We study the diffusive instability subject to a solid–liquid interface and a Kolmogorov flow using modal, non-modal analyses and energy analysis. It is found that the phase boundary has different effects on the threshold of diffusive convection for weak and strong salinity stratification. In the context of oceanography where the salinity Rayleigh number RS is very high, the ice–water interface has negligible influence on the onset of diffusive convection. In the presence of shear, the diffusive convection for RS < 106 tends to be inhibited and with the increase of shear intensity, the oscillatory, steady convective and Kelvin–Helmholtz instabilities will be successively dominant. For RS > 106, the shear has a destabilizing effect on the diffusive convection, due to the generation of thermohaline-sheared instability found by Radko (J. Fluid Mech., vol. 805, 2016, pp. 147–170). Non-modal analysis indicates that for realistic parameters of high-latitude oceans, with the transition of the ultimate energy source of instability from the density gradient to background current, the thermohaline-shear instability is expected to transition from oscillatory to steady instability. The initial transient amplification due to double diffusion has also been studied, which is due to the generation of overstable instability at the initial phase. For RS < 106, the optimal initial condition to achieve the maximum transient growth favours longitudinal rolls. For thermohaline-shear instability, however, it favours transverse rolls and specifically, in oscillatory thermohaline-shear instability, the transient amplification can be enhanced by the shear by one order of magnitude, thus having important influence on the stability of the system.

When a droplet coalesces with a flat liquid–air interface, a secondary drop may be left behind resulting in only a partial coalescence rather than complete coalescence. In this paper, we employ an arbitrary Lagrangian–Eulerian method to demonstrate that applying an electric field favours the occurrence of partial coalescence. To understand this phenomenon, we systematically study the effect of an external electric field on the coalescence process between a droplet and a liquid–air interface. In an electric field, the induced electric stresses can overcome the downward flow of the droplet, thus lifting it upwards. As a result, the positive Laplace pressure in the neck region squeezes the droplet towards pinch-off. We observe that both the initial neck expansion and neck shrinkage are suppressed by the electric field. These effects become weaker as the Ohnesorge number $Oh$ increases. Based on the scaling analysis, we report a critical Ohnesorge number $O{h_c} = 14.39{\varGamma ^{3/2}} + 0.029$ to quantify the transition from partial coalescence to complete coalescence in the presence of an electric field, where $\varGamma $ represents the dimensionless electric Bond number. Finally, a relationship between the secondary droplet size and the two key dimensionless numbers of $Oh$ and $\varGamma $ has been developed, which could be useful for producing droplets of desired sizes in microfluidic applications.