Abstract
The linear stability of CouettePoiseuille flow of two superposed fluid layers in a horizontal channel is considered. The lower fluid layer is populated with surfactants that appear either in the form of monomers or micelles and can also get adsorbed at the interface between the fluids. A mathematical model is formulated which combines the Navier Stokes equations in each fluid layer, convectiondiffusion equations for the concentration of monomers (at the interface and in the bulk fluid) and micelles (in the bulk), together with appropriate coupling conditions at the interface. The primary aim of this study is to investigate when the system is unstable to arbitrary wavelength perturbations, and in particular, to determine the influence of surfactant solubility and/or sorption kinetics on the instability. A linear stability analysis is performed and the growth rates are obtained by solving an eigenvalue problem for Stokes flow, both numerically for disturbances of arbitrary wavelength and analytically using longwave approximations. It is found that the system is stable when the surfactant is sufficiently soluble in the bulk and if the fluid viscosity ratio m and thickness ratio n satisfy the condition m < n2. On the other hand, the effect of surfactant solubility is found to be destabilising if m n2. Both of the aforementioned results are manifested for low bulk concentrations below the critical micelle concentration; however when the equilibrium bulk concentration is sufficiently high (and above the critical micelle concentration) so that micelles are formed in the bulk fluid, the system is stable if m < n2 in all cases examined.
Original language  English 

Pages (fromto)  1848 
Number of pages  31 
Journal  Journal of Fluid Mechanics 
Volume  873 
Early online date  18 Jun 2019 
DOIs  
Publication status  Published  25 Aug 2019 
Profiles

Mark Blyth
 School of Mathematics  Professor of Applied Mathematics
 Fluid and Solid Mechanics  Member
Person: Research Group Member, Academic, Teaching & Research