Analysis of transient mixed convection in a vertical channel filled with nanofluids: combined analytical and numerical approaches

This study investigates the transient mixed convection of nanofluids in a vertical channel, focusing on the time-dependent evolution of velocity, temperature, and nanoparticle concentration profiles. While steady-state mixed convection in vertical channels has been extensively studied, the transient behavior of nanofluids under time-dependent boundary conditions remains relatively unexplored. This study aims to bridge this gap by developing a mathematical model that incorporates the effects of Brownian motion and thermophoresis, which are critical in understanding the dynamics of nanofluids. The governing equations for momentum, energy, and nanoparticle concentration are solved using both analytical and numerical methods. Analytical solutions are derived using separation of variables and integration techniques for both the steady-state case and the transient components. The numerical solutions for the governing equations are analyzed using an implicit finite difference method. The numerical solutions are validated against the steady-state analytical results, demonstrating excellent agreement. The study examines the influence of key parameters, including the buoyancy ratio parameter $$Nr$$ Nr , Brownian motion parameter ( $$Nb$$ Nb ), and thermophoresis parameter ( $$Nt$$ Nt ), on the transient behavior of the system. Results reveal that the transient velocity, temperature, and nanoparticle concentration profiles evolve significantly over time, with flow reversal observed near the channel walls during the transient phase. The buoyancy ratio parameter ( $$Nr$$ Nr ) and thermophoresis parameter ( $$Nt$$ Nt ) are found to have a pronounced impact on the flow and heat transfer characteristics. As time progresses, the system stabilizes, and the transient profiles approach the steady-state solution, with well-defined thermal and concentration boundary layers forming near the walls. Results show $$Nt=0.5$$ N t = 0.5 maximizes $$Nu$$ Nu by $$30%$$ 30 % , while $$Nb>0.7$$ N b > 0.7 degrades thermal performance.