#### 2.2. Improved MRT-LBM

Standard MRT-LBM in two-dimensional space (D2Q9 model) with a forcing term can be expressed as [

27]:

where

**r** is the displacement vector,

**e**_{i} is the discrete velocity,

**F′** is an external force, and

δt is the time step size, which is always set as

δt = 1. The second item on the right side relates to the external force.

**f** is a distribution function and

**f**^{eq} is an equilibrium distribution function, which is defined as:

where

c_{s} represents the lattice sound speed, where

${c}_{s}^{2}={c}^{2}/3$;

**u** is the velocity vector;

ω_{i} is the weight coefficient, given by

ω_{0} = 4/9,

ω_{i} = 1/9 for

i = 1–4, and

ω_{i} = 1/36 for

i = 5–8.

**M** is a transformation matrix:

The discrete velocity

**e**_{i} is expressed as:

The lattice speed c = δx/δt, δx represents the lattice step size, whose value is always set as 1.

The principal diagonal symmetric matrix

$\overline{\mathit{S}}$ mainly has an important effect on the relaxation process. The specific expression is given by:

where

s_{0},

s_{3},

s_{5},

s_{7}, and

s_{8} are common relaxation factors,

s_{3} and

s_{5} have the same meaning, and

s_{7} and

s_{8} have the same meaning.

s_{1} is related to the viscosity, and

s_{2} can be used to adjust the stability of numerical simulation.

s_{4} and

s_{6} can be used to improve the accuracy of MRT-LBM. The factors have the following relations:

s_{3} =

s_{5}, where they have no influence on macroscopic process;

s_{4} =

s_{6}, where they affect the accuracy of MRT model; and

s_{7} =

s_{8}, where they represent relaxation factors. Then,

s_{0} is an important related factor to the density, and it also has no effect on the macroscopic process.

s_{1} is usually related to viscosity. The stability can also be improved by adjusting it. The factor

s_{2} also has an impact on the stability. Here,

s_{1} and

s_{2} are set to be 0.8. According to the above description,

s_{0},

s_{3}, and

s_{5} are selected arbitrarily as 0.

s_{4} and

s_{6} are equal to 1.9.

s_{7} and

s_{8} will be discussed further below.

With MRT-LBM, kinematic viscosity is defined as:

For a standard single relaxation time LBM (LBM mentioned below refers to a standard single relaxation time LBM), the strain rate tensor is described as:

where

ρ is the density, and

τ is the relaxation time, where is

τ = 1/

s_{8}. However, the strain rate tensor in standard MRT-LBM is different from LBM, which is expressed as:

Then, the second invariant of strain rate tensor can be acquired using Equation (11) as:

With Equation (12), the shearing rate can be calculated as:

According to the equality of two different expressions in Reference [

21], then substituting viscosity into the related equation, the stress tensor

σ_{αβ} can be described as:

where

δ_{αβ} is a Kronecker delta,

K is the viscosity coefficient, and

P is the pressure. In the following section, the forcing term will be used to explain the non-Newtonian effect. For standard MRT-LBM with external force, the forcing term is expressed as [

23]:

Further, the item

$\overline{\mathit{F}}$ in Equation (15) can be calculated using:

The Navier–Stokes equation would be expanded by Chapman–Enskog at incompressible limit as:

where

u_{α} and

u_{β} are the components of

u, and

μ is the dynamic viscosity. The relation between kinematic viscosity and dynamic viscosity is

ν =

μ/

ρ. Also, the above equation can also be described using the following equation, which includes the forcing term:

Combining Equations (3) and (14), with Equation (17), the force related term

**F** in Equation (18) can be obtained as:

Thus, the non-Newtonian effect described as a special external force will be achieved. Combining the shearing rate obtained from Equation (13) with Equation (3), the parameter s_{8} for next iteration could be calculated. The specific programming process based on software MATLAB 2014a (MathWorks, Natick, MA, USA) can be conducted according to the following steps:

- (1)
Conduct the physical transformation according to the dimension theory. The Reynolds number is taken as the key criteria, and then the other parameters used in simulation can be obtained.

- (2)
Set the simulation domain and initial conditions, especially the initial value of the factor s_{8}.

- (3)
Calculate the equilibrium distribution function according to Equation (5).

- (4)
Conduct the collision step, where the specific function is shown as:

where

ϱ(

**r**,

t) =

**Mf**(

**r**,

t),

ϱ^{eq}(

**r**,

t) =

**Mf**^{eq}(

**r**,

t), and the force term is mainly used to explain the non-Newtonian effect of yielding fluids. The calculations can be seen in Equations (15), (16), and (19).

- (5)
Calculate the strain rate tensor and shearing rate by using Equations (11)–(13), then the kinetic viscosity can be obtained using Equation (3) and the relaxation factor s_{8} will be updated.

- (6)
Conduct the streaming step, where the streaming function is shown as:

- (7)
Process the boundary condition, where the non-equilibrium bounce-back scheme is taken as the boundary method.

- (8)
Conduct the next calculation from Step (3).

- (9)
Calculate the density and velocity, where the corresponding equations are: