A high-order perturbation solution for the steady sedimentation of a sphere in a viscoelastic fluid

被引:22
作者
Housiadas, Kostas D. [1 ]
Tanner, Roger I. [2 ]
机构
[1] Univ Aegean, Dept Math, Karlovassi 83200, Samos, Greece
[2] Univ Sydney, Sch Aerosp Mech & Mechatron Engn, Sydney, NSW 2006, Australia
关键词
Sphere sedimentation; Phan-Thien and Tanner model; Negative wake; Drag force; FINITE BOUNDARIES; ELASTIC FLUID; NEGATIVE WAKE; FLOW; MOTION; DRAG; SIMULATIONS; PARTICLE; MEDIA; SHEAR;
D O I
10.1016/j.jnnfm.2016.03.014
中图分类号
O3 [力学];
学科分类号
08 ; 0801 ;
摘要
The steady, creeping and isothermal sedimentation of a rigid sphere in an incompressible viscoelastic fluid which follows the exponential Phan-Thien and Tanner constitutive equation is studied analytically. The solution of the governing equations is expanded as a regular perturbation series for small values of the Deborah number, and the resulting sequence of two-dimensional partial differential equations is solved analytically up to eighth order. Although the domain is unbounded, the solution is able to resolve features of the flow that cannot be revealed by a low-order theoretical analysis, such as very fine flow structures and a stress boundary layer close to the surface of the sphere. The calculation of the drag force exerted on the sphere by the fluid was done by developing two formulas. The first is based on the flow field close to the sphere and the second is based on the far flow-field. Both formulas produce the same analytical expressions verifying the correctness and consistency of the series solution. At small Deborah numbers, a decrease of the drag force is predicted (i.e., an increase of the sedimentation velocity), followed by a significant drag enhancement at higher Deborah numbers. Investigation of the solution for the existence of a negative wake close to the rear stagnation point did not reveal such a phenomenon when physical constraints on the solutions were posed. On the contrary, a negative wake is predicted only as an artifact due to the loss of positive definiteness of the conformation tensor. It is also demonstrated that the effect of viscoelasticity is maximized in a region around the sphere with a radius which is about ten times the radius of the sphere. Last, it is shown that the fluid disturbances due to the viscoelasticity of the matrix fluid decay very slowly with the distance from the sphere, depending on the magnitude of the Deborah number. Streamlines, extra-stress and pressure contours as well as the extension of the polymer molecules are also presented and discussed. (C) 2016 Elsevier B.V. All rights reserved.
引用
收藏
页码:166 / 180
页数:15
相关论文
共 27 条
[1]  
[Anonymous], CAMBRIDGE PHILOS T
[2]   THE SEDIMENTATION OF A SPHERE THROUGH AN ELASTIC FLUID .1. STEADY MOTION [J].
ARIGO, MT ;
RAJAGOPALAN, D ;
SHAPLEY, N ;
MCKINLEY, GH .
JOURNAL OF NON-NEWTONIAN FLUID MECHANICS, 1995, 60 (2-3) :225-257
[3]   THE CREEPING MOTION OF A NON-NEWTONIAN FLUID PAST A SPHERE [J].
CASWELL, B ;
SCHWARZ, WH .
JOURNAL OF FLUID MECHANICS, 1962, 13 (03) :417-426
[4]   EFFECT OF FINITE BOUNDARIES ON MOTION OF PARTICLES IN NON-NEWTONIAN FLUIDS [J].
CASWELL, B .
CHEMICAL ENGINEERING SCIENCE, 1970, 25 (07) :1167-&
[5]  
Chhabra R.P., 1993, Bubbles, drops, and particles in non-Newtonian fluids
[6]   CREEPING FLOW OF DILUTE POLYMER-SOLUTIONS PAST CYLINDERS AND SPHERES [J].
CHILCOTT, MD ;
RALLISON, JM .
JOURNAL OF NON-NEWTONIAN FLUID MECHANICS, 1988, 29 (1-3) :381-432
[7]   ANALYSIS OF THE BRINKMAN EQUATION AS A MODEL FOR FLOW IN POROUS-MEDIA [J].
DURLOFSKY, L ;
BRADY, JF .
PHYSICS OF FLUIDS, 1987, 30 (11) :3329-3341
[8]   Negative wake behind a sphere rising in viscoelastic fluids: A lattice Boltzmann investigation [J].
Frank, Xavier ;
Li, Huai Z. .
PHYSICAL REVIEW E, 2006, 74 (05)
[9]   The falling sphere problem and capturing enhanced drag with Boger fluids [J].
Garduno, I. E. ;
Tamaddon-Jahromi, H. R. ;
Webster, M. F. .
JOURNAL OF NON-NEWTONIAN FLUID MECHANICS, 2016, 231 :26-48
[10]   Unexpected phenomena observed in particle settling in non-Newtonian media [J].
Gheissary, G ;
vandenBrule, BHAA .
JOURNAL OF NON-NEWTONIAN FLUID MECHANICS, 1996, 67 :1-18