Existence and Uniqueness of Weak Solutions and Error Analysis of the Galerkin Finite Element Method for Time-Dependent Convective Nanofluid Poiseuille Flow Problems

Abstract

A finite element analysis of the plane Poiseuille nanofluid flow and heat transfer based on the time-dependent Buongiorno
model equations is performed. A suitable weak formulation of the sequentially-linearized governing equations is first constructed.
Then, the spatial discretization of the weak form is done using the Galerkin finite element formulation, while a Backward-Euler finite
difference scheme is used for the temporal discretization. Existence, uniqueness, and stability of the weak, semi-discrete and fullydiscrete forms are discussed. Furthermore, L^2-error estimates for the semi-discrete and fully-discrete forms are obtained. Moreover,numerical computations are performed to verify the theoretical results and estimate the rate of convergence.