**Appendix 1**

### **2.1. Convection-Diffusion-Deposition of the particles**

### **1) The computation of the deposition rate coefficients of the particles,** *kf i*

The mean deposition rate coefficients of the particles are expressed by the following relation [10–16]:

$$k\_{\uparrow}^{\perp} = \frac{3(1 - \varepsilon\_{\circ})}{2 \,\varepsilon\_{\circ} \, d\_{\circ}} \eta\_{\iota}^{\vee} \upsilon\_{\iota^{\vee}} \text{ (i = 1, 2)}\tag{A1.1}$$

The following non dimensional coefficients are defined [13]:

and fluid velocity on particle deposition rate due to interception;

 = (1 − *ε<sup>i</sup>* )

found the following correlation equation:

*<sup>α</sup><sup>i</sup>* <sup>=</sup> exp[\_\_1

+ 0.222 *As i*

*NE*<sup>2</sup>

<sup>36</sup> *<sup>π</sup> <sup>μ</sup> <sup>R</sup>*<sup>2</sup> *<sup>U</sup>* is London number;

The expression of *α<sup>C</sup>*−*<sup>C</sup>*

The expression of *α<sup>B</sup>*−*<sup>T</sup>*

*α<sup>B</sup>*−*<sup>T</sup>*

ficients: *NLO*, *NE*<sup>1</sup>

**7.** *NLO* <sup>=</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>4</sup>*<sup>A</sup>*

*εr ε*0(*ζ<sup>p</sup>* <sup>2</sup> <sup>+</sup> *<sup>ζ</sup><sup>c</sup>* 2 ) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

**8.** *NE*<sup>1</sup> <sup>=</sup>

*α<sup>C</sup>*−*<sup>C</sup>*

—Peclet number which defines the ratio of the convective transport to diffusive

Modeling of the Temperature Field in the Magnetic Hyperthermia

http://dx.doi.org/10.5772/intechopen.71364

319

*kT*—van der Waals number—the ratio of van der Waals interaction energy to the par-

<sup>12</sup> *<sup>R</sup>*<sup>2</sup> *<sup>U</sup>*—attraction number—combined influence of van der Waals attraction forces

Deposition of the particles on the pore wall is influenced by the electrostatic repulsive forces. The *attachment (collision) efficiency coefficient* (filter coefficient) *α<sup>i</sup>* (*i* = 1, 2) represents the fractional reduction in deposition rate of the particles due to the presence of the electrostatic repulsive energies [15]. Bai and Tien (1999), Chang and Chan [16] computed the expression

. The analytical expression was compared successfully with the experimental data. They

*<sup>i</sup>* ) + ln(*α<sup>B</sup>*−*<sup>T</sup>*

<sup>2</sup>(ln(*α<sup>C</sup>*−*<sup>C</sup>*

*<sup>i</sup>* is given by Chang and Chan [15, 16]:

*<sup>i</sup>* = 0.024 (*NDL*)0.969 (*NE*1)<sup>−</sup>0.423 (*NE*2)2.880 (*NLO*)1.5 + 3.176 (*As*

*<sup>i</sup>* is given by Bai and Tien [15, 16]:

and *NDL* have the following expressions:

In the absence of the electrostatic double layer forces, *α<sup>i</sup>*

<sup>6</sup> <sup>π</sup> <sup>μ</sup> <sup>R</sup> *<sup>U</sup>* is the first electrokinetic parameter;

*<sup>U</sup>* —gravity number—the ratio of Stokes particle settling velocity to approach

1/3—porosity dependent parameter of Happel's model.

*i*

(*NR*)3.041 (*NPe*)<sup>−</sup>0.514 (*NLO*)0.125 + (*NR*)<sup>−</sup>0.24 (*NG*)1.11(*NLO*) (A1.4)

*<sup>i</sup>* = 2.527 ∙ 10<sup>−</sup><sup>3</sup> (*NLO*)0.7031 (*NE*1)<sup>−</sup>0.3121 (*NE*2)3.5111 (*NDL*)1.352 (A1.5)

*<sup>i</sup>* ))], (i = 1, 2). (A1.3)

)1/3 (*NR*)<sup>−</sup>0.081 (*NPe*

become 1. The nondimensional coef-

*<sup>i</sup>* )<sup>−</sup>0.715 (*NLO*)2.687

—the aspect ratio;

**1.** *NR* <sup>=</sup> \_\_ *D dc*

**2.** *NPe* <sup>=</sup> *<sup>U</sup> <sup>d</sup>* \_\_\_\_*<sup>c</sup> Dw*

**3.** *Nvdw* <sup>=</sup> \_\_\_ *<sup>A</sup>*

**5.** *NG* <sup>=</sup> \_\_2 9 *R*2

**6.** *As <sup>i</sup>* =

of *α<sup>i</sup>*

transport;

**4.** *NA* <sup>=</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>A</sup>*

ticle's energy;

(*ρMNP* <sup>−</sup> *<sup>ρ</sup>ferrof*)*<sup>g</sup>* \_\_\_\_\_\_\_\_\_\_\_\_

velocity of the fluid;

2(1 − *γ<sup>i</sup>* 5 ) \_\_\_\_\_\_\_\_\_\_\_ 2 <sup>−</sup> <sup>3</sup> *<sup>γ</sup><sup>i</sup>* + 3 *γ<sup>i</sup>* <sup>5</sup> <sup>−</sup> <sup>2</sup> *<sup>γ</sup><sup>i</sup>* <sup>6</sup>; *γ<sup>i</sup>*

The porous media contains the spherical collector cells with diameters *dc* ranged from 0.05 to 0.50 mm. The coefficients *k f i* depend on the particle diameter D, collector diameter *dc* , porosities of the malignant and healthy tissues *ε<sup>i</sup>* , and the radial velocity *vr* . The *collector efficiency <sup>η</sup><sup>s</sup> i* describes the ratio of the particles captured by the solid surface to those brought into a unit structural cell of the porous medium [8, 13]. This coefficient is given by the expression:

$$
\eta\_s^\vee = \alpha^\vee \eta\_v^\vee. \tag{A1.2}
$$

The *single collector contact efficiency η*<sup>0</sup> *i* is the fraction of the particles brought to the collector surface by the Brownian diffusion, interception and/or gravitational sedimentation. This coefficient was computed by N. Tufenkji and M. Elimelech considering the superposition of the effects developed by the hydrodynamic forces, van der Waals interactions and gravity effect [13]:

$$
\eta\_0^\iota = \underbrace{\eta\_D^\iota}\_{\textit{transport by}} + \underbrace{\eta\_I^\iota}\_{\textit{transport by}} + \underbrace{\eta\_G^\iota}\_{\textit{transport by}}
$$

with:

$$
\eta\_D^i = \text{2.4} \left( A\_s^\dagger \right)^{13} N\_\text{R}^{-0.081} \left( N\_{\text{Pe}} \right)^{-0.715} N\_{\text{vdW}}^{0.032}; \\
\eta\_I^i = 0.55 A\_s^i N\_\text{R}^{1.875} \left( N\_A \right)^{0.125};
$$

$$
\eta\_{\mathcal{G}} = \ 0.22 \ N\_{\mathbb{R}}^{-0.24} \ \{N\_{\mathcal{G}}\}^{1.11} N\_{\text{vdW}}^{0.053} .
$$

The following non dimensional coefficients are defined [13]:

\*\*1.\*\*  $N\_{\kappa} - \frac{D}{d\_{\prec}}$ —the aspect ratio;

on larger distances on radial direction from the injection site within tumor. As a result, the particles which not remain in the vicinity of the injection site are distributed in the tumor volume. This important effect determines a temperature field with small temperature gradients. The model developed in this paper can be used as a planning tool to compute the temperature

The mean deposition rate coefficients of the particles are expressed by the following relation

describes the ratio of the particles captured by the solid surface to those brought

into a unit structural cell of the porous medium [8, 13]. This coefficient is given by the

*<sup>i</sup>* = *α<sup>i</sup> η*<sup>0</sup> *i*

tor surface by the Brownian diffusion, interception and/or gravitational sedimentation. This coefficient was computed by N. Tufenkji and M. Elimelech considering the superposition of the effects developed by the hydrodynamic forces, van der Waals interactions and gravity

> + *η<sup>I</sup> i transport* ⏟ *by interception*

> > 0.052 ; *η<sup>I</sup>*

<sup>−</sup>0.24 (*NG*)1.11 *NvdW*

+ *η<sup>G</sup> transport* ⏟ *by gravitation*

*<sup>i</sup>* = 0.55 *As*

0.053.

*<sup>i</sup> NR*

1.675 (*NA*)0.125 ;

*i*

ranged from 0.05

. The *collector* 

,

*<sup>i</sup> vr* , (i = 1, 2) (A1.1)

. (A1.2)

depend on the particle diameter D, collector diameter *dc*

, and the radial velocity *vr*

is the fraction of the particles brought to the collec-

field for different parameters.

318 Numerical Simulations in Engineering and Science

*kf*

to 0.50 mm. The coefficients *k*

**2.1. Convection-Diffusion-Deposition of the particles**

**1) The computation of the deposition rate coefficients of the particles,** *kf*

*<sup>i</sup>* <sup>=</sup> 3(1 <sup>−</sup> *<sup>ε</sup><sup>i</sup>* ) \_\_\_\_\_\_\_\_\_\_ 2 *ε<sup>i</sup> dc ηs*

The porous media contains the spherical collector cells with diameters *dc*

*i*

*<sup>i</sup>* = *η<sup>D</sup> i transport* ⏟ *by diffusion*

<sup>−</sup>0.081 (*NPe*)<sup>−</sup>0.715 *NvdW*

*f i*

porosities of the malignant and healthy tissues *ε<sup>i</sup>*

*η<sup>s</sup>*

The *single collector contact efficiency η*<sup>0</sup>

*η*<sup>0</sup>

*<sup>i</sup>* = 2.4 (*As*

*η<sup>G</sup>* = 0.22 *NR*

*i* )1/3 *NR*

**Appendix 1**

[10–16]:

*efficiency <sup>η</sup><sup>s</sup>*

expression:

effect [13]:

with:

*η<sup>D</sup>*

*i*


Deposition of the particles on the pore wall is influenced by the electrostatic repulsive forces. The *attachment (collision) efficiency coefficient* (filter coefficient) *α<sup>i</sup>* (*i* = 1, 2) represents the fractional reduction in deposition rate of the particles due to the presence of the electrostatic repulsive energies [15]. Bai and Tien (1999), Chang and Chan [16] computed the expression of *α<sup>i</sup>* . The analytical expression was compared successfully with the experimental data. They found the following correlation equation:

$$a^\iota = \exp\left[\frac{1}{2} (\ln \langle a^\iota\_{\subset \cdot \subset} \rangle + \ln \langle a^\iota\_{\succeq \cdot \subset} \rangle) \right], (\text{i } = 1, 2). \tag{A1.3}$$

The expression of *α<sup>C</sup>*−*<sup>C</sup> <sup>i</sup>* is given by Chang and Chan [15, 16]:

$$\begin{aligned} \alpha^{i}\_{\text{-C}} &= 0.024 \, \text{(N}\_{\text{DL}}\text{)}^{0.989} \, \text{(N}\_{\text{L}}\text{)}^{-0.423} \, \text{(N}\_{\text{L}}\text{)}^{2.880} \, \text{(N}\_{\text{LO}}\text{)}^{15} + 3.176 \, \text{(A}^{\dagger}\_{\text{s}}\text{)}^{1/3} \, \text{(N}\_{\text{R}}\text{)}^{-0.081} \, \text{(N}\_{\text{R}}\text{)}^{-0.715} \, \text{(N}\_{\text{LO}}\text{)}^{2.887} \\ &+ 0.222 \, \text{A}^{i}\_{\text{s}} \, \text{(N}\_{\text{R}}\text{)}^{3.041} \, \text{(N}\_{\text{R}}\text{)}^{-0.514} \, \text{(N}\_{\text{LO}}\text{)}^{0.125} + \, \text{(N}\_{\text{R}}\text{)}^{-0.24} \, \text{(N}\_{\text{G}}\text{)}^{111} \, \text{(N}\_{\text{LO}}\text{)} \end{aligned} \tag{A1.4}$$

The expression of *α<sup>B</sup>*−*<sup>T</sup> <sup>i</sup>* is given by Bai and Tien [15, 16]:

$$a\_{\rm B-T}^{\prime} = 2.527 \cdot 10^{-3} \,\mathrm{\{N\_{LO}\}}^{0.7031} \,\mathrm{\{N\_{t1}\}}^{0.3121} \,\mathrm{\{N\_{t2}\}}^{3.511} \,\mathrm{\{N\_{t3}\}}^{1.332} \,\mathrm{\} \tag{A1.5}$$

In the absence of the electrostatic double layer forces, *α<sup>i</sup>* become 1. The nondimensional coefficients: *NLO*, *NE*<sup>1</sup> *NE*<sup>2</sup> and *NDL* have the following expressions:

**7.** *NLO* <sup>=</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>4</sup>*<sup>A</sup>* <sup>36</sup> *<sup>π</sup> <sup>μ</sup> <sup>R</sup>*<sup>2</sup> *<sup>U</sup>* is London number; **8.** *NE*<sup>1</sup> <sup>=</sup> *εr ε*0(*ζ<sup>p</sup>* <sup>2</sup> <sup>+</sup> *<sup>ζ</sup><sup>c</sup>* 2 ) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>6</sup> <sup>π</sup> <sup>μ</sup> <sup>R</sup> *<sup>U</sup>* is the first electrokinetic parameter; **9.** *NE*<sup>2</sup> <sup>=</sup> <sup>2</sup> *<sup>ζ</sup><sup>p</sup> <sup>ζ</sup><sup>c</sup>* \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ *ζp* <sup>2</sup> <sup>+</sup> *<sup>ζ</sup><sup>c</sup>* <sup>2</sup> is the second electrokinetic parameter;

**10.***NDL* = 2 *κ R* is the double layer force parameter;

*κ* is Debye length for the colloidal suspension; *ζ<sup>p</sup>* is particle zeta potential; *ζ<sup>c</sup>* is collector zeta potential (**Table 1**) and *<sup>U</sup>* <sup>=</sup> *<sup>Q</sup>*\_\_\_\_\_*<sup>v</sup> Sneedle* is the ferrofluid velocity at the top of the needle.

The repulsive electrostatic double layer (EDL) forces appear in the liquid medium due to the ionic conditions measured by pH and ionic strength.

### **2. The computation of the MNP concentrations** *Ci* **=***Ci (r)* **as solution of Eq. (5)**

At equilibrium, in the steady-state: <sup>∂</sup> *<sup>C</sup>*\_\_\_*<sup>i</sup>* <sup>∂</sup>*<sup>t</sup>* <sup>=</sup> <sup>0</sup> and Eq. (5) becomes:

$$\nabla \cdot \left< \vec{\boldsymbol{\upsilon}} \, \mathbf{C}\_{i} \right> = \nabla \cdot \left< \mathbf{D}\_{i}^{\*} \nabla \mathbf{C}\_{i} \right> - k\_{j}^{i} \, \mathbf{C}\_{i} \, \tag{A1.6}$$

which is equivalent with

C<sup>0</sup>

Bessel *<sup>I</sup>*[<sup>√</sup>

order <sup>√</sup>

(**const1**) i

**i.** *C*<sup>2</sup>

i(r) =

\_\_\_\_\_\_ 1 − 4 *mi* ,

*C<sup>i</sup>*

and (**const2**)

lowing four boundary conditions:

*<sup>D</sup>*<sup>1</sup>

 = Cmax.

i

expression C<sup>1</sup>

The constants (**const1**)

**2.2. The temperature model**

*r* 2 \_\_∂ <sup>∂</sup>*r*[*<sup>r</sup>* <sup>2</sup> <sup>∂</sup>*T*\_\_\_*<sup>i</sup>*

At the thermal equilibrium, Eq. (7) are:

<sup>∂</sup>*<sup>r</sup>* ] <sup>+</sup> *<sup>β</sup><sup>i</sup>* 2 (*Te i* (r) − *Ti*

 = *rTi*

**Appendix 2**

\_\_1

Using substitution *Ri*

\_\_\_\_\_\_ 1 − 4 *mi*

\_\_<sup>∂</sup>

(**const1**) e

*A* \_\_*i*

+ (**const2**)

i

\_\_ Ai

2r Bessel I[√

<sup>2</sup>*<sup>r</sup>* ] and Bessel *<sup>K</sup>*[<sup>√</sup>

. The expressions \_\_

2r ] \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>√</sup> \_

<sup>∂</sup>r{*<sup>r</sup>* <sup>2</sup> <sup>∂</sup>*C*<sup>0</sup> *i* \_\_\_ <sup>∂</sup><sup>r</sup> } <sup>+</sup> *Ai* ( ∂*C*<sup>0</sup> *i* \_\_\_ <sup>∂</sup><sup>r</sup> ) <sup>+</sup> *mi <sup>C</sup>*<sup>0</sup>

The solutions of Eq. (A1.13) are given by the following expressions:

*Ai*

Ai \_\_\_\_\_\_\_2r √ \_

(*r*) <sup>=</sup> ( <sup>e</sup><sup>−</sup>\_\_

 = 0 on the external boundary of the geometry (r = R<sup>2</sup>

**ii.** Neumann boundary condition at the all inner interfaces;

*C*1(r = R1) = *C2*(r = R1)

**iii.** at the injection site (IS), at the top of the needle (r = r<sup>o</sup>

and (**const2**)

i

\_\_\_\_\_\_ <sup>1</sup> <sup>−</sup> <sup>4</sup> mi , \_\_ Ai

<sup>r</sup> +

\_\_\_\_\_\_ 1 − 4 *mi* ,

Eq. (5) are computed using the expressions (A1.14) in the expression (A1.11):

<sup>r</sup> )((**const1**)

<sup>∗</sup> <sup>∂</sup> <sup>C</sup> \_\_\_1 <sup>∂</sup><sup>r</sup> |r=R1

) = 0 or

, these equations become:

∂<sup>2</sup> *T* \_\_\_\_*<sup>i</sup>* <sup>∂</sup>*<sup>r</sup>* <sup>2</sup> <sup>+</sup> \_\_2 *r* ∂*T*\_\_\_*<sup>i</sup>* <sup>∂</sup>*<sup>r</sup>* <sup>−</sup> *<sup>β</sup><sup>i</sup>*

= *D*<sup>2</sup> <sup>∗</sup> <sup>∂</sup> <sup>C</sup> \_\_\_2 <sup>∂</sup><sup>r</sup> |r=R1

*A* \_\_*i*

<sup>i</sup> Bessel K[√

(**const2**) e

<sup>i</sup> Bessel I[√

\_\_\_\_\_\_ <sup>1</sup> <sup>−</sup> <sup>4</sup> mi , \_\_

are the four integration constants which are determined from the fol-

);

were computed in the Wolfram Mathematica 10 software.

<sup>2</sup> *Ti* + *β<sup>i</sup>* <sup>2</sup> *Te i*

\_\_\_\_\_\_\_\_ A<sup>i</sup>

2r Bessel K[√

2r ] \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>√</sup> \_

<sup>2</sup>*<sup>r</sup>* are the variables of these functions. The general solutions of

\_\_\_\_\_\_ <sup>1</sup> <sup>−</sup> <sup>4</sup> mi , \_\_

Ai

*<sup>i</sup>* = 0 (A1.13)

http://dx.doi.org/10.5772/intechopen.71364

321

<sup>r</sup> (A1.14)

2r ]) (A1.15)

) the concentration has the particular

(r) = 0 (A2.1)

\_\_\_\_\_\_ <sup>1</sup> <sup>−</sup> <sup>4</sup> mi , \_\_ Ai

Modeling of the Temperature Field in the Magnetic Hyperthermia

<sup>2</sup>*<sup>r</sup>*] are modified Bessel functions I and K of the

Ai 2r ]

where the deposition rate coefficients of the particles *k f i* are given by the relations (A1.1):

$$\frac{1}{\mathbf{r}^2} \frac{\partial}{\partial \mathbf{r}} \{\mathbf{v}\_r r^2 \: \mathbb{C}\} - \frac{1}{\mathbf{r}^2} \frac{\partial}{\partial \mathbf{r}} \cdot \left(D\_i^\* \, r^2 \frac{\partial \mathbb{C}\_i}{\partial r}\right) = -k\_f^i \mathbb{C}\_i \tag{A1.7}$$

$$\frac{\partial}{\partial \mathbf{r}} \left( \mathbf{v}\_r \, r^2 \, \mathbf{C}\_i - \mathbf{D}\_i^\* \, r^2 \frac{\partial \mathcal{L}\_i}{\partial r} \right) = \mathbf{M}\_i \, \mathbf{C}\_i \tag{A1.8}$$

$$\frac{\partial}{\partial \mathbf{r}} \left[ C\_i r^2 \left| \frac{A\_i}{r^2} - \frac{1}{C\_i} \frac{\partial C\_i}{\partial r} \right| \right] = \left. m\_i \right| \mathbf{C}\_i \tag{A1.9}$$

with the following constants:

$$M\_{i} = -\frac{3(1-\varepsilon\_{i})}{2} \eta\_{s}^{\perp} B; m\_{i} = \frac{M\_{i}}{D\_{i}^{\prime\prime}}; A\_{i} = \frac{B}{D\_{i}^{\prime\prime}}; B = \frac{Q\_{v}}{\pi} \tag{A1.10}$$

Solution of Eq. (A1.9) has the following form:

$$C\_{l}(r) = C\_{0}(r) \exp\left[-\frac{A\_{i}}{r}\right] \tag{A1.11}$$

Considering *C*<sup>0</sup> *<sup>i</sup>* <sup>=</sup> *<sup>C</sup>*<sup>0</sup> *i* (*r*),Eq. (A1.10) can be written as:

$$-\frac{\partial}{\partial \mathbf{r}} \left| \left( r^2 \exp\left[ -\frac{A\_i}{r} \right] \right) \frac{\partial C\_0}{\partial \mathbf{r}} \right| \tag{A1.12}$$

or:

$$r^2 \left(\frac{\partial^2 C\_0^i}{\partial \mathbf{r}^2}\right) + \left(2r + A\_i\right)\left(\frac{\partial C\_0^i}{\partial \mathbf{r}}\right) + m\_i C\_0^i = \mathbf{0}$$

which is equivalent with

**9.** *NE*<sup>2</sup> <sup>=</sup>

<sup>2</sup> *<sup>ζ</sup><sup>p</sup> <sup>ζ</sup><sup>c</sup>* \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ *ζp* <sup>2</sup> <sup>+</sup> *<sup>ζ</sup><sup>c</sup>*

potential (**Table 1**) and *<sup>U</sup>* <sup>=</sup> *<sup>Q</sup>*\_\_\_\_\_*<sup>v</sup>*

320 Numerical Simulations in Engineering and Science

At equilibrium, in the steady-state:

∇ ∙(*v*

\_\_1

\_\_<sup>∂</sup>

\_\_<sup>∂</sup>

with the following constants:

*Mi* = −

Considering *C*<sup>0</sup>

or:

Solution of Eq. (A1.9) has the following form:

*C<sup>i</sup>*

*<sup>i</sup>* <sup>=</sup> *<sup>C</sup>*<sup>0</sup> *i*

−\_\_<sup>∂</sup>

*r* <sup>2</sup>

<sup>2</sup> is the second electrokinetic parameter;

is particle zeta potential; *ζ<sup>c</sup>*

*(r)* **as solution of Eq. (5)**

*<sup>i</sup> Ci* , (A1.6)

*<sup>i</sup> Ci* (A1.7)

*Qv<sup>π</sup>* (A1.10)

*<sup>r</sup>* ] (A1.11)

*<sup>r</sup>* ] (A1.12)

are given by the relations (A1.1):

<sup>∂</sup>*<sup>r</sup>* ) <sup>=</sup> *Mi <sup>C</sup><sup>i</sup>* (A1.8)

<sup>∂</sup>*<sup>r</sup>* )} <sup>=</sup> *mi <sup>C</sup><sup>i</sup>* (A1.9)

is the ferrofluid velocity at the top of the needle.

**=***Ci*

*f i*

<sup>∗</sup>; *Ai* <sup>=</sup> \_\_\_*<sup>B</sup> Di*

<sup>∗</sup>; *B* = \_\_\_

*<sup>i</sup>* exp[−\_\_ *Ai*

*<sup>i</sup>* = 0

<sup>∂</sup>*<sup>r</sup>* ) <sup>=</sup> <sup>−</sup>*kf*

<sup>∗</sup> *<sup>r</sup>* <sup>2</sup> <sup>∂</sup><sup>C</sup> \_\_\_*<sup>i</sup>*

<sup>∂</sup>*<sup>t</sup>* <sup>=</sup> <sup>0</sup> and Eq. (5) becomes:

∗ ∇*Ci* ) − *kf*

<sup>∗</sup> *<sup>r</sup>* <sup>2</sup> <sup>∂</sup>*<sup>C</sup>* \_\_\_*<sup>i</sup>*

*<sup>i</sup> <sup>B</sup>*; *mi* <sup>=</sup> \_\_\_ *Mi Di*

> (*r*) exp[−\_\_ *Ai*

<sup>∂</sup><sup>r</sup> } <sup>=</sup> *mi <sup>C</sup>*<sup>0</sup>

)( ∂*C*<sup>0</sup> *i* \_\_\_ <sup>∂</sup><sup>r</sup> ) <sup>+</sup> *mi <sup>C</sup>*<sup>0</sup>

(*r*) = *C*<sup>0</sup> *i*

> *Ai <sup>r</sup>* ]) <sup>∂</sup>*C*<sup>0</sup> *i* \_\_\_

<sup>∂</sup>r2 ) <sup>+</sup> (2*<sup>r</sup>* <sup>+</sup> *Ai*

) = ∇ ∙(*Di*

The repulsive electrostatic double layer (EDL) forces appear in the liquid medium due to the

is collector zeta

**10.***NDL* = 2 *κ R* is the double layer force parameter;

*κ* is Debye length for the colloidal suspension; *ζ<sup>p</sup>*

ionic conditions measured by pH and ionic strength.

**2. The computation of the MNP concentrations** *Ci*

where the deposition rate coefficients of the particles *k*

r2 \_\_∂ <sup>∂</sup>r(*vr <sup>r</sup>* <sup>2</sup> *<sup>C</sup><sup>i</sup>*

*Sneedle*

<sup>∂</sup> *<sup>C</sup>*\_\_\_*<sup>i</sup>*

<sup>→</sup> *Ci*

) <sup>−</sup> \_\_1 r2 \_\_∂ <sup>∂</sup><sup>r</sup> <sup>∙</sup>(*Di*

<sup>∂</sup>r(*vr <sup>r</sup>* <sup>2</sup> *<sup>C</sup><sup>i</sup>* <sup>−</sup> *Di*

<sup>∂</sup>r{*C<sup>i</sup> <sup>r</sup>* <sup>2</sup> (\_\_ *Ai <sup>r</sup>* <sup>2</sup> <sup>−</sup> \_\_1 *Ci* <sup>∂</sup>*<sup>C</sup>* \_\_\_*<sup>i</sup>*

3(1 − *ε<sup>i</sup>* ) \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ 2 *ε<sup>i</sup> dc ηs*

(*r*),Eq. (A1.10) can be written as:

<sup>∂</sup>r{(*<sup>r</sup>* <sup>2</sup> exp[−\_\_

( ∂<sup>2</sup> *C*<sup>0</sup> *i* \_\_\_\_

$$\frac{\partial}{\partial \mathbf{r}} \Big| r^2 \frac{\partial C\_0}{\partial \mathbf{r}} \Big| + A \Big| \frac{\partial C\_0}{\partial \mathbf{r}} \Big| + m\_i C\_0^i = 0 \tag{A1.13}$$

The solutions of Eq. (A1.13) are given by the following expressions:

The solutions of Eq. (A1.13) are given by the following expressions:

$$\mathbf{C}\_{0}(\mathbf{r}) = \frac{(\mathbf{const1}) \stackrel{\triangle}{\text{e}^{2}} \text{ Bessel} \mathbf{I} \left[ \sqrt{1 - 4 \, \mathbf{m}\_{i}}, \, \frac{\mathbf{A}\_{i}}{2 \overline{x}} \right]}{\sqrt{\overline{\mathbf{r}}}} + \frac{(\mathbf{const2}) \stackrel{\triangle}{\text{e}^{2}} \text{ Bessel} \mathbf{K} \left[ \sqrt{1 - 4 \, \mathbf{m}\_{i}}, \, \frac{\mathbf{A}\_{i}}{2 \overline{x}} \right]}{\sqrt{\overline{\mathbf{r}}}} \tag{A1.14}$$

Bessel *<sup>I</sup>*[<sup>√</sup> \_\_\_\_\_\_ 1 − 4 *mi* , *A* \_\_*i* <sup>2</sup>*<sup>r</sup>* ] and Bessel *<sup>K</sup>*[<sup>√</sup> \_\_\_\_\_\_ 1 − 4 *mi* , *A* \_\_*i* <sup>2</sup>*<sup>r</sup>*] are modified Bessel functions I and K of the order <sup>√</sup> \_\_\_\_\_\_ 1 − 4 *mi* . The expressions \_\_ *Ai* <sup>2</sup>*<sup>r</sup>* are the variables of these functions. The general solutions of Eq. (5) are computed using the expressions (A1.14) in the expression (A1.11):

$$\mathcal{L}\_{\text{i}}(r) = \left(\frac{\mathbf{e}^{\frac{\lambda}{2\mathbf{r}}}}{\sqrt{\mathbf{r}}}\right) \left(\mathbf{(const1)}\_{\text{i}}\text{ Bessel}\,\mathbf{I}\left[\sqrt{1-4\,\mathbf{m}}\_{\text{i}}\,\prime,\frac{\mathbf{A}\_{\text{i}}}{2\mathbf{r}}\right]\right)$$

$$+ \left(\mathbf{const2}\right)\_{\text{i}}\text{ Bessel}\,\mathbf{K}\left[\sqrt{1-4\,\mathbf{m}}\_{\text{i}}\,\prime,\frac{\mathbf{A}\_{\text{i}}}{2\mathbf{r}}\right]\right) \tag{A1.15}$$

(**const1**) i and (**const2**) i are the four integration constants which are determined from the following four boundary conditions:


$$\begin{aligned} \mathbf{C}\_1(\mathbf{r} = \mathbf{R}\_1) &= \left. \mathbf{C}\_2(\mathbf{r} = \mathbf{R}\_1) \right|\_{\mathbf{r} = \mathbf{R}\_1} \\\\ \left. D\_{\mathbf{r}}^\* \frac{\partial \mathbf{C}\_1}{\partial \mathbf{r}} \right|\_{\mathbf{r} = \mathbf{R}\_1} &= \left. D\_{\mathbf{r}}^\* \frac{\partial \mathbf{C}\_2}{\partial \mathbf{r}} \right|\_{\mathbf{r} = \mathbf{R}\_1} \end{aligned}$$

**iii.** at the injection site (IS), at the top of the needle (r = r<sup>o</sup> ) the concentration has the particular expression C<sup>1</sup>  = Cmax.

The constants (**const1**) i and (**const2**) i were computed in the Wolfram Mathematica 10 software.
