^{1,a)}and O. K. Matar

^{2}

### Abstract

The three-dimensional 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. We focus on the parameter ranges for which Squire’s theorem for the two-layer Newtonian problem does not exist. The modified Orr–Sommerfeld and Squire equations in each layer are derived and solved using an efficient spectral collocation method. Our results demonstrate the presence of three-dimensional instabilities for situations where the square root of the viscosity ratio is larger than the thickness ratio of the two layers; these “interfacial” mode instabilities are also present when density stratification is destabilizing. These results may be of particular interest to researchers studying the transient growth and nonlinear stability of two-fluid non-Newtonian flows. We also show that the “shear” modes, which are present at sufficiently large Reynolds numbers, are most unstable to two-dimensional disturbances.

The authors would like to acknowledge the support of the Indian Institute of Technology Hyderabad, India. O.K.M. also acknowledges the support of the Technology Strategy Board of the U.K. through Grant No. TP//ZEE/6/1/21191.

I. INTRODUCTION

II. FORMULATION

A. Base state

B. Linear stability analysis

C. Numerical procedure and validation

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Flow instabilities
- 21.0
- Channel flows
- 15.0
- Poiseuille flow
- 15.0
- Viscosity
- 14.0
- Reynolds stress modeling
- 11.0

## Figures

Schematic of a two-layer flow in a channel of height , where represents the thickness of the lower, non-Newtonian fluid.

Schematic of a two-layer flow in a channel of height , where represents the thickness of the lower, non-Newtonian fluid.

Basic state profiles of the steady, streamwise velocity component for , (a), and (b), respectively. The rest of the parameter values are , , and .

Basic state profiles of the steady, streamwise velocity component for , (a), and (b), respectively. The rest of the parameter values are , , and .

The effect of increasing the order of Chebyshev polynomials in each layer, , on the variation of normalized growth rate, , with , where is the growth rate associated with the corresponding two-dimensional disturbance . The rest of the parameter values are , , , , , , , , and .

The effect of increasing the order of Chebyshev polynomials in each layer, , on the variation of normalized growth rate, , with , where is the growth rate associated with the corresponding two-dimensional disturbance . The rest of the parameter values are , , , , , , , , and .

The dispersion curves ( vs ) for different values of ; (a) and (b) . The rest of the parameter values are , , , , , , and . This corresponds to a case where both the layers are Newtonian fluids. The labels in (b) are used to designate the maxima in the dispersion curves; the energy budgets associated with these points are provided in Table II.

The dispersion curves ( vs ) for different values of ; (a) and (b) . The rest of the parameter values are , , , , , , and . This corresponds to a case where both the layers are Newtonian fluids. The labels in (b) are used to designate the maxima in the dispersion curves; the energy budgets associated with these points are provided in Table II.

The dispersion curves ( vs ) for different values of ; (a) and (b) . The rest of the parameter values are the same as in Fig. 4 but with .

The dispersion curves ( vs ) for different values of ; (a) and (b) . The rest of the parameter values are the same as in Fig. 4 but with .

The dispersion curves ( vs ) for different values of ; (a) and (b) . Here, and the rest of the parameter values are the same as in Fig. 4. The labels and are used to designate maxima in the dispersion curves in (a) and (b), respectively; the energy budgets associated with these points are provided in Tables III and IV, respectively.

The dispersion curves ( vs ) for different values of ; (a) and (b) . Here, and the rest of the parameter values are the same as in Fig. 4. The labels and are used to designate maxima in the dispersion curves in (a) and (b), respectively; the energy budgets associated with these points are provided in Tables III and IV, respectively.

The dispersion curves ( vs ) for different values of Bn; (a) and (b) . Here, and the rest of the parameter values are the same as in Fig. 4. The labels and are used to designate maxima in the dispersion curves in (a) and (b), respectively; the energy budgets associated with these points are provided in Tables V and VI, respectively.

The dispersion curves ( vs ) for different values of Bn; (a) and (b) . Here, and the rest of the parameter values are the same as in Fig. 4. The labels and are used to designate maxima in the dispersion curves in (a) and (b), respectively; the energy budgets associated with these points are provided in Tables V and VI, respectively.

The dispersion curves ( vs ) for different values of ; (a) and (b) . Here, and the rest of the parameter values are the same as in Fig. 7.

The dispersion curves ( vs ) for different values of ; (a) and (b) . Here, and the rest of the parameter values are the same as in Fig. 7.

The effect of varying on the neutral stability curves of the shear mode, (a) and (b), the cross-stream structure of the real and imaginary parts of associated with the most dangerous interfacial and shear modes in panel (e) are shown in panels (c) and (d); the dispersion curves ( vs ) are shown in panels (e) and (f). Panels (a), (c), and (e), and (b), (d), and (f) are associated with and , respectively. The rest of the parameter values are , , , , , , and .

The effect of varying on the neutral stability curves of the shear mode, (a) and (b), the cross-stream structure of the real and imaginary parts of associated with the most dangerous interfacial and shear modes in panel (e) are shown in panels (c) and (d); the dispersion curves ( vs ) are shown in panels (e) and (f). Panels (a), (c), and (e), and (b), (d), and (f) are associated with and , respectively. The rest of the parameter values are , , , , , , and .

The effect of varying on the neutral stability curves of the shear mode (a), (c), (e), (b), (d), and (f). The rest of the parameter values in the panels are , , , , , , , and .

The effect of varying on the neutral stability curves of the shear mode (a), (c), (e), (b), (d), and (f). The rest of the parameter values in the panels are , , , , , , , and .

## Tables

Maximum normalized growth rate, in Fig. 3 for different values of .

Maximum normalized growth rate, in Fig. 3 for different values of .

Energy budgets for the points labeled , , , , and in Fig. 4(b).

Energy budgets for the points labeled , , , , and in Fig. 4(b).

Energy budgets for the points labeled , , , and in Fig. 6(a).

Energy budgets for the points labeled , , , and in Fig. 6(a).

Energy budgets for the points labeled , , , and in Fig. 6(b).

Energy budgets for the points labeled , , , and in Fig. 6(b).

Energy budgets for the points labeled , , , and in Fig. 7(a).

Energy budgets for the points labeled , , , and in Fig. 7(a).

Energy budgets for the points labeled , , , and in Fig. 7(b).

Energy budgets for the points labeled , , , and in Fig. 7(b).

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