Numerical Simulation and the Universality Class of the KPZ Equation for Curved Substrates
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The Kardar-Parisi-Zhang (KPZ) equation for surface growth has been analyzed for over three decades, and its properties and universality classes are well established. The vast majority of all the past studies were, however, concerned with surface growth that started from flat substrates. In several natural phenomena, as well as technological processes, interface growth occurs on curved surfaces. Examples include tumour and bacterial growth, as well as the interface between two fluid phases during injection of a fluid into a porous medium in which it moves radially. Since in growth on flat substrates the linear size of the system remains constant, whereas it increases in the case of growth on curved substrates, the universality class of the resulting growth process has remained controversial. We present the results of extensive numerical simulations in (1+1)-dimensions of the KPZ equation, starting from an initial circular substrate. We find that unlike the KPZ equation for flat substrates, the transition from linear to nonlinear universality classes is not sharp. Moreover, the interface width exhibits logarithmic growth with the time, instead of saturation, in the long-time limit. Furthermore, we find that evaporation dominates the growth process when the coefficient of the nonlinear term in the KPZ equation is small, and that average radius of interface decreases with time and reaches a minimum but not zero value.
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