Lie group analysis and general forms of self-similar parabolic equations for fluid flow, heat and mass transfer of nanofluids

On the basis of symmetry group analysis applied to fluid flow, heat and mass transfer equations for nanofluids in parabolic approximation, their symmetries (i.e., the Lie groups) were obtained. Using these groups, self-similar forms for independent variables and functions were derived, which describe velocity, temperature and concentration fields. These forms enabled obtaining generalized self-similar ordinary differential equations for parabolic flows of nanofluids. In these equations, physical properties of nanofluids (viscosity, thermal conductivity, diffusion coefficients and density) were specified in general form as functions of the temperature and nanoparticle concentration. Therefore, the proposed equations by their nature are universal and free from any specific form of the functional dependence of the physical properties of nanofluids on the temperature and nanoparticle concentration. Self-similar equations take into account different additional effects that arise in nanofluid flows. The paper includes particular examples of the description of specific problems (flows with streamwise pressure gradient, impingement, turbulence, free convection, boiling, etc.) on the basis of generalized transport equations. A novel self-similar solution for heat transfer in nanofluids accounting for dissipation effects was derived in the paper. This solution demonstrates that adding of nanoparticles causes earlier conversion of the cooling mode into the heating mode.