Contact angle hysteresis can modulate the Newtonian rod climbing effect
The present work investigates the role of contact angle hysteresis at the liquid-liquid-solid interface (LLS) on the rod climbing effect of two immiscible Newtonian liquids using experimental and numerical approaches. Experiments revealed that the final steady state contact angle, $\theta_{w}$ at the LLS interface varies with the rod rotation speed, $\omega$. For the present system, $\theta_{w}$ changes from $\sim$69$^{\circ}$ to $\sim$83$^{\circ}$ when the state of the rod is changed from static condition to rotating at 3.3 Hz. With further increase in $\omega$, the $\theta_{w}$ exceeds 90$^{\circ}$ which cannot be observed experimentally. It is inferred from the simulations that the input value of $\theta_{w}$ saturates and attains a constant value of $\sim$120$^{\circ}$ for $\omega>$ 5 Hz. Using numerical simulations, we demonstrate that this contact angle hysteresis must be considered for the correct prediction of the Newtonian rod climbing effect. Using the appropriate values of the contact angle in the boundary condition, an excellent quantitative match between the experiments and simulations is obtained in terms of: the climbing height, the threshold rod rotation speed for onset of climbing, and the shape of liquid-liquid interface. This resolves the discrepancy between the experiments and simulations in the existing literature where a constant value of the contact angle has been used for all speeds of rod rotation.