Analytical solutions of upper-convected Maxwell fluid flow with exponential dependence of viscosity on the pressure
Starting from basic models according to [\textit{S. Bair} and \textit{P. Kottke}, ``Pressure-viscosity relationships for elastohydrodynamics'', Tribol. Trans. 46, 289--295 (2003); \textit{P. W. Bridgman}, The physics of high pressure. New York, NY: MacMillan Company (1931); \textit{G. G. Stokes}, ``On the theories of the internal friction of fluids in motion, and motion of elastic solids'', Trans. Camb. Phil. Soc. 8, 287--305 (1845)] and from a model according to [\textit{S. Karra} et al., ``On Maxwell fluids with relaxation time and viscosity depending on the pressure'', Int. J. Non-Linear Mech. 46, No. 1, 819--827 (2011; \url{doi:10.1016/j.ijnonlinmec.2011.02.013})], the authors considered the following model, more specifically, they consider the constitutive equations of incompressible upper-convected Maxwell (UCM) fluids with pressure-dependent viscosity given by the relations \[ T=-pI+S, \quad S+\lambda((\delta S)/(\delta t))=\eta(p)A, \tag{1}\] where \(S\) is the extra-stress tensor, \(\lambda\) is the relaxation time, \(\eta(p)\) is the pressure-dependent viscosity, and the upper convected derivative \((\delta/(\delta t))\) is defined by \[ ((\delta S)/(\delta t))=((dS)/(dt))-LS-SL^{T}. \] The tensor \(L\) is the gradient of the velocity vector \(v\), and the Laplace multiplier \(p\) is not the same as the mean normal stress, as shown by Karra et al. [loc. cit.]. They will study some unsteady motions of incompressible UCM fluids whose pressure-dependent viscosity \(\eta(p)\) is of the exponential form \[ \eta(p)=\mu e^{\alpha(p-p_0)}. \] If \(\alpha\to 0\) in this relation, then \(\eta(p)\to\mu\), and Equations (1) reduce to the constitutive equations of ordinary imcompressible UCM fluids. Exact and simple expressions for the stationary solutions corresponding to two oscillatory motions of incompressible upper-convected Maxwell fluids with exponential dependence of viscosity on the pressure between parallel plates have been established using suitable changes of the spatial variable and the unknown function and the Laplace transform technique. The solutions that have been obtained satisfy the boundary conditions, and governing equations are independent of the initial conditions. The convergence of starting solutions to the corresponding stationary components that has been graphically proved confirms the correctness of obtained results. The oscillation amplitude is an increasing function with respect to the dimensionless pressure-viscosity coefficient and the Weissenberg number We. Some examples of applications and numerical simulations are given at the end of this work.
scientific article; zbMATH DE number 7436719
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