Page 142 - IJB-10-6

International Journal of Bioprinting Fluid mechanics of extrusion bioprinting

forces can be effective. Two main regimes describe the channel with hydraulic diameter d and length l, the
dynamics of coaxial flow in a flow-focusing microfluidic residence time (t ) is given by:
device: (I) the dripping regime, in which the breakup of res
the core flow creates droplets dispersed in the sheath t ≈ d 2 (XXX)
flow, and (ii) the jetting flow with continuous core flow res Q
(Figure 11A). 157,158 For successful coaxial bioprinting, a The mixing time (t ) for diffusion mixing of fluids is
stable jetting flow should be maintained inside the needle. mix
calculated by:
Utada et al. analyzed the transition between dripping
157
and jetting flow regimes based on the balance of capillary, d 2
viscous shear, and inertial forces for Newtonian fluids. t mix ≈ mix (XXXI)
Derzsi et al. extended this analysis to coaxial flow of D
158
Newtonian core and viscoelastic sheath flows, similar where D is diffusion coefficient, and the mixing
to hollow fiber printing, in a flow-focusing microfluidic distance (d ) that components must diffuse to make a
mix
chip. Their experiments, covering a wide range of viscosity homogenous mixture is proportional to the hydraulic
and flow rate ratios, demonstrated that the Weissenberg diameter of the channel (d). Therefore, for efficient mixing
number governs the dynamics of coaxial flows with a inside the head, 159
viscoelastic sheath layer. They defined the Weissenberg
.
number based on extensional strain rate γ and extensional t < t ⇒ Pe = Q < (XXXII)
e
relaxation time (λ ): mix res dD d
e
where Pe denotes the dimensionless Péclet number,
.
Wi = γ λ e (XXVIII) which is defined as the ratio of the advective transport rate
e
of a component to its diffusive transport rate. To enhance
the mixing process of biomaterials inside a multi-material
Figure 11B illustrates the results of Derzsi et al. for head, the process can be optimized by reducing the total
158
the dripping-jetting transition, plotting the dimensionless flow rate, which corresponds to printing at lower speeds,
group of volumetric flow rate (Q) and viscosity in the core or by shortening the mixing distance through increased
and sheath flow (Q η ) against the critical Weissenberg interdigitation between the streamlines of precursor
c c
numbers (Wi trans ) at which the transition from dripping flows. Interdigitation can be achieved by passive mixing
to jetting flow regime occurs. Their results are based on techniques, without external energy requirement, or active
the elasticity number (El) of the focusing stream, which methods that involve external energy supply for their
compares elastic and inertial forces to represent the dynamic mechanics. While active mixing methods can
strength of elasticity in the focusing stream : provide rapid mixing of high-viscosity biomaterials, 160
158
passive mixing is more commonly used in multi-material
heads because of its simplicity and low stress levels on
λη the bioink.
El = e (XXIX)
ρ w 2 4.2.1. Multi-material heads with static mixers
Static mixers with diverse geometries have been developed
where w is the channel width. While the jetting regime to meet the requirements of various applications. While
stabilizes the continuous coaxial flow, the Rayleigh-Plateau highly efficient static mixers can effectively mix fluids,
instability 56–58 can lead to breakup of the core stream into they often generate high levels of shear stress due to their
droplets (Figure 11A). Therefore, the length of needle complex geometry and narrow flow paths. However,
should be shorter than the continuous jet region to prevent helical mixers with alternating left- and right-hand twisted
the core stream breakup. elements, known as Kenics type mixers (Figure 13A), offer
a good balance between mixing efficiency and shear stress.
4.2. Multi-material bioprinting with mixing Their simple geometry allows for effective mixing without
The complete mixing of two fluids inside a multi-material 161
head requires that the shortest residence time (t ) of the excessively high shear stress.
res
precursors streams inside the mixer exceeds the mixing In the context of bioprinting, with creeping flow inside
time (t ) of components. For the parallel (or coaxial) flow the dispensing nozzle (Re << 1), the mixing potential of
mix
of two precursor fluids inside the head, the only effective vortices is limited. Helical mixers with twisted elements
mechanism of mixing is molecular diffusion. For a straight facilitate mixing through chaotic advection. 156,162

Volume 10 Issue 6 (2024) 134 doi: 10.36922/ijb.3973