^{1}, P. Valluri

^{1}, P. D. M. Spelt

^{1}and O. K. Matar

^{1}

### Abstract

The linear stability characteristics of pressure-driven two-layer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a Herschel-Bulkley fluid. A pair of coupled Orr-Sommerfeld eigenvalueequations are derived and solved using an efficient spectral collocation method for cases in which unyielded regions are absent. An asymptotic analysis is also carried out in the long-wave limit, the results of which are in excellent agreement with the numerical predictions. Our analytical and numerical results indicate that increasing the dimensionless yield stress, prior to the formation of unyielded plugs below the interface, is destabilizing. Increasing the shear-thinning tendency of the lower fluid is stabilizing.

The authors would like to acknowledge the support of the DTI, through Grant No. TP//ZEE/6/1/21191. Useful discussions with Professor C. J. Lawrence are also acknowledged.

I. INTRODUCTION

II. FORMULATION

A. Governing equations

B. Linear stability analysis

C. Numerical procedure and validation

D. Asymptotic analysis:, “long-wave” limit

III. RESULTS

A. Interfacial” mode

B. Shear mode

IV. CONCLUSIONS

### Key Topics

- Viscosity
- 20.0
- Yield stress
- 17.0
- Eigenvalues
- 10.0
- Flow instabilities
- 9.0
- Poiseuille flow
- 9.0

## Figures

Schematic of a two-layer flow in a channel of height , where represents the thickness of the lower, non-Newtonian fluid. Also shown here are profiles of the steady, streamwise velocity component generated with and .

Schematic of a two-layer flow in a channel of height , where represents the thickness of the lower, non-Newtonian fluid. Also shown here are profiles of the steady, streamwise velocity component generated with and .

The effect of increasing the order of Chebyshev polynomials, , on the variation of with with , , , , , , , and .

The effect of increasing the order of Chebyshev polynomials, , on the variation of with with , , , , , , , and .

Eigenvalue distribution generated for the Newtonian case of South and Hooper (Ref. 23) with , , , , , , and .

Eigenvalue distribution generated for the Newtonian case of South and Hooper (Ref. 23) with , , , , , , and .

Comparison of the numerical solution (solid lines) with the predictions of the long-wave analysis (dotted lines) in terms of the variation of the complex part of the phase speed, , with the wavenumber, . (a) , , , ; (b) , , , . The rest of the parameters are , , and .

Comparison of the numerical solution (solid lines) with the predictions of the long-wave analysis (dotted lines) in terms of the variation of the complex part of the phase speed, , with the wavenumber, . (a) , , , ; (b) , , , . The rest of the parameters are , , and .

The effect of varying the Bingham number, , on the dispersion curves; (a) and (b) . The rest of the parameters are , , , , , and . The labels A, B, C, D, and E are used to designate maxima in the dispersion curves; the energy “budgets” associated with these points are provided in Table I. The dotted line in (a) shows the dispersion curve obtained by linearizing about the base state for and setting in Eqs. (11) and (12).

The effect of varying the Bingham number, , on the dispersion curves; (a) and (b) . The rest of the parameters are , , , , , and . The labels A, B, C, D, and E are used to designate maxima in the dispersion curves; the energy “budgets” associated with these points are provided in Table I. The dotted line in (a) shows the dispersion curve obtained by linearizing about the base state for and setting in Eqs. (11) and (12).

The effect of varying the flow index, , on the dispersion curves; (a) and (b) . Here, and the rest of the parameters are the same as in Fig. 5. The dotted line in (a) shows the dispersion curve obtained by linearizing about the base state for and setting and in Eqs. (11) and (12).

The effect of varying the flow index, , on the dispersion curves; (a) and (b) . Here, and the rest of the parameters are the same as in Fig. 5. The dotted line in (a) shows the dispersion curve obtained by linearizing about the base state for and setting and in Eqs. (11) and (12).

Effect of varying the density ratio on the dispersion curves for small, (a), and large, (b), . The rest of the parameters are , , , , , , and . The labels A, B, and C are used to designate global maxima in the dispersion curves associated with , respectively; the energy “budgets” associated with these points are provided in Table II.

Effect of varying the density ratio on the dispersion curves for small, (a), and large, (b), . The rest of the parameters are , , , , , , and . The labels A, B, and C are used to designate global maxima in the dispersion curves associated with , respectively; the energy “budgets” associated with these points are provided in Table II.

Effect of varying the thickness ratio , (a), and viscosity ratio , (b), on the growth rate associated with the most dangerous mode, in the presence and absence of a yield stress. The rest of the parameters are , , , , , , and .

Effect of varying the thickness ratio , (a), and viscosity ratio , (b), on the growth rate associated with the most dangerous mode, in the presence and absence of a yield stress. The rest of the parameters are , , , , , , and .

Effect of varying the Reynolds number, , on the dispersion curves generated by plotting the imaginary parts of the two leading eigenvalues against with (dashed lines) and (solid lines). The rest of the parameters are , , , , , and .

Effect of varying the Reynolds number, , on the dispersion curves generated by plotting the imaginary parts of the two leading eigenvalues against with (dashed lines) and (solid lines). The rest of the parameters are , , , , , and .

The effect of varying on the neutral stability curves of the “shear” mode; (a) and (b) . The rest of the parameters are , , , , and .

The effect of varying on the neutral stability curves of the “shear” mode; (a) and (b) . The rest of the parameters are , , , , and .

Effect of varying on the dispersion curves with ; (a) and (b) . The rest of the parameters are the same as in Fig. 10. The curves associated with the “shear” modes are shown in the insets of panels (a) and (b).

Effect of varying on the dispersion curves with ; (a) and (b) . The rest of the parameters are the same as in Fig. 10. The curves associated with the “shear” modes are shown in the insets of panels (a) and (b).

The cross-stream structure of the real and imaginary parts of associated with the most dangerous “interfacial” and “shear” modes in Fig. 11(a) shown in (a) and (b), respectively.

The cross-stream structure of the real and imaginary parts of associated with the most dangerous “interfacial” and “shear” modes in Fig. 11(a) shown in (a) and (b), respectively.

The effect of varying the flow index, , on the neutral stability curve of the “shear” mode; (a) and (b) . The rest of the parameters are , , , , and . The dotted line in (a) was obtained by linearizing about a base state with and and by setting and in Eqs. (11) and (12).

The effect of varying the flow index, , on the neutral stability curve of the “shear” mode; (a) and (b) . The rest of the parameters are , , , , and . The dotted line in (a) was obtained by linearizing about a base state with and and by setting and in Eqs. (11) and (12).

The effect of varying on the dispersion curves with ; (a) and (b) . The rest of the parameters are the same as Fig. 13.

The effect of varying on the dispersion curves with ; (a) and (b) . The rest of the parameters are the same as Fig. 13.

The effect of varying on the neutral stability curves of the “shear” mode; (a) and (b) . The rest of the parameters are , , , , and .

## Tables

Energy “budgets” for the points labelled A, B, C, D, and E in Fig. 5.

Energy “budgets” for the points labelled A, B, C, D, and E in Fig. 5.

Energy “budgets” for the points labelled A, B, and C in Fig. 7.

Energy “budgets” for the points labelled A, B, and C in Fig. 7.

Energy “budgets” for , and the parameter values used to generate Fig. 9.

Energy “budgets” for , and the parameter values used to generate Fig. 9.

Energy budgets for the points labelled A–D in Fig. 11.

Energy budgets for the points labelled A–D in Fig. 11.

Energy budgets for the points labelled A–D in Fig. 14.

Energy budgets for the points labelled A–D in Fig. 14.

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