is a coefficient. Because the total interparticle interaction forces cannot be optionally added in the lattice Boltzmann equation, we introduce an unknown coefficient in the total interparticle interaction forces. In order to enable the lattice Boltzmann equation including the total interparticle interaction forces to recover to the Navier-Stokes equation, based on the mass and momentum conservation, we used multi-scale technique to deduce the unknown coefficient which is equal to . Due to the very long derivation process, we directly gave the final result in the paper. The weight coefficient B α is given

as: (4) For the two-dimensional nine-velocity LB model (D2Q9) considered herein, the discrete velocity BGJ398 set for each component α is: (5) The density equilibrium distribution function is chosen as follows: (6) (7) where is the lattice’s sound MAPK Inhibitor Library screening velocity, and w α is the weight coefficient. The macroscopic temperature field is simulated using the temperature distribution

function. (8) where τ T is the dimensionless collision-relaxation time for the temperature field. The temperature equilibrium distribution function is chosen as follows: (9) In the case of no internal forces and external forces, the macroscopic temperature, density and velocity are respectively calculated as follows: (10) (11) (12) Considering the internal and external forces, the macroscopic velocities for nanoparticles and base fluid are modified to: (13) (14) where F p represents the total forces acting on the nanoparticles, F w represents the total forces acting on the base fluid, and L x L y represents the total number of lattices. When the internal forces and external forces are considered, energy between nanoparticles and base fluid is exchanged, and the macroscopic temperature for nanoparticles and base fluid is then given as: (15) where Φ αβ is the energy exchange between nanoparticles and base fluid, ,

and h αβ is the convective heat transfer coefficient of the nanofluid. The corresponding kinematic viscosity and thermal Alectinib mouse diffusion coefficients are respectively defined as follows: (16) (17) The dimensionless collision-relaxation times τ f and τ T are respectively given as follows: (18) (19) where Ma = 0.1, H = 1, c = 1, δt = 1, and the other parameters equations are given as follows: (20) (21) From Equations 18 and 19, the collision-relaxation time for the flow field and the temperature field can be calculated. For water phase, the τ f collision-relaxation times are respectively 0.51433 and 0.501433 at Ra = 103 and Ra = 105, and the collision-relaxation time τ T is 0.5. For nanoparticle phase, the τ f collision-relaxation times are respectively 0.50096 and 0.500096 at Ra = 103 and Ra = 105, and the collision-relaxation time τ T is 0.500025. Interaction forces between base fluid and nanoparticles As noted before, a nanofluid is, in reality, a kind of two-phase fluid.