To the theory of magnetically induced flow in a ferrofluid cloud: effect of the cloud initial shape

Abstract

We present results of mathematical modeling of a macroscopic circulating flow induced by nonuniform oscillating magnetic field in a ferrofluid cloud subjected in a cylinder filled with a nonmagnetic fluid. The aim of this work is development of a scientific basis for a progressive method of an address drug delivery in thrombosed blood vessels with the help of the magnetically induced circulation flow. Our results show that the initial shape of the ferrofluid cloud crucially influences on the flow structure and velocity.

Appendix A

In this appendix, we use the finite difference scheme [9] for the direct numerical solution of the third equation in (5). Due to the incompressibility of the fluid, it will be convenient to rewrite this equation in the symmetric form:

$$\begin{aligned} \frac{\partial \varphi }{\partial t}+\frac{{{\mathbf {u}}\nabla }\varphi }{2}+\frac{{\mathrm {div}}\left( {{\mathbf {u}}}\varphi \right) }{2}-D{\Delta }\varphi =0. \end{aligned}$$

(A.1)

The boundary conditions for Eq. (A.1) correspond to the impenetrability of the channel wall for the particles and the no-flux condition far from the drop:

$$\begin{aligned}&\frac{\partial \varphi }{\partial \rho }=0,\quad \text {at}\ \rho =0,\nonumber \\&\frac{\varphi v_{p}}{kT}\frac{\partial U}{\partial \rho }+\frac{\partial \varphi }{\partial \rho }=0,\quad \text {at } \rho =a,\nonumber \\&\varphi u_{z}-D\frac{\partial \varphi }{\partial z}\rightarrow 0,\quad \text {at } z\rightarrow \pm \infty . \end{aligned}$$

(A.2)

The second equation in (A.2) is written considering the magnetophoretic phenomenon. The calculations performed in [5] show that the magnetophoretic phenomenon is negligible and can be ignored in the boundary conditions. Then the second equation in (A.2) can be rewritten as follows:

$$\begin{aligned} \frac{\partial \varphi }{\partial \rho }=0,\quad \text {at}\ \rho =a. \end{aligned}$$

Note that the convective flux is not considered in the first relation of (A.2) (condition of the wall impenetrability for the particles) because of the no-penetration conditions ($u_{\rho } = 0$ at $\rho = a)$ on the channel wall. The relation $\partial \varphi$/$\partial \rho = 0$, at $\rho = 0$ provides not infinitive value of the term $\Delta \varphi$, therefore, not infinite rate of change the concentration at the channel axis at finite values of $\varphi$.

Since the infinity long distance cannot be used in the numerical scheme, we have changed the last relation in (A.2) to

$$\begin{aligned} \varphi u_{z}-D\frac{\partial \varphi }{\partial z}=0,\quad \text {at}\ z=\pm L. \end{aligned}$$

(A.3)

The strong inequality $L \gg \sigma _{z}$ is supposed here.

We will write the non-stationary convection–diffusion problem as the following differential-operator equation:

$$\begin{aligned}&\frac{\partial \varphi }{\partial t}+A_{O}\varphi =0,\quad A_{O}\varphi =C_{O}\varphi +D_{O}\varphi ,\nonumber \\&\quad C_{O}\varphi =C_{O}^{\left( 1 \right) }\varphi +C_{O}^{\left( 2 \right) }\varphi ,\nonumber \\&D_{O}\varphi =D_{O}^{\left( 1 \right) }\varphi +D_{O}^{\left( 2 \right) }\varphi . \end{aligned}$$

(A.4)

Here, $C_{O}$ is the convective transfer operator, and $D_{O}$ is the diffusion transfer operator. These operators are defined by the following relations.

Let us introduce a uniform grid as follows:

$$\begin{aligned}&z_{i}=-L+\left( \frac{1}{2}+i \right) h_{z},\quad \rho _{j}=\left( \frac{1}{2}+j \right) h_{\rho },\nonumber \\&\quad i=0,1,\ldots ,N_{z}-1,\ j=0,1,\ldots ,N_{\rho }-1, \nonumber \\&\quad h_{z}=\frac{2L}{N_{z}},\ h_{\rho }=\frac{D}{2N_{\rho }}. \end{aligned}$$

(A.5)

Here, $N_{z}$ and $N_{\rho }$ are the number of nodes along the coordinate z and $\rho$, respectively.

For all nodes j, except those close to the boundary, in the case of symmetric convection operator, we have

$$\begin{aligned}&C_{O}^{\left( 1 \right) }\varphi _{i,j}=\frac{u_{\rho \left( i,j \right) }}{2\rho _{j}}\varphi _{i,j}+\frac{u_{z\left( i,j \right) }+u_{z\left( i,j+1 \right) }}{4h_{\rho }}\varphi _{i,j+1}\nonumber \\&\quad -\frac{u_{\rho \left( i,j \right) }+u_{\rho \left( i,j-1 \right) }}{4h_{\rho }}\varphi _{i,j-1},\quad j=1,\ldots ,N_{\rho }-2.\nonumber \\ \end{aligned}$$

(A.6)

Table 1 The maximum value of the modulus of velocity vs. the time step $\tau$

Table 2 The maximum value of the modulus of velocity vs. number of nodes along the z coordinate $N_{z}$

Table 3 The maximum value of the modulus of velocity vs. number of nodes along the $\rho$ coordinate $N_{\rho }$

For all nodes, except for the extreme ones, the grid operator $D_{O}^{\left( 1 \right) }$ can be written in the following form:

$$\begin{aligned}&D_{O}^{\left( 1 \right) }\varphi _{i,j}=\frac{2D}{h_{\rho }^{2}}\varphi _{i,j}-\frac{D}{h_{\rho }}\left( \frac{1}{h_{\rho }}-\frac{1}{2\rho _{j}} \right) \varphi _{i,j-1}\nonumber \\&\quad -\frac{D}{h_{\rho }}\left( \frac{1}{h_{\rho }}+\frac{1}{2\rho _{j}} \right) \varphi _{i,j+1},\quad j=1,\ldots ,N_{\rho }-2.\nonumber \\ \end{aligned}$$

(A.7)

The mathematical expressions for $C_{O}^{\left( 2 \right) }\varphi _{i,j}$ and $D_{O}^{\left( 2 \right) }\varphi _{i,j}$ can be obtained in the same way. The convection–diffusion equation in (5) can be solved numerically using an implicit difference scheme:

$$\begin{aligned} \frac{{\hat{\varphi }}_{i,j}-\varphi _{i,j}}{\tau }+A_{O}{\hat{\varphi }}_{i,j}=0. \end{aligned}$$

(A.8)

Here, $\tau$ is time step; ${\hat{\varphi }}_{i,j}$ is the concentration on the upper layer over time.

To determine the optimal parameters, calculations were carried out for a different number of nodes and a time step. The calculation results are presented in Tables 1, 2 and 3. The results show that for $\tau \ge 10^{-4}$, $N_{z} \ge 400$ and $N_{\rho } \ge 50$, the maximum value of the modulus of velocity remains practically unchanged. Therefore, all calculations were given for $\tau = 10^{-4}$, $N_{z} = 400$ and $N_{\rho } = 50.$