## 2 Foundations
### Hagen-Poiseuille Flow  
Artificial tears density, viscosity, and Reynolds numbers are the properties relevant for the design of the microinjector. The density of artificial tears is practically the same as water since they are aqueous solutions with different soluble molecules at very low concentrations (e.g., 0.02%). For practical purposes and theoretical estimations, it is taken as 1.0 gr/cm\({}^{3}\).  
Artificial tears viscosity, unlike density, is highly dependent on chemical composition and varies considerably between different commercial formulas. [PERSON] et al. [32] reported measured values from 1 up to \(\sim\)600 Pa-s of dynamic viscosity for common high viscosity and low viscosity artificial tears at temperatures from 25 to 37 \({}^{\circ}\)C. [PERSON] et al. [35] and [PERSON] et al. [32] presented studies to determine the effect of tears viscosity on well-being in dry eye patients, and reported a slight improvement with the increase of viscosity, while some others reported no improvements at all.  
The present microinjector emulates the flow produced by the human tear system, normally ranging from 2 to 7 mL/day of organic lubricant [36]. Taking into account the range of experimental viscosities reported by [PERSON] [32], the flows handled by the microinjector, the density of artificial tears, and the internal diameter of convenient commercially available capillaries (\(\sim\)275 to \(\sim\)800 \(\upmu\)m), the Reynolds numbers involved are very small, \(\sim\)5.0 \(\times\) 10\({}^{-7}\) to \(\sim\)3.0 \(\times\) 10\({}^{-4}\), so the flow of artificial tears from our microinjection system is laminar.  
In the microinjector, the shape of all tubes and capillaries involved are cylindrical and the flow is pressure driven only. Under these conditions, the flow of artificial tears is described by the well known Hagen-Poiseuille flow. The solution in cylindrical coordinates \(\left(r,\theta,z\right)\) of the Navier-Stokes equations for this flow may be found elsewhere [33; 37]. The radial and angular components of the velocity are independent of both \(\theta\) and \(z\) and null, then \(v_{r}\left(r\right)=0\), \(v_{\theta}\left(r\right)=0\) and \(v_{z}=v_{z}\left(r\right)\). Velocity gradients in \(\theta\) and \(z\) are also zero. Under these conditions, the final result for \(v_{z}(r)\) and the pressure driven flow, \(Q\), are:  
\[v_{z}=-\frac{1}{4\mu}\frac{\partial p}{\partial z}\left(R^{2}-r^{2}\right) \tag{1}\]  
and  
\[Q=-\frac{\pi R^{2}}{8\mu}\frac{\partial p}{\partial z} \tag{2}\]  
where \(\mu\) is the dynamic viscosity, \(p\) is pressure, \(R\) is the capillary radius, and \(\partial p/\partial z\) is the pressure gradient. This is the well known result empirically found by [PERSON] [38] and [PERSON] [39].  
For practical purposes, it will be the dry eye patient whom, according to his needs, establishes the flow \(Q\) of artificial tears that he needs to feel better in the microinjector, as noticed, the human lubricating system works with \(Q=1\) to 5 \(\upmu\)L/min. Then Equation (2) is useful to calculate the exact minimum pressure gradient needed to keep the user-required flow, which will be applied by the engine of the device.