Understanding time scales of diffusive fluxes and the implication for steady state and steady shape conditions

The diffusion equation is one of the most commonly used models for describing environmental problems involving heat, solute, and water transport. A diffusive system can be either transient or steady state . When a system is transient, the dependent variable (e.g., temperature, concentration, or hydraulic head) varies with time; whereas at steady state, the temporal variations are negligible. Here we consider an intermediate state, called steady shape , corresponding to the situation where temporal variations in diffusive fluxes are negligible but the dependent variable may remain transient. We present a general theoretical framework for identifying steady shape conditions and propose a novel method for evaluating the time scale needed for a diffusive system to approach both steady shape and steady state conditions.