## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### Pressure distribution in gas microbearing modeled with fractional derivatives for all rarefaction degrees

**ABSTRACT**

Velocity slip boundary condition for all Knudsen number values in microbearing gas flow is modeled by fractional derivative. For this purpose, a variant of Caputo derivative with the variable order α which depends on the local value of Knudsen number is applied. Such a universal boundary condition is implemented in the solving procedures of continuity and momentum equations, which leads to the general corrected Reynolds lubrication equation for all rarefaction degrees. An appropriate transformation of the variables enabled obtaining an analytical solution for mass flow rate and pressure distribution in the microbearing. The presented solution is in an excellent agreement with the solutions based on the kinetic theory for all regimes: continuum, slip, transition, and free molecular flow.

**KEYWORDS**

PAPER SUBMITTED: 2024-04-14

PAPER REVISED: 2024-06-01

PAPER ACCEPTED: 2024-06-09

PUBLISHED ONLINE: 2024-06-22

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