Combined pressure-driven and electroosmotic flow of Casson fluid through a slit microchannel

https://doi.org/10.1016/j.jnnfm.2013.03.003Get rights and content

Highlights

  • Analytical solutions for electroosmotic flow of Casson fluid in a microchannel.

  • Various combinations of pressure and electrokinetic forcings.

  • Asymptotic approximations for integrals of nonlinear coupling terms.

  • Flow-rate shown to be sensitively affected by the yield stress.

Abstract

This study aims to develop analytical solutions for steady electroosmotic (EO) flow of a viscoplastic material, namely Casson fluid, through a parallel-plate microchannel. The flow is driven by electric as well as pressure forcings. A very thin electric double layer is assumed, and the Debye–Hückel approximation is used. The Casson yield stress makes the present problem distinct from existing studies on EO flow of other kinds of non-Newtonian such as power-law fluids. A first step of the analysis is to locate the yield surface, which divides the flow section into sheared and unsheared regions, where the stress is larger and smaller in magnitude than the yield stress, respectively. Different combinations of the electric and pressure forcings can lead to different types of distribution of stress relative to the yield stress. In this study, integrals of the nonlinear coupling terms of the two forcings are analytically expressed by uniformly valid approximations derived using the boundary-layer theory. It is shown that even a small value of the Casson yield stress, characteristic of that of blood, can considerably reduce the rate of flow of the fluid through a microchannel by electroosmosis. The decreasing effect of the yield stress on the flow is intensified by the presence of a pressure gradient, whether favorable or adverse.

Introduction

The rheology of fluid is an important consideration in microflows: Newtonian as well as non-Newtonian fluids need to be considered in engineering flows through micro-devices [1]. In these small devices, polymeric solutions are often used as the media for DNA and protein separation [2]. Colloidal systems involving blood are also often implemented on a microscale. A Newtonian model will fall short in describing the rheology of these systems.

The need to understand the effect of non-Newtonian rheology in microflows has spurred a growing interest in this topic in recent years. Flow through a microchannel is commonly driven by a pressure gradient, an electric field, or a combination of both. Flow driven by electric field works on the basis of electroosmosis, and is therefore known as electroosmotic (EO) flow. A body force that drags fluid into motion arises when the electric field interacts with the unbalanced charge in an electric double layer (EDL), which is formed at the contact interface of an electrolyte and a solid surface.

While pressure-driven non-Newtonian flow has long been studied, the EO flow of a non-Newtonian fluid was not theoretically investigated until recently. Motivated to study biofluids in micro-systems, Das and Chakraborty [3], and Zimmerman et al. [4] were among the first to present theoretical work on non-Newtonian electrokinetic flow and transport in microchannels. The non-Newtonian model adopted by Das and Chakraborty [3] is the power-law model, while that by Zimmerman et al. [4] is the Carreau model. The power-law (also known as Ostwald-de Waele) model is one of the simplest models to describe nonlinear viscous behaviors. It is a two-parameter model, where the shear-thinning or shear-thickening behaviors can be conveniently represented by the flow index being less than or larger than 1, respectively. For sufficiently simple geometry and in the absence of other forcings, it is possible to determine EO flow of a power-law fluid analytically.

EO flow of a power-law fluid in microchannels of various shapes (single surface, parallel-plate, rectangular, cylindrical, or annular) has since then been intensively studied. For relatively simple flows (e.g., one-dimensional flow, no pressure forcing, and linearized Poisson–Boltzmann equation), analytical or semi-analytical solutions (exact or approximate) have been obtained by Chakraborty [5], Berli and Olivares [6], Zhao et al. [7], Olivares et al. [8], Zhao and Yang [9], [10], [11], Berli [12], Vasu and De [13], and Sadeghi et al. [14]. More complicated flows (e.g., two-dimensional flow, unsteady flow, nonlinear coupling of the EO and pressure forcings, nonlinear Poisson–Boltzmann equation) can only be solved or simulated numerically. Such numerical studies have been performed by Bharti et al. [15], Tang et al. [16], Vasu and De [17], Babaie et al. [18], Hadigol et al. [19], Cho et al. [20], [21], Deng et al. [22], Shamshiri et al. [23], and Vakili et al. [24].

Because of its simplicity, the power-law model has been the most chosen model to describe non-Newtonian rheology in EO flow through microchannels. Much fewer existing studies are based on other models. The Bingham, Herschel–Bulkley, and Carreau models have been selected only in a limited number of studies (e.g., Berli and Olivares [6], Tang et al. [25], Zhao and Yang [26]).

The present study aims to look into flow of a particular kind of viscoplastic material, viz. Casson fluid, under the combined action of electrokinetics and pressure gradient in a parallel-plate microchannel. Polymeric solutions and colloidal suspensions are often classified as viscoplastic materials. They exhibit a yield stress, a critical value of stress below which the materials do not flow or flow like a rigid body. Most viscoplastic materials can be described by the Bingham, Casson, or Herschel–Bulkley models [27]. Electrokinetic driven flow of Casson fluids in microcapillaries has been studied by Liu and Yang [28] and Liu et al. [29], who applied a two-phase Newtonian/non-Newtonian model to describe blood flow.

The model proposed by Casson [30] can be generalized intoμγ̇=1-τ0/|τ|2τfor|τ|>τ00for|τ|τ0,where τ is the stress tensor, γ̇=u+(u)T is the rate of deformation tensor, μ is the plastic viscosity, τ0  0 is the Casson yield stress, and|τ|=12τ:τ,is the magnitude of the stress. The Casson model reduces to the Newtonian model when the yield stress vanishes τ0 = 0, by which μ becomes the Newtonian viscosity. Even with a non-zero yield stress, a Casson fluid will asymptotically behave like a Newtonian fluid as the stress increases to a level much higher than the yield stress. The viscosity μ is also known as the limiting Newtonian (high-shear-rate) viscosity.

The Casson model was originally developed for pigment-oil suspensions of the printing ink. It is now commonly used to describe the rheology of blood [31], [32]. The yield stress of blood is a strong function of the volume fraction of red blood cells, or the hematocrit. At hematocrit of 40%, the yield stress for normal blood is typically 0.04 dynes/cm2 or 4 mPa [32]. Such a small value of yield stress is often regarded as physiologically insignificant for blood flow in large vessels. The significance of the Casson yield stress, however, remains to be determined for EO flow in microchannels. This has motivated the present study. The behaviors of a Casson fluid under EO pumping in a microchannel are yet to be fully understood.

A complication arises in the analysis of flow of a yield-stress fluid. It needs to divide the flow domain into regions where the stress is smaller or larger in magnitude than the yield stress. The two regions are accordingly called unsheared (or unyielded) and sheared (or yielded) regions. The location of the yield surface, which is the interface between the two regions, is in general not known a priori, and has to be found as part of the final solution. The problem then becomes highly nonlinear, and can only be solved numerically. To study the physics as analytical as possible, we shall consider in this study strictly one-dimensional flow in a uniform channel, for which the stress distribution is known prior to finding the velocity. With a known stress distribution, the position of the yield surface can be readily determined. The velocity can then be found straightforwardly.

Another complication arises when the flow is simultaneously driven by two forcings. Owing to nonlinear rheology, linear superposition of solutions due to individual forcings is not applicable. A nonlinear coupling term of the two forcings is bound to appear when the Cauchy momentum equation is solved for the velocity. As a result, the velocity may not be analytically expressed, unless some approximations are made (e.g., Zhao and Yang [9]).

In the present problem, both pressure and electric forcings are considered, and they can independently affect the stress distribution. Different combinations of the two forcings will lead to qualitatively different stress distributions, thereby different types of the partitioning into sheared and unsheared regions. We shall first identify all the possible cases of stress distribution, and then derive expressions for the velocity in each case. To obtain analytical expressions for the velocity, we shall use the boundary-layer theory to approximate the integral of the nonlinear coupling term. If the electric double layer (EDL) is very thin compared with the channel height, the stress distribution will have two components. Across the entire channel section is a slowly varying component due to the pressure forcing, while near the wall is a rapidly varying component due to the electric forcing. By virtue of this boundary-layer structure, an asymptotically uniform approximation can be deduced for the integral.

In the modeling of flow of non-Newtonian polymeric solutions in microchannels, a two-phase distribution of rheology is often considered: near the wall is a thin layer which is depleted of polymer molecules and is hence Newtonian, while outside this layer the fluid is non-Newtonian. This two-phase configuration also accounts for the well-known Fahraeus–Lindqvist effect in blood flow in a capillary: the apparent viscosity of blood decreases as the tube diameter decreases. This is due to the fact that adjacent to the capillary wall is a plasma skimming layer, which is depleted of red blood cells and is hence a region of reduced viscosity. Such a near-wall depletion effect has been incorporated in the EO flow models by, among others, Zimmerman et al. [4], Berli and Olivares [6], Olivares et al. [8], Berli [12], Liu and Yang [28], and Liu et al. [29]. In the present problem, we shall, however, not consider the depletion layer. We simply consider the EO flow of a one-phase fluid that is homogeneously non-Newtonian. This will enable us to look into the coupling effect of the electrokinetic forcing and the non-Newtonian rheology, as has been done by Babaie et al. [18] and Hadigol et al. [19]. We leave it to a future study to extend the problem to a two-phase model.

Our problem is further described in Section 2, where assumptions are stated as the problem is mathematically formulated. The various possible cases of stress distributions are then described in Section 3. For each case of stress distribution, the position(s) of the yield surface(s) and the velocity are analytically derived. We shall then examine in Section 4, through some numerical results, the effect of the yield stress on the flow rate as a function of the pressure gradient. Velocity profiles are also presented for illustration of the yield stress effects. Some concluding remarks are finally given in Section 5.

Section snippets

Combined pressure-driven and electroosmotic flow

We consider steady flow through a microchannel bounded by two parallel plates, which are uniformly charged with a zeta potential ζ, and at a distance 2h apart. Fig. 1 shows a definition sketch of our problem, where (x, y) are the streamwise and transverse coordinates, respectively. The channel is filled with a non-Newtonian liquid electrolyte, which can be modeled as a Casson fluid. Because of symmetry of the flow about the mid-plane, it suffices for us to consider only the lower half of the

Possible stress and velocity profiles

It is necessary that, in order to determine the velocity, the flow domain is divided into sheared and unsheared regions, where the stress is larger and smaller in magnitude than the yield stress, respectively. The interface between the two regions, known as the yield surface, along which the stress is equal in magnitude to the yield stress, needs to be located before the flow in individual regions can be determined. In the present one-dimensional problem, for which the stress distribution is

Discussion

The present problem depends on three parameters: the normalized pressure gradient α, the normalized yield stress β, and the normalized Debye parameter κ. In the following, we shall examine results to find out how the EO flow may be affected by the yield stress of a Casson fluid under a favorable or adverse pressure gradient. A sufficiently large Debye parameter κ = 100, for a very thin EDL, is used in all our calculations. Let us first have an order of magnitude estimate about the yield stress

Concluding remarks

In this paper, we have presented analytical solutions for one-dimensional steady flow of a Casson fluid, under the combined action of pressure gradient and electroosmotic (EO) pumping, through a parallel-plate channel. The problem is simplified by the assumptions of very small potentials (or the Debye–Hückel approximation) and a very thin electric double layer (EDL) compared with the channel height. The flow analysis starts with locating the yield surfaces in order to partition the flow section

Acknowledgment

The work was financially supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, through Project HKU 715510E.

References (34)

Cited by (62)

  • Heat transfer of power-law fluids with slip-dependent zeta potential

    2022, Colloids and Surfaces A: Physicochemical and Engineering Aspects
    Citation Excerpt :

    They found that Deborah number can enhance the flow velocity. For the combined electroosmotic and pressure-driven flow of Casson fluids, Ng [12] obtained the exact solution of flow velocity in a parallel-plate channel. Results showed that flow rate can be reduced with the increase of the Casson yield stress and such reduction is more obvious when the external pressure exists.

View all citing articles on Scopus
View full text