, Volume 54, Issue 1–2, pp 123–134 | Cite as

Viscous flow induced by plates moving in opposite directions: a finite-breadth case

  • Chi-Min LiuEmail author


Ignited by and based on the studies (Liu in Math Probl Eng 2008:754262, 2008; J Eng Math 89(1):1–11, 2014) on viscous flows generated by two infinite-breadth plates moving in opposite directions, which are also named as extended Stokes’ problems, a finite-breadth case of Newtonian flow is examined in present study. Solutions of both cases of infinite-depth and finite-depth flows are separately derived. Main mathematical methods including integral transforms in time and spatial domains are applied to acquire the exact solution of flow velocity. Based on the derived solutions, evolution of velocity profiles in time as well as in space, and velocity gradient are examined. Solutions derived in past studies for the infinite-breadth cases are verified to be limiting cases of present results. Present study not only provides mathematical methods for analyzing viscous flow driven by moving boundaries, but also can be applied to examine many practical applications such as flows in mechanical manufacturing, flows induced by faulting, and even heat-transfer problems.


Extended Stokes’ problem Finite-breadth plates Integral transform 



The author is indebted to the reviewer’s comments and detailed suggestions to the derivation and analysis. Financial support from Ministry of Science and Technology of Taiwan with the Grant MOST 106-2221-E-270-002-MY2 is also acknowledged.


  1. 1.
    Liu CM (2008) Complete solutions to extended Stokes’ problems. Math Probl Eng 2008:754262MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Liu CM (2014) Extended Stokes’ problems for Maxwell fluids. J Eng Math 89(1):1–11MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Stokes GG (1851) On the effect of the internal friction of fluids on the motion of pendulums. Trans Camb Philos Soc 9:8–106ADSGoogle Scholar
  4. 4.
    Panton R (1968) The transient for Stokes’ oscillating plate: a solution in terms of tabulated functions. J Fluid Mech 31:819–825ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Erdogan ME (2000) A note on an unsteady flow of a viscous fluid due to an oscillating plane wall. Int J Nonlinear Mech 35:1–6MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Liu CM, Liu IC (2006) A note on the transient solution of Stokes’ second problem with arbitrary initial phase. J Mech 22:349–354CrossRefGoogle Scholar
  7. 7.
    Zeng Y, Weinbaum S (1995) Stokes problems for moving half-planes. J Fluid Mech 287:59–74ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Riccardi G (2011) Remarks on the solution of extended Stokes’ problems. Int J Nonlinear Mech 46:958–970CrossRefGoogle Scholar
  9. 9.
    Oberhettinger F, Badii L (1973) Tables of Laplace transforms. Springer, BerlinCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Division of Mathematics, General Education CenterChienkuo Technology UniversityChanghua CityTaiwan

Personalised recommendations