One of the novel CFD methods which has solved the Boltzmann equation to simulate the flow instead of solving the Navier–Stokes equations is called the LBM. LBM has several advantages over other conventional CFD methods, such as a simple calculation procedure and efficient implementation for parallel computation, because of its particulate nature and local dynamics. The LB model uses one distribution function, *f*, for the flow. It uses modelling of movement of fluid particles to capture macroscopic fluid quantities, such as velocity and pressure. In this approach, the fluid domain discretizes to uniform Cartesian cells. Each cell holds a fixed number of distribution functions which represent the number of fluid particles moving in these discrete directions. The D_{2}Q_{9} model [14] was used and values of *w*_{0}=4/9 for |*c*_{0}|=0 (for the static particle), *w*_{1–4}=1/9 for |*c*_{1–4}|=1 and *w*_{5–9}=1/36 for $$\left|{c}_{5-9}\right|\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sqrt{2}$$ are assigned in this model [Figure 1(b)]. The density and distribution functions are calculated by solving the lattice Boltzmann equation which is a special discretization of the kinetic Boltzmann equation. After introducing the Bhatnagar–Gross–Krook (BGK) approximation, the general form of the lattice Boltzmann equation with external force is as follows. For the flow field

$${f}_{i}\mathrm{(}x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{c}_{i}\Delta t,t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\Delta t\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{f}_{i}\mathrm{(}x,t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{\Delta t}{{\tau}_{v}}[{f}_{i}^{eq}\mathrm{(}x,t\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{f}_{i}\mathrm{(}x,t\mathrm{)}]\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\Delta t{c}_{i}{F}_{k}\text{\hspace{1em}(4)}$$(4)

where Δ*t* denotes lattice time step, *c*_{i} is the discrete lattice velocity in direction *i, F*_{k} is the external force in direction of lattice velocity, *τ*_{v} denotes the lattice relaxation time for the flow. The kinetic viscosity *υ* is defined in terms of its respective relaxation times, i.e., $$\upsilon \text{\hspace{0.17em}}=\text{\hspace{0.17em}}{c}_{s}^{2}\mathrm{(}{\tau}_{v}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1/2\mathrm{)}.$$ Note that the limitation 0.5<*τ* should be satisfied for both relaxation times to ensure that viscosity and thermal diffusivity are positive. Furthermore, the local equilibrium distribution function determines the type of problem that needs to be solved. It also models the equilibrium distribution functions, which are calculated with (4) for flow field, respectively.

$${f}_{i}^{eq}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{w}_{i}\rho \left[1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{{c}_{i}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}u}{{c}_{s}^{2}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{1}{2}\frac{{\mathrm{(}{c}_{i}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}u\mathrm{)}}^{2}}{{c}_{s}^{4}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{1}{2}\frac{{u}^{2}}{{c}_{s}^{2}}\right]\text{\hspace{1em}(5)}$$(5)

where *w*_{i} is a weighting factor and *ρ* is the lattice fluid density.

In order to incorporate buoyancy forces and magnetic forces into the model, the force term in (4) need to calculate as follows:

$$\begin{array}{c}F\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{F}_{x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{F}_{y}\\ {F}_{x}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}3{w}_{i}\rho \left[A\mathrm{(}-{H}_{y}^{2}u\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{H}_{x}{H}_{y}v\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}B\mathrm{(}H\frac{\partial H}{\partial x}\mathrm{)}\right],\\ {F}_{y}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}3{w}_{i}\rho \left[A\mathrm{(}-{H}_{x}^{2}v\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{H}_{x}{H}_{y}u\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}B\mathrm{(}H\frac{\partial H}{\partial y}\mathrm{)}\right]\end{array}\text{\hspace{1em}(6)}$$(6)

where *A* and *B* are $$A\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{H{a}^{2}\upsilon}{{L}^{2}}$$ and $$B\text{\hspace{0.17em}}=\text{\hspace{0.17em}}M{n}_{F}{u}_{in}^{2},$$ respectively. $$Re\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{D{u}_{in}}{\upsilon},\text{\hspace{0.17em}}Ha\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\mu}_{0}{H}_{0}D\sqrt{\frac{\sigma}{\mu}}$$ and $$M{n}_{F}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\mu}_{0}\chi {H}_{0}^{2}}{\rho {u}_{in}^{2}}$$ are Reynolds number, Hartmann number, and magnetic number arising from ferrohydrodynamic (FHD) [18].

Finally, macroscopic variables calculate with the following formula:

$$\begin{array}{l}\text{Flow\hspace{0.17em}density}:\text{\hspace{0.17em}}\rho \text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{i}{f}_{i}},\\ \text{Momentum}:\rho \text{u}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle \sum _{i}{\text{c}}_{i}{f}_{i}},\end{array}\text{\hspace{1em}(7)}$$(7)

In order to simulate the nanofluid by the LBM, because of the interparticle potentials and other forces on the nanoparticles, the nanofluid behaves differently from the pure liquid from the mesoscopic point of view and is of higher efficiency in energy transport, as well as better stabilised than the common solid-liquid mixture. For modelling the nanofluid because of changing in the fluid density, viscosity and electrical conductivity some of the governed equations should change. The effective density (*ρ*_{nf}), the effective viscosity (*μ*_{nf}) and electrical conductivity (*σ*)_{nf} of the nanofluid are defined [10] as:

$${\rho}_{n\text{\hspace{0.17em}}f}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\rho}_{f}\mathrm{(}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\varphi \mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\rho}_{s}\varphi ,\text{\hspace{1em}(8)}$$(8)

$${\mu}_{nf}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\mu}_{f}}{{\mathrm{(}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\varphi \mathrm{)}}^{2.5}}\text{\hspace{1em}(9)}$$(9)

$$\frac{{\sigma}_{nf}}{{\sigma}_{f}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3\mathrm{(}\frac{{\sigma}_{s}}{{\sigma}_{f}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1\mathrm{)}\varphi}{\mathrm{(}\frac{{\sigma}_{s}}{{\sigma}_{f}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\mathrm{(}\frac{{\sigma}_{s}}{{\sigma}_{f}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1\mathrm{)}\varphi}\text{\hspace{1em}(10)}$$(10)

with *ϕ* being the volume fraction of the nanoscale ferromagnetic particle (Fe_{3}O_{4}) and subscripts *nf, s*, and *f* standing for the mixture, nanoscale ferromagnetic particle (Fe_{3}O_{4}) and the carrier fluid (blood plasma), respectively.

The local and average skin friction coefficient is defined as follows:

$${C}_{f}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{2}{\mathrm{Re}}\frac{{\rho}_{f}}{{\rho}_{nf}}\frac{1}{{\mathrm{(}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\varphi \mathrm{)}}^{2.5}}{\frac{\partial U}{\partial Y}|}_{X\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0}\text{\hspace{0.17em}and\hspace{0.17em}}\overline{{C}_{f}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\displaystyle {\int}_{0}^{L}{C}_{f}}dX\text{\hspace{1em}(11)}$$(11)

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