Simulation of Ferrofluid Flow for Magnetic Drug Targeting Using the Lattice Boltzmann Method

1 Introduction

Magnetic drug targeting [1] is a way of moving a drug to an ill person through a method that augments the concentration of the drug in some regions of the body relative to others. The aim of such a system is to delay, localize, target, and have a protected drug interaction with the diseased tissue. The advantages of this system are the reduction in the frequency of the drug amounts taken by the patient, reduction of drug side effects, and decreased fluctuation in circulating drug levels. The disadvantage of the system is high price. The magnetic drug targeting system is highly joined and requires cooperation of various disciplines, such as biology, chemistry, and engineering [2]. Magnetic nanoparticles have three advantages in biomedicine. First, they have controllable dimensions which locate them at sizes that are smaller than a cell. Second, the nanoparticles are magnetic, so they can be influenced by an external magnetic field gradient. Third, a time-varying magnetic field can affect the magnetic nanoparticles.

Sheikholeslami et al. [3] studied the magnetic field effect on CuO–water nanofluid flow and heat transfer in an enclosure which is heated from below. They found that the effect of the Hartmann number and heat source length is more pronounced at a high Rayleigh number. Sheikholeslami et al. [4] used control volume based finite element method (CVFEM) in order to simulate the magnetohydrodynamic (MHD) effect on natural convection heat transfer of Cu–water nanofluid. Hatami et al. [5] simulated flow and heat transfer of a non-Newtonian third grade nanofluid in porous medium of a hollow vessel in the presence of a magnetic field. They found that by increasing the MHD parameter, velocity profiles decreased due to magnetic field effect. Rashidi et al. [6] studied MHD biorheological transport phenomena in a porous medium. Blood flow through a tapered artery with a stenosis was analysed by Nadeem et al. [7]. They assumed that the flow is steady and blood is treated as non-Newtonian power law fluid model. Ellahi [8] studied the MHD flow of non-Newtonian nanofluid in a pipe. Ellahi observed that the MHD parameter decreases the fluid motion and that the velocity profile is larger than that of temperature profile, even in the presence of variable viscosities. Magnetohemodynamic laminar viscous flow of a conducting physiological fluid in a semiporous channel under a transverse magnetic field was investigated by Basiri Pars et al. [9]. Ferrohydrodynamic (FHD) and MHD effects on ferrofluid flow and convective heat transfer was studied by Sheikholeslami and Domiri Ganji [10]. They proved that the magnetic number has a different effect on the Nusselt number corresponding to the Rayleigh number. Some numerical and experimental studies on nanofluids are listed in [11–13].

For more than one decade, the lattice Boltzmann method (LBM) has been demonstrated to be a very effective numerical tool for a broad variety of complex fluid flow phenomena that are problematic for conventional methods. The kinetic nature of the LBM distinguishes it from other numerical methods, mainly in three aspects. First, the convection operator of the LBM is linear in velocity space, so computational efforts are greatly reduced compared to those of some macroscopic computational fluid dynamics (CFD) methods, such as the Navier–Stokes equation solvers. Second, the pressure of the LBM can be directly calculated using an equation of state, unlike the direct numerical simulation of the incompressible Navier–Stokes equation, in which the pressure must be obtained from the Poisson equation. Third, the LBM utilises a minimal set of velocities in phase space; therefore, the transformation relating the microscopic distribution function and macroscopic quantities is greatly simplified. Mohamad and Kuzmin [14] used the LBM to present a detailed analysis of a natural convection problem. They showed the high efficiency of the LBM in simulating natural convection. Free convection heat transfer in a concentric annulus between a cold square and heated elliptic cylinders in the presence of a magnetic field was investigated by Sheikholeslami et al. [15]. They found that the enhancement in heat transfer increases as the Hartmann number increases, but it decreases with the increase of the Rayleigh number. Free convection of ferrofluid in a cavity heated from below in the presence of an external magnetic field was studied by Sheikholeslami and Bandpy [16]. They found that particles with a smaller size have better ability to dissipate heat, and a larger volume fraction would provide a stronger driving force which leads to an increase in temperature profile. Entropy generation of nanofluid in the presence of a magnetic field was investigated by Sheikholeslami and Ganji [17]. They found that the heat transfer rate and dimensionless entropy generation number increased with increase of the Rayleigh number and nanoparticle volume fraction but decreased with the increase of the Hartmann number.

The main objective of this article is to study the effect of an external magnetic source on Fe₃O₄-plasma flow in a channel. The LBM is used to solve this problem. Effects of the Reynolds number and magnetic number on the flow characteristics have been considered.

2 Problem Definition and Mathematical Model

2.1 Problem Statement

The steady, two-dimensional, incompressible laminar flow is considered taking place in a channel. The width of the channel is D, and the length of the channel is L, such that L / D=25. The flow at the entrance is assumed to be fully developed (uuin = 4yD(1 − yD)) [see Fig. 1(a)]. The flow is subjected to a magnetic source which is placed very close to the lower plate and below it (a=0.25 L, b=–0.1 D) (see Fig. 2). The fluid in the vessel is considered a homogenous mixture of Fe₃O₄-blood plasma. Thermophysical properties of the nanoscale ferromagnetic particle and blood plasma are assumed to be constant (Tab. 1).

Figure 1

(a) Geometry of the vessel; (b) discrete velocity set of two-dimensional nine-velocity.

Figure 2

Contours of the magnetic field strength H.

Table 1

Thermo-physical properties of blood plasma and Magnetite.

	ρ (kg/m3)	σ (Ω/m)	μ (Pa·s)
Blood plasma	1025	0.7	0.0015045
Magnetite (Fe3O4)	5000	25,000	–

The components of the magnetic field intensity (H_x,H_y) and the magnetic field strength (H) can be considered [17] as

(1)Hx = |b|(x − a)2 + (y − b)2(y − b) (1)

(2)Hy = −|b|(x − a)2 + (y − b)2(x − a) (2)

(3)H = Hx2 + Hy2 = |b|(x − a)2 + (y − b)2 (3)

2.2 Lattice Boltzmann Method

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 D₂Q₉ model [14] was used and values of w₀=4/9 for |c₀|=0 (for the static particle), w_1–4=1/9 for |c_1–4|=1 and w_5–9=1/36 for |c5−9| = 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

(4)fi(x + ciΔt,t + Δt) = fi(x,t) + Δtτv[fieq(x,t) − fi(x,t)] + ΔtciFk (4)

where Δt 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., υ = cs2(τv − 1/2). 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.

(5)fieq = wiρ[1 + ci ⋅ ucs2 + 12(ci ⋅ u)2cs4 − 12u2cs2] (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:

(6)F = Fx + FyFx = 3wiρ[A(−Hy2u + HxHyv) + B(H∂H∂x)],Fy = 3wiρ[A(−Hx2v + HxHyu) + B(H∂H∂y)] (6)

where A and B are A = Ha2υL2 and B = MnFuin2, respectively. Re = Duinυ, Ha = μ0H0Dσμ and MnF = μ0χH02ρuin2 are Reynolds number, Hartmann number, and magnetic number arising from ferrohydrodynamic (FHD) [18].

Finally, macroscopic variables calculate with the following formula:

(7)Flow density: ρ = ∑ifi,Momentum:ρu = ∑icifi, (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:

(8)ρn f = ρf(1 − ϕ) + ρsϕ, (8)

(9)μnf = μf(1 − ϕ)2.5 (9)

(10)σnfσf = 1 + 3(σsσf − 1)ϕ(σsσf + 2) − (σsσf − 1)ϕ (10)

with ϕ being the volume fraction of the nanoscale ferromagnetic particle (Fe₃O₄) and subscripts nf, s, and f standing for the mixture, nanoscale ferromagnetic particle (Fe₃O₄) and the carrier fluid (blood plasma), respectively.

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

(11)Cf = 2Reρfρnf1(1 − ϕ)2.5∂U∂Y|X = 0 and Cf¯ = ∫0LCfdX (11)

3 Grid Testing and Code Validation

In order to reach the grid independency, numerical experiments were performed as shown in Table 2. Different mesh sizes were used for the case of Re=400, ϕ=0.04, Mn_F=100 and Ha=20. The present code is tested for grid independence by calculating the average skin friction coefficient on the upper wall. It is found that a grid size of 100×2500 ensure the grid independent solution for the present case. The convergence criterion for the termination of all computations is:

(12)maxgrid|Γn + 1 − Γn| ≤ 10−7 (12)

where n is the iteration number and Γ stands for the independent variables (U,V). The present numerical solution is validated by comparing the present code results to the results of Mohamad et al. [19] for viscous flow (ϕ=0) (see Fig. 3). This comparison indicates the accuracy of the present LBM code.

Figure 3

Comparison between present work with previous other work [14] when Ra=400.

4 Results and Discussion

A targeted drug delivery system under the influence of an applied magnetic field is studied. The working fluid is homogenies mixture of magnetite in blood plasma. The LBM scheme was utilized to obtain the numerical simulation. Effect of the Reynolds number (Re=50, 100, 200 and 400) and magnetic number (Mn_F=0, 20, 60 and 100) on streamline, velocity profile, and skin friction coefficient are examined for constant values of volume fraction of nanoparticle (ϕ=0.04) and the Hartmann number (Ha=20).

Effects of the magnetic number and Reynolds number on streamlines are shown in Figures 4–7. In the absence of the magnetic field (Mn_F=0) the streamlines are straight lines. As the Reynolds number increases, velocity boundary layer thickness decreases. In the presence of the magnetic field a clockwise vortex appears at the area where the magnetic source is placed. This vortex causes perturbation in the flow pattern, and nanofluid flow recirculation may occur in this region. This phenomenon is more pronounced for higher values of the magnetic number.

Figure 4

Effect of magnetic number on streamlines when Re=50, Ha=20, ϕ=0.04.

Figure 5

Effect of magnetic number on streamlines when Re=100, Ha=20, ϕ=0.04.

Figure 6

Effect of magnetic number on streamlines when Re=200, Ha=20, ϕ=0.04.

Figure 7

Effect of magnetic number on streamlines when Re=400, Ha=20, ϕ=0.04.

Figure 8 shows the effects of the magnetic number and Reynolds number on velocity profile at x=0.25 L At x=0.25 L flow is separated from the bottom wall and then reattaches with the lower plate. Finally, the flow at the outlet is again reverted to fully developed. This phenomenon is more pronounced for higher values of the Reynolds number. Effects of the Magnetic number and Reynolds number on local skin friction coefficient along the lower and upper walls are shown in Figure 9. The value of local skin friction coefficient is the lower wall increases rapidly near x=0.2 L, where it reaches its maximum value. Very close to the region where the source is located, x=0.25 L, a corresponding decrement takes place, and at x=0.25 L this parameter takes its minimum negative value. C_f|_{Y= 1} increases near the area of the magnetic source in a smoother way, and its sign does not change as happens with the lower plate where reverse of the flow occurs. The wall shear is more influenced on the lower plate, below which the magnetic source is located. It is an interesting observation that there are two points with zero values of skin friction for the lower plate. In these points there is no skin friction, and this result may be exciting in the case of creation of fibrinoid. Figure 10 shows the effects of the magnetic number and Reynolds number on average skin friction coefficient. The drag acting on the upper plate is greater than that on the lower. Average skin friction coefficient decreases with increase of the Reynolds number and magnetic number.

Figure 8

Effects of magnetic number and Reynolds number on velocity profile at x=0.25 L when Ha=20, ϕ=0.04.

Figure 9

Effects of magnetic number and Reynolds number on local skin friction coefficient along the (a) lower and (b) upper plates when Ha=20, ϕ=0.04.

Figure 10

Effects of magnetic number and Reynolds number on average skin friction coefficient along the (a) lower and (b) upper plates when Ha=20, ϕ=0.04.

5 Conclusions

In this study, the LBM is applied in order to simulate a targeted drug delivery system under the influence of an applied magnetic field. The mathematical model used for the formulation of the problem is consistent with the principles of FHD and MHD. Effects of the Reynolds number and magnetic number on streamline, velocity profile, and skin friction coefficient are examined. Results indicate that the presence of a magnetic field influences considerably the flow field. Skin friction coefficient decreases with increase of the Reynolds number and magnetic number.