In this section the theoretical model for the magnetic nanoparticle dynamics is presented briefly. This model is not the main scope of the paper, but its description is necessary to understand the build-up of the two-phase model. A detailed discussion of the micro-domain model is going to be published in another paper.
In this model each magnetic nanoparticle is considered separately (discrete particle method). Detailed description and material parameters of the MNP can be found in the experimental section. Here we only summarize the details necessary for the modeling. The nanoparticles have a magnetite core with , and the core is coated with a 20 thick silica shell. The particle size therefore is . The core is treated as a linear magnetic material with the magnetic susceptibility of .
2.2. Forces Effecting the Particle
Suppose now that the particle is placed in the fluid flow in a microchannel. The fluid is water and its relative permeability is assumed to be 1. Next we put a neodymium magnet over the channel. The non-homogeneous field of the magnet acts with a force on the particle:
where
is the vacuum permeability. The equality of the last two phrases in Equation (
5) can be done, as
.
The drag force on the particle is the Stokes force:
where
is the dynamic viscosity of the fluid,
is the particle radius and the last term is the velocity difference between the fluid and the particle. Besides this, if the particle rotates in the fluid, a stopping torque from the viscous fluid appears
where
is the particle’s angular velocity [
19].
If two particles are close, a magnetic force appears, as one particle is at the other particle’s non-uniform magnetic dipole field. The magnetic force between the close particles
i and
j is
where
and
are the magnetic moment’s of the particles,
is the distance between their centers and
is the unit vector of the distance [
14]. This force is assumed to be the main reason for the aggregation. It causes the particles to arrange into chains; the direction of these is parallel to the main magnetic field. The particles in the chain are attracted each other. It can be shown that in the field of the neodymium magnet, which is presented in the numerical results section, the attractive force between two magnetized particles in the chain is approximately three orders of magnitude higher than the force of the neodymium magnet on the particle. If the chain is bent as a result of other forces, it tries to rotate back to be parallel with the magnetic field.
The gravitational and buoyancy forces at this size range are negligible compared to the magnitude of the previously discussed forces.
2.3. Chain Formulation
As was mentioned in the previous section, in a magnetic field the magnetic nanoparticles are magnetized and due to the magnetic force between the particles they aggregate into a chain, which is parallel to the magnetic field. The chain formulation of the magnetized particles was also observed experimentally; see, e.g., [
20]. The chain formulation is also widely investigated in magnetorheological (MR) fluids [
21].
In the following we focus on the particle chain which is placed in a fluid flow with homogeneous strain rate. The idea originated from the electrorheological (ER) particle chain investigations in [
22]. Our goal was to find the net impact of the magnetic interactions of the particles on the fluid dynamics.
The arrangement is shown in
Figure 1a. The fluid flows to the direction
x, and has a homogeneous strain rate of
; i.e.,
. The magnetic field is vertical and uniform, noted as
, which causes the particles to arrange into a vertically oriented chain.
Suppose that initially the chain is vertical and the particles have zero velocity. The fluid then starts to move the chain to the direction
x due to the drag force. However, as the fluid has a strain rate, the drag force at the top of the chain is higher than at the bottom section, which causes the chain to start to rotate. As the chain rotates, a counter magnetic torque between the neighboring particle pairs appears due to the magnetic particle–particle interaction. This can be seen more clearly if Equation (
8) is rewritten with its tangential and normal components:
where
is the angle between the magnetic field and the particle-pair direction; see
Figure 1c. The magnetic moments can be expressed with the magnetic field. Based on Equation (
4) the moments are proportional to the magnetic field; therefore, the pair force is
.
An excellent description of this force and the angle-dependency is presented in [
14]. Here only the main consequences are collected based on [
14]:
The normal force between the particles is attractive until . This attractive force is the main reason for the chain formulation.
The tangential force is 0 at , and it approximately linearly increases with the increasing at small angles.
The chain rotates in the non-homogeneous flow until the torque from the magnetic forces counter-balances the torque of the fluid. Finding the exact rotated shape which is shown in
Figure 1b is complicated; therefore, in the follow the chain is considered linear, similarly to a model in [
22]. Now the torque of the flow is calculated on the linear chain. First consider the case when the number of particles in the chain is odd. In this case the axis is fixed to the center particle; see
Figure 1. We will concentrate to the effect of the upper section of the particles on the center particle. The torque of the
ith particle’s drag from the center particle is:
where
as all particles have the same velocity, which is equal to the fluid velocity at the center due to the symmetry. In other words, the relative velocity between the particle and the fluid phase is zero at the center. The total torque of the fluid on the chain based on the last two equations is:
The multiplication by 2 has to be done as the bottom section of the chain is also rotated.
The counter torque from the magnetic pair interactions between the neighboring particles can be calculated with Equation (
9). For one particle pair the torque is caused by the tangential magnetic force pair, which is shown in
Figure 1c. As there are
pairs in the chain, the total magnetic torque is:
In the steady rotated chain the net torque is zero; therefore,
. By comparing Equation (
11) with Equation (
12) the tangent of the angle of the steady chain can be calculated as:
The presented equation shows that an increasing strain rate increases the angle. Increasing the magnetic field in turn decreases the angle, as , and in this case the magnetic counter torque becomes higher. Longer chains have a higher rotation angle.
If
N is even, i.e., there is no center particle, the torque of the drag in Equation (
11) slightly changes, and the summation for
i changes to
. This has to be applied also for Equation (
13). It should be noted that the long chains can be broken over a critical angle; that problem is presented in detail in [
22]. In the current work this phenomenon is not investigated.
Using the approximation of
, in Equation (
13) the magnetic torque for the chain can be rewritten as:
when
N is odd, and for the even case, the summation should be changed as described above. This equation shows that we can determine the magnetic torque of the chain from the chain length. The equation can be used to calculate the increased viscosity of the magnetic nanoparticle phase (see later). It can be noted that because of
with the increasing chain length, the steady-state magnetic torque is increasing rapidly.
Finally, we also investigated the case when the chain is close to a horizontal wall and one of the chain ends is fixed to it. The fluid velocity is parallel to the wall and its profile is
, where
z is the distance from the wall. The chain becomes bent due to the drag and the total magnetic torque can be identified with a similar derivation, which is shown above. In this case the total magnetic torque is approximately four times higher for a chain with
N particles compared to Equation (
14).