A two-dimensional Darcy–Forchheimer mathematical model for bioconvection flow with microbial exoprotein activity

Microbes play a pivotal role in heat and mass transfer in boundary layer flow. These microbes are added to nanofluids to promote efficiency and the desired fluid properties. Some microbes excrete exoproteins into the solution, which significantly affect chemical reactions, heat and mass transfer. Based on an extensive literature review, we found no models that consider the existence of exoprotein dynamics when modelling boundary layer flow. In our model, we developed a system of partial differential equations and included exoprotein dynamics. The system of partial differential equations was transformed into a system of ordinary differential equations and then solved numerically using the spectral quasilinearisation method to a high degree of accuracy. Our results show that thermophoresis and the Brownian motion parameters increase the exoprotein concentration. The increase in growth rate, carrying capacity, and exoprotein production rate causes a significant increase in exoprotein production, and eventually, the product increases. The degradation of the exoprotein decreases the product concentration. The increase in the growth rate and the thermophoresis parameters also causes the Nusselt number to increase.