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Chand, et al.
has the highest outlet velocity. This result is contrary to the Figure 3E), the strand was broken and caused coagulation
result expected from the steady-state simulations, but it can at the outlet, which caused the fluctuation. In practical
be explained by the choice of constant inlet mass flow rate applications, the variations in the velocities are in terms
used for the transient simulations contrary to the constant of a fraction of millimeters, so it is unlikely that it would
inlet pressure for the steady simulations. For the lower make a significant difference.
printing speed (1 mms and 5 mms ), the outlet velocity
−1
−1
plateaued at a constant velocity throughout the simulation 3.6. Empirical relationship
across all three nozzles. However, the variation in velocity As cell viability and survivability are closely related to
was greater for the higher printing speed especially in the the shear stress experienced by the cell in the nozzle,
latter half of the simulations. A similar trend was observed predicting the amount of shear stress experienced by
in the case of the outlet pressure across the three nozzles, as the cell can serve to increase cell viability. Nair et al.
[31]
shown in Figure 11B. These variations can be explained by developed an empirical model from experimental data to
looking at the strand profile for the individual simulations.
Due to the higher translational velocity of the bottom predict the degree of survivability of cells as a function
wall, the strands moved further away from the nozzle, of inlet pressure and outlet diameter. Since the dispensing
causing the thread profile to move upwards. For instance, pressure and diameters are independent variables, a
in the case of the conical nozzle at 10 mms (Appendix, complete second-order model with two independent
−1
variables can be expressed as shown in Equation 2 .
[31]
x +
( , y = x )  0  + 11  2 2  3 1 2  4 1 2  5 2 2 (2)
x +
x +
x
xx +
Based on this, we fitted a second-degree curve to the
results of our computational simulations to get an empirical
relationship for ink6040 and estimate the MWSS (z)
experienced in the nozzle as a function of the inlet pressure
(x) used for extrusion and the outlet diameter of the nozzle
(y). Empirical equations can be calculated similarly for
other bioinks and could be used to compare the MWSS
and, in turn, survivability of cells in different bioinks.
The surface plot for the observed relationship is
shown in Figure 12 and the empirical relation is given
Figure 9. Line graph of outlet mass flow rate against inlet pressure by Equations 3, 4, and 5 for the tapered conical, conical,
for the four bioinks. and cylindrical nozzles, respectively. For the cylindrical

A B C

D E

Figure 10. Contour of volume fraction for extruded bioink in conical nozzle with printing speed 5 mms−1 at (A) 2 s, (B) 4 s, (C) 6 s,
(D) 8 s, and (E) 10 s.

International Journal of Bioprinting (2022)–Volume 8, Issue 2 53