Elsevier

Applied Mathematics and Computation

Volume 250, 1 January 2015, Pages 636-649
Applied Mathematics and Computation

Reynolds number effects in the flow of an electrorheological fluid of a Casson type between fixed surfaces of revolution

https://doi.org/10.1016/j.amc.2014.10.112Get rights and content

Abstract

Many electrorheological fluids (ERFs) as fluids with micro-structure demonstrate viscoplastic behaviours. Rheological measurements indicate that the flows of these fluids may be modelled as the flows of a Casson fluid. Our concern in the paper is to examine the pressurized laminar flow of an ERF of a Casson type in a narrow clearance between to fixed surfaces of revolution. In order to solve this problem the boundary layer equations are used. The Reynolds number effects (the effects of inertia forces) on pressure distribution are examined by using the averaged inertia method. Numerical examples of externally flows in the clearance between parallel disks and concentric spherical surfaces are presented.

Introduction

Electrorheological fluids (abbreviated to ERFs) are a kind of novel intelligent soft matter as a two-phase suspension system formed by dielectric solid particles dispersing in the insulating medium oil [1], [2], [3], [4], [5], [6], [7].

Electrorheological fluids have long held promise for use in vibration control and torque transmission devices, based on the characteristic dependence of their viscosity on the applied electric field strength. Since their discovery by Winslow [1] many kinds of organic or inorganic particles and their composites have been used and reported as the ER materials. When the external electric field is imposed to an ERF, it behaves as a viscoplastic fluid [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], displaying a field-dependent yield shear stress which is widely variable. Without the electric field, the ERF has a reversible and a constant viscosity so that it flows as a Newtonian fluid. Another salient feature of the ERF is that the time required for the variation is very short (<0.001 s). These attractive characteristics of the ERF provide the possibility of the appearance of new engineering technology (e.g.: nuclear and space engineering, mechatronics, etc.). Recently, the application of the ERF to rotor-bearing systems has been initiated Dimarogonas and Kollias [4] and Jung and Choi [6] and developed by Basaravaja et al. [12] and El Wahed [13], but its application to vibration control has been initiated by Lee et al. [14], Hong et al. [15], [16] and Choi et al. [17].

To describe the rheological behaviour of viscoplastic fluids in complex geometries the Bingham model is used [2], [18]. Recently, the non-linear model of Shulman [19] has been successfully applied. The constitutive equation of this model is given as follows:τ=τ01/n+(μγ̇)1/mn.where τ is the shear stress, τ0 is the yield shear stress, μ is the coefficient of plastic viscosity, γ̇ is the shear strain rate, n and m are the non-linearity indices. By reducing the coefficients in the Shulman equation one can obtain simpler models describing the flow of a viscoplastic fluid. One of these models is a Casson model [20]:τ=τ01/n+(μγ̇)1/nn.

The most popular model of the ERF is the Bingham model for which m = n = 1 in Eq. (1). An analysis of many investigations [13], [21], [22], [23], [24], [25], [26] indicates that other models may be used – such as Casson (m = n), Vočadlo (m = 1) and Herschel–Bulkley (n = 1) in Eq. (1) – to describe ERFs flows.

This paper deals with a laminar flow of ERFs modelled as a Casson fluid in the clearance between two surfaces of revolution shown in Fig. 1. Using the method of averaged inertia [27], [28] the influence of inertia terms of the equations of motion and viscoplastic behaviour on the pressure distribution is analysed. The final results are presented for a “simple” Casson fluid (n = 2) and are also presented for a Newtonian fluid because for some physical conditions the insulating oil (or other solvents for ERFs) may manifest Newtonian behaviours.

Section snippets

Equations of motion of the ERF

The yield shear stress for the ERF varies with respect to the electric field. According to the experimental results reported in [2], [3], [5], [29], [8], [30], [31] the relation between the yield shear stress τ0 and the electric field strength E is given as follows:τ0(E)=αU2hβwhere U and 2h are the applied voltage and the film thickness, respectively. Both parameters α and β are the experimental constants of which the range of the exponent β is 1–2.4. Other experimental data [32] suggest that

Solution for the flow of a Casson ER fluid

Taking into account Eq. (12) one can rearrange Eq. (13) writing it in the form:ρ1R(Rυx2)x+y(υxυy)=-dpdx+ySτ01n+μυxy1nn.

Then, averaging the left-hand side of Eq. (20) across the clearance thickness and taking into account boundary conditions (17) we obtain the following equation:ySτ01n+μυxy1nn=f(x)where f(x) is defined as:f(x)=dpdx+ρhRxR0hυx2dy.Integrating Eq. (21) we get [23], [33], [34], [35]:

  • for shear flow:

υxs(y)=h2μ(-f)i=0nFi1-yh2n-inwhere |Tyx| > τ0 or |y| > h0;

  • for core flow

υxc=υ

Graphic presentation of some results

Taking into account the results obtained in the previous section, we will present the pressure distribution in the clearance of constant thickness between two parallel disks shown in Fig. 2 and between two concentric spherical surfaces shown in Fig. 3; to this aim we shall introduce the following dimensionless parameters:

  • for the clearance between parallel disks:

x̃=xxo,R̃=RRo=x̃,

  • for the clearance between concentric spherical surfaces:

x̃=xRs=φ,R̃=RRs=sinφand:Rλ=ρQhoπμRo2,p̃=ρ(p-po)(2ho)4μ2Ro2Rλ,K

Conclusions

In this work the relative inertia effect as a function of the Reynolds number was investigated in a clearance flow of an ERF of the Casson type between two surfaces of revolution. It was also investigated the Newtonian fluid flow for the reason that under some physical condition ERFs may manifest Newtonian behaviour. All these theoretical results was obtained by using the method of averaged inertia.

From the general considerations, formulae and graphs presented here for the flows in the narrow

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