Interaction of micro-fluid structure in a pressure-driven duct flow with a nearby placed current-carrying wire: A numerical investigation

Hua Bian; Kashif Ali; Sohail Ahmad; Hina Bashir; Wasim Jamshed; Kashif Irshad; Mohammed K. Al Mesfer; Mohd Danish; Sayed M. El Din

doi:10.1515/rams-2023-0134

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Interaction of micro-fluid structure in a pressure-driven duct flow with a nearby placed current-carrying wire: A numerical investigation

Hua Bian , Kashif Ali , Sohail Ahmad , Hina Bashir , Wasim Jamshed , Kashif Irshad , Mohammed K. Al Mesfer , Mohd Danish and Sayed M. El Din

Published/Copyright: November 15, 2023

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From the journal REVIEWS ON ADVANCED MATERIALS SCIENCE Volume 62 Issue 1

Abstract

High population density in major cities has led to compact designs of residential multi-story buildings. Consequently, it is a natural choice of the architects to suggest the location of high-voltage wires close to the ducts with contaminated air. This observation results in the motivation for this study, i.e., the understanding of the complicated interaction of the Lorentz force (due to the current-carrying wire) with the micropolar flow in the vertical direction in the duct, with polluted air (containing dust particles) being modeled as a micropolar fluid, which is driven by some external pressure gradient. Therefore, this study focuses on an incompressible and electrically conducting micropolar fluid flow through a rectangular vertical duct, in the presence of a current-carrying wire placed outside the flow regime. The governing equations, after being translated into a dimensionless form, are solved numerically using a finite volume approach. The velocity, microrotation, and temperature fields thus obtained are examined. It has been noted that the strong magnetic force caused by the wire may distort the flow symmetry and slows down the flow. Furthermore, in the absence of wire, particles spinning in clockwise and counter-clockwise directions occupy the same amount of space in the duct, thus incorporating a sort of equilibrium in the duct. However, the imposed variable magnetic field adds to the spinning of particles in one part of the duct, while simultaneously suppressing it in the other region.

Keywords: pressure gradient; current-carrying wire; micro-fluid structure; duct; finite volume approach

1 Introduction

Microstructured fluids are one of the categories in non-Newtonian fluids, which possess a microstructure that consists of rigid homogeneously suspended particles. Therefore, the fluid motion is described by two vectors (known as the classical velocity and the microrotation) that arise from the fluid movement and rotation of these fine particles. There are many technical applications of magnetohydrodynamic (MHD) effects in electrically conducting fluids. Many researchers have discussed fluid flows through circular pipe, within rectangular duct, and between parallel plates with the effect of a transverse magnetic field. Pyatnitskaya et al. [1] used two distinct approaches to study the transverse magnetic field interaction in a submerged flat jet flow. The position of channel was adjusted in such a way that the jet plane was normal to the magnetic field. The results were established for the waveforms and velocity by the swivel-type probe. Jeyanthi and Ganesh [2] explained that how much a MHD flow was affected by a duct having electrically conducting walls. In this analysis, suitable inclination angles of inclined magnetic field were suggested and discussed. An axisymmetric MHD flow of viscous fluid was considered in a circular pipe by Nagaraju and Garvandha [3]. To find the temperature as well as velocity fields, the Navier–Stokes equations were taken into account in the cylindrical form. The solution of velocity components was achieved by the homotopy analysis method.

Chen et al. [4] worked on the Williamson fluids involved in a MHD peristaltic flow and determined the effects of the Grashof thermal number, the Darcy number, and the magnetic parameter on the concentration and temperature distribution. The Couette flow occurring between two parallel plates with the effect of inclined magnetic field was elaborated by Guria et al. [5]. This work mainly depicted the theoretical analysis, which was based on the results obtained from the Laplace transform. Sulochana [6] investigated the viscous flow through porous medium between two parallel plates under the influence of the inclined magnetic field. The lower plate was taken at rest, while the upper one was facing non-torsional oscillations.

An analytical analysis of electrically conducting fluid was presented by Sezgin [7]. The flow was assumed in a rectangular duct. He numerically solved the problem that involved the Fredholm integral equation of second kind. Sai et al. [8] investigated the suction injection effects on laminar incompressible fluid flow taken in a duct. The transverse magnetic field was taken normal to the non-conducting walls of the duct. They have given analytical solutions for magnetic field and velocity, which are helpful for finding the electric field strength and current density. Yakhot et al. [9] discussed numerically the viscous laminar incompressible pulsating fluid flow in a rectangular duct in which motion was caused by pressure difference. Non-dimensional parameters were used for different flow regimes, which were based on the width of the duct and frequency of the imposed pressure gradient oscillations.

Krasilnikov [10] investigated the electro-dynamical design of antennas placed on a plane interface. It was also found that electric and induction fields of horizontal dipole lying in the interface plane were the same as those of a dipole in a vacuum. Mousavi et al. [11] numerically investigated a non-uniform magnetic field, which was induced by a wire carrying electric current on bio-magnetic fluid flow in a duct under the influence of the MHDs and ferro-hydrodynamics. The effect of magnetic field on bio-magnetic fluid flow in different percentages of the duct has been discussed with the variation of the strength of magnetic field. A magnetic field has important effects on the wall of shear stress as magnetic field has sufficient strength, which causes the recirculation areas downstream of contraction to become smaller. Ferdows et al. [12] discussed bio-magnetic fluid’s heat transfer and flow in a stretching cylinder, which dealt with the principle of MHD. The blood was taken to be an electrically conducting fluid that simultaneously exhibited polarization. Recent numerical investigations on several fluid flows can be studied from previously published studies [13–26].

Ducts are commonly required for supply, return or exhaust of air, used for heating, and also considered necessary for ventilation or air conditioning in massive building structures. There is a tradition to construct skyscrapers of tens of story because of the rapidly increase in population particularly in the industry concentrated region. They are normally used vertical ducts for removing air pollution having dust particles. To the author’s best knowledge, no researcher has investigated the complex interaction of micropolar flow in a vertical duct with a current-carrying wire placed nearby. Therefore, this study is focused at presenting a primary understanding of the impact of variable magnetic field on the micropolar nature of the flow, for which Eringen’s model has been used. The governing equations have been re-casted into the dimensionless form and are solved numerically by incorporating a finite volume methodology. The relevant work [27–34] also describes the duct flows under different assumptions. It has been observed, in the concerned work, that in the absence of a magnetic field, the flow is symmetric in x and y directions, and the velocity becomes maximum at the center of the rectangular-shaped duct, while the symmetry is distorted with strengthened magnetic field, generated by the current-carrying wire.

2 Problem formulation

We assume the steady, two-dimensional, and incompressible micropolar flow (driven by an external pressure gradient) through a rectangular duct with uniform cross-section, under the influence of an external magnetic field generated by a current-carrying wire placed nearby duct, as shown in Figure 1. Naturally, the Cartesian coordinate system is our convenient choice for writing the governing equations, as the sides of the duct coincide with the coordinate planes.

$Figure 1 (a) Schematic diagram for the micropolar flow in a vertical duct with a current-carrying wire located at ( 0 , − ε 0 , z ) (0,\hspace{.25em}-{\varepsilon }_{0},\hspace{.25em}z) , (b) a cross-section of the physical model showing the magnetic field generated due to wire, and (c) physical domain computational domain.$

Figure 1

(a) Schematic diagram for the micropolar flow in a vertical duct with a current-carrying wire located at ( 0 , − ε 0 , z ) , (b) a cross-section of the physical model showing the magnetic field generated due to wire, and (c) physical domain computational domain.

We adopt the Eringen theory [35–38] to develop the mathematical model. According to this theory, micro-motions and structure of the fluid elements show the non-Newtonian behavior. This theory basically describes the non-Newtonian nature of the micropolar fluid. The body as well as stress moments and spin inertia is strengthened by this fluid. However, micropolar theory is complicated as compared to the constitutive linear theory. Micropolar fluid not only demonstrates the microrotational inertia but it also incorporates microrotational effects. Mathematical framework is not so much complicated for such type of fluids.

The mathematical model may be described as:

(1) ∂ p ∼ ∂ ξ = 0 ,

(2) ∂ p ∼ ∂ η = 0 ,

(3) ∂ p ∼ ∂ z + κ ∂ ω 1 ∼ ∂ ξ − ∂ ω 2 ∼ ∂ η + ( μ + κ ) ∂ 2 w ∼ ∂ ξ 2 + ∂ 2 w ∼ ∂ η 2 − σ B ¯ 2 w ∼ = 0 ,

(4) − 2 κ ω 1 ∼ + κ ∂ w ∼ ∂ η − γ ∂ ∂ η ∂ ω 2 ∼ ∂ ξ − ∂ ω 1 ∼ ∂ η + ( α + β + γ ) ∂ ∂ ξ ∂ ω 1 ∼ ∂ ξ + ∂ ω 2 ∼ ∂ η = 0 ,

(5) − 2 κ ω 2 ∼ − κ ∂ w ∼ ∂ ξ + γ ∂ ∂ ξ ∂ ω 2 ∼ ∂ ξ − ∂ ω 1 ∼ ∂ η + ( α + β + γ ) ∂ ∂ η ∂ ω 1 ∼ ∂ ξ + ∂ ω 2 ∼ ∂ η = 0 .

An obvious consequence of the aforementioned Eqs. (1) and (2) is that p ∼ = p ∼ ( z ) . For the fully developed flow, it is well known that the pressure gradient is invariant along the axial direction, i.e.,

∂ p ∼ ∂ z = Constant = p 0 ( say ) .

Furthermore, the magnetic field induction B ¯ is given as: B ¯ = μ ¯ 0 H ¯ , with H ¯ = γ ˜ 2 π 1 ( ξ − ξ 0 ) 2 + ( η − η 0 ) 2 ,

where ( ξ 0 , η 0 ) is the location of the wire in the cross-section shown in Figure 1a. For the present problem, the wire is located at ( 0 , − ε 0 ) , which leads to H ¯ = γ ˜ 2 π 1 ξ 2 + ( η + ε 0 ) 2 .

It is to point out that μ , κ , α , β , and γ are the material constants and satisfy the following relations:

κ ≥ 0 , 2 μ + κ ≥ 0 , 3 α + β + γ ≥ 0 , γ ≥ | β | .

Following the dimensionless coordinates are introduced: x = ξ a , y = η a , z = ς a , w = − μ p 0 a 2 w ∼ , ω 1 = − μ p 0 a ω 1 ∼ , ω 2 = − μ p 0 a ω 2 ∼ , H = H ¯ H ¯ 0 with H ¯ 0 = γ ˜ 2 π ε 0 ,

which lead to:

(6) − 1 + N 1 − N − ∂ ω 2 ∂ x + ∂ ω 1 ∂ y − 1 1 − N ∂ 2 w ∂ x 2 + ∂ 2 w ∂ y 2 + M n H 2 w = 0 ,

(7) ω 1 − 1 2 ∂ w ∂ y + 2 − N m 2 ∂ ∂ y ∂ ω 2 ∂ x − ∂ ω 1 ∂ y − 1 l 2 ∂ ∂ x ∂ ω 1 ∂ x + ∂ ω 2 ∂ y = 0 ,

(8) ω 2 − 1 2 ∂ w ∂ x − 2 − N m 2 ∂ ∂ x ∂ ω 2 ∂ x − ∂ ω 1 ∂ y − 1 l 2 ∂ ∂ y ∂ ω 1 ∂ x + ∂ ω 2 ∂ y = 0 ,

where l 2 = 2 a κ α + β + γ , N = κ μ + κ ( 0 ≤ N ≤ 1 ), and m 2 = a 2 κ ( 2 μ + κ ) γ ( μ + κ ) are the micropolar parameters, whereas H ( x , y ) = 1 x 2 + ( y + ε ) 2 with ε = ε 0 a .

Figure 1c shows that there exists a linear relationship between the dimensions of the physical and computational domains. Furthermore, the appropriate boundary conditions in the dimensionless form are given by:

(9) w = 0 at x = ± 1 and y = ± 1 ,

(10) ω 1 = ω 2 = 0 at x = ± 1 and y = ± 1 .

The aforementioned boundary constraints together with governing equations are solved numerically, using a finite volume approach, which is discussed below in detail.

3 Finite volume method

The general form of governing Eqs. (6)–(8) is as follows:

(11) a ∂ 2 f ∂ 2 x + b ∂ 2 f ∂ 2 y + c ∂ 2 f ∂ x ∂ y = g ( x , y ) ,

where a , b , and c are the constants and g ( x , y ) is a known function. For the numerical solution, a structured mesh is considered that describes the domain of vortex-based finite volume method (Figure 2). We integrate Equation (10) at general point P by assuming a rectangular-shaped control volume about P , as follows:

(12) ∬ Ω a ∂ 2 f ∂ x 2 + b ∂ 2 f ∂ y 2 + c ∂ 2 f ∂ x ∂ y d A = ∬ Ω g ( x , y ) d A ,

(13) a ∬ Ω ∂ 2 f ∂ x 2 d A + b ∬ Ω ∂ 2 f ∂ y 2 d A + c ∬ Ω ∂ 2 f ∂ x ∂ y d A = ∬ Ω g ( x , y ) d A .

Figure 2

Rectangular-shaped control volume.

The rectangular-shaped region Ω is characterized as:

(14) x w ≤ x ≤ x e , y s ≤ y ≤ y n .

The aforementioned equation becomes:

(15) a ∫ x w x e ∫ y s y n ∂ 2 f ∂ x 2 d y d x + b ∫ x w x e ∫ y s y n ∂ 2 f ∂ y 2 d y d x + c ∫ x w x e ∫ y s y n ∂ 2 f ∂ x ∂ y d y d x = ∫ x w x e ∫ y s y n g ( x , y ) d y d x .

Now, the aforementioned integrals are evaluated one by one, as follows:

(16) ∫ x w x e ∫ y s y n ∂ 2 f ∂ x 2 d yd x = ∫ y s y n ∫ x w x e ∂ 2 f ∂ x 2 d x d y ,

(17) = ∫ y s y n ∂ f ∂ x x e − ∂ f ∂ x x w d y ,

(18) = ∫ y s y n f E − f P Δ x d y − ∫ y s y n f P − f W Δ x d y ,

(19) = f E − f P Δ x ( y n − y s ) − f P − f W Δ x ( y n − y s ) ,

(20) = Δ y Δ x { f E − f P − f P + f W } ,

(21) = Δ y Δ x { f E − 2 f P + f W } .

Now, the second term

(22) ∫ x w x e ∫ y s y n ∂ 2 f ∂ y 2 d y d x = ∫ x w x e ∂ f ∂ y y n − ∂ f ∂ y y s d x = ∫ x w x e f N − f P Δ y − f P − f S Δ y d x ,

(23) = ∫ x w x e f N − 2 f P + f s Δ y d x ,

(24) = f N − 2 f P + f s Δ y ( x e − x w ) ,

(25) = Δ x Δ y { f N − 2 f P + f S } ,

and

(26) ∫ x w x e ∫ y s y n ∂ 2 f ∂ x ∂ y d y d x = ∫ x w x e ∂ f ∂ x y n − ∂ f ∂ x y s d x = ∫ x w x e f ne − f nw Δ x − f se − f sw Δ x d x ,

(27) = 1 Δ x 1 4 f NE − 1 4 f NW − 1 4 f SE + 1 4 f SW Δ x ,

(28) = 1 4 ( f NE − f NW − f SE + f SW ) .

Finally,

a Δ y Δ x { f E − 2 f P + f W } + b Δ x Δ y { f N − 2 f P + f S } + c 1 4 ( f NE − f NW − f SE + f SW ) = g P Δ x Δ y .

Multiplying both sides by 1 / Δ x Δ y refers to:

(29) a f E − 2 f P + f W Δ x 2 + b f N − 2 f P + f S Δ y 2 + c f NE − f NW − f SE + f SW 4 Δ x Δ y = g P .

The system of algebraic equations, on the pattern of the aforementioned difference equation, is finally solved numerically. For this purpose, we have many options, for example, the successive over relaxation (SOR) method, the conjugate gradient method, and the steepest descent method. But, in this study, we have opted the SOR method due to its inherent simplicity, elegance in programming, and reasonable efficiency. Figure 3 represents the schematic representation of the algorithm.

Figure 3

Schematic diagram of the numerical algorithm.

The efficiency of the code is examined by comparing the results (in Table 1) obtained from the finite volume method (present case) with those obtained from the spectral method (Ali et al. [39]). The results are compared in limiting cases when there are no magnetic field and no current-carrying wire. In order to compare the results, we assumed the following solution of the problem:

(30) θ ( ζ 1 , ζ 2 ) = ∑ j = 1 n 2 ∑ i = 1 n 1 b i , j sin ( i π ζ 1 ) sin ( j π ζ 2 ) , W ( ζ 1 , ζ 2 ) = ∑ j = 1 n 2 ∑ i = 1 n 1 a i , j sin ( i π ζ 1 ) sin ( j π ζ 2 ) .

Table 1

Comparison with spectral method

ζ ₁	Temperature distribution along the line ζ ₂ = 0.5
ζ ₁	Spectral method [22]	Present results
0.1	−0.10526785	−0.10526707
0.2	−0.19630316	−0.19629774
0.3	−0.26491723	−0.26490668
0.4	−0.30715132	−0.30713710
0.5	−0.32137073	−0.32135521

The unknown coefficients a _i,j, and b _i,j are then calculated by exploiting the orthogonality of the polynomials, which is used to calculate the unknown coefficients a i , j and b i , j .

4 Results and discussion

The impacts of different parameters such as coupling parameter N, micropolar fluid parameter m, and the magnetic interaction parameter M _n have been discussed. Flow, microrotation, and temperature fields are calculated numerically by taking the region as a grid of mesh points (x _i, y _i), and the computational outcomes are expressed graphically. We have considered the following values for the non-dimensional parameters during our numerical simulations: N = 0.1, m = 5, L = 1, and ε = 0.2 unless otherwise stated.

The influence of the magnetic interaction parameter on the component w of velocity can be seen in Figure 4. It can be seen that in the absence of a magnetic field, the flow is symmetric in x and y directions, and at the center of the rectangle, the velocity is maximum. As the magnetic field is strengthened, the symmetry is distorted and flow becomes to slow down gradually. In order to gain further insight into the influence of the magnetic field on the velocity component, contours of the respective surfaces are also drawn (Figure 5). In this figure, it can be noted that the velocity profile varied from negative to positive as we strengthened the magnetic field with zero exactly at the dipole location. Initially, symmetry in the streamlines about the line y = 0.5 is noted; then, the imposed magnetic field not only shifts the velocity gradient to the extreme right corner away from the dipole but also makes the velocity field non-symmetric.

Figure 4

Surfaces of velocity component for (a) M _n = 0, (b)M _n = 10, (c) M _n = 20, (d) M _n = 30, (e) M _n = 40, and (f) M _n = 50.

Figure 5

Contours of velocity component for (a) M _n = 0, (b) M _n = 10, (c) M _n = 20, (d) M _n = 30, (e) M _n = 40, and (f) M _n = 50.

The influence of the magnetic field strength on the first component of microrotation u may be seen in Figure 6. It is easy to see that the rotation is positive in one part of the domain and negative in the remaining part. The distribution is obviously symmetric across the line x = 0.5 in the absence of the magnetic field. However, as the magnetic field is strengthened, the symmetry is distorted, with positive components being reduced significantly and occupying a vast area of the domain. On the other hand, the negative component is magnified and is pushed toward the wall x = 1. In order to gain further insight into the influence of the magnetic field on the first component of the microrotation, contours of the respective surfaces are also drawn (Figure 7). In this figure, the red and blue contours represented the positive and negative parts of the component of microrotation, with green lines reflecting the area where the microrotation is almost zero. For the positive part, the maxima is shifted away from the wall x = 0, with the magnetic field. Moreover, the region with almost zero rotation is also localized near the wall x = 1 .

Figure 6

Surfaces of the first microrotation component for (a) M _n = 0, (b) M _n = 10, (c) M _n = 20, (d) M _n = 30, (e) M _n = 40, and (f) M _n = 50.

Figure 7

Contours of the first microrotation component for (a) M _n = 0, (b) M _n = 10, (c) M _n = 20, (d) M _n = 30, (e) M _n = 40, and (f) M _n = 50.

The effect of magnetic field strength on the second component of microrotation v may be seen in Figure 8. It can be seen that the rotation is negative in one part of the domain and positive in the remaining part. The distribution is symmetric across the line y = 0.5 in the absence of the magnetic field. However, as the magnetic field is strengthened, the symmetry is distorted with the negative component being reduced and pushed toward the wall y = 1. On the other hand, the positive component reduced significantly and occupied a vast area of the domain. In addition, the influence of the magnetic field on the second component of microrotation, contours of the respective surfaces, are also drawn (Figure 9). In this figure, the red and blue contours represented the positive and negative parts of the component of microrotation, with green lines reflecting the area where the microrotation is almost zero. For the positive part, the maxima are shifted away from the wall y = 0, with the magnetic field. Moreover, the region with almost zero rotation is also localized near the wall y = 1.

Figure 8

Surfaces of the second microrotation component for (a) M _n = 0, (b) M _n = 10, (c) M _n = 20, (d) M _n = 30, (e) M _n = 40, and (f) M _n = 50.

Figure 9

Contours of the second microrotation component for (a) M _n = 0, (b) M _n = 10, (c) M _n = 20, (d) M _n = 30, (e) M _n = 40, and (f) M _n = 50.

Figure 10(a)–(c) portrays the velocity and microrotation for various estimations of magnetic field. Velocity got reduced for large values of M _n. Similarly, the first microrotation component u first increases along the line y = 0.5 (Figure 10(b)) and then decreases. It has been noted that microrotation v decreases or increases symmetrically near the both sides of rectangle.

Figure 10

(a) Velocity component along the line y = 0.5, (b) first microrotation component along the line y = 0.5, and (c) second microrotation component along the line y = 0.5.

The effect of the magnetic field M _n on the microrotation and velocity is shown in Figure 11(a)–(c). The velocity component decreases and the first microrotation component u (Figure 11(b)) enhances near to one side of the rectangle and depreciates near to the other side of the rectangle symmetrically. Similarly, the second microrotation component v first increases along the line x = 0.5 and then decreases for large values of a magnetic field.

Figure 11

(a) Velocity component along the line x = 0.5, (b) first microrotation component along the line x = 0.5, and (c) second microrotation component along the line x = 0.5.

5 Conclusion

An inclusive analysis of electrically conducting microstructured fluid flow in a rectangular duct is presented in the recent work. Two dielectric regions are placed in such a way that the electric dipole is situated between these two regions. The finite volume method is developed to determine the numerical solution of the problem. The influences of magnetic field on microrotation and velocity have been discussed through surfaces and contours. It is shown that in the absence of a magnetic field, the flow is symmetric in x and y directions and velocity is maximum at the center of the rectangle, while symmetry is distorted with strengthened magnetic field and flow becomes to slow down gradually. The impact of magnetic field on the first and second microrotation components has been discussed as rotations are positive in one part of the domain and negative in the remaining part. The imposed variable magnetic field adds to the spinning of particles in one part of the duct, while simultaneously suppressing it in the other region. In the future, the existing method might be used in a number of physical and technical obstacles [40–46].

Parameter expressions

a and b are the lengths of the rectangle sides, D = 2 ab a + b , which is the hydraulic diameter, σ = b a , which is the aspect ratio, B 0 is the magnetic induction, N = κ μ + κ , which is the coupling number, m ² = a 2 κ ( 2 μ + κ ) γ ( μ + κ ) , which is the micropolar parameter, and p is the fluid pressure.

Acknowledgments

This research was funded by Deanship of Scientific Research (Project no. RGP. 2/108/43), King Khalid University, Abha, KSA.

Funding information: This research was funded by Deanship of Scientific Research (Project no. RGP. 2/108/43), King Khalid University, Abha, KSA.
Author contributions: Kashif Ali and Sohail Ahmad formulated the problem. Kashif Ali, Sohail Ahmad, Hina Bashir, and Wasim Jamshed solved the problem. Hua Bian, Kashif Ali, Sohail Ahmad, Hina Bashir, Wasim Jamshed, Kashif Irshad, Mohammed K. Al Mesfer, Mohd Danish, and Sayed M El Din computed and scrutinized the results. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2023-01-31

Accepted: 2023-09-19

Published Online: 2023-11-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/rams-2023-0134

Keywords for this article

pressure gradient; current-carrying wire; micro-fluid structure; duct; finite volume approach

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