having a porous membrane, the couple stress fluid flows across a fixed permeable sphere and divides the entire region into flow region-I external to the sphere and region-II internal to the sphere.

The basic equations governing the flow of an incompressible couple stress fluid as proposed by V.K. Stokes are

Neglecting convective terms, on the basis of Stokes assumption that flow 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 flow direction is considered. Velocity field and pressure suitable for this oscillating flow are of the form,

where the scale factors for spherical coordinate system are h₁ = 1, h₂ = R and h₃ = Rsinθ and Ψ the stream function is taken to satisfy equation (1).

By the choice of equation for velocity in (4), we note that

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

and E02=E2a2 and E2=∂2∂r2−cot⁡θR2∂∂θ+1R2∂2∂θ2= Stokes stream function operator

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 non-dimensional quantities and the capital letters on LHS indicate dimensional quantities.

By this non-dimensional scheme the equation (7) reduces to

Let

Now we find the solution of the equation (10) for ψ under the following conditions:

Condition (i) of (13) represents the uniform flow condition (after removing oscillation term e^iωt) far away from the sphere and finite 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 fluids satisfy type A condition and suspension like fluids satisfy type B condition.

Condition (iv) represents continuity condition for normal velocity which is equal to suction velocity V₀ on the surface.

3 Solution for the problem undertaken

The solution for (10) is sought in the form,

for r≥ 1

for r≤ 1

For external flow

f₀(r) = r² +a₁/r ; f₁(r) = √rK_3/2(λ₁r) and f₂(r) = √rK_3/2(λ₂r)

For internal flow

f₀(r) = a₂r² ; f₁(r) = √rI_3/2(λ₁r) and f₂(r) = √rI_3/2(λ₂r)

The suitable solutions for regularity condition are given by

Now from the condition (i) – (iv) we obtain the equations as given below:

b′₁, b′₂, c′₁ and c′₂ are defined as given below:

Now from (16.e), we get the constants as,

where

By solving the equations (16.a – d), in the following form, we get the constants

where

and

The arbitrary constants a₁, a₂, b₁, b₂, c₁, c₂ in (16) are expressed as a₁, a₂, b′₁, b′₂, c′₁, c′₂ in (17). The arbitrary constants are six (6) in number, but the number of equations are five (5). b′₁, b′₂, are expressed in terms of c′₁, c′₂. 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 a₂. Hence τ need not take real values and need not start from zero value. Now τ is defined as permeability parameter.

4 Pressure distribution

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 flows this reduces to,

i.e.

i.e.

5 Bounds for permeability parameter τ

On the surface the filtration velocity V₀ = 0 gives τ = 0 And ΔP = P_e – P_i = 0 gives τ=3Δ3Δ43Δ3Δ4+2(1−ϵ)(Δ4−Δ3) 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.

6 Drag on the sphere

The drag D on the sphere is given by the formula

Stress tensor for couple stress fluid is given by the constitutive equation

The constitutive equation for Couple stress tensor M is given by

The strain tensor E in (20) is calculated by

This strain in non-dimensional form in terms of stream function is given by

The couple stress tensor M in (21) is given by

where ϵ = η′/η.+-

Now the following quantities can be evaluated.

and I×∇⋅M=−ρηU∞2a22E4ψ(ereθ−eθer)

From this we get that the non -dimensional drag D′_ex due to external flow and D′_in.

Drag due to external flow D_ex is given by,

By taking f′ = 0 and f = V₀ on r = 1, the drag simplifies to

Similarly Drag due to internal flow =

The non-dimensional drag D^* is obtained by comparing the drag with Stokes drag.

7 Results and discussion

The geometric parameters λ₁² and λ₂² of equation (8) are computed by solving the quadratic equation

Then, the constants in the stream function ψ in (14) and (15) for internal flow and external flow are obtained by using the equations (17).

Filtration velocityV₀on the surface: Then the permeability parameter τ is fixed 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 fix a real value which is less than the maximum bound in (18). Now filtration velocity V₀ in (16.4) can be computed. This filtration velocity V₀ is presented in figure2. With an increase of S there is an increase in the filtration velocity also. But for any value of S, filtration velocity is less than 50% of the velocity at infinity.

Figure 1

Flow geometry

Figure 2

Filtration velocity V₀ vs. permeability parameter τ

Radial velocity: The stream function ψ in terms of radial function f(r) is shown in Figure 3 at different 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 flow, the radial velocity will be less than that in couple stress fluid flow (since as S→ ∞, the flow reduces to viscous flow).

Figure 3

Stream function F for various values of a) permeability parameter τ; b) couple stress parameter S

Drag: Drag on the sphere because of the flow of couple stress fluid 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 a fixed quantity. (in the figure near to 45 at σ = 100 and Re = 0.5). This indicates that the drag for viscous fluid flow will be less than the case of couple-stress fluids.

Figure 4

Drag vs. couple stress parameter

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.

Figure 5

Drag vs. time for time period 2π/σ at a) σ = 20; b) σ = 50; c) σ = 100

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 different permeability parameter τ. The flow is as perceived by an observer travelling with the flow. 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 flow 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 flowing 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.

Figure 6

Stream lines without time factor e^iσt at different permeabilities; ψ= 0.01, ψ= 0.05 and ψ= 0.12 are passing through the sphere

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 differ completely. Now the stream lines take both negative and positive values. The flow is as per the observations of an observer fixed in space. It is exciting to note that near the sphere there is another fluid spherical region in which flow circulations take place. Within this fluid sphere ψ takes negative values. Outside this fluid sphere ψ is positive and flow is same as that of flow past an impermeable sphere. As τ increases, the circulation is more prominent and the centre of the circulation is below the top pole near to it.

Figure 7

Stream lines with time factor e^iσt. Stream lines with circulations are negative.

8 Conclusions

The following observations are made in the study of the oscillating flow of couple stress fluid past a permeable sphere.

1. As permeability parameter τ increases, filtration velocity increases.

2. As couple-stress parameter increases, there is an increase in filtration velocity.

3. The observer with the flow (flow pattern excluding e^iσt), observes a small circulation near the pole at small permeabilities.

4. The observer fixed in space (flow pattern including e^iσt) observes a circulation of fluid within a fluid sphere which passes through the permeable sphere. The center of circulation is below the pole of the sphere.