Theoretical and experimental study of low conducting fluid MHD flow in an open annular channel

doi:10.1016/j.ijheatmasstransfer.2018.08.047

International Journal of Heat and Mass Transfer

Volume 127, Part C, December 2018, Pages 322-331

https://doi.org/10.1016/j.ijheatmasstransfer.2018.08.047 Get rights and content

Highlights

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Three dimensional laminar MDH flow in an open annular channel.
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Quasi-analytical solution for velocity profile and flow rate.
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The maximum experimental velocity is determined using Particle Image Velocimetry.
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The flow rate is maximized when the radii ratio is about fifteen percent.

Abstract

In this paper, we research theoretically and experimentally the 3D laminar flow of an electrolyte in an annular open channel driven by a Lorentz force. The annular duct is formed by two concentric electrically conducting cylinders and limited by an insulating bottom wall. The interaction between a uniform magnetic field along the axial direction and a radial electric current produces a Lorentz force that drives the fluid in the azimuthal direction, producing a Magnetohydrodynamic (MHD) flow. The effect of the three walls is analyzed in detail by considering the gap between the cylinders as well as the depth of the channel. The steady flow is solved using a Galërkin method with orthogonal Bessel-Fourier series. A quasi-analytic approximation for velocity is found. Velocity profiles are explored by varying the depth of the channel and the gap between the cylinders in order to show the effect of the walls on the flow configuration. Results are compared with experimental data provided by PIV technique as well as in those found in scientific literature.

Introduction

Recent advances in Micro-Electromechanical Systems (MEMS) include improvements done in the propulsion of fluids, either through mechanical or dynamics pumps. Because of the advantage of dynamics pumps over mechanical ones, a lot of micropumps have been designed in order to drive fluids involving no moving parts. These have been particularly useful in microfluidic applications where a conducting fluid needs to be transported for several operations in micro Total Analysis Systems ( $μ$ TAS). Operations in $μ$ TAS include control, mixing, separation and chemistries processes, such as operations in a small laboratory, for this reason, are sometimes called Lab-On-Chip (LOC) systems. The flow induced by the MHD effects, as a result of the Lorentz force produced by the interaction between an electric current within the fluid and an external magnetic field, is a feasible way to pump fluid in microfluidic and laboratory devices. This phenomenon is particularly important in processes in microreactors where Polymerase Chain Reaction (PCR) is used for the amplification of Deoxyribonucleic Acid (DNA) [1] and mass micromixing processes [2], [3]. Some devices perform PCR for nucleic acid detection and amplification utilizes MHD for fluid propulsion and sensing [4]. Electrolytes are fluids of small conductivity present in microfluidics, this property allows them to be propelled without the flow induces a considerable change in the applied electric field, in consequence, the velocity is uncoupled from the potential difference; this does not occur for liquid metal where the electric potential and velocity are coupled and the induced current cannot be ignored causing an additional drag force that becomes an important factor into the flow’s configurations. However, additional drag force can appear due to friction with the boundaries delimiting the fluid. In open channels, the presence of the bottom wall influences the flow configuration acting like an additional drag force in the fluid. Studies of these walls have been done in straight channels and annular ones. The problem of the flow in the annular channel driven by a Lorentz force has been studied with 2D quasi-analytical approximation including the effect of the bottom wall [5]. However, a 3D flow must be considered taking into account the effects of the walls. Analytical solutions for 2D flow (ignoring the effects of the bottom and lateral walls) and many studies with infinite cylinders have been done including stability. However, when the depth of the channel is finite, the effect of the wall is particularly important in determining the basic flow and the influence of its stability. The case of the closed channel has been dealt by Ortiz et al. [6] including a general slip in the four walls, but the particular case of free surface on the top deserves special attention because the walls’ effects can be translated to the flow on the surface of the liquid, been the open channel one of the configurations available in micromixers [3] which allows the application of Particle Image Velocimetry (PIV) or particle tracking, due to that this configuration facilities the observation of the free surface on the annular channel.

In the present contribution, we analyse the steady laminar flow of an electrolyte in an open annular channel driven by an azimuthal Lorentz force. In Section 2, the formulation of the problem and the governing equations are presented. In Section 3, the method of solution and the use of Tracker and PIV packages are described. In Section 4, the main results are presented regarding the validation of the model with obtained measurements from PIV and its comparison with data from previous scientific works. Finally, some concluding remarks are stated in Section 5.

Section snippets

Formulation of the problem

An open annular channel of depth h, limited laterally by two concentric electrically conducting cylinders of radius $R_{1}$ and $R_{2}$ ( $R_{1} < R_{2}$ ) and by an insulating wall at the bottom is considered. A low conductivity liquid of uniform electric conductivity $σ$ , mass density $ρ$ and kinematic viscosity $ν$ , is contained in the channel, while a uniform magnetic field of strength $B_{0}$ is applied vertically along the cylinders’ axis. A potential difference $Δ ϕ$ is applied between the cylindrical electrodes so that an

Experimental setup

The cylinders are formed with a sheet of copper of 1 mm of thickness, the electrodes are connected internally the 10 × 10 cm prototype made of PSP polymer from a 3D print. The conducting fluid is distilled water (5 parts-per-million and electrical conductivity of $σ = 0.001$ S/m) mixed with 10  $μ$ m covered with silver coated hollow glass spheres (S-HGS-10 Dantec Dynamics seeding particles, Item No. 80A7001) that makes the water electrically more conductive (water + particles: $σ = 0.00287$ S/m) and less

Model results

The common no slip boundary condition is applied in the cylinders’ surfaces. For bottom wall and free surface on the top, the eigenvalues of the trigonometric eigenfunctions take the limit values $λ_{n} \to (2 n - 1) π / 2$ . The eigenvalues of the Bessel functions have to be determined for every value of $η$ . Then, the velocity solution becomes: $v_{θ} = \sum_{m = 1}^{M} \sum_{n = 1}^{N} \frac{8 {(- 1)}^{n + 1} α_{m}}{(α_{m}^{2} + \frac{{[(2 n - 1) π]}^{2}}{4 ∊^{2}}) (v'_{1; m}^{2} (1) - η^{2} v'_{1; m}^{2} (η)) | \ln η |} \frac{v_{0; m} (1) - v_{0; m} (η)}{[(2 n - 1) π]} v_{1, m} (r) g_{n} (z)$ The dimensionless flow rate is given by $q = - \sum_{m = 1}^{M} \sum_{n = 1}^{N} \frac{16}{(α_{m}^{2} + \frac{{[(2 n - 1) π]}^{2}}{4 ∊^{2}}) (v'_{1; m}^{2} (1) - η)}$

Conclusions

In this investigation, new quasi-analytic results of the flow in an open annular channel have been presented. The results calculated with the Galërkin method with Fourier-Bessel series have been checked with modelled - experimental data found in the literature, also with experimental measurements obtained from PIVLAB toolbox and with the Tracker Software, they are in agreement, in special PIVLAB toolbox results between 2 and 4.5 V, presenting lower deviation from Galërkin results than the

Conflict of interest

We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

A.S. Ortiz-Pérez thanks the L. of L., K. of K. and G. of G. Technical support of Jorge Miramón, Jonhatan Rivera, Dannyel Farías, Salvador Melchor León, Jesús Yosef Galaviz Medina and Kevin Rojo Aguilera is also acknowledged. A.S. Ortiz-Pérez thanks the UABC project 3064 from 19th internal convocatory.

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View full text

Theoretical and experimental study of low conducting fluid MHD flow in an open annular channel

Highlights

Abstract

Introduction

Section snippets

Formulation of the problem

Experimental setup

Model results

Conclusions

Conflict of interest

Acknowledgments

Sensors Actuat. B: Chem.

Sensors Actuat. B: Chem.

Sensors Actuat. A: Phys.

Application of magnetohydrodynamic actuation to continuous flow chemistry

Lab Chip

Transient micromixing: examples of laminar and chaotic stirring

Phys. Fluids

Analysis of an Annular MHD Stirrer for Microfluidic Applications

Magnetohydrodynamic flow with slippage in an annular duct for microfluidic applications

Microfluid. Nanofluid.