On the Modified Stokes Second Problem for Maxwell Fluids with Linear Dependence of Viscosity on the Pressure
Open Access
- 24 January 2022
- Vol. 14 (2), 219
- https://doi.org/10.3390/sym14020219
Abstract
The modified Stokes second problem for incompressible upper-convected Maxwell (UCM) fluids with linear dependence of viscosity on the pressure is analytically and numerically investigated. The fluid motion, between infinite horizontal parallel plates, is generated by the lower wall, which oscillates in its plane. The movement region of the fluid is symmetric with respect to the median plane, but its motion is asymmetric due to the boundary conditions. Closed-form expressions are found for the steady-state components of start-up solutions for non-dimensional velocity and the corresponding non-trivial shear and normal stresses. Similar solutions for the simple Couette flow are obtained as limiting cases of the solutions corresponding to the motion due to cosine oscillations of the wall. For validation, it is graphically proved that the start-up solutions (numerical solutions) converge to their steady-state components. Solutions for motions of ordinary incompressible UCM fluids performing the same motions are obtained as special cases of present results using asymptotic approximations of standard Bessel functions. The time needed to reach the permanent or steady state is also determined. This time is higher for motions of ordinary fluids, compared with motions of liquids with pressure-dependent viscosity. The impact of physical parameters on the fluid motion and the spatial–temporal distribution of start-up solutions are graphically investigated and discussed. Ordinary fluids move slower than fluids with pressure-dependent viscosity.Keywords
This publication has 27 references indexed in Scilit:
- Role of pressure dependent viscosity in measurements with falling cylinder viscometerInternational Journal of Non-Linear Mechanics, 2012
- Flow of fluids with pressure- and shear-dependent viscosity down an inclined planeJournal of Fluid Mechanics, 2012
- On Maxwell fluids with relaxation time and viscosity depending on the pressureInternational Journal of Non-Linear Mechanics, 2011
- Revisiting Stokes first and second problems for fluids with pressure-dependent viscositiesInternational Journal of Engineering Science, 2010
- A semi-inverse problem of flows of fluids with pressure-dependent viscositiesInverse Problems in Science and Engineering, 2008
- Parallel shear flows of fluids with a pressure-dependent viscosityJournal of Non-Newtonian Fluid Mechanics, 2003
- Pressure-Viscosity Relationships for ElastohydrodynamicsTribology Transactions, 2003
- On the unsteady unidirectional flows generated by impulsive motion of a boundary or sudden application of a pressure gradientInternational Journal of Non-Linear Mechanics, 2002
- Simple flows of fluids with pressure–dependent viscositiesProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2001
- Note on the Dependence of Viscosity on Pressure and TemperatureProceedings of the American Academy of Arts and Sciences, 1891