^{1,a)}and Onofrio Semeraro

^{2,b)}

### Abstract

A complete original study of the linear temporal instability analysis of two viscous and immiscible fluids enclosed in a rigid cylinder rotating about its axis and separated by a cylindrical interface is performed for the case of higher density fluid located in the annulus. The results of the present contribution fill the lack of an overall assessment of the system behavior due to the increase of both the analytical difficulties and the number of the governing parameters when the several physical effects are all included. The analysis is carried out numerically by discretizing the equations of the evolution of disturbances separately in the two phases formulated in a rotating reference frame. Normal mode analysis leads to a generalized eigenvalue problem which is solved by means of a Chebyshev collocation spectral method. The investigation of the preferred modes of instability is carried out over wide ranges of the parameters space. The behavior of the system is physically discussed and is compared to inviscid asymptotic limits and to viscous approximate solutions of the previous literature.

I. INTRODUCTION

II. GOVERNING EQUATIONS AND PARAMETERS

A. Problem formulation

B. Linearized equations

C. Normal mode analysis

III. NUMERICAL TREATMENT

IV. REVIEW OF MEANINGFUL STABILITY RESULTS

A. Stability criteria

B. Maximum growth rates evaluations

C. Physical considerations on the onset of the instability

V. RESULTS

A. The viscous rotating annulus

B. The viscous two-fluid rotating system

VI. CONCLUSIONS

### Key Topics

- Viscosity
- 56.0
- Coriolis effects
- 21.0
- Surface tension
- 21.0
- Reynolds stress modeling
- 17.0
- Viscous flow instabilities
- 12.0

## Figures

Geometric configuration: inner surface of radius *R* _{1} is the base flow interface, outer surface of radius *R* _{2} is the rigid boundary.

Geometric configuration: inner surface of radius *R* _{1} is the base flow interface, outer surface of radius *R* _{2} is the rigid boundary.

(Color online) Growth rate vs wavenumber for the rotating annulus at different values of the Hocking number L_{2} and of the radius ratio *b*. (a): *b* = 1.2, L_{2} = 1.5, (b): *b* = 1.2, L_{2} = 10, (c): *b* = 12, L_{2} = 1.5, and (d): *b* = 12, L_{2} = 10. Dashed lines correspond to the inviscid solution of Weidman *et al.* (Ref. 8).

(Color online) Growth rate vs wavenumber for the rotating annulus at different values of the Hocking number L_{2} and of the radius ratio *b*. (a): *b* = 1.2, L_{2} = 1.5, (b): *b* = 1.2, L_{2} = 10, (c): *b* = 12, L_{2} = 1.5, and (d): *b* = 12, L_{2} = 10. Dashed lines correspond to the inviscid solution of Weidman *et al.* (Ref. 8).

(Color online) Variation with the Reynolds number Re_{2} of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the radius ratio *b* and at L_{2} = 1.5. Dashed line corresponds to the prediction of the theory of de Hoog and Lekkerkerker (Ref. 10).

(Color online) Variation with the Reynolds number Re_{2} of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the radius ratio *b* and at L_{2} = 1.5. Dashed line corresponds to the prediction of the theory of de Hoog and Lekkerkerker (Ref. 10).

(Color online) Variation with the Reynolds number Re_{2} of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the radius ratio *b* and at L_{2} = 10. Dashed line corresponds to the prediction of the theory of de Hoog and Lekkerkerker (Ref. 10).

(Color online) Variation with the Reynolds number Re_{2} of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the radius ratio *b* and at L_{2} = 10. Dashed line corresponds to the prediction of the theory of de Hoog and Lekkerkerker (Ref. 10).

Variation with the Reynolds number Re_{2} of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the radius ratio *b* for L_{2} = 2. Lines denoted with + symbols are relative to computations in which Coriolis terms are artificially suppressed.

Variation with the Reynolds number Re_{2} of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the radius ratio *b* for L_{2} = 2. Lines denoted with + symbols are relative to computations in which Coriolis terms are artificially suppressed.

(Color online) Variation with the Reynolds number Re_{2} of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the Ohnesorge number Oh_{2} and for *b* = 20. Continuous line is the prediction of Tomotika (Ref. 4) theory. Dashed lines are the predictions of de Hoog and Lekkerkerker theory (Ref. 10) for the first three values of Oh_{2}.

(Color online) Variation with the Reynolds number Re_{2} of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the Ohnesorge number Oh_{2} and for *b* = 20. Continuous line is the prediction of Tomotika (Ref. 4) theory. Dashed lines are the predictions of de Hoog and Lekkerkerker theory (Ref. 10) for the first three values of Oh_{2}.

(Color online) Variation with the Hocking number L_{2} of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the Reynolds number Re_{2}. (a) and (b): *b* = 20, (c) and (d): *b* = 1.2. Dashed lines are the predictions of de Hoog and Lekkerkerker theory (Ref. 10) for the first three values of Re_{2}. Dash-dotted line is the prediction of Rayleigh (Ref. 2) theory.

(Color online) Variation with the Hocking number L_{2} of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the Reynolds number Re_{2}. (a) and (b): *b* = 20, (c) and (d): *b* = 1.2. Dashed lines are the predictions of de Hoog and Lekkerkerker theory (Ref. 10) for the first three values of Re_{2}. Dash-dotted line is the prediction of Rayleigh (Ref. 2) theory.

(Color online) Variation with the radius ratio *b* of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the Reynolds number Re_{2}. (a) and (b): L_{2} = 1.5, (c) and (d): L_{2} = 10. Dashed lines are the inviscid solutions of Weidman *et al.* (Ref. 8).

(Color online) Variation with the radius ratio *b* of the maximum growth rate and wavenumber of maximum growth rate for the rotating annulus at different values of the Reynolds number Re_{2}. (a) and (b): L_{2} = 1.5, (c) and (d): L_{2} = 10. Dashed lines are the inviscid solutions of Weidman *et al.* (Ref. 8).

(Color online) Maximum growth rate and critical wavenumber as functions of Re_{2} for the two-fluid rotating system. λ = χ = 1, *b* = 12. Continuous lines are the predictions of Ashmore and Stone’s (Ref. 11) large-Taylor number asymptotic theory. Thick lines are the predictions of Tomotika (Ref. 4) theory.

(Color online) Maximum growth rate and critical wavenumber as functions of Re_{2} for the two-fluid rotating system. λ = χ = 1, *b* = 12. Continuous lines are the predictions of Ashmore and Stone’s (Ref. 11) large-Taylor number asymptotic theory. Thick lines are the predictions of Tomotika (Ref. 4) theory.

(Color online) Maximum growth rate and critical wavenumber as functions of Re_{2} for the two-fluid rotating system. λ = 0.5, χ = 1, *b* = 12. Continuous lines are the predictions of Ashmore and Stone’s (Ref. 11) large-Taylor number asymptotic theory, thick line are the predictions of Tomotika (Ref. 4) theory, and dashed lines are the prediction of de Hoog and Lekkerkerker theory (Ref. 10) for the first three values of Oh_{2}.

(Color online) Maximum growth rate and critical wavenumber as functions of Re_{2} for the two-fluid rotating system. λ = 0.5, χ = 1, *b* = 12. Continuous lines are the predictions of Ashmore and Stone’s (Ref. 11) large-Taylor number asymptotic theory, thick line are the predictions of Tomotika (Ref. 4) theory, and dashed lines are the prediction of de Hoog and Lekkerkerker theory (Ref. 10) for the first three values of Oh_{2}.

Maximum growth rate and critical wavenumber as functions of Re_{2} for the two-fluid rotating system for χ = 0.5, *b* = 5 and various λ. (a) and (b): L_{2} = 1, (c) and (d): L_{2} = 5.

Maximum growth rate and critical wavenumber as functions of Re_{2} for the two-fluid rotating system for χ = 0.5, *b* = 5 and various λ. (a) and (b): L_{2} = 1, (c) and (d): L_{2} = 5.

(Color online) Maximum growth rate and critical wavenumber as functions of λ for the two-fluid rotating system for χ = 0.5, L_{2} = 5, *b* = 5 and various Re_{2}. Dashed lines are the inviscid solution of Weidman *et al.* (Ref. 8).

(Color online) Maximum growth rate and critical wavenumber as functions of λ for the two-fluid rotating system for χ = 0.5, L_{2} = 5, *b* = 5 and various Re_{2}. Dashed lines are the inviscid solution of Weidman *et al.* (Ref. 8).

(Color online) Maximum growth rate and critical wavenumber as functions of L_{2} for the two-fluid rotating system for χ = 0.5, *b* = 2 and various λ. Continuous lines are computed for Re_{2} = 10, dashed lines are the inviscid solution of Weidman *et al.* (Ref. 8).

(Color online) Maximum growth rate and critical wavenumber as functions of L_{2} for the two-fluid rotating system for χ = 0.5, *b* = 2 and various λ. Continuous lines are computed for Re_{2} = 10, dashed lines are the inviscid solution of Weidman *et al.* (Ref. 8).

(Color online) Variation of L_{2c } (a) and σ_{ Mc } (b) as functions of Re_{2}. Symbol legends are: Δ: *b* = 1.2, ○: *b* = 5. Continuous lines are for χ = 1, dashed lines are for χ = 0.5, and dash-dotted lines are for χ = 0.1.

(Color online) Variation of L_{2c } (a) and σ_{ Mc } (b) as functions of Re_{2}. Symbol legends are: Δ: *b* = 1.2, ○: *b* = 5. Continuous lines are for χ = 1, dashed lines are for χ = 0.5, and dash-dotted lines are for χ = 0.1.

Maximum growth rate and critical wavenumber as functions of χ for the two-fluid rotating system for Re_{2} = 1, L_{2} = 5, *b* = 5 and various λ.

Maximum growth rate and critical wavenumber as functions of χ for the two-fluid rotating system for Re_{2} = 1, L_{2} = 5, *b* = 5 and various λ.

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