Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow
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APPLIED MATHEMATICS AND MECHANICS (ENGLISH EDITION) https://doi.org/10.1007/s10483-020-2663-7
Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow∗ Chen YIN1,† , Chunwu WANG1 ,
Shaowei WANG2
1. College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; 2. Department of Engineering Mechanics, School of Civil Engineering, Shandong University, Jinan 250061, China (Received Jun. 9, 2020 / Revised Jul. 19, 2020)
Abstract The thermal convection of a Jeffreys fluid subjected to a plane Poiseuille flow in a fluid-porous system composed of a fluid layer and a porous layer is studied in the paper. A linear stability analysis and a Chebyshev τ -QZ algorithm are employed to solve the thermal mixed convection. Unlike the case in a single layer, the neutral curves of the two-layer system may be bi-modal in the proper depth ratio of the two layers. We find that the longitudinal rolls (LRs) only depend on the depth ratio. With the existence of the shear flow, the effects of the depth ratio, the Reynolds number, the Prandtl number, the stress relaxation, and strain retardation times on the transverse rolls (TRs) are also studied. Additionally, the thermal instability of the viscoelastic fluid is found to be more unstable than that of the Newtonian fluid in a two-layer system. In contrast to the case for Newtonian fluids, the TRs rather than the LRs may be the preferred mode for the viscoelastic fluids in some cases. Key words
viscoelastic fluid, thermal convection, Poiseuille flow, fluid-porous system
Chinese Library Classification O35 2010 Mathematics Subject Classification
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76A10, 76Exx, 76R10
Introduction
The onset of thermal convection in a two-layer system comprising a fluid layer overlying a porous medium has received much attention recently due to its crucial significant industrial and geophysical applications[1–4] , such as solidification of alloys, hydrothermal synthesis in the growth of crystalline materials, mixing in ice-covered lakes, and a flow of oil in underground reservoirs. In the past decades, a large number of investigations on the thermal instability in the ∗ Citation: YIN, C., WANG, C. W., and WANG, S. W. Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow. Applied Mathematics and Mechanics (English Edition) (2020) https://doi.org/10.1007/s10483-020-2663-7 † Corresponding author, E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos. 11702135, 11271188, and 11672164), the Natural Science Foundation of Jiangsu Province of China (No. BK20170775), the China Postdoctoral Science Foundation (No. 2016M601798), and the Jiangsu Planned Project for Postdoctoral Research Funds (No. 1601169B) ©The Author(s) 2020
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Chen YIN, Chunwu WANG, and Shaowei WANG
fluid-porous system have been published[5–14] . Nield[5] first constructed a two-domain model to investigate the thermal instability in a fluid-porous system heated from below. A simple case with a constant fl
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