Cell models for viscous flow past a swarm of Reiner–Rivlin liquid spherical drops
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- 作者：B. R. Jaiswal ; B. R. Gupta
- 关键词：Cell models ; Drag coefficient ; Fictitious surface ; Normalized hydrodynamic drag force ; Reiner–Rivlin fluid ; Stream function
- 刊名：Meccanica
- 出版年：2017
- 出版时间：January 2017
- 年：2017
- 卷：52
- 期：1-2
- 页码：69-89
- 全文大小：
- 刊物类别：Physics and Astronomy
- 刊物主题：Classical Mechanics; Civil Engineering; Automotive Engineering; Mechanical Engineering;
- 出版者：Springer Netherlands
- ISSN：1572-9648
- 卷排序：52
文摘
This paper presents an analytical study of Stokes flow of an incompressible viscous fluid through a swarm of immiscible Reiner–Rivlin liquid droplets-in-cell using the cell model technique. The stream function solution of Stokes equation is obtained for the flow in the fictitious envelope region, while for the inner flow field within the liquid drop, the solution is obtained by expanding the stream function in a power series of S. The proper boundary conditions are taken on the surface of the liquid sphere, while the appropriate conditions applied on the fictitious boundary of the fluid envelope vary depending on the kind of cell-model. The analytical solution of the problem for four models: Happel’s, Kuwabara’s, Kvashnin’s and Mehta–Morse’s model (usually referred to as Cunningham’s) is derived. The velocity profile and the pressure distribution outside of the droplet are shown in numerous graphs for different values of the parameters. Numerical results for the normalized hydrodynamic drag force \(W_{C}\) acting, in each case, on the spherical droplet-in-cell obtained for different values of the parameters characterizing volume fraction \(\gamma ,\) the relative viscosity \(\lambda\), and the cross-viscosity, i.e., S are presented in tabular and graphical forms as well. It is found that normalized hydrodynamic drag force \(W_{C}\) is a monotonic increasing function of particle volume fraction \(\gamma .\) It is also observed that solid sphere in-cell experiences greater drag force \(C_{D}\), whereas spherical bubble experiences smaller. One of the important findings of the present investigation is that the cross-viscosity \(\mu _{c}\) of Reiner–Rivlin fluid decreases \(W_{C}\) on the liquid droplet-in-cell. Further, the drag coefficient \(C_{D}\) get reduced to analytical results obtained earlier.