Fig 1.
Viscosity and flow profile interpolation.
(a) The viscosity-shear rate relationship of an arbitrary shear thinning fluid obtained e. g. from a rheometer measurement is interpolated by power-law intervals. The bounds of the intervals (vertical lines) are given by the intermediate shear rates, . By using a large number of intervals, any arbitrary viscosity-shear rate relationship can be approximated as closely as desired. (b) A long cylinder with uniaxial, stationary flow is used as a model for the flow of a bioink through a printer nozzle. The flow profile is split into radial intervals Ri determined implicitly via the intermediate shear rates
.
Fig 2.
Validation with a mathematical solution.
Flow profiles for the simplified CY model: the global mathematical solution and the prediction by our algorithm agree very well. The parameters are N = 1000, η0 = 100 Pa s, K = 1.0 s and G = −1.95 × 106 Pa m−1.
Fig 3.
Validation with Lattice Boltzmann calculations.
Flow profiles of a chitosan hydrogel with a pressure gradient of G = −7.0 × 107 Pa m−1 and N = 1000.
Fig 4.
Validation with experimental flow measurements.
Experimental measurement of the flow profile of a 2% alginate solution in a 190 μm × 190 μm microchannel. (a) Example micrograph of the bead tracking procedure. The velocity with respect to the lateral position is obtained as the length (yellow circles and labels) of the streaks divided by the exposure time. (b) The measured flow profile is in excellent agreement with our Lattice Boltzmann simulations.
Fig 5.
Experimental capillary rheometer setup.
Schematic of the experimental setup using a custom-made bioprinter as capillary rheometer: the bioink is driven through a syringe under defined pressure, and the flow rate of the extruded alginate is measured with a precision scale.
Fig 6.
Comparison between capillary rheometer and cone-plate rheometer results.
(a) Viscosity versus shear rate for a 2.5% alginate solution as measured with a cone-plate rheometer (data from 4 independent measurements, red squares) shows the pronounced shear thinning of a CY fluid that is well characterized by 3 fit parameters (black line) according to Eq (10). This shear thinning behavior can be predicted (blue line) from an independent capillary rheometry experiment using our Python tool. (b) Flow rate versus pressure relationship of the alginate solution when extruded through a 28 mm long 551 μm diameter capillary (red squares). This relationship follows our numerical solution using 3 fit parameters (blue line). The flow rate versus pressure relationship can similarly be predicted (black line) from the viscosity values obtained from an independent cone-plate rheometer experiment shown in the upper panel, showing significant deviations with increasing pressure.
Fig 7.
Alginate flow profile from capillary rheometer and cone-plate rheometer data.
Flow profiles of 2.5% alginate hydrogel with a pressure difference of Δp = −105 Pa and N = 150. There is only a slight difference between the flow profiles calculated from the viscosity parameters obtained with a cone-plate rheometer (black line) and our capillary rheometer (blue line).