Odd-viscosity-induced instability of a falling thin film with an external electric field

The influence of odd viscosity of Newtonian fluid on the instability of thin film flowing along an inclined plane under a normal electric field is studied. By odd viscosity, we mean apart from the well-known coefficient of shear viscosity, a classical liquid with broken time-reversal symmetry is endowed with a second viscosity coefficient in biological, colloidal, and granular systems. Under the long wave approximation, a nonlinear evolution equation of the free surface is derived by the method of systematic asymptotic expansion. The effects of the odd viscosity and external electric field are considered in this evolution equation and an analytical expression of critical Reynolds number is obtained. It is interesting to find that, by linear stability analysis, the critical Reynolds number increases with odd viscosity and decreases with external strength of electric field. In other words, odd viscosity has a stable effect and electric field has a destabilized effect on flowing of thin film. In addition, through nonlinear analysis, we obtain a Ginsburg-Landau equation and find that the film has not only the supercritical stability zone and the subcritical instability zone but also the unconditional stability zone and the explosive zone. The variations of each zone with related parameters, such as the strength of electric field, odd viscosity, and Reynolds number, etc., are investigated. The results are conducive to the further development of related experiments.

Figure 1

Liquid film flow down an inclined parallel plate.

Figure 2

Dispersion relation for the long-wave model (${\mathrm{Re}}_{c1}=1.25$, $E_c=0.2$, $\beta ={45}^{\circ }$, $d=2$): (a) $E=0$, (b) $\eta =0$.

Figure 3

Stability curves $k$ vs Re for different η ($d=2$, $E$ = 0, β = 45): (a) $\eta =0.0$, (b) η = 0.5, (c) η = 1.0, (d) η = 1.5.

Figure 4

Stability curves $k$ vs Re for different $E$ ($d=2$, η = 0, β = 45 °): (a) $E$ = 0.0, (b) $E$ = 0.5, (c) $E$ = 1.0, (d) $E$ = 5.0.

Figure 5

Variations of critical amplitude with wave number $k$ for value of different η (β = 45 °, $d=2$, Re = 10): (a) $E$ = 0.0, (b) $E$ = 0.5, (c) $E$ = 1, (d) $E$ = 5.