Effects of magnetic fields on capillary-gravity waves in the presence of magnetic surfactants

. The stability of capillary-gravity wave motion on horizontal free surface of viscous non-compressible fluid in the presence of magnetic surfactant in an external magnetic field was studied. It is shown that for normal as well as for tangential external magnetic field the horizontal free liquid surface is unstable for field strength exceeding some critical value that does not depend on the elastic constant of the surfactant film. However, for oblique external magnetic field the stability of the free surface depends not only on the field value but also on the surfactant elastic constant.


Governing equations
Recently, new surfactant molecules with magnetic properties have been synthesized [1]. This makes it possible the magnetic control over liquid surface properties by sufficiently strong (0.4...1 T) external magnetic fields. The surface tension tensor for these media is anisotropic and depends on magnetic field strength. The analogous dependence was found for magnetic fluid -water interfaces subjected to moderate-intensity (60...110 G) magnetic fields [2]. We study the stability of capillary-gravity wave motion on horizontal free surface of viscous non-compressible fluid in the presence of magnetic surfactant in external magnetic field.
Let D − be the domain occupied by liquid, D + be the external domain and Σ be the interfacial boundary, defined by the equation In equilibrium 0 η = . The z axis is directed upwards. The normal vector n on the boundary is defined as the external normal to D − .
Inside the domain D − the Navier-Stokes equations are valid: (0,0, ); Here ρ is the density and µ is the dynamic viscosity of the liquid, i v are the velocity components, p is the pressure, ( ) f ik p is the stress tensor (excluding the magnetic field stress) and g is the acceleration of gravity.
The following conditions are valid at the interface [2]: The constant β characterizes the dependence of surface tension on surfactant concentration [4]. The dependence of ab σ on magnetic field strength is taken from [2]. The following stability conditions are necessary: 0, 0 n t λ λ ≥ ≤ [3]. The dynamics of surfactant surface density is described by the following equations: Нere the surfactant diffusion effects and mass transfer effects to or from the dividing surface are neglected and the surfactant surface velocity a u Σ is taken equal to the tangential fluid velocity [4]; ab a are the components of first quadratic form on the surface.
The Maxwell equations for magnetic field in nonmagnetizable media are div 0, rot 0, The conditions at the infinity take the form : 0; : Here , t n H H are the given values of external magnetic field strength in tangential and normal directions to the horizontal unperturbed surface.

Linearization and dispersion equation
To study capillary-gravity waves we perform the linearization of the equations (1)

Conclusions
The stability of capillary-gravity wave motion on horizontal free surface of viscous non-compressible fluid in the presence of magnetic surfactant in an external magnetic field was studied taking into account the anisotropy of surface tension tensor and the dependence of surface tension on surfactant surface density and on magnetic field strength. It is shown that for normal as well as for tangential external magnetic field the horizontal free liquid surface is unstable for field strength exceeding some critical value that does not depend on the elastic constant of the surfactant film. However, for oblique external magnetic field the stability of the free surface depends not only on the field value but also on the surfactant elastic constant. The damping decrement of capillary-gravity waves depends significally on surfactant elastic constant. Support by the Russian Foundation for Basic Research (projects Nos. 16-01-00157 and 17-01-00037) is acknowledged.