# Laser-Induced Motion of a Nanofluid in a Micro-Channel

^{*}

## Abstract

**:**

_{2}, Fe

_{2}O

_{3}, Al

_{2}O

_{3}MgO, and SiO

_{2}nanoparticles with water as the base fluid. The particle diameter is 50 nm and the laser beam is a 4 W continuous beam of 6 mm diameter and 532 nm wavelength. The results indicate that, as the particle moves, a significant volume of the surrounding water (up to about 8 particle diameters away from the particle surface) is disturbed and dragged along with the moving particle. The results also show the effect of the particle refractive index on the particle velocity and the induced volume flow rate. The velocity and the volume flowrate induced by the TiO

_{2}nanoparticle (refractive index n = 2.82) are about 0.552 mm/s and 9.86 fL, respectively, while those induced by SiO

_{2}(n = 1.46) are only about 7.569 μm/s and 0.135, respectively.

## 1. Introduction

^{−15}m

^{3}to 10

^{−6}m

^{3}. Such devices have been found to be of significant importance in a variety of technologies such as cooling of electronic chips, inkjet printers, drug delivery, biomedical diagnostics, and biochemical processing, etc.

_{2}, Fe

_{2}O

_{3}, Al

_{2}O

_{3}∙MgO, and SiO

_{2}) with water as the liquid medium. We use these typical particles because they are commonly used as the solid component in many nanofluids.

## 2. Forces Acting on a Single Particle

**u**is the velocity vector,

**b**the body force, and ${\mathit{F}}_{\mathit{E}}$. represent the various forces acting on the particle which can include the terms on the right hand side of Equation (2) and also forces due to the electric current, magnetic field interactions (see Kim et al. [21]), and the radiation due to the laser, etc. In this paper, we ignore all the forces except that due to the laser. In the next section we discuss this force.

## 3. Forces Due to the Laser

_{o}at the focal point located at x = 0. Its wavelength λ is very long compared to the suspended particle size and it is chosen so that the fluid base with the refractive index n

_{f}is transparent to it. The suspended nanoparticle of radius a is considered to be non-absorptive relative to the laser wavelength and it is located at a position r = (x, y, z) relative to the beam waist center. The particle refractive index is n

_{p}. Since the particle is non-absorptive, the momentum transfer during refraction and reflection of the photon becomes dominant. In this case, the resulting forces are the scattering force, F

_{sctr}, and the force due to the gradient of intensity, often called the gradient force. The scattering force acts in the direction of the photon propagation. In contrast, the gradient force has, in a Cartesian coordinate system, three components: two transverse components (F

_{grad,y}, and F

_{grad,z}) which act to restore the particle to the beam center and a longitudinal component, F

_{grad,x}which acts against the scattering force to pull the particle toward the beam waist. Since the size of the nano-particle is very small compared to the wavelength of the laser beam, these forces are in the Rayleigh scattering regime and they are given as [22,23]:

_{p}/n

_{f}is the particle relative refractive index, P

_{o}is the beam power, w

_{o}is the beam waist, $\overline{x}=x/k{w}_{o}^{2},$ $\overrightarrow{y}=y/{w}_{o},$ and $\overline{z}=z/{w}_{o}$ are the dimensionless coordinates. If the focal region of the beam is assumed to be cylindrical in shape, the focal spot size, in terms of the beam waist w

_{o}is given as

_{grad,y}and F

_{grad,z}where they are pushed along the direction of the beam propagation or pulled back to the beam waist depending on the magnitudes of the scattering force and the longitudinal component of the gradient force. The magnitudes of these forces depend significantly on the particle size, as shown in Figure 3.

_{grad,y}and F

_{grad,z}are zero and the remaining non-zero forces are the longitudinal component of the force due to the gradient of intensity, F

_{grad,x}and the scattering force, F

_{scat}. The longitudinal component F

_{grad,x}is acting toward the beam waist, and its magnitude increases negatively from zero at the beam waist to a maximum value at $\overline{x}=0.288$ ($x=k{w}_{o}^{2}/2\sqrt{3}$) away from the beam waist. The scattering force is acting along the direction of the propagation of the laser beam. Its magnitude has a maximum value at the beam waist and decreases rapidly as the particle location moves in the direction of the laser propagation. For the particle sizes shown here, the longitudinal component of the gradient force is significantly larger than the scattering force for the particle size of ≤5 nm and it becomes less significant for the particle sizes larger than 10 nm. Thus, smaller particles are pulled back to the beam waist and the larger particles are pushed along the laser beam. Therefore, in order to induce the flow of a nanofluid in a micro-channel by a laser beam, the nanofluid must contain nanoparticles whose sizes are determined by the ratio of the scattering force F

_{sctr}to the maximum backward longitudinal component of the gradient force. For a Gaussian beam of wavelength λ, a focal spot size w

_{o}, the maximum longitudinal component of the gradient force occurs at y = z = 0 and $x=k{w}_{o}^{2}/2\sqrt{3}$ where this ratio becomes

_{sctr}is greater than that of F

_{grad,x}. Thus, the required particle size to move in the direction of the laser propagation is

## 4. Results

_{f}= 1.332 and viscosity μ

_{f}= 0.98 × 10

^{−3}kg/m-s. The particle velocity is calculated using Equation (11) with F

_{sctr}and F

_{grad,x}given by Equations (4) and (5), respectively. Referring to Figure 1, the velocity profile of the surrounding water, v

_{x}(in the direction of the particle motion), across the flow field is calculated from the velocity field described by Equations (13) and (14). The flowrate due to the motion of the particle is $\dot{Q}=2\mathsf{\pi}{\displaystyle {\int}_{0}^{R}R{v}_{x}dR}$ where $R=a\mathrm{tan}\mathsf{\theta}$. We use the Simpson’s 1/3 rule to calculate its value from θ = 0 to θ where v

_{x}→ 0. Typical results for the axial forces, the particles velocities, and the flowrate are shown in Table 1.

_{2}nanoparticle, Fe

_{2}O

_{3}nanoparticle, and SiO

_{2}nanoparticle are shown in Figure 4.

_{2}nanoparticle, (refractive index n = 2.82), are about 0.552 mm/s and 9.86 fL, respectively, while those of SiO

_{2}(n = 1.46) are only about 7.569 μm/s and 0.135, respectively. Thus, using a laser beam to activate the flow of a nanofluid in a microchannel is more effective if the fluid contains high refractive index nanoparticles.

_{2}, Fe

_{2}O

_{3}, Al

_{2}O

_{2}, and MgO nanoparticles; these are shown in Figure 5 and Figure 6, respectively. With the typical conditions used here, the results show that a volume up to 4 × 10

^{−11}L/s could be transported which depends significantly on the laser power, the particle size, and the beam waist. The effects of these parameters on the fluid average velocity can be seen in Figure 6. To consider the electroosmotic effect on the velocities Sinton and Li [25] and Sinton et al. [26] showed that velocities increased in a linear fashion in 100 μm and 200 μm diameter circular channels and in 100 μm and 50 μm width square channels from about 0.2 mm/s when the external applied field was 5 kV/m; whereas the velocity was 1.2 mm/s when the external applied field was 20 kV/m. A similar velocity range is also obtained in our present calculations for TiO

_{2}and Fe

_{2}O

_{3}particles. For example, for a 5 W laser power the fluid average velocity increases up to about 2.1 mm/s with a 150 nm particle while for the same laser power with a 1 μm beam waist, the velocity is about 5 mm/s for a 100 nm particle. However, for Al

_{2}O

_{3}and MgO particles, due to their low refractive indexes, the average velocity up to 0.5 mm/s is calculated.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Representation of the streamlines and the x-component velocity, v

_{x}(velocity along with the direction of the particle motion) for flow of the surrounding fluid due to the moving particle, r, θ, are the spherical coordinates of the surrounding fluid described by Equations (13) and (14), U: particle velocity, x: particle flow direction, R: radius of the flow channel relative to the centerline.

**Figure 2.**Representation of the interaction between a Gaussian continuous laser beam and a nanofluid in a micro-channel. The laser beam is slightly focused using a focusing lens, w

_{o}is half of the beam waist which is at the channel entrance (x = 0), a is the particle radius, r is the location of the particle relative to center of the beam waist.

**Figure 3.**Scattering force, F

_{sctr}and the longitudinal component of the gradient force F

_{grad,x}as a function of the dimensionless $\overline{x}$ acting on a particle in water located on the beam center $(\overline{y}=\overline{x}=0)$, n

_{p}= 1.592, n

_{f}= 1.332, P

_{o}= 100 mW, lens focal length f = 75 mm, beam diameter D = 4 mm, wavelength λ= 532 nm.

**Figure 4.**Velocity profiles in axial plane of the flow of water induced by the moving particle (P

_{o}= 4 W, focal length = 75 mm, beam diameter d = 6 mm, λ = 532 nm, nm (w

_{o}= 4.2375 μm, Equation (8)), refractive index of water n

_{f}= 1.332, water viscosity μ

_{f}= 0.98 × 10

^{−3}kg/m-s, a = 50 nm, the particle is at the beam waist).

**Figure 5.**Effects of the laser power, P

_{o}, beam waist, W

_{o}, and particle size on the fluid volume flowrate (focal length = 75 mm, beam diameter d = 6 mm, λ = 532 nm, nm (w

_{o}= 4.2375 μm, Equation (8)), refractive index of water n

_{f}= 1.332, water viscosity μ

_{f}=0.98 × 10

^{−3}kg/m-s, the particle is at the beam waist).

**Figure 6.**Effects of the laser power, P

_{o}, beam waist, W

_{o}, and particle size on the fluid average velocity (focal length = 75 mm, beam diameter d = 6 mm, λ = 532 nm (w

_{o}= 4.2375 μm, Equation (8)), refractive index of water n

_{f}= 1.332, water viscosity μ

_{f}= 0.98 × 10

^{−3}kg/m-s, the particle is at the beam waist).

**Table 1.**Typical axial forces, particle velocities, and volume flowrate transported by nanoparticles under laser action, (P

_{o}= 4 W, focal length = 75 mm, beam diameter d = 6 mm, λ = 532 nm, refractive index of water n

_{f}= 1.332, water viscosity μ

_{f}=0.98 x 10

^{-3}kg/m

^{-s}, a = 50 nm, the particle is at the beam waist, v

_{p}: particle velocity, v

_{f,ave}: fluid average velocity, Q: fluid volume flowrate).

Particle | Refractive Index | F_{x}(pN) | v_{p}(mm/s) | Q(fl/s) | v_{f,ave}(mm/s) |
---|---|---|---|---|---|

TiO_{2} | 2.82 | 0.4628 | 0.55182 | 9.86 | 0.009 |

Fe_{2}O_{3} | 1.986 | 0.1345 | 0.16039 | 2.867 | 0.0028 |

Al_{2}O_{3} | 1.773 | 0.0671 | 0.08006 | 1.431 | 0.0014 |

MgO | 1.73 | 0.0557 | 0.06637 | 1.186 | 0.0015 |

SiO_{2} | 1.46 | 0.00635 | 0.007569 | 0.135 | 0.00013 |

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**MDPI and ACS Style**

Phuoc, T.X.; Massoudi, M.; Wang, P.
Laser-Induced Motion of a Nanofluid in a Micro-Channel. *Fluids* **2016**, *1*, 35.
https://doi.org/10.3390/fluids1040035

**AMA Style**

Phuoc TX, Massoudi M, Wang P.
Laser-Induced Motion of a Nanofluid in a Micro-Channel. *Fluids*. 2016; 1(4):35.
https://doi.org/10.3390/fluids1040035

**Chicago/Turabian Style**

Phuoc, Tran X., Mehrdad Massoudi, and Ping Wang.
2016. "Laser-Induced Motion of a Nanofluid in a Micro-Channel" *Fluids* 1, no. 4: 35.
https://doi.org/10.3390/fluids1040035