The multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) is used in a suddenly expanded channel to demonstrate the flow of viscoplastic Bingham nanofluid with Al2O3 nanoparticles. The geometry has two sections namely, inlet and outlet, and the corresponding heights are denoted by h and H, respectively. The 1 length of the entire channel is 20H, and the expanded channel has a height of 16H. The purpose of the MRT-LBM simulation is to investigate the impact of changing the Bingham number (0 ≤ Bn ≤ 200), keeping the Reynolds number (Re) fixed for different volume fractions (φ = 0.00 and 0.04). In addition, the consequences of variations in the Reynolds number (50 ≤ Re ≤ 1000) at constant Bingham number (Bn) are also studied for those two different volume fractions. The results demonstrate that with fixed Bn = 2, Re = 400 is the point where the flow pattern and recirculation regions are exactly the same for both volume fractions. An increase in Re causes the recirculation regions to grow for a fixed Bn for both volume fractions as Re’s rise increases the velocity and decreases the viscous force. Bn’s increment with Re and volume fraction unchanged lowers the recirculation region’s size due to a rise in viscous force. Higher Re and lower Bn cause the more significant recirculation regions to break down into smaller areas. Incrementing the volume fraction lowers size of the recirculation region. An unstable flow was observed for higher Bn (e.g., Bn ≥ 100) and lower Bn (e.g., 0 ≤ Bn ≤ 10) when Re ≥ 500 for both volume fractions in maximum cases. Unstable flow for lower Bn makes the recirculation regions asymmetric, and when 2 Re is high, the recirculation regions break down for the base fluid (φ = 0.00). When Re = 300 and Bn = 2, the length of the recirculation region of the upper wall decreases by 28.58%, and the length of the lower wall falls a 3 bit less by 26.37% when φ is increased from 0.00 to 0.04. For x/h = 2, the nanoparticle mixed fluid’s velocity 4 (φ = 0.04) never gets a negative magnitude till the final position for Re = 700. In most situations, an increased volume fraction increases the skin-friction effect on both walls. 

The Galerkin weighted residual finite element method (GFEM) has been used to numerically investigate the natural convection flow of two-phase Bingham hybrid nanofluid in a square cavity with a corrugated cylinder at the center. The base fluid here is water. The hybrid nanofluid is made of Fe3O4-CoFe2O4 nanoparticles. The flow inside the enclosure is considered laminar, non-Newtonian, and incompressible. The corrugated cylinder is taken heated. The top and bottom walls of the outer square are considered adiabatic, and the left and right walls are considered cold. Several non-dimensional parameters, including the Rayleigh number (Ra = 10^4, 10^5, 10^6), the Bingham number (Bn = 0, 0.5, 1), the volume fraction (ϕ = 0.00, 0.04), the Brownian parameter (Nb = 0.1, 0.2, 0.3), thermophoresis parameter (Nt = 0.1, 0.2, 0.3), buoyancy ratio (Nr = 0.1, 0.2, 0.3), corrugation (N = 5, 6, 7), the Prandtl number (Pr = 6.2), and the Lewis number (Le = 1000) have been numerically simulated. The main focus of this work is to witness streamlines, isotherm, and nanoparticles' volume fraction distribution and understand heat transmission with the help of average Nusselt number and local Nusselt number for several parameters for a two-phase Bingham hybrid nanofluid. The results show that an upsurge in Ra increases the extent of the upper two vortices and decreases the extent of the lower two vortices. All four vortices are seen to increase in dimension for an increase in Bn. The same observation is found for an increase in ϕ. The lower vortices are growing, and the top ones are nearly eliminated for base fluid, while those are visible for nanofluid when corrugation is increased. A further expansion completely destroyed the higher ones and increased the size of the bottom ones. The average Nusselt number increases when Ra is incremented. The average Nusselt number rises with a rise in ϕ. An increment in ϕ to 0.04 results in a 4.43% gain in the average Nusselt number. An increase in Bn is causing the average Nusselt number to drop by 18.02%. For Ra = 10^5 and Bn = 2, ϕ = 0.02 has the maximum heat transfer enhancement between 0.00 ≤ ϕ ≤ 0.04. In the future, we plan to use mixed convection in a three-dimensional domain along with the impact of magnetohydrodynamics (MHD).