This study presents closed-form analytical and numerical solutions for both steady and transient free convection flow of a nanofluid in a vertical channel. The mathematical model is based on the Buongiorno model, incorporating the effects of Brownian motion and thermophoresis. Exact solutions for the steady-state velocity, temperature, and nanoparticle concentration profiles are derived using the method of separation of variables. For the transient regime, a semi-analytical solution is obtained using the Laplace transform technique after a linearization of the governing equations. Numerical validation is performed using an Implicit Finite Difference Method (IFDM), with stability assured by Von Neumann analysis. The influence of key dimensionless parameters such as Grashof number (Gr), buoyancy ratio (Nr), Brownian motion (Nb), and thermophoresis (Nt) are investigated comprehensively. Results indicate that increasing Gr from 1 to 10 enhances the maximum velocity by 637%, while increasing Nr from 0.5 to 2.0 dampens the flow intensity by 23%. Transient analysis reveals significant overshoots in Nusselt (Nu) and Sherwood (Sh) numbers, reaching up to 40% and 28% above their steady-state values, respectively, highlighting enhanced heat and mass transfer during the initial stages. The analytical solutions provide excellent benchmarks, showing perfect agreement with numerical results and deviations of less than 1% from established literature.
Keywords: Laplace transform., analytical solution., free convection, heat transfer enhancement, nanofluid, transient flow, vertical channel
