Abstract
Based on the thermovibrational convection equations, we have investigated the structure of the averaged plane-parallel convective flow in a plane vertical layer of Williamson fluid executing high-frequency linearly polarized vibrations along the layer. We show that as the vibrations are intensified, the nonlinear viscous properties of a pseudoplastic fluid cease to affect the structure and intensity of its main flow, and it becomes similar to a flow of ordinary Newtonian fluid. The linear problem of stability of an averaged plane-parallel flow of pseudoplastic Williamson fluid has been formulated and solved for the case of longitudinal high-frequency linearly polarized vibrations for small periodic perturbations along the layer. Numerical calculations have shown that, as in a Newtonian fluid, the monotonic hydrodynamic perturbations are most dangerous at low Prandtl numbers. As the Prandtl number increases, the thermal instability modes begin to exert an undesirable effect. An enhancement of pseudoplastic fluid properties leads to destabilization of the main flow for both types of perturbations. Similarly to a Newtonian fluid, an additional vibrational instability mode to which small Grashof numbers correspond appears in the presence of vibrations. The influence of this vibrational mode on the stability of the main flow is determined by the vibration frequency and the temperature gradient. An intensification of the vibrations destabilizes the flow for all of the investigated instability modes. For a given set of rheological parameters of the Williamson model, there are critical values of the modified and vibrational Grashof numbers at which the averaged flow completely loses its stability with respect to the types of perturbations under consideration. Absolute destabilization of the main flow in a pseudoplastic fluid occurs at higher values of the vibrational Grashof number than those in a Newtonian fluid.
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References
Gershuni, G.Z. and Zhukhovitskii, E.M., On two types of instability of convective motion between parallel vertical planes, Izv. Vyssh. Uchebn. Zaved., Ser. Fiz., 1958, no. 4, pp. 43–47.
Rudakov, R.N., On small perturbations of convective motion between vertical parallel planes, J. Appl. Math. Mech., 1967, vol. 30, no. 2, pp. 439–445. https://doi.org/10.1016/0021-8928(67)90192-X
Birikh, R.V., Gershuni, G.Z., Zhukhovitskii, E.M., and Rudakov, R.N., On oscillatory instability of plane-parallel convective motion in a vertical channel, J. Appl. Math. Mech., 1972, vol. 36, no. 4, pp. 707–710. https://doi.org/10.1016/0021-8928(72)90121-9
Gershuni, G.Z., Zhukhovitskii, E.M., and Nepomnyashchy, A.A., Ustoichivost’ konvektivnykh techenii (Stability of Convective Flows), Moscow: Nauka, 1989.
Perminov, A.V. and Lyubimova, T.P., Stability of the stationary plane-parallel flow of pseudoplastic fluids in a plane vertical layer, Vychisl. Mekh. Splosh. Sred, 2014, vol. 7, no. 3, pp. 270–278. https://doi.org/10.7242/1999-6691/2014.7.3.27
Lyubimova, T.P. and Perminov, A.V., Stability of stationary plane-parallel flow of viscoplastic fluid between two differentially heated vertical plates, J. Non-Newton. Fluid, 2015, vol. 224, pp. 51–60. https://doi.org/10.1016/j.jnnfm.2015.08.003
Lyubimova, T.P., Convection of non-Newtonian liquids in closed cavities heated from below, Fluid Dyn., 1974, vol. 9, no. 2, pp. 319–322. https://doi.org/10.1007/BF01092673
Semakin, I.G., Hydrodynamic stability of convective flow of a non-Newtonian fluid in a vertical layer, J. Eng. Phys., 1977, vol. 32, no. 6, pp. 690–693. https://doi.org/10.1007/BF00862576
Semakin, I.G., Vibrational instability of stationary convection of a non-Newtonian fluid, J. Eng. Phys., 1978, vol. 35, no. 2, pp. 969–972. https://doi.org/10.1007/BF00860223
Subbotin, E.V., Trufanova, N.M., and Shcherbinin, A.G., Numerical study of polymer fluid flows in the channel of a screw extruder using one-and two-dimensional models, Vychisl. Mekh. Splosh. Sred, 2012, vol. 5, no. 4, pp. 452–460. https://doi.org/10.7242/1999-6691/2012.5.4.53
Lyubimova, T.P., Numerical investigation of convection in a viscoplastic liquid in a closed region, Fluid Dyn., 1977, vol. 12, no. 1, pp. 1–5. https://doi.org/10.1007/BF01074616
Lyubimova, T.P., Convective motions of a viscoplastic fluid in a rectangular region, Fluid Dyn., 1979, vol. 14, no. 5, pp. 747–750. https://doi.org/10.1007/BF01409817
Lyubimova, T.P., On steady solutions of the equations of convection of visco-plastic fluid heated from below with temperature-dependent rheological parameters, Izv. Akad. Nauk BSSR, Fiz.-Mat. Nauki, 1986, no. 1, pp. 91–96.
Perminov, A.V. and Shulepova, E.V., Influence of high-frequency vibrations on convective motion of non-newtonian fluid, Nauch.-Tekh. Vedom. SPbGPU, Fiz.-Mat. Nauki, 2011, vol. 3, no. 129, pp. 169–175.
Gershuni, G.Z. and Zhukhovitskii, E.M., Free thermal convection in a vibration field under conditions of weightlessness, Dokl. Akad. Nauk SSSR, 1979, vol. 249, no. 3, pp. 580–584.
Gershuni, G.Z. and Zhukhovitskii, E.M., Convective instability of a fluid in a vibration field under conditions of weightlessness, Fluid Dyn., 1981, vol. 16, no. 4, pp. 498–504. https://doi.org/10.1007/BF01094590
Zen’kovskaya, S.M. and Simonenko, I.B., Effect of high frequency vibration on convection initiation, Fluid Dyn., 1966, vol. 1, no. 5, pp. 35–37. https://doi.org/10.1007/BF01022147
Gershuni, G.Z. and Lyubimov, D.V., Thermal Vibrational Convection, New York: Wiley, 1998.
Sharifulin, A.N., Stability of convective motion in a vertical layer in the presence of longitudinal vibrations, Fluid Dyn., 1983, vol. 18, no. 2, pp. 335–338. https://doi.org/10.1007/BF01091136
Sharifulin, A.N., Wave instability of free-convective motion in a vibration field, in Nestatsionarnye protsessy v zhidkostyakh i tverdykh telakh (Unsteady Processes in Liquids and Solids), Sverdlovsk: Ural. Nauch. Tsentr AN SSSR, 1983, pp. 58–62.
Perminov, A.V., Stability of the rigid state of a generalized Newtonian fluid, Fluid Dyn., 2014, vol. 49, no. 2, pp. 140–148. https://doi.org/10.1134/S0015462814020033
Tetel’min, V.V. and Yazev, V.A., Reologiya nefti (Oil Rheology), Moscow: Granitsa, 2009.
Lyubimov, D.V., Lyubimova, T.P., and Morozov, V.A., Software package for numerical investigation of linear stability of multi-dimensional flows, Bull. Perm Univ., Inform. Syst. Technol., 2001, no. 5, pp. 74–81.
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Original Russian Text © A.V. Perminov, T.P. Lyubimova, 2018, published in Vychislitel’naya Mekhanika Sploshnykh Sred, 2017, Vol. 10, No. 1, pp. 78–89.
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Perminov, A.V., Lyubimova, T.P. Stability of Thermovibrational Convection of a Pseudoplastic Fluid in a Plane Vertical Layer. J Appl Mech Tech Phy 59, 1167–1178 (2018). https://doi.org/10.1134/S0021894418070118
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DOI: https://doi.org/10.1134/S0021894418070118