Microfluidics/Fluid dynamics equation

The motion of fluid is described by applying the fundamental law of mechanics, known as Newton's second law, not on a point particle but on a fluid.

Fluid particle


A fluid particle is a volume, large enough to contain smooth molecular variations, but small compared to the system size. It has a mesoscopic character, intermediate between a microscopic (molecular) and macroscopic description. A continuum approach is possible.

In liquids the fluid particle size can be larger than molecules and smaller than micrometric channels.

In gases the mean free path $$\lambda$$ can be not smaller than the microsystem size, a specific approach is necessary.

The continuum approach allows to define, as a function of the position in space $$\mathbf{r}$$ and the time $$t$$:
 * $$\rho (\mathbf{r},t)$$, the mass density (unit kg/m^3)
 * $$\mathbf{u}(\mathbf{r},t)$$, the velocity (unit m/s)

Mass conservation
It writes
 * $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$$

For incompressible fluids (which is almost the case with liquids, and gases at velocities small in front of the speed of sound), $$\rho=\mathrm{cst}$$, and mass conservation implies a specific flow field that is divergence free,  $$\scriptstyle div(\mathbf{u})=0$$.

Forces within a fluid


Two kind of forces are present in a fluid:
 * Isotropic normal force: the pressure. They provide a component
 * $$d\mathbf{f}=-p\, \mathbf{n}\,dS$$on a small element of surface $$dS$$, $$\mathbf{n}$$ being the outward normal to volume element.


 * Shear forces: due to internal friction they provide a component
 * $$d\mathbf{f}=\mathbf{\tau}\, .\, \mathbf{n}\, dS$$, with $$\mathbf{\tau}$$ the shear stress tensor.

For a Newtonian fluid the shear stress tensor components are given by
 * $$\tau_{ij}=\mu\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i} \right)$$

where $$\mu$$ is the viscosity.

Navier-Stokes equation
For a Newtonian fluid the Navier-stokes equation writes
 * $$\rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}$$

The second term on the left hand side comes from the inertia of the fluid. It is a non-linear term that produces history dependent effect very present at the human scale or larger scale:
 * swirls, tornados
 * turbulence
 * propulsive forces

However at small scales inertia is damped by viscous effects (second term on the right) that are preponderant. The Reynolds number provides an estimation of the ratio inertial forces/viscous forces: if the typical velocity of the fluid is $$u$$, and the typical size $$l$$
 * $$R\!e=\frac{\rho u l}{\mu}$$.

At small scales $$R\!e\ll1$$, and the second term $$\mathbf{u} \cdot \nabla \mathbf{u}$$ can be neglected, the Navier-Stokes equation becomes the Stokes equation. This equation is linear and easier to solve.
 * $$\rho \frac{\partial \mathbf{u}}{\partial t} = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}$$

The velocity field solution of these equations is linear in applied stresses meaning:
 * the solution is unique (whereas the full Navier-Stokes equation gives rise to turbulence and instabilities)
 * the solution is reversed when the forces are reversed: it is impossible to create a fluid "diode" at small scales

It can also be shown that the solution minimizes the total dissipated power.

Boundary conditions
The equations of motions can be solved when boundary conditions are given at the surface of the fluid.

For usual liquids
there is a no-slip boundary condition, $$\mathbf{u}=\mathbf{0}$$at the surface

For gases, and for some non-wetting liquids
For gases, the Knudsen number compares the mean free path to the typical lengthscale of the flow.
 * $$Kn=\frac{\lambda}{l}$$

For $$Kn<0.003$$ the continuum approach still holds with no slip at the interface. For $$Kn>0.01$$ a molecular approach is necessary. For intermediate numbers $$0.003<Kn<0.01$$, Navier-Stokes still holds but there is a slip at the solid surface, defined by the Navier length $$L_s$$, such that the slip velocity of the fluid is
 * $$u_s=L_s\frac{\partial u_x}{\partial z}$$

with $$z$$ the distance to the surface. Usually $$L_s\simeq \lambda$$

For liquids, the debate is still open...