Analytical Solutions to the Unsteady Poiseuille Flow of a Second Grade Fluid with Slip Boundary Conditions

Author:

Baranovskii Evgenii S.1ORCID

Affiliation:

1. Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, 394018 Voronezh, Russia

Abstract

This paper deals with an initial-boundary value problem modeling the unidirectional pressure-driven flow of a second grade fluid in a plane channel with impermeable solid walls. On the channel walls, Navier-type slip boundary conditions are stated. Our aim is to investigate the well-posedness of this problem and obtain its analytical solution under weak regularity requirements on a function describing the velocity distribution at initial time. In order to overcome difficulties related to finding classical solutions, we propose the concept of a generalized solution that is defined as the limit of a uniformly convergent sequence of classical solutions with vanishing perturbations in the initial data. We prove the unique solvability of the problem under consideration in the class of generalized solutions. The main ingredients of our proof are a generalized Abel criterion for uniform convergence of function series and the use of an orthonormal basis consisting of eigenfunctions of the related Sturm–Liouville problem. As a result, explicit expressions for the flow velocity and the pressure in the channel are established. The constructed analytical solutions favor a better understanding of the qualitative features of time-dependent flows of polymer fluids and can be applied to the verification of relevant numerical, asymptotic, and approximate analytical methods.

Publisher

MDPI AG

Subject

Polymers and Plastics,General Chemistry

Reference48 articles.

1. Bird, R.B., Curtiss, C., Amstrong, R., and Hassager, O. (1987). Dynamics of Polymeric Liquids, Wiley. [2nd ed.].

2. Doi, M., and Edwards, S.F. (1988). The Theory of Polymer Dynamics, Oxford University Press.

3. Pokrovskii, V.N. (2010). The Mesoscopic Theory of Polymer Dynamics, Springer.

4. Micro-macro models for viscoelastic fluids: Modelling, mathematics and numerics;Sci. China Math.,2012

5. Stress-deformation relations for isotropic materials;Rivlin;J. Ration. Mech. Anal.,1955

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3