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Double-diffusive instability in core–annular pipe flow

Published online by Cambridge University Press:  27 January 2016

Kirti Chandra Sahu
Kirti Chandra Sahu*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Kandi, Sangareddy - 502285 Telangana, India
*
†Email address for correspondence: [email protected]
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Abstract

The instability in a pressure-driven core–annular flow of two miscible fluids having the same densities, but different viscosities, in the presence of two scalars diffusing at different rates (double-diffusive effect) is investigated via linear stability analysis and axisymmetric direct numerical simulation. It is found that the double-diffusive flow in a cylindrical pipe exhibits strikingly different stability characteristics compared to the double-diffusive flow in a planar channel and the equivalent single-component flow (wherein viscosity stratification is achieved due to the variation of one scalar) in a cylindrical pipe. The flow which is stable in the context of single-component systems now becomes unstable in the presence of two scalars diffusing at different rates. It is shown that increasing the diffusivity ratio enhances the instability. In contrast to the single fluid flow through a pipe (the Hagen–Poiseuille flow), the faster growing axisymmetric eigenmode is found to be more unstable than the corresponding corkscrew mode for the parameter values considered, for which the equivalent single-component flow is stable to both the axisymmetric and corkscrew modes. Unlike single-component flows of two miscible fluids in a cylindrical pipe, it is shown that the diffusivity and the radial location of the mixed layer have non-monotonic influences on the instability characteristics. An attempt is made to understand the underlying mechanism of this instability by conducting the energy budget and inviscid stability analyses. The investigation of linear instability due to the double-diffusive phenomenon is extended to the nonlinear regime via axisymmetric direct numerical simulations. It is found that in the nonlinear regime the flow becomes unstable in the presence of double-diffusive effect, which is consistent with the predictions of linear stability theory. A new type of instability pattern of an elliptical shape is observed in the nonlinear simulations in the presence of double-diffusive effect.

JFM classification

Instability: Absolute/convective instability Convection: Double diffusive convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 789 , 25 February 2016 , pp. 830 - 855
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Copyright
© 2016 Cambridge University Press 

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References

Bhagat, K. D., Tripathi, M. K. & Sahu, K. C. 2016 Instability due to double-diffusive phenomenon in pressure-driven displacement flow of one fluid by another in an axisymmetric pipe. Eur. J. Mech. (B/Fluids) 55, 63–70.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1987 Spectral Methods in Fluid Dynamics, 1st edn. Springer.Google Scholar
Ern, P., Charru, F. & Luchini, P. 2003 Stability analysis of a shear flow with strongly stratified viscosity. J. Fluid Mech. 496, 295–312.CrossRefGoogle Scholar
Govindarajan, R. 2004 Effect of miscibility on the linear instability of two-fluid channel flow. Intl J. Multiphase Flow 30, 1177–1192.Google Scholar
Govindarajan, R. & Sahu, K. C. 2014 Instabilities in viscosity-stratified flows. Annu. Rev. Fluid Mech. 46, 331–353.Google Scholar
Hinch, E. J. 1984 A note on the mechanism of the instability at the interface between two shearing fluids. J. Fluid Mech. 144, 463–465.CrossRefGoogle Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core–annular flows. Annu. Rev. Fluid Mech. 29, 65–90.CrossRefGoogle Scholar
Lajeunesse, E., Martin, J., Rakotomalala, N. & Salin, D. 1997 3D instability of miscible displacements in a Hele-Shaw cell. Phys. Rev. Lett. 79, 5254–5257.Google Scholar
Lajeunesse, E., Martin, J., Rakotomalala, N., Salin, D. & Yortsos, Y. C. 1999 Miscible displacement in a Hele-Shaw cell at high rates. J. Fluid Mech. 398, 299–319.Google Scholar
Malik, S. V. & Hooper, A. P. 2005 Linear stability and energy growth of viscosity stratified flows. Phys. Fluids 17, 024101.Google Scholar
Mishra, M., De Wit, A. & Sahu, K. C. 2012 Double diffusive effects on pressure-driven miscible displacement flows in a channel. J. Fluid Mech. 712, 579–597.Google Scholar
Mishra, M., Trevelyan, P. M. J., Almarcha, C. & De Wit, A. 2010 Influence of double diffusive effects and miscible viscous fingering. Phys. Rev. Lett. 105, 204501.CrossRefGoogle ScholarPubMed
Nasr-Azadani, M. & Meiburg, E. 2014 Turbidity currents interacting with three-dimensional seafloor topography. J. Fluid Mech. 745, 409–443.Google Scholar
d’Olce, M., Martin, J., Rakotomalala, N., Salin, D. & Talon, L. 2008 Pearl and mushroom instability patterns in two miscible fluids core annular flows. Phys. Fluids 20, 024104.Google Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572–600.Google Scholar
Pritchard, D. 2009 The linear stability of double-diffusive miscible rectilinear displacements in a Hele-Shaw cell. Eur. J. Mech. (B/Fluids) 28 (4), 564–577.Google Scholar
Ranganathan, B. T. & Govindarajan, R. 2001 Stabilisation and destabilisation of channel flow by location of viscosity-stratified fluid layer. Phys. Fluids 13 (1), 1–3.CrossRefGoogle Scholar
Rayleigh, L. 1880 On the stability of certain fluid motions. Proc. Lond. Math. Soc. 11, 57–70.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a finger into a porous medium in a Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Sahu, K. C. 2013 Double diffusive effects on pressure-driven miscible channel flow: influence of variable diffusivity. Intl J. Multiphase Flow 55, 24–31.Google Scholar
Sahu, K. C., Ding, H., Valluri, P. & Matar, O. K. 2009 Linear stability analysis and numerical simulation of miscible channel flows. Phys. Fluids 21, 042104.Google Scholar
Sahu, K. C. & Govindarajan, R. 2005 Stability of flow through a slowly diverging pipe. J. Fluid Mech. 531, 325–334.CrossRefGoogle Scholar
Sahu, K. C. & Govindarajan, R. 2011 Linear stability of double-diffusive two-fluid channel flow. J. Fluid Mech. 687, 529–539.Google Scholar
Sahu, K. C. & Govindarajan, R. 2012 Spatio-temporal linear stability of double-diffusive two-fluid channel flow. Phys. Fluids 24, 054103.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Scoffoni, J., Lajeunesse, E. & Homsy, G. M. 2001 Interface instabilities during displacement of two miscible fluids in a vertical pipe. Phys. Fluids 13, 553–556.Google Scholar
Selvam, B., Merk, S., Govindarajan, R. & Meiburg, E. 2007 Stability of miscible core–annular flows with viscosity stratification. J. Fluid Mech. 592, 23–49.Google Scholar
Selvam, B., Talon, L., Lesshafft, L. & Meiburg, E. 2009 Convective/absolute instability in miscible core–annular flow. Part 2. Numerical simulations and nonlinear global modes. J. Fluid Mech. 618, 323–348.Google Scholar
Swernath, S. & Pushpavanam, S. 2007 Viscous fingering in a horizontal flow through a porous medium induced by chemical reactions under isothermal and adiabatic conditions. J. Chem. Phys. 127, 204701.CrossRefGoogle Scholar
Taghavi, S. M., Séon, T., Martinez, D. M. & Frigaard, I. A. 2009 Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639, 1–35.CrossRefGoogle Scholar
Talon, L. & Meiburg, E. 2011 Plane Poiseuille flow of miscible layers with different viscosities: instabilities in the Stokes flow regime. J. Fluid Mech. 686, 484–506.Google Scholar
Tripathi, M. K., Sahu, K. C. & Govindarajan, R. 2015 Dynamics of an initially spherical bubble rising in quiescent liquid. Nat. Commun. 6, 6268.Google Scholar
Turner, J. S. 1974 Double-diffusive phenomena. Annu. Rev. Fluid Mech. 6, 37–54.Google Scholar
Yih, C. S. 1967 Instability due to viscous stratification. J. Fluid Mech. 27, 337–352.Google Scholar