Fuzzy modelling to describe the pollutant concentration in fluids

Abstract

The study of the concentration dynamics of a pollutant substance in a fluid is a classic problem of fluid mechanics given by the transport equation $u_{t}+cu_{x}=0$, where $u=u(x,t)$ denotes the pollutant concentration along a horizontal pipe of a fixed cross-section in the positive $x$ direction at he time $t>0$ and $c$ represents the fluid propagation velocity. In view of that, the velocity of propagation of the fluid is a physical quantity, obtained, generally in an approximate form, which makes such quantity uncertain. In this paper, we propose to obtain the concentration when the constant $c$ represents the fuzzy set. The concentration was obtained by using the Zadeh's Extension Principle. Through the concentration obtained, we analyze the influence of uncertainty on the fluid propagation velocity in the concentration dynamics and explore possible practial applications in case-studies of engineering, environmental and soil sciences.