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We study the melting process of a solid under microgravity, driven solely by lateral vibrations that are perpendicular to the applied temperature gradient due to the absence of gravity-induced convection. Using direct numerical simulations with the phase-field method, we examine two-dimensional vibration-induced melting in a square cavity over four orders of magnitude of vibrational Rayleigh numbers, 
$10^5\le Ra_{{vib}}\le 10^9$. Our results show that as melting progresses, the flow structure transitions from a periodic-circulation regime with diffusion-dominated heat transfer to a columnar regime with vibroconvection. The mean height of the liquid–solid interface follows a power-law dependency with time, 
$\bar {\xi } \sim \tilde t^{1/(2-2\alpha )}$, where 
$\alpha = 0$ in the periodic-circulation regime and 
$\alpha = 1/2$ in the columnar regime. We further observe that within the columnar regime, the morphological evolution of the liquid–solid interface is influenced by the interaction of columnar thermal plumes in the central regions and the peripheral flow near the sidewalls. Specifically, we offer a comprehensive analysis of the plume merging behaviour, which is governed by the aspect ratio (
$\bar {\xi }$) of the liquid layer and the intensity of vibration, quantified by the effective vibrational Rayleigh number 
$Ra_{vib}^{eff}$. We identify the relationship between the number of columnar plumes 
$K_m$ and 
$Ra_{vib}^{eff}$, finding that 
$K_m \sim \bar {\xi }^{-1} (Ra_{vib}^{eff})^{\gamma }$ with the fitting scaling exponent 
$\gamma = 0.150 \pm 0.025$. We subsequently quantify the characteristics of the interface roughness amplitude evolution in microgravity vibroconvection. Our results indicate that the roughness amplitude exhibits a power-law dependence on the mean height of the liquid layer. Drawing from the Stefan boundary condition, we theoretically deduce this dependence under the assumption of a non-uniform heat flux distribution at the interface, where the theory is corroborated by our numerical simulations.
A positive even number is said to be a Maillet number if it can be written as the difference between two primes, and a Kronecker number if it can be written in infinitely many ways as the difference between two primes. It is believed that all even numbers are Kronecker numbers. We study the division and multiplication of Kronecker numbers and show that these numbers are rather abundant. We prove that there is a computable constant k and a set D consisting of at most 720 computable Maillet numbers such that, for any integer n, 
$kn$ can be expressed as a product of a Kronecker number and a Maillet number in D. We also prove that every positive rational number can be written as a ratio of two Kronecker numbers.
