# Orr-sommerfeld Stability Analysis of Two-fluid Couette Flow

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**ORR-SOMMERFELD STABILITY ANALYSIS OF TWO-FLUID COUETTE FLOW WITH SURFACTANT**

__V.P.T.N.C.Srikanth BOJJA__, Maria FERNANDINO^{1*}^{1}, Roar SKARTLIEN^{2}

## ABSTRACT

In the present work, the surfactant induced instability of a sheared two fluid system is examined. The linear stability analysis of two-fluid couette system with an amphiphilic surfactant is carried out by developing Orr-Sommerfeld type stability equations along with surfactant transport equation and the system of ordinary differential equations are solved by Chebyshev Collocation method[1,2]. Linear stability analysis reveals that the surfactant either induces Marangoni instability or significantly reduces the rate at which small perturbations decay.

## Keywords:Linear stability, Orr-Sommerfeld, Marangoni mode, Amphiphilic surfactant.

NOMENCLATURE

A complete list of symbols used, with dimensions, is given.

*Greek Symbols*

Growth rate

Surfactant concentration

Mass density, [kg/m^{3}].

Dynamic viscosity, [kg/m.s].

Height of perturbed inteface

Surface tension

Wave number

,Stream functions

*Latin Symbols*

Capillary number

Marangoni number

Number of Collocation pints

Reynolds number

Plate/Wall velocity

Complex wave spped

Width of fluid layer

Amplitude of Pressure disturbance

Amplitude of surfactant concentration disturbance

Amplitude of interface perturbation

Viscosity ratio

Depth ratio

Shear of basic velocity

Velocity, [m/s].

*Sub/superscripts*

Index *i.*

Index *j*.

Perturbed quantities

Base state quantities

INTRODUCTION

Two layer channel flows and flows with and without surfactants have been given considerable importance because of its numerous industrial applications. Oil recovery[3], lubricated pipelining[4], liquid coating processes[5] are typical industrial situations where Two layer channel flows are often seen. Surfactants also have wide range of industrial applications for example in enhanced oil recovery[6].

Using Perturbation analysis, the primary instability of the two-layer plane Couette–Poiseuille flow was studied by Yih[7] and his studies revealed that even at small Reynolds numbers, the interface is susceptible to long-wave instability associated with viscosity stratification. Yiantsios & Higgins[8] later extended this study for small to large values of wavenumber and confirmed the existence of the shear mode instability. Boomkamp & Miesen[9] came up with the method of an energy budget for studying instabilities in parallel two-layer flows, where energy is supplied from the primary flow to the perturbed flow and instability appears at sufficiently long wave numbers through the increase of kinetic energy of an infinitesimal disturbance with time. In the presence of surfactant at the sheared interface, Frenkel & Halpern[10,11] discovered that even in the stokes flow limit, the interface is unstable as the surfactant induces Marangoni instability, which was later confirmed by Blyth & Pozrikidis[12]. In the case of Stokes flow, they identified two normal modes, the Yih mode due to viscosity stratification inducing a jump in the interfacial shear, and the Marangoni mode associated with the presence of the surfactant. In contrast, at finite Reynolds numbers, infinite number of normal modes are possible and by parameter continuation with respect to the Reynolds number the most dangerous Yih and Marangoni modes can be identified.

In this article, the effect of an insoluble surfactant on the stability of two-layer couette channel flow is studied in detail for low to moderate values of the Reynolds number. To isolate the Marangoni effect, gravity was suppressed in this problem and this was done by considering equal density fluids. Linear stability analysis was carried out by formulating Orr–Sommerfeld boundary value problem, which was solved numerically using Chebyshev collocation method[1][2] for all wavenumbers. Both Marangoni mode and Shear mode are detected and utmost focus is given to Marangoni mode as Shear mode is always stable at moderate to long wavenumbers under the influence of inertia.

The rest of the paper is organized as follows. In § Model description, the governing equations for the system in question are laid out, Normal mode analysis of the physical system is carried out, Orr–Sommerfeld boundary value problem is formulated. General description of Chebyshev collocation method and detailed description of numerical simulation of Orr–Sommerfeld boundary value problem by Chebyshev collocation method and validation of numerical method with literature data is given in § Numerical method. Detailed discussion of results done in § Results. The concluding remarks and outlook for further-work in § Conclusions. Finally acknowledgements in and §Acknowledgemnts.

Model description

Consider two super-imposed immiscible liquid layers between two infinite parallel plates located at, as in Fig. 1. Let the basic flow be driven only by steady motion of plates. It is well known that the basic ‘‘Couette’’ velocity profiles are steady and vary only in the span-wise direction and in the basic state, the unperturbed interface between the liquids is flat and is located at. The gravity is suppressed in this problem by considering equal densities in order to investigate the effects of surfactant and inertia on the stability of the system under consideration. The subscripts *1* and *2* refer to the lower or upper fluid, respectively and channel walls move in the horizontal direction, x, with velocities and. The interface is occupied by an insoluble surfactant with surface concentration which is only convected and diffused over the interface, but not into the bulk of the fluids thus locally changing the surface tension .

## Governing equations

The mass and momentum conservation equations governing the two-layer system are

, (1)

Where subscript represents lower and upper liquid layers respectively.

Here ,

**Figure** **1**: Schematic sketch of Couette-Poiseuille flow with surfactant laden interface. The perturbed interface is shown as sinusoidal curve. is the concentration of insoluble surfactant.

The associated boundary conditions for the system are no slip and no penetration boundary conditions at the walls.

,at and

,at

The associated interface conditions are continuity of velocity, tangential stress and normal stress.

Continuity of velocity at the interface

, at

The tangential and normal stress conditions at the interface are given by

(2)

Where are stress tensors, is unit normal, is unit tangent and

Kinematic interfacial condition is

The surfactant transport equation[13] at the interface is given by

(3)

Where is surface molecular diffusivity of surfactant. is usually negligible and neglected in this case.

We introduce dimension less variables as follows

, , , ,

The dimensionless variables in base state for the couette flow with flat interface and uniform surfactant concentration are given by

, ( ) and

, ()

Where is shear of basic velocity at interface and is given by

We consider the perturbed state with small deviation from the base state:

,,,,

Now we represent disturbance velocity in disturbance stream-functions and such that

,,,

Performing normal mode analysis by substituting

Where is wave number of the disturbance, and are constants, and is the complex wave speed. Linearizing the kinematic boundary condition yields

.

Linerarizing the dimensionless *x* and *y-*components of Navier-Stokes equation (2) followed by subtraction from the corresponding base state equations and elimination of pressure terms, yields two 4^{th} order Orr-Sommerfield ODEs in stream-functions, one for each fluid.

(4a)

(4b)

Where is the Reynolds number and . (when,)

Boundary conditions at wall in terms of stream-functions are

(5a)

(5b)

Continuity of velocity at interface gives

, (5c)

Linearization of normal stress condition gives

(5d)

Linearization of surfactant transport equation gives

Linearization of tangential stress balance condition gives

Where is the Marangoni number.

By substituting the value of from linearized surfactant transport equation in linearized tangential stress balance condition gives

(5e)

For each value of Eqs. (4),(5) forms a eigen value problem, which was numerically solved using chebyshev collocation method[1,2] and QZ algorithm for determining the complex phase velocity .

Numerical method

The two Orr-Sommerfield equations eqs. (4) along with eight boundary conditions eqs. (5) are solved numerically using Pseudo-spectral Chebyshev collocation method[1,2]. To implement the Chebyshev method, we transformed each of the two fluid domains into standard Chebyshev domain that is

*Fluid 1* domain is mapped to and *Fluid 2* domain is mapped to by substituting and respectively.

Next, we represent each stream function as truncated summation of orthogonal Chebyshev polynomials by setting.

and(6)

Where and are unknown Chebyshev coefficients and *N* is the number of Cheyshev collation points in each domain. Upon substituting eq. (6) in eq. (4) and projecting them on to arbitrary orthogonal functions and respectively by taking the Chebyshev inner product,

.

these two Chebyshev inner products forms *N-3* equations each summing up to *2N-6* equations and *N+1* coefficients in and *N+1* coefficients . *2N-6* equations along with 8 boundary conditions obtained by substituting eq. (6) in eq. (5) and *2N+2* coefficients forms a linear system

Where, and,are square matrices of size *2N+2*.

This generalized eigen value problem was solved by QZ algorithm to obtain and subsequently growth rate, .We used, above which the eigen values are independent of number of collocation points.

The accuracy of the Numerical method is checked by comparing current results with published literature[10] for the Two layer couette flow with an insoluble surfactant in stokes flow limit. To make this comparison, growth rates are calculated by muting the inertial terms by settingin the our code and with same parameters as in Halpern’s[10] Fig 2a and Fig 2b, where growth rates are predicted by long-wave evolution equation. Fig xxx shows excellent agreement between two numerical procedures.

**Figure 2**: Dispersion curves for the most (a)Unstable

**Figure 3**: Dispersion curve for the (solid line), (dashed line), at, ,,

## RESULTS and discussions

Blyth and Pozrikidis[14] observed that in the Stoke’s flow limit, there exists two modes that govern the stability of a two-layer couette flow system with surfactant: the Marangoni mode and the Yih mode associated with surfactant and the clean liquid-liquid interface respectively. But on the other hand, in flows with inertia, there exists more than two normal modes. From Fig. 3, the broken line corresponding to is above the solid line, which corresponds to , it is evident that the surfactant in the presence of inertia has significantly reduced the rate at which small perturbations decay. Earlier stability analysis for stoke flow in presence of surfactant opens up a range of unstable wave numbers extending from zero up to the critical wavenumber .The neutral stability curve Fig. 4 for values (,, and ) is in accordance with the earlier stokes flow stability analysis and in addition at , a second small window of stable wave numbers appears to form an island of stable modes, with the island tip located at . In Fig. 5 we plot the growth rate of the Marangoni mode against the Reynolds number, up to and beyond, for , corresponding to the stable island tip. At, linear stability for Stokes flow predicts the growth rate, for the Marangoni mode. The present results confirms that the Marangoni mode at marks the inauguration of the lower stable loop.

In Fig. 6 for a fixed Reynolds number , we show the dependence of the growth rates of the Marangoni mode on the wave number. The close-up near , presented in Fig. 6(b), shows that the Marangoni mode has negative growth rate for small band of wave numbers ranging from and has positive growth rate thereafter up-to , beyond which the Marangoni mode is stable again. These results clearly demonstrate the crucial role of the surfactant, which either provokes instability or significantly lowers the rate of decay of infinitesimal perturbations.

**Figure 4**: Neutral stability curves for ,, and

**Figure 5**: Growth rate vs. Reynolds number for the Marangoni mode for, , , , ,

**Figure 6**: Dispersion curve for the Marangoni mode (solid line) for,, , ,, (b) Zoom-in of (a) around

**Figure 7**: Neutral stability curves for , , and (a) (b) (c) (d) (e) (f)

Further, we investigated the effect of Marangoni number on the stability of the system under consideration via Fig 7(a) and this shows that in the devoid of surfactant that is at there is very small band of wavenumbers where the system is unstable for any Reynolds number. Moreover around the band of unstable wavenumbers is slightly larger than at any arbitrary Re. In presence of surfactant, Fig. 7(b)-7(e) a second small window of stable wave numbers appears to form an island of stable modes. By further increasing the Marangoni number, the upper curve doesn’t change but the island of secondary stable modes gets thinner and eventually disappears at around Fig. 7(f), leaving space for more unstable modes and thus making system more unstable.

Changing the viscosity ratio, keeping other parameters at (, , ), has significant impact on the topology of the neutral stability curve. By increasing, the upper neutral stability curve shrinks towards low wave numbers and bulges outwards close to , while the lower island of stable modes moves to the left Fig. 8(a)-8(c), until they meet point as shown in Fig 8(d). Upon on further raising, the two curves split leaving small island of unstable modes close to the origin and even bigger island of unstable mode to its upper right Fig. 8(e)-8(g). The small island of unstable modes around the origin shrinks and finally disappears at Fig 8(h) and this confirms the findings of Halpern[10] and Blyth[14] at creeping flow limit. Further increase in does not change the shape of neutral stability, but the thumb shaped unstable region moves in the direction of increasing as shown in Fig. 8(i)-8(j).

**Figure 8**: Neutral stability curves for , , and (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

## CONCLUSION

Early studies of Yih[7] revealed that even at small Reynolds numbers, the interface is susceptible to long-wave instability. With both inertia and surfactant present, the surfactant can either induce Marangoni instability or significantly reduce the rate at which small perturbations decay. The influence of surfactant on stability of two-layer couette flow was investigated by performing linear stability analysis. By normal mode analysis, Orr-Sommerfeld equations were formulated and growth rate was computed for different values of parameters using Pseudo-spectral Chebyshev collocation method. The results obtained for small wave numbers in Stoke’s flow limit agrees with those reported by Halpern et al[10]. Neutral stability curves are plotted and critical values of Reynolds number and Wave numbers are found. Beyond the critical Reynolds number a small island of stables modes are found and also a band of wave numbers with positive growth rates for Reynolds number less than critical value are found. The influence of Marangoni number on stability is investigated and we found that by increasing Marangoni number, the small island of stable modes shrink in the direction of increasing Reynolds number and eventually disappear. By increasing the viscosity ratio, multiple regions of stable and unstable modes are found and on further increasing viscosity ratio, the small island of unstable regions around the origin shrinks and finally disappears, leaving large region of unstable modes and further increase in viscosity ratio, shrinks the large regions of unstable modes in the direction of increasing Reynolds number.

For the future work, it would be interesting to compare results of the linear stability analysis with direct numerical simulation of two-layer flow in presence of surfactant using our Lattice Boltzmann based flow simulator.

## Acknowledgement

This work was performed as part of FACE, a research co-operation between NTNU, IFE and SINTEF, funded by The Research Council of Norway, and by the following industrial partners: StatoilHydro ASA, Norske ConocoPhillips AS, GE Oil & Gas, Scandpower Petroleum Technology AS, FMC, CD-adapco, ENI Norge AS, Shell Technology Norway AS.

We would also like to thank Prof. Sahu for providing early insights on linear stability analysis.

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