Pseudo spectral solution of extended Graetz problem for combined pressure-driven and electroosmotic flow in a triangular micro-duct

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Abstract

Thermally developing mixed electroosmotically and pressure-driven flow in a triangular micro-duct with a step change wall temperature is investigated. Both axial conduction (Graetz problem extended) and Joule heating effects are taken into consideration. A new higher order accurate method is proposed. The proposed method is based on pseudo spectral Galerkin approach and the eigenfunctions of the Helmholtz equation for a triangular geometry. Convergence of the method is investigated and the effect of the Péclet (Pe) number, Joule heating and Debye–Hückel parameter on the temperature distribution and Nusselt number (Nu) are discussed in detail.

Introduction

Since the advent of microfluidics, electroosmosis becomes an important mechanism for flow generation in modern microscale laboratories, which is capable of performing medical diagnoses, known as lab-on-a-chip. Various generations of flow methodologies have been explored by the researchers for efficient design of such microfluidic systems. In many practical applications, electroosmotic flow actuation mechanisms [1], [2], [3], [4] have been found to offer extremely efficient methodologies for manipulating the transport of polar solvents (often carrying different types of biological samples).

In recent years, a number of authors studied electro-osmotic microflows in various shape of microstructures [5], [6]. Electro-osmotic flows mainly originate from the interaction between an electric double layer (EDL) and an externally applied electric field. This interaction of the electric field and the charged ions results in thermal energy generation such as Joule heating.

Forced convection heat transfer from the wall of a heated tube to the fluid passing through in a laminar, hydrodynamically developed flow with negligible axial conduction, and no internal heating is known as the Graetz problem [7]. Later on, many authors have investigated the effect of the axial-conduction for energy equations when the Peclet number is less than 100 [8], [9], [10], [11], [12], [13]. This problem is known as the extended Greatz problem. However, solving the extended Graetz problem is mathematically challenging, since the associated eigenvalue problem is non-self-adjoint when compared to the standard Sturm–Liouville problem. This problem was analysed first by Papoutsakis et al. [11], [14]. They found that a matrix operator acting upon a two-component temperature/longitudinal gradient vector could provide a symmetric operator to the extended Graetz problem, which includes axial conduction. Mathematical properties of the matrix operator for the extended Graetz problem are given in Fehrenbach et al. [15] and De Gourney et al. [16]. They also show the existence and uniqueness of the solution for the Graetz problem. Pierre and Plouraoue provide numerical analysis of mixed formulation for eigenvalue convection–diffusion problems in [17], they used finite element method and justify a separation of variable solution approach by defining the eigenvalue/eigenfunction decomposition of an appropriate mixed operator, but, this technique cannot be applied for our geometry, therefore our study is not included in that study. The analytical study of the extended Graetz problem has received increasing attention after the innovative study of Papoutsakis et al. [14], where they address the problem in round tubes. A thermally developing electrokinetic flow is in fact extended Graetz problem since the axial conduction and internal heating are both equally significant. The first investigations of the extended Graetz problem for electroosmotic flow were explored by Horiuchi and Dutta [18] for microchannel and Maynes and co-workers [19] for rectangular microchannel geometry. They both derive the analytical solution for a vanishing EDL (electric double layer) to allow considering a uniform velocity. Later Horiuchi and Dutta in [20] and [21] extended their works for mixed flow case and vanishing EDL. Finally, Sadeghi et al. [22] consider the combined electroosmotic and pressure driven flow in a rectangular microchannel by considering the actual variations of the velocity within EDL and applying a step change to the wall temperature, they used variational formulation, actually, they used Galerkin method to obtain the semi analytical solution of the problem.

Regarding the triangular geometry, Zhang [23] considered the laminar forced flow and heat transfer in plate-fin isosceles triangular duct and obtained the Nusselt number for various apex angle configurations. Heris et al. [24] investigated the same problem with nanofluids and found that Nusselt number increases with increasing concentration of nanoparticles and decreasing diameter. Axial heat conduction was neglected in all these studies. More recently, Siginer et al. [25] considered the classical Graetz problem neglecting axial heat conduction in tubes of arbitrary cross section and found that the heat exchange is faster in the zone near the vertices due to the increased area of contact with the wall.

The aim of the present work is twofold both theoretical and numerical to investigate the thermally developing combined electroosmotic and pressure driven flow in a triangular micro-duct by considering the actual variations of the velocity within EDL and applying a step change to the wall temperature. Both axial conduction (Graetz problem extended) and Joule heating effects are taken into consideration. The approach is based on the pseudo spectral Galerkin method with eigenfunctions of the Helmholtz equation used to solve the energy equation. These results are interesting since finite domains represent the most relevant configurations for applications such as convective heat duct; also, analytical as well as semi analytical results are rare and important in applied fluid mechanics. Furthermore, we show that our solution is higher order accurate. The problem is considered in two semi-infinite regions of the channel, i.e., upstream in which x<0 and downstream in which x<0 and the solutions of the two regions are then matched at x=0.

There are other considerations for spectral method on triangular region, for example Dubiner [26] considered the system of polynomials derived from Jacobi polynomials: gl,m=2l+321ylJ10,0ζJm2l+1,0η,ζ=2x+11y,η=2y1which are L2T-orthogonal on the triangle T=x,y:0x1,0y1,0x+y1

However, the main drawbacks of this approach lie in that the interpolation points are unfavourably clustered near one vertex of the triangle, and there is no corresponding nodal basis, making it difficult to implement. This problem was also considered by Y. Li et al. [27] and J. Li et al. [28] who defined a new rectangle-to-triangle mapping and they show that this mapping provides reasonable grid distributions and efficient implementation than the usual approach based on the collapsed transform. The error analysis based on Dubiner mapping is given in the work of Guo and Wang [29].

We note here that all these works basically map the triangular region into a rectangular duct and apply the spectral method to solve the resulting differential equation. But, in this study, mapping of triangular regions into rectangular regions is not used. Instead we use eigenfunctions of the Helmholtz equation in triangular regions. This approach has not been considered before. There are many advantages to the technique advocated in this paper. For example, energy equation in triangular regions is much simpler than transformed energy equation in rectangular regions.

The outline of this paper is as follows. In Section 2, we present problem formulation and existence of the solution in general form for energy equation. Pseudo spectral Galerkin method is introduced for two-dimensional elliptic equation in Section 3. In Section 4, we provide extensive numerical results to assess the convergence and accuracy of the method. In Section 5, we summarize the main features of our method, and briefly comment on the extensions of it.

Section snippets

Problem definition

We consider a combined electroosmotically and pressure-driven fluidic transport through a micro triangular prism with a characteristic cross-sectional length L. The physical model is sketched in Fig. 1, the origin of the coordinate system is at the centre of the triangular. The flow is assumed to be hydrodynamically fully developed. The wall temperature is the constant T0 when x<0 and it is the constant Tw when x>0. In the analysis, the following assumptions are considered:

  • (1)

    The fluid is

Temperature distribution, existence of the solution and pseudo spectral Galerkin method

In order to obtain temperature distribution, we need to concern fundamental physical law of energy conservation. The mathematical representation of this law appropriate to a fully developed mixed pressure and electroosmotic driven flow is given by ρcpwx,yTz=k2x2+2y2+2z2T+Ez2σwhere k is thermal conductivity, cp is the specific heat, i is the ionic current density, and σ is the electrical conductivity, last term represents the effects of Joule heating. Introducing the following

Discussion

We first justify using of Debye–Hückel approximation for small zeta potential. Since the Poisson–Boltzmann equation (PB) is not linear, it seems that analytical solution is not possible, in this report, we used collocation method with the base functions for right triangle region, where we used 100 base functions and we compare the numerical solution of PB equation with analytical solution of Debye–Hückel approximation, Fig. 5 -a–b show the difference between analytical solution of Debye–Hückel

Conclusion

We develop a new solution methodology for three-dimensional elliptic energy equation in micro triangular cross-sectional tubes. The problem of the hydrodynamically fully developed and thermally developing laminar flow of an incompressible fluid in a triangular duct with axial conduction, the extended Graetz problem, is solved via a new and efficient pseudo spectral Galerkin method based on the eigenfunctions of the Helmholtz equation for a triangular geometry. Convergence of the algorithm is

CRediT authorship contribution statement

Fehaid Salem Alshammari: Software, Visualization. F. Talay Akyildiz: Formal analysis, Writing - review & editing.

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