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Pressure distribution in gas microbearing modeled with fractional derivatives for all rarefaction degrees

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.
PAPER REVISED: 2024-06-01
PAPER ACCEPTED: 2024-06-09
  1. Burgdorfer, A., The influence of the molecular mean free path on the performance of hydrodynamic gas lubricated bearing, J. Basic Eng. Trans., 81 (1959), pp. 94-100
  2. Mitsuya, Y., Modified Reynolds equation for ultra-thin film gas lubrication using 1.5-order slip-flow model and considering surface accommodation coefficient, ASME J. Trib. 115, (1993), pp. 289-294
  3. Hsia, Y., Domoto, G., An experimental investigation of molecular rarefaction effects in gas-lubricated bearings at ultra low clearances, J. Lubr. Technol. 105 (1983), pp. 120-130
  4. Lockerby, D. A., et al., Velocity boundary condition at solid wall in rarefied gas calculation, Phys. Rev. E 70, 017303 (2004)
  5. Barber, R. E., Emerson, D. R., Challenges in modelling gas-phase flow in microchannels: from slip to transition, Heat Transfer Eng. 27 (2006), pp. 3-12
  6. Chen, D., Bogy, D. B., Comparisons of Slip-Corrected Reynolds Lubrication Equations for the Air Bearing Film in the Head-Disk Interface of Hard Disk Drives, Tribol. Lett. 37 (2010), pp. 191-201
  7. Bahukudumbi, P., Beskok, A., A Phenomenological lubrication model for the entire Knudsen Regime, J. Micromech. Microeng. 13 (2003), 6, pp. 873-884
  8. Sun, Y.H., et al., A slip model for gas lubrication based on an effective viscosity concept, J. Eng. Tribol. 217 (2003), pp. 187-195
  9. Struchtrup H., Torrilhon, M., Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids 15 (2003), 9, pp. 2668-2680
  10. Gu, X.J., Emerson, D.R., A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions, J. Comput. Phys. 225 (2007), 1, pp. 263-283
  11. Torrilhon, M., Struchtrup, H., Boundary conditions for regularized 13-moment-equations for micro-channel-flows, J. Comput. Phys. 227 (2008), pp. 1982-2011
  12. Gu, X.J., Emerson, D.R., A high-order moment approach for capturing non-equilibrium phenomena in the transition regime, J Fluid Mech. 636 (2009), pp. 177-216
  13. Yang, Q., et al., Improved modified Reynolds equation for thin-film gas lubrication from an extended slip velocity boundary condition, Microsyst Technol. 22 (2016), pp. 2869-2875
  14. Gu, X.J., et al., A new extended Reynolds equation for gas bearing lubrication based on the method of moments, Microfluidics and Nanofluidics 20:23 (2016)
  15. Alexander, F.J., et al., Direct simulation Monte Carlo for thin-film bearings, Phys. Fluids, 6 (1994), 12, pp. 3854-3860
  16. Liu, N., Ng, E.Y.K., The posture effects of a slider air bearing on its performance with a direct simulation Monte Carlo method, J. Micromech. Microeng. 11 (2001), 5, pp. 463-473
  17. John, B., Damodaran, M., Computation of head-disk interface gap micro flowfields using DSMC and continuum-atomistic hybrid methods, Int. J. Numer. Meth. Fluids, 61 (2009), pp. 1273-1298
  18. Fukui, S., Kaneko, R., Analysis of ultra-thin gas film lubrication based on linearized Boltzmann equation: First report-derivation of a generalized lubrication equation including thermal creep flow, J. Tribol. 110 (1988), pp. 253-262
  19. Fukui, S., Kaneko, R., A database for interpolation of Poiseuille flow rates for high Knudsen number lubrication problems, J. Tribol. 112 (1990), pp. 78-83
  20. Stevanovic, N. D., Djordjevic, V. D., The exact analytical solution for the gas lubricated bearing in the slip and continuum flow regime, Publications de l'Institute mathematique 91 (2012), pp. 83-93
  21. Stevanovic, N. D., Djordjevic, V. D., An exact analytical solution for the second order slip-corrected Reynolds lubrication equation, FME Transactions, 43 (2015), pp. 16-20
  22. Szeri, A. Z., Fluid film lubrication: theory and design, Cambridge University Press, Cambridge, 1998
  23. Aghdam, Y. E., et al., Numerical approach to simulate diffusion model of a fluid-flow in porous media, Thermal Sciense, 25 (2021), 2, pp. 251-261
  24. Gundogdu, H. and Gozukizil, O. F., On the approximate numerical solutions of fractional heat equation with heat source and heat loss, 26 (2022), 5, pp. (3773-3776)
  25. Djordjevic, V., et al., Fractional Derivatives Embody Essential Features of Cell Rheological Behavior, Annals of Biomedical Engineering, 31 (2003), pp. 692-699
  26. Podlubny, I., Fractional Differential Equations, Academic Press, 1999
  27. Holey, H., et al., Height‑Averaged Navier-Stokes Solver for Hydrodynamic Lubrication, Tribol. Lett., 70:36 (2022)