Systematic analysis of the extrusion process in 3D bioprinting is mandatory for process optimization concerning production speed, shape fidelity of the 3D construct and cell viability. In this study, we applied numerical and analytical modeling to describe the fluid flow inside the printing head based on a Herschel–Bulkley model. The presented analytical calculation method nicely reproduces the results of Computational Fluid Dynamics simulation concerning pressure drop over the printing head and maximal shear parameters at the outlet. An approach with dimensionless flow parameter enables the user to adapt rheological characteristics of a bioink, the printing pressure and needle diameter with regard to processing time, shear sensitivity of the integrated cells, shape fidelity and strand dimension. Bioinks consist of a blend of polymers and cells, which lead to a complex fluid behavior. In the present study, a bioink containing alginate, methylcellulose and agarose (AMA) was used as experimental model to compare the calculated with the experimental pressure gradient. With cultures of an immortalized human mesenchymal stem cell line and plant cells (basil) it was tested how cells influence the flow and how mechanical forces inside the printing needle affect cell viability. Influences on both sides increased with cell (aggregation) size as well as a less spherical shape. This study contributes to a systematic description of the extrusion-based bioprinting process and introduces a general strategy for process design, transferable to other bioinks.