Modeling of liquid-gas meniscus dynamics for arbitrary nozzle geometries

An apparatus for determining a damping rate of unforced oscillations of a meniscus, comprising:

• a memory;
• and
• a processing device operatively coupled to the memory, wherein the processing device is to:
• receive input describing a shape of a container, physical parameters of a liquid inside the container, and an equilibrium shape of the meniscus;
• generate a mesh conforming to the shape of the container, wherein the mesh comprises a plurality of nodes, each node representing a point inside the liquid;
• generate a discrete version of a continuous eigenvalue problem based on the mesh;
• compute nodal values for pressure, velocity components, and meniscus surface deformation corresponding to n least-damped late-time oscillation modes of the liquid and computing angular frequency and damping rate of the n least-damped late-time oscillation modes from the discrete version of the continuous eigenvalue problem;
• identify a mode of the n least-damped late-time oscillation modes that has a lowest damping rate;
• compute a liquid relaxation time by inverting the damping rate of the identified mode, wherein the liquid relaxation time is an amount of time expected to pass before the liquid becomes quiescent;
• and
• output the liquid relaxation time.

A non-transitory computer-readable storage medium having instructions stored thereon that, when executed by a processing device for determining a damping rate of unforced oscillations of a meniscus, cause the processing device to:

• receive input describing a shape of a container, physical parameters of a liquid inside the container, and an equilibrium shape of the meniscus;
• generate a mesh conforming to the shape of the container, wherein the mesh comprises a plurality of nodes, each node representing a point inside the liquid;
• generate a discrete version of a continuous eigenvalue problem based on the mesh;
• compute nodal values for pressure, velocity components, and meniscus surface deformation corresponding to n least-damped late-time oscillation modes of the liquid and computing angular frequency and damping rate of the n least-damped late-time oscillation modes from the discrete version of the continuous eigenvalue problem;
• identify a mode of the n least-damped late-time oscillation modes that has a lowest damping rate;
• compute a liquid relaxation time by inverting the damping rate of the identified mode, wherein the liquid relaxation time is an amount of time expected to pass before the liquid becomes quiescent;
• and
• output the liquid relaxation time.

A method of determining the damping rate of unforced oscillations of a meniscus of liquid print material formed in a nozzle of a drop-on-demand printer, comprising:

• receiving input describing a shape of the nozzle, physical parameters of the liquid print material inside the nozzle, and an equilibrium shape of the meniscus;
• generating a mesh conforming to the shape of the nozzle, wherein the mesh comprises a plurality of nodes, each node representing a point inside the liquid print material;
• generating a discrete version of a continuous eigenvalue problem based on the mesh;
• computing nodal values for pressure, velocity components, and meniscus surface deformation corresponding to n least-damped late-time oscillation modes of the liquid print material and computing angular frequency and damping rate of the n least-damped late-time oscillation modes from the discrete version of the continuous eigenvalue problem;
• identifying a mode of the n least-damped late-time oscillation modes that has a lowest damping rate;
• computing a liquid relaxation time by inverting the damping rate of the identified mode, wherein the liquid relaxation time is an amount of time expected to pass before the liquid print material becomes quiescent;
• and
• outputting the liquid relaxation time.