Non-Newtonian rivulet flows on an inclined planar surface applying the 2nd Stokes problem

doi:10.1007/s10483-026-3336-7

摘要/Abstract

Abstract:

The newly formulated non-Newtonian rivulet flows streaming down an inclined planar surface, with additional periodic perturbations arising from the application of the 2nd Stokes problem to the investigation of rivulet dynamics, are demonstrated in the current research. Hereby, the 2nd Stokes problem assumes that the surface, with a thin shared layer of the fluid on it, oscillates in a harmonic manner along the $x$ -axis of the rivulet flow, which coincides with the main flow direction streaming down the underlying surface. We obtain the exact extension of the rivulet flow family, clarifying the structure of the pressure field, which fully absorbs the arising perturbation. The profile of the velocity field is assumed to be Gaussian-type with a non-zero level of plasticity. Hence, the absolutely non-Newtonian case of the viscoplastic flow solution, which satisfies the motion and continuity equations, is considered (with particular cases of exact solutions for pressure). The perturbed governing equations of motion for rivulet flows then result in the Riccati-type ordinary differential equation (ODE), describing the dynamics of the coordinate $x (t) .$ The approximated schematic dynamics are presented in graphical plots.

Key words: rivulet flow, non-Newtonian fluid, creeping viscoplastic flow, 1st/2nd Stokes problem

中图分类号:

O357

. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(1): 153-164.

S. V. ERSHKOV, E. S. BARANOVSKII, A. V. YUDIN. Non-Newtonian rivulet flows on an inclined planar surface applying the 2nd Stokes problem[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(1): 153-164.

图/表 3

参考文献 72

[1]	SINZ, D. K. N., HANYAK, M., ZEEGERS, J. C. H., and DARHUBER, A. A. Insoluble surfactant spreading along thin liquid films confined by chemical surface patterns. Physical Chemistry Chemical Physics, 13(20), 9768–9777 (2011)
[2]	DARHUBER, A. A., CHEN, J. Z., DAVIS, J. M., and TROIAN, S. M. A study of mixing in thermocapillary flows on micropatterned surfaces. Philosophical Transactions of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 362(1818), 1037–1058 (2004)
[3]	FRAYSSE, N. and HOMSY, G. M. An experimental study of rivulet instabilities in centrifugal spin coating of viscous Newtonian and non-Newtonian fluids. Physics of Fluids, 6(4), 1491–1504 (1994)
[4]	KABOV, O. A. Heat transfer from a small heater to a falling liquid film. Heat Transfer Research, 27(1), 221–226 (1996)
[5]	KABOV, O. A. and CHINNOV, E. A. Heat transfer from a local heat source to a subcooled falling liquid film evaporating in a vapor-gas medium. Journal of Engineering Thermophysics, 7(1-2), 1–34 (1997)
[6]	SLADE, D., VEREMIEIEV, S., LEE, Y. C., and GASKELL, P. H. Gravity-driven thin film flow: the influence of topography and surface tension gradient on rivulet formation. Chemical Engineering and Processing, 68, 7–12 (2013)
[7]	SCHEID, B., ORON, A., COLINET, P., THIELE, U., and LEGROS, J. C. Nonlinear evolution of nonuniformly heated falling liquid films. Physics of Fluids, 14(12), 4130–4151 (2002)
[8]	CHO, H. C. and CHOU, F. C. Rivulet instability with effect of Coriolis force. Journal of Mechanics, 22(3), 221–227 (2006)
[9]	BRUN, K. and KURZ, R. Analysis of secondary flows in centrifugal impellers. International Journal of Rotating Machinery, 2005(1), 45–52 (2005)
[10]	KALLIADASIS, S., KIYASHKO, A., and DEMEKHIN, E. A. Marangoni instability of a thin liquid film heated from below by a local heat source. Journal of Fluid Mechanics, 475, 377–408 (2003)
[11]	HOOKE, R. L. B. Principles of Glacier Mechanics, Cambridge University Press, Cambridge (2005)
[12]	PATERSON, W. S. B. The Physics of Glaciers, Pergamon Press, Oxford (1994)
[13]	WU, Z., ZHANG, H. W., LIU, S. Y., REN, D., BAI, X. J., XUN, Z., and MA, Z. T. Fluctuation analysis in the dynamic characteristics of continental glacier based on full-Stokes model. Scientific Reports, 9, 20245 (2019)
[14]	SLADE, D. R. J. Gravity-Driven Thin Liquid Films: Rivulets and Flow Dynamics, Ph. D. dissertation, University of Leeds (2013)
[15]	TOWELL, G. D. and ROTHFELD, L. B. Hydrodynamics of rivulet flow. AIChE Journal, 12, 972–980 (1966)
[16]	KLIMOV, D. M., PETROV, A. G., and GEORGIEVSKY, D. V. Viscous-Plastic Flows: Dynamical Chaos, Stability, and Confusion (in Russian), Nauka, Moscow (2005)
[17]	ERSHKOV, S. V., PROSVIRYAKOV, E. Y., and LESHCHENKO, D. Marangoni-type of nonstationary rivulet-flows on inclined surface. International Journal of Non-Linear Mechanics, 147, 104250 (2022)
[18]	ERSHKOV, S. V. and LESHCHENKO, D. D. Non-Newtonian pressure-governed rivulet flows on inclined surface. Mathematics, 12, 779 (2024)
[19]	ERSHKOV, S. V. and LESHCHENKO, D. Note on semi-analytical nonstationary solution for the rivulet flows of non-Newtonian fluids. Mathematical Methods in the Applied Sciences, 45(12), 7394–7403 (2022)
[20]	ERSHKOV, S. V., BARANOVSKII, E. S., PROSVIRYAKOV, E. Y., and YUDIN, A. V. Non-Newtonian rivulet-flows on unsteady heated plane surface. International Journal of Non-Linear Mechanics, 170, 104984 (2025)
[21]	ERSHKOV, S. and LESHCHENKO, D. Revisiting glacier dynamics for stationary approximation of plane-parallel creeping flow. Mathematical Modelling of Engineering Problems, 8(5), 721–726 (2021)
[22]	ERSHKOV, S. V. Non-stationary creeping flows for incompressible 3D Navier-Stokes equations. European Journal of Mechanics: B, 61(1), 154–159 (2017)
[23]	ERSHKOV, S. V. Non-stationary Riccati-type flows for incompressible 3D Navier-Stokes equations. Computers & Mathematics with Applications, 71(7), 1392–1404 (2016)
[24]	ERSHKOV, S. V. A procedure for the construction of non-stationary Riccati-type flows for incompressible 3D Navier-Stokes equations. Rendiconti del Circolo Matematico di Palermo, 65(1), 73–85 (2016)
[25]	PUKHNACHEV, V. V. and ZHURAVLEVA, E. N. Viscous flows with flat free boundaries. The European Physical Journal Plus, 135(7), 554 (2020)
[26]	STOKES, G. G. On the effect of the internal friction of fluid on the motion of pendulums. Transactions of the Cambridge Philosophical Society, 9, 8–106 (1851)
[27]	FARINA, A., FUSI, L., MIKELIĆ, A., SACCOMANDI, G., SEQUEIRA, A., and TORO, E. F. Non-Newtonian Fluid Mechanics and Complex Flows, Springer International Publishing, Switzerland (2018)
[28]	BARANOVSKII, E. S. Optimal boundary control of nonlinear-viscous fluid flows. Sbornik: Mathematics, 211(4), 505–520 (2020)
[29]	BARANOVSKII, E. S. and ARTEMOV, M. A. Model for aqueous polymer solutions with damping term: solvability and vanishing relaxation limit. Polymers, 14, 3789 (2022)
[30]	ASTARITA, G. and MARUCCI, G. Principles of Non-Newtonian Fluid Hydromchanics, McGraw Hill, Maidenhead (1974)
[31]	PAVLOVSKII, V. A. On the theoretical description of weak water solutions of polymers. Doklady Akademii Nauk SSSR, 200, 809–812 (1971)
[32]	BURMISTROVA, O. A., MELESHKO, S. V., and PUKHNACHEV, V. V. Exact solutions of boundary layer equations in polymer solutions. Symmetry, 13, 2101 (2021)
[33]	LADYZHENSKAYA, O. A. On the global unique solvability of some two-dimensional problems for the water solutions of polymers. Journal of Mathematical Sciences, 99, 888–897 (2000)
[34]	BARANOVSKII, E. S. Global solutions for a model of polymeric flows with wall slip. Mathematical Methods in the Applied Sciences, 40, 5035–5043 (2017)
[35]	BARANOVSKII, E. S. and ARTEMOV, M. A. Optimal control for a nonlocal model of non-Newtonian fluid flows. Mathematics, 9, 275 (2021)
[36]	PUKHNACHOV, V. V. On a problem of viscous strip deformation with a free boundary. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics, 328(4), 357–362 (1999)
[37]	BARANOVSKII, E. S. Flows of a polymer fluid in domain with impermeable boundaries. Computational Mathematics and Mathematical Physics, 54(10), 1589–1596 (2014)
[38]	BARANOVSKII, E. S. An optimal boundary control problem for the motion equations of polymer solutions. Siberian Advances in Mathematics, 24(3), 159–168 (2014)
[39]	WILSON, S. K., DUFFY, B. R., and ROSS, A. B. On the gravity-driven draining of a rivulet of a viscoplastic material down a slowly varying substrate. Physics of Fluids, 14(2), 555–571 (2002)
[40]	WILSON, S. K., DUFFY, B. R., and HUNT, R. A slender rivulet of a power-law fluid driven by either gravity or a constant shear stress at the free surface. The Quarterly Journal of Mechanics and Applied Mathematics, 55(3), 385–408 (2002)
[41]	YATIM, Y. M., WILSON, S. K., and DUFFY, B. R. Unsteady gravity-driven slender rivulets of a power-law fluid. Journal of Non-Newtonian Fluid Mechanics, 165(21-22), 1423–1430 (2010)
[42]	YATIM, Y. M., DUFFY, B. R., WILSON, S. K., and HUNT, R. Similarity solutions for unsteady gravity-driven slender rivulets. The Quarterly Journal of Mechanics and Applied Mathematics, 64(4), 455–480 (2011)
[43]	YATIM, Y. M., DUFFY, B. R., and WILSON, S. K. Similarity solutions for unsteady shear-stress-driven flow of Newtonian and power-law fluids: slender rivulets and dry patches. Journal of Engineering Mathematics, 73(1), 53–69 (2012)
[44]	PATERSON, C., WILSON, S. K., and DUFFY, B. R. Pinning, de-pinning and re-pinning of a slowly varying rivulet. European Journal of Mechanics: B, 41, 94–108 (2013)
[45]	AL MUKAHAL, F. H. H., DUFFY, B. R., and WILSON, S. K. A rivulet of a power-law fluid with constant contact angle draining down a slowly varying substrate. Physics of Fluids, 27(5), 052101 (2015)
[46]	AL MUKAHAL, F. H. H., WILSON, S. K., and DUFFY, B. R. A rivulet of a power-law fluid with constant width draining down a slowly varying substrate. Journal of Non-Newtonian Fluid Mechanics, 224, 30–39 (2015)
[47]	AL MUKAHAL, F. H. H., DUFFY, B. R., and WILSON, S. K. Advection and Taylor-Aris dispersion in rivulet flow. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473, 20170524 (2017)
[48]	AL MUKAHAL, F. H. H., DUFFY, B. R., and WILSON, S. K. Rivulet flow of generalized Newtonian fluids. Physical Review Fluids, 3(8), 083302 (2018)
[49]	ALSHAIKHI, A. S., WILSON, S. K., and DUFFY, B. R. Rivulet flow over and through a permeable membrane. Physical Review Fluids, 6(10), 104003 (2021)
[50]	WANG, M., ZHAO, J. H., and DUAN, R. Q. Rivulet formulation in the flow of film down a uniformly heated vertical substrate. Engineering Applications of Computational Fluid Mechanics, 13(1), 396–416 (2019)
[51]	FEDOTKIN, I. M., MEL’NICHUK, G. A., KOVAL’, F. F., and KLIMKIN, E. V. Hydrodynamics of rivulet flow on a vertical surface. Journal of Engineering Physics, 46(1), 9–14 (1984)
[52]	SCHMUKI, P. and LASO, M. On the stability of rivulet flow. Journal of Fluid Mechanics, 215, 125–143 (1990)
[53]	BENTWICH, M., GLASSER, D., KERN, J., and WILLIAMS, D. Analysis of rectilinear rivulet flow. AIChE Journal, 22(4), 772–779 (1976)
[54]	MYERS, T. G., LIANG, H. X., and WETTON, B. The stability and flow of a rivulet driven by interfacial shear and gravity. International Journal of Non-Linear Mechanics, 39(8), 1239–1249 (2004)
[55]	DRENCKHAN, W., GATZ, S., and WEAIRE, D. Wave patterns of a rivulet of surfactant solution in a Hele-Shaw cell. Physics of Fluids, 16(8), 3115–3121 (2004)
[56]	WEILAND, R. H. and DAVIS, S. H. Moving contact lines and rivulet instabilities, part 2: long waves on flat rivulets. Journal of Fluid Mechanics, 107(1), 261 (2006)
[57]	YOUNG, G. W. and DAVIS, S. H. Rivulet instabilities. Journal of Fluid Mechanics, 176(1), 1–31 (1987)
[58]	DONIEC, A. Laminar flow of a liquid rivulet down a vertical solid surface. The Canadian Journal of Chemical Engineering, 69(1), 198–202 (2009)
[59]	PATERSON, C., WILSON, S. K., and DUFFY, B. R. Pinning, de-pinning and re-pinning of a slowly varying rivulet. European Journal of Mechanics: B, 41, 94–108 (2013)
[60]	SINGH, R. K., GALVIN, J. E., and SUN, X. Three-dimensional simulation of rivulet and film flows over an inclined plate: effects of solvent properties and contact angle. Chemical Engineering Science, 142, 244–257 (2016)
[61]	AKTERSHEV, S., ALEKSEENKO, S., and BOBYLEV, A. Waves in a rivulet falling down an inclined cylinder. AIChE Journal, 67(1), e17002 (2020)
[62]	BATISHCHEV, V. A., KUZNETSOV, V. V., and PUKHNACHEV, V. V. Marangoni boundary layers. Progress in Aerospace Sciences, 26(4), 353–370 (1989)
[63]	BIRIKH, R. V. Thermocapillary convection in a horizontal layer of liquid. Journal of Applied Mechanics and Technical Physics, 7(3), 43–44 (1966)
[64]	PUKHNACHEV, V. V. Non-stationary analogues of the Birikh solution (in Russian). Izvestiya Altaiskogo Gosudarstvennogo Universiteta, 1-2, 62–69 (2011)
[65]	PARDO, R., HERRERO, H., and HOYAS, S. Theoretical study of a Bénard-Marangoni problem. Journal of Mathematical Analysis and Applications, 376, 231–246 (2011)
[66]	TUKMAKOV, D. A. and TUKMAKOVA, N. A. Numerical study of the effect of droplet coagulation on the dynamics of a two-fraction aerosol in an acoustic resonator. Russian Technological Journal, 9(2), 96–104 (2021)
[67]	JOSEPH, S. P. New classes of periodic and non-periodic exact solutions for Newtonian and non-Newtonian fluid flows. International Journal of Engineering Science, 180, 103740 (2022)
[68]	HAYAT, T., ASHRAF, B., SHEHZAD, S. A., and ABOUELMAGD, E. Three-dimensional flow of Eyring Powell nanofluid over an exponentially stretching sheet. International Journal of Numerical Methods for Heat & Fluid Flow, 25(3), 593–616 (2015)
[69]	FETECAU, C., VIERU, D., and FETECAU, C. A note on the second problem of Stokes for Newtonian fluids. International Journal of Non-Linear Mechanics, 43, 451–457 (2008)
[70]	FETECAU, C. and FETECAU, C. The first problem of Stokes for an Oldroyd-B fluid. International Journal of Non-Linear Mechanics, 38, 1539–1544 (2003)
[71]	NAZAR, M., FETECAU, C., VIERU, D., and FETECAU, C. New exact solutions corresponding to the second problem of Stokes for second grade fluids. Nonlinear Analysis: Real World Applications, 11, 584–591 (2010)
[72]	PUKHNACHEV, V. V. and FROLOVSKAYA, O. A. Influence of vibration on the onset of convection in second grade fluid. Computational Mathematics and Mathematical Physics, 65, 1142–1149 (2025)