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To understand the complex problem of surface growth phenomena, we developed a model in which the regular diffusion equation is coupled to the Kardar–Parisi–Zhang (KPZ) equation. The fundamental or Gaussian solution of the regular diffusion equation is considered as an external noise or source term in the KPZ equation. The obtained system of partial differential equations is analytically solved by a self-similar Ansatz and expressed the solution as a combination of elementary and special functions. Using this solution, the effects of the physical parameters were explicitly investigated. Our recent explicit numerical method, the leapfrog-hopscotch algorithm, is also tested for this problem. It is shown that this method can be safely used with orders of magnitude larger time step sizes than the usual explicit (Euler) scheme as well if we are far from the cusp-like solutions. We pointed out that the cusps themselves cannot be properly simulated by any method that we know.

Magneto-thermomechanical interaction between heated viscous incompressible ferrofluid and a cold wall in the presence of a spatially varying magnetic field is investigated. A model for two dimensional ferrofluid flows is constructed. The influence of magnetic dipole in a Newtonian ferrofluid is characterized. The dynamic response of heated ferrofluid in a non-uniform magnetic field is studied along a nonlinearly stretching sheet with changing surface temperature. The nonlinearity is considered as power law relationship. Similarity transformation is applied to convert system of partial differential equations into ordinary differential equations and then solved numerically. The impact of the parameters on the velocity and temperature are explored numerically. It is found that the magneto-thermomechanical interaction slows down the motion of fluid particles. At the surface, the skin friction coefficient decreases and the rate of heat transfer increases with increasing ferromagnetic parameter or with increasing power law exponent.