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# Physics of Microfluidics Basic Properties of Microfluidic Flows

The physics of microfluidics is a branch of fluid dynamics in which we consider fluids that are confined in structures of micrometer scales, i.e structures that have at least one dimension (width, length or depth) lower than a millimeter. At microscales, the dominant forces are different from those we experience at the macroscale, and flows typically exhibit properties that can be exploited in microfluidic devices.

## The Navier-Stokes equations in microfluidic devices

The velocity field of an incompressible fluid can be derived from the Navier-Stokes equations : where is the fluid velocity field, is its density, the pressure field, its kinematic viscosity and is an external acceleration field (due to gravity or electrostatic forces for example). Boundary conditions also apply but are not shown here.

These equations physically describe the space-time evolution of the velocity field of given fluid in a given domain. They are respectively obtained from simplified laws of conservation of momentum and mass, with the assumption that the fluid is incompressible (density is constant over space and time) and newtonian (viscosity is uniform).

We set the non-dimensionnal parameters and . If L is the characteristic length of the geometry (also called hydraulic diameter in a microchannel), U the typical velocity of the fluid, and the typical intensity of the acceleration field, each of these non-dimensional parameters are of the same order of magnitude . Furthermore, Equation (1) can be written in the following non-dimensional form  : Re is called the Reynolds number. This non-dimensional number gives the ratio between inertial and viscous forces. Scaling down in dimension and/or flow velocity tends to decrease this number. It is a powerful indicator of the flow regime.

## Flow regime as a function of the Reynolds number in microfluidics

### 1. Turbulent Flow

From Equation (3), we can see that there are two types of flow regimes depending on the magnitude of the Reynold number.

At high Re, the left part of the equation dominates, and the flow is governed by the equation: which is unsteady and non-linear, and then instrinsicaly chaotic. This regime is called turbulent flow.

### 2. Laminar Flow

From Equation (3), we can see that there are two types of flow regimes depending on the magnitude of the Reynold number.

At low Re, the right side of equation (3) dominates, and the flow is governed by the Stokes equation: This equation is intrinsically regular, leading to flow with parallel streamlines. This regime, occurring when the viscous forces dominate over inertial forces, is called laminar flow. This is typically the kind of regime that occurs in microfluidics.

In practice, it is shown that the limit between turbulent and laminar flows is:

• If Re < 2300, the flow can be considered as laminar
• If Re >2300, the flow will be turbulent.

In between these two regimes, there exists what is called transient regime, when streamlines remain parallel but start weaving.

## Pressure driven steady flow in a circular microchannel : Poiseuille flow

Let’s consider a microchannel of diameter d=200µm (radius a), of length l>>d, filled with water traveling at an average speed of U=1mm/s (corresponding to a flow rate of 1.2 µl/min) in a direction perpendicular to the gravity. The flow is only driven by a pressure gradient: a pressure is applied to the left side, and a pressure to the right side (we define ). We assume that there is no velocity at the boundary (no-slip wall condition).

The Reynolds number is . Therefore, the flow is Equation (5) applies and it can be written in a cylindrical coordinate system  : with r the radial coordinate and z the central axis of the microchannel.

The solution of equation (7) shows the velocity is parabolic: This type of flow, that commonly occurs in microfluidics, is called Poiseuille flow. For this type of flow, the volumetric flow rate is directly related to the pressure drop applied at the extremities of a microchannel.
Other kind of flow may occur also depending on the external force field applied. Electro-osmotic flow is an example flow that occurs when the fluid is submitted in a high electric field.

## Microfluidic resistance

Interestingly, properties of flows at microscales have similarities with other fields of physics. We can show that there exists an equivalent « Ohm’s law » in microfluidics similar to the law in electronics.

Starting from Equations (7) and (8), we can derive the volumetric flow rate Q noticing that: with U the mean velocity obtained by integrating the velocity over the section : This linear relationship between the pressure drop at the extremities of a circular microchannel and the volumetric flow rate is strictly similar to Ohm’s law, where Rh is called hydrodynamic resistance.
The higher the hydrodynamic resistance, the higher the pressure drop to induce a given flow rate. That it is possible to evaluate the resistance knowing the geometrical characteristics of the microchannel and the fluid viscosity.

## Diffusion of mass transport in a microchannel

Similarly to the flow regime, a scaling effect also occurs with mass transport. Starting from the diffusion-advection equation expressing the space-time evolution of the concentration field of a given species : where is the diffusion coefficient, one can derive, in a way similar to the Reynolds number, the one called Peclet number Pe: This number compares the convective transport to the diffusion transport. Taking the same example of microchannel used above, using, for example, the diffusivity of ethanol in water (D=0.84 10-5 cm2/s), we get Pe=1.9 10-3.

Such low Peclet number indicates that in microchannels, at reasonably low flow rate, mixing mainly occurs through diffusion process, which is quite a slow phenomenon. Mixing of two species in a microchannel is quite difficult and tricks  have to be used to obtain a satisfactory mixing in a reasonable channel length. This property can also be used for particle extraction  or sample injection .

## Surface tension in physics of microfluidics

The physics of microfluidics is not only about flows in microchannels with a single type of liquids but also about the interface between fluids and/or surfaces at micro scales.

At the human scale, surface tension of liquid is rather low. That is why immersing an object in water is pretty easy with human force. However, capillary forces are difficult to overcome at micro scales. Small object with low mass, such as water spiders, cannot permeate the surface and that is why they are floating.

Surface tension is responsible for phenomena that can be exploited in microfluidic systems, such are capillary pumping  or droplet generation.

## References

 C. R. Doering, J.D. Gibbon, « Applied analysis of the Navier-Stokes equations », Cambridge University Press, 2001, ISBN 9780521445689

 D. J. Tritton, « Physical fluid dynamics », Clarenton Press, 1988

 « MEMS Introduction and Fundamentals », page 10.7-10.8, CRC Press, Edited by M. Gad-el-Hak, 2005, ISBN 9781420036572

 H. Bruus, « Theorical microfluidics » lecture notes, DTU (Technical University of Denmark) Fall, 2006

 D. L. Young et al., Eng. Anal. Bound. Elem., 24, 2000, 449-457

 K. Ward, Z. H. Fan, J. Micromech. Microeng. 25 (9), 2015

 M. F. Shafique et al, J. Eur. Ceram. Soc, 31 (13), 2013

 D. Sinton, L. Ren, D. Li, J. Colloid Interface Sci., 266 (2), 2003

 W. R. Jong et al., Int. Commun. Heat Mass, 34, 2007, 186-196

 W. R. Jong et al., Int. Commun. Heat Mass, 34, 2007, 186-196

 R. Epifania, Sens. Actuat. B : chemical, 265, 2018