Measurement of the oscillatory flow field inside tapered cylindrical inkjet nozzles using micro-particle image velocimetry

Abstract

The flow field within a tapered, cylindrical, piezoelectrically actuated glass inkjet nozzle is captured using fluorescence micro-PIV (μPIV) assisted by a novel, custom-designed PDMS micro-fabricated nozzle holder and a microsecond-resolution cyclic triggering system. The presented work overcomes key imaging challenges such as distortion from the refractive index mismatched curved glass–air interface and the typically large depth of field/correlation found in inkjet imaging set-ups. The PDMS holder permits fluorescence imaging of the seeded flow tracing particles with minimal distortion as the holder is refractive-index-matched with the glass nozzle. The cyclic triggering system allows visualization of the transient phases of a periodic droplet ejection event. The system utilizes an inverted microscope with an objective lens capable of producing a low depth of correlation of 12.25 µm. Double-frame images for µPIV were acquired beginning from the onset of droplet formation to study the flow field evolution during droplet formation and after droplet break-off. An oscillatory flow field was observed within the nozzle during the droplet ejection process which closely correlates with modelling results.

Appendices

Appendix 1

The governing equation for the fluid flow inside the tapered nozzle is the Navier–Stokes equation as:

$$\frac{{\partial u_{z} }}{\partial t} + u_{r} \frac{{\partial u_{z} }}{\partial r} + u_{z} \frac{{\partial u_{z} }}{\partial z} = - \frac{1}{{\rho_{f} }}\frac{\partial p}{\partial z} + \upsilon \left( {\frac{{\partial^{2} u_{z} }}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial u_{z} }}{\partial r} + \frac{{\partial^{2} u_{z} }}{{\partial z^{2} }}} \right) ,$$

(3)

where u _z and u _r are the z- and r-component of velocity, respectively, $\rho_{\text{f}}$ is the fluid density and υ is the fluid kinetic viscosity. In this analysis, it is assumed that fluid is incompressible and the walls are rigid. Moreover, it is assumed that pressure gradient in r-direction, convective terms and $\frac{{\partial^{2} u_{z} }}{{\partial z^{2} }}$ are negligible (Shin et al. 2005). Therefore, the linearized Navier–Stokes equation for unidirectional flow becomes

$$\frac{{\partial u_{z} }}{\partial t} = - \frac{1}{{\rho_{f} }}\frac{\partial p}{\partial z} + \upsilon \left( {\frac{{\partial^{2} u_{z} }}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial u_{z} }}{\partial r}} \right)$$

(4)

It can be shown that the analytical solution to the linearized Navier–Stokes equation for an oscillatory pressure gradient of

$$\frac{\partial p}{\partial z} = \exp (i\omega t)$$

(5)

is (Shin et al. 2005)

$$u_{z} \left( {r,z,t} \right) = u\left( {r,z} \right)\exp \left( {i\omega t} \right) ,$$

(6)

and

$$u\left( {r,z} \right) = \frac{1}{{\rho_{f} i\omega }}\left( {1 - \frac{{J_{0} \left( {\lambda r} \right)}}{{J_{0} \left( {\lambda R\left( z \right)} \right)}}} \right) ,$$

(7)

where

$$R\left( z \right) = R_{1} - z\tan \theta$$

(8)

and

$$\lambda = \sqrt {{{ - i\omega } \mathord{\left/ {\vphantom {{ - i\omega } \upsilon }} \right. \kern-0pt} \upsilon }} .$$

(9)

In these equations, J ₀ is the Bessel function of first kind, θ is the tapered angle and R ₁ is the inner radius at the upstream of nozzle. On the other hand, the equation for pressure, p(z), can be obtained from the continuity equation as:

$$\int_{0}^{R\left( z \right)} {\frac{\partial }{\partial r}} \left( {ru_{r} } \right)dr + \frac{\partial }{\partial z}\left( {\int_{0}^{R\left( z \right)} {ru_{z} dr} } \right) = 0,$$

(10)

which simplifies to

$$\frac{{\partial^{2} p\left( z \right)}}{{\partial z^{2} }} + F_{3} \left( z \right)\frac{\partial p\left( z \right)}{\partial z} = 0,$$

(11)

where F ₃(z) ≡ F ₂(z)/F ₂(z)F ₁(z).F ₁(z), and F ₁(z) and F ₂(z) are defined as

$$F_{1} \left( z \right) = \int_{0}^{R\left( z \right)} {r - \frac{{J_{0} \left( {\lambda r} \right)}}{{J_{0} \left( {\lambda R\left( z \right)} \right)}}{\text{d}}r}$$

(12)

$$F_{2} \left( z \right) = \int_{0}^{R\left( z \right)} {rJ_{0} \left( {\lambda r} \right){\text{d}}r} \frac{{\lambda \tan \theta \,J_{1} \left( {\lambda R\left( z \right)} \right)}}{{\left( {J_{0} \left( {\lambda R\left( z \right)} \right)} \right)^{2} }}$$

(13)

As a result, the pressure and velocity fields can be obtained as:

$$p\left( z \right) = C_{1} + C_{2} \int_{{z_{1} }}^{z} {\exp \left( { - \int_{{z_{1} }}^{\zeta } {F_{3} \left( \xi \right){\text{d}}\xi } } \right){\text{d}}\zeta }$$

(14)

$$u\left( {r,z} \right) = - \frac{{C_{2} }}{{\rho_{f} i\omega }}\left( {1 - \frac{{J_{0} \left( {\lambda r} \right)}}{{J_{0} \left( {\lambda R\left( z \right)} \right)}}} \right)\exp \left( { - \int_{{z_{1} }}^{z} {F_{3} \left( \xi \right){\text{d}}\xi } } \right)$$

(15)

Due to the linear nature of the simplified Navier–Stokes equation, in most of the previous studies (Shin et al. 2005; Shin and Smith 2008), the pressure resulting from the piezoelectric actuator is expressed as a sum of oscillating pressure using Fourier series, and the velocity field is found by superposition. However, in this study, the velocity field is measured directly. Therefore, using Fourier series, the measured velocity at the upstream of the flow (at z = 105 µm) can be expressed as a sum of oscillating velocity terms, and as a result, the pressure field and velocity at other points can be found.

Appendix 2

The error due to distortion from the liquid–glass interface is calculated by projecting the refracted light ray back to the imaging plane and finding the difference of the projected position relative to the original particle position (Fig. 10). This is first done with Snell’s law.

$$n_{1} \sin (\theta_{1} ) = n_{2} \sin (\theta_{2} )$$

(16)

$$\theta_{2} = \sin^{ - 1} \left( {\frac{{n_{1} }}{{n_{2} }}\sin (\theta_{2} )} \right)$$

(17)

The angle in which the refracted light is deviated from the undistorted path (θ _d ) is then identified by

$$\theta_{d} = \theta_{1} - \theta_{2}$$

(18)

$$\theta_{d} = \theta_{1} - \sin^{ - 1} \left( {\frac{{n_{1} \cdot d}}{{n_{2} \cdot r}}} \right)$$

(19)

$$\theta_{d} = \sin^{ - 1} \left( \frac{d}{r} \right) - \sin^{ - 1} \left( {\frac{{n_{1} \cdot d}}{{n_{2} \cdot r}}} \right)$$

(20)

Once θ _d was identified, the radial error (e) can be easily calculated from simple trigonometric identities as depicted by the inset in Fig. 10.

$$e = \sqrt {r^{2} - d^{2} } \tan \left[ {\sin^{ - 1} \left( \frac{d}{R} \right) - \sin^{ - 1} \left( {\frac{{n_{1} \cdot d}}{{n_{2} \cdot R}}} \right)} \right]$$

(21)

The radial error (e) and distance of the particle (d) can then be normalized by the total cross-sectional radius (R) to produce a profile of the expected radial error measurements along R. As expected, the error is zero during the case of d = 0 and d = R with the error generally increasing with d.