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