Nonlinear frictiophoresis is the unidirectional drift of a particle in a medium caused by periodic driving force with zero mean. The effect is possible due to nonlinear dependence of the friction-drag force on the particle's velocity. It was discovered theoretically., and is mainly known as nonlinear electrofrictiophoresis
. At first glance, a periodic driving force with zero mean is able to entrain a particle into an oscillating movement without unidirectional drift, because integral momentum provided to the particle by the force is zero. The possibility of unidirectional drift can be recognized if one takes into account that the particle itself loses momentum through transferring it further to the medium it moves in/at. If the friction is nonlinear, then it may so happen that the momentum loss during movement in one direction does not equal to that in the opposite direction and this causes unidirectional drift. For this to happen, the driving force time-dependence must be more complicated than it is in a single sinusoidal harmonic.
A simple example - Bingham plastic
Nonlinear friction The simplest case of friction-velocity dependence law is the Stokes's one:
( 1 ) F d r ( v ) = λ v , {\displaystyle (1)\qquad F_{dr}(v)=\lambda v,}
where F d r ( v ) {\displaystyle F_{dr}(v)} is the friction/drag force applied to a particle moving with velocity v {\displaystyle v} in a medium. The friction-velocity law (1) is observed for a slowly moving spherical particle in a Newtonian fluid. It is linear, see Fig. 1, and is not suitable for nonlinear frictiophoresis to take place. The characteristic property of the law (1) is that any, even a very small driving force is able to get particle moving. This is not the case for such media as Bingham plastic. For those media, it is necessary to apply some threshold force, d {\displaystyle d} , to get the particle moving. This kind of friction-velocity (dry friction) law has a jump discontinuity at v = 0 {\displaystyle v=0} :
( 2 ) F d r ( v ) = λ v + d ⋅ s i g n ( v ) . {\displaystyle (2)\qquad F_{dr}(v)=\lambda v+d\cdot \mathrm {sign} (v).}
It is nonlinear, see Fig. 2, and is used in this example.
Periodic driving force Let T > 0 {\displaystyle T>0} denote the period of driving force. Chose a time value t 1 {\displaystyle t_{1}}
such that 0 < t 1 < T {\displaystyle 0<t_{1}<T}
and two force values, F + > 0 {\displaystyle F^{+}>0} , F − < 0 {\displaystyle F^{-}<0}
such that the following relations are satisfied:
F + > d , | F − | < d , {\displaystyle \qquad \qquad F^{+}>d,\quad |F^{-}|<d,}
( 3 ) F + t 1 + F − ( T − t 1 ) = 0. {\displaystyle (3)\qquad F^{+}t_{1}+F^{-}(T-t_{1})=0.}
The periodic driving force f ( t ) {\displaystyle f(t)}
used in this example is as follows:
( 4 ) f ( t ) = { F + , if 0 < t ≤ t 1 , F − , if t 1 < t ≤ T , f ( t + T ) = f ( t ) . {\displaystyle (4)\qquad f(t)={\begin{cases}F^{+},{\text{ if }}0<t\leq t_{1},\\F^{-},{\text{ if }}t_{1}<t\leq T,\quad f(t+T)=f(t).\end{cases}}}
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