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The influence of the inertially dominated outer region on the rheology of a dilute dispersion of low-Reynolds-number drops or rigid particles | |
Alternative Title | J. Fluid Mech. |
Subramanian, Ganesh1; Koch, Donald L.2; Zhang, Jingsheng3; Yang, Chao3 | |
2011-05-01 | |
Source Publication | JOURNAL OF FLUID MECHANICS |
ISSN | 0022-1120 |
Volume | 674Issue:674Pages:307-358 |
Abstract | We calculate the rheological properties of a dilute emulsion of neutrally buoyant nearly spherical drops at O(phi Re(3/2)) in a simple shear flow(u(infinity) =(gamma) over dot x(2)1(1),(gamma) over dot being the shear rate) as a function of the ratio of the dispersed-and continuous-phase viscosities (lambda = (mu) over cap/mu). Here, phi is the volume fraction of the dispersed phase and Re is the micro-scale Reynolds number. The latter parameter is a dimensionless measure of inertial effects on the scale of the dispersed-phase constituents and is defined as Re =(gamma) over dot a(2) rho/mu, a being the drop radius and rho the common density of the two phases. The analysis is restricted to the limit f, Re << 1, when hydrodynamic interactions between drops may be neglected, and the velocity field in a region around the drop of the order of its own size is governed by the Stokes equations at leading order. The dominant contribution to the rheology at O(phi Re(3/2)), however, arises from the socalled outer region where the leading-order Stokes approximation ceases to be valid. The relevant length scale in this outer region, the inertial screening length, results from a balance of convection and viscous diffusion, and is O(aRe(-1/2)) for simple shear flow in the limit Re << 1. The neutrally buoyant drop appears as a point-force dipole on this scale. The rheological calculation at O(phi Re(3/2)) is therefore based on a solution of the linearized Navier-Stokes equations forced by a point dipole. The principal contributions to the bulk rheological properties at this order arise from inertial corrections to the drop stresslet and Reynolds stress integrals. The theoretical calculations for the stresslet components are validated via finite volume simulations of a spherical drop at finite Re; the latter extend up to Re approximate to 10. Combining the results of our O(phi Re(3/2)) analysis with the known rheology of a dilute emulsion to O(phi Re) leads to the following expressions for the relative viscosity (mu(e)), and the non-dimensional first (N(1)) and second normal stress differences (N(2)) to O(phi Re(3/2)): mu(e) = 1+phi[(5 lambda+ 2)/(2(lambda+ 1))+ 0.024Re(3/2)(5 lambda+2)(2)/(lambda+1)(2)]; N(1) = phi[-Re4(3 lambda(2)+3 lambda+1)/(9(lambda+ 1)(2))+0.066Re(3/2)(5 lambda+2)(2)/(lambda+1)(2)] and N(2) = phi[Re 2 (105 lambda(2)+96 lambda+35)/(315(lambda+1)(2))-0.085Re(3/2)(5 lambda+2)(2)/(lambda+1)(2)]. Thus, for small but finite Re, inertia endows an emulsion with a non-Newtonian rheology even in the infinitely dilute limit, and in particular, our calculations show that, aside from normal stress differences, such an emulsion also exhibits a shearthickening behaviour. The results for a suspension of rigid spherical particles are obtained in the limit lambda ->infinity.; We calculate the rheological properties of a dilute emulsion of neutrally buoyant nearly spherical drops at O(phi Re(3/2)) in a simple shear flow(u(infinity) =(gamma) over dot x(2)1(1),(gamma) over dot being the shear rate) as a function of the ratio of the dispersed-and continuous-phase viscosities (lambda = (mu) over cap/mu). Here, phi is the volume fraction of the dispersed phase and Re is the micro-scale Reynolds number. The latter parameter is a dimensionless measure of inertial effects on the scale of the dispersed-phase constituents and is defined as Re =(gamma) over dot a(2) rho/mu, a being the drop radius and rho the common density of the two phases. The analysis is restricted to the limit f, Re << 1, when hydrodynamic interactions between drops may be neglected, and the velocity field in a region around the drop of the order of its own size is governed by the Stokes equations at leading order. The dominant contribution to the rheology at O(phi Re(3/2)), however, arises from the socalled outer region where the leading-order Stokes approximation ceases to be valid. The relevant length scale in this outer region, the inertial screening length, results from a balance of convection and viscous diffusion, and is O(aRe(-1/2)) for simple shear flow in the limit Re << 1. The neutrally buoyant drop appears as a point-force dipole on this scale. The rheological calculation at O(phi Re(3/2)) is therefore based on a solution of the linearized Navier-Stokes equations forced by a point dipole. The principal contributions to the bulk rheological properties at this order arise from inertial corrections to the drop stresslet and Reynolds stress integrals. The theoretical calculations for the stresslet components are validated via finite volume simulations of a spherical drop at finite Re; the latter extend up to Re approximate to 10. Combining the results of our O(phi Re(3/2)) analysis with the known rheology of a dilute emulsion to O(phi Re) leads to the following expressions for the relative viscosity (mu(e)), and the non-dimensional first (N(1)) and second normal stress differences (N(2)) to O(phi Re(3/2)): mu(e) = 1+phi[(5 lambda+ 2)/(2(lambda+ 1))+ 0.024Re(3/2)(5 lambda+2)(2)/(lambda+1)(2)]; N(1) = phi[-Re4(3 lambda(2)+3 lambda+1)/(9(lambda+ 1)(2))+0.066Re(3/2)(5 lambda+2)(2)/(lambda+1)(2)] and N(2) = phi[Re 2 (105 lambda(2)+96 lambda+35)/(315(lambda+1)(2))-0.085Re(3/2)(5 lambda+2)(2)/(lambda+1)(2)]. Thus, for small but finite Re, inertia endows an emulsion with a non-Newtonian rheology even in the infinitely dilute limit, and in particular, our calculations show that, aside from normal stress differences, such an emulsion also exhibits a shearthickening behaviour. The results for a suspension of rigid spherical particles are obtained in the limit lambda ->infinity. |
Keyword | Emulsions Particle/fluid Flow Rheology |
Subtype | Article |
WOS Headings | Science & Technology ; Technology ; Physical Sciences |
DOI | 10.1017/jfm.2010.654 |
URL | 查看原文 |
Indexed By | SCI |
Language | 英语 |
WOS Keyword | PLANE COUETTE-FLOW ; FINITE-AMPLITUDE SOLUTIONS ; SHEAR-FLOW ; CIRCULAR-CYLINDER ; NUMERICAL-SOLUTION ; ROTATIONAL FLOW ; SPHERE ; EMULSION ; DIFFUSION ; SUSPENSION |
WOS Research Area | Mechanics ; Physics |
WOS Subject | Mechanics ; Physics, Fluids & Plasmas |
WOS ID | WOS:000290112900014 |
Citation statistics | |
Document Type | 期刊论文 |
Version | 出版稿 |
Identifier | http://ir.ipe.ac.cn/handle/122111/6374 |
Collection | 研究所（批量导入） |
Affiliation | 1.Jawaharlal Nehru Ctr Adv Sci Res, Engn Mech Unit, Bangalore 560064, Karnataka, India 2.Cornell Univ, Sch Chem & Biomol Engn, Ithaca, NY 14853 USA 3.Chinese Acad Sci, Inst Proc Engn, Beijing 100190, Peoples R China |
Recommended Citation GB/T 7714 | Subramanian, Ganesh,Koch, Donald L.,Zhang, Jingsheng,et al. The influence of the inertially dominated outer region on the rheology of a dilute dispersion of low-Reynolds-number drops or rigid particles[J]. JOURNAL OF FLUID MECHANICS,2011,674(674):307-358. |
APA | Subramanian, Ganesh,Koch, Donald L.,Zhang, Jingsheng,&Yang, Chao.(2011).The influence of the inertially dominated outer region on the rheology of a dilute dispersion of low-Reynolds-number drops or rigid particles.JOURNAL OF FLUID MECHANICS,674(674),307-358. |
MLA | Subramanian, Ganesh,et al."The influence of the inertially dominated outer region on the rheology of a dilute dispersion of low-Reynolds-number drops or rigid particles".JOURNAL OF FLUID MECHANICS 674.674(2011):307-358. |
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