Study of weakly nonlinear double-diffusive magneto-convection with throughflow under concentration modulation

Atul Jakhar; Anand Kumar; Priyanka Joshi

doi:10.1515/nleng-2024-0013

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Study of weakly nonlinear double-diffusive magneto-convection with throughflow under concentration modulation

Atul Jakhar , Anand Kumar and Priyanka Joshi

Published/Copyright: August 9, 2024

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From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

This article aims to study double-diffusive magneto-convective flow of electrically conducting and Newtonian fluid in the presence of throughflow and concentration modulation. Here, two infinite horizontal plates have been considered with heated from below and cooled and salted from above. The flow is also influenced by the induced magnetic field for which a constant magnetic field is applied in the perpendicular direction to the plates and vertically upward direction. A weakly nonlinear analysis is used to obtain the expression of heat and mass transport rate using Ginzburg–Landau equation. The influence of various physical parameters on Nusselt and Sherwood numbers is presented by graphs. From the numerical outcome, it is found that Péclet, Chandrasekhar, and magnetic Prandtl numbers enhance the mass and heat transport rate, while Lewis number increases only the rate of mass transport. The major result of this study is that the onset of convection postpones in the presence of throughflow and magnetic field.

Keywords: heat transfer; mass transfer; double diffusive convection; concentration modulation; MHD; horizontal plates; throughflow

Nomenclature

A: amplitude of convection
g →: gravitational acceleration
H →: induced magnetic field vector
Le: Lewis number
p →: pressure
Pm: magnetic Prandtl number
q →: velocity vector
Q: Chandrasekhar number
Ra T: thermal Rayleigh number
Ra S: solute Rayleigh number
S: concentration of fluid
Sc: Schmidt number
T: temperature
Pe: Péclet number
( x , y , z ): coordinate axis
: subscripts
δ 1: amplitude of modulation
b: basic state
ρ: density
Ω: frequency of modulation
ε: perturbation parameter
ϕ: phase angle
μ m: magnetic permeability
ν m: magnetic diffusivity
β S: mass expansion coefficient
0: reference value
τ: slow time
κ S: solute diffusivity
ψ: stream function
κ T: thermal diffusivity
β T: thermal expansion coefficient
δ: total wave number
μ: viscosity
: superscripts
*: dimensionless quantities
′: perturbed quantities

1 Introduction

In numerous physical scenarios, the gradient of solute concentration exhibits variations across both temporal and spatial dimensions. This gradient, referred to as concentration modulation, can be achieved by solving the concentration equation using selected time-dependent solute boundary conditions. This modulation has the potential to influence and manage the mass and heat transfer within convective systems. The impact of concentration modulation on magneto-convection, with its relevance and utility spanning various domains of science, engineering, and technology, holds significance. Magneto-convection finds extensive industrial applications as well. Some of them are in production of magnetic fluids and in oil reservoirs, where the magnetic field influences the geothermal area of the earth (Wallace et al. [1]). This article will also offer assistance in examining convective issues within the Earth’s core, where molten fluids containing solutes exhibit electrical conductivity. Exploring the combined effects of magnetic fields and concentration modulation holds potential value for material science studies.

Bejan [2] extensively explored the intricacies of heat transfer in fluids. Griffiths [3], on the other hand, conducted experiments on double diffusive convection (DDC) in a Hele Shaw cell. His observations included the identification of a thin diffusive interface within the Hele Shaw cell and verified later through experiments in a laboratory for porous medium. The substances under investigation in Griffiths’ research were sugar and salt. Baig et al. [4] outlined how the Rayleigh number influences DDC in a porous annulus. They employed the finite element method to explain the solution with a focus on convergence. Additionally, their findings indicated that reducing the Rayleigh number enhances the rate of convergence. Iqbal et al. [5] studied the flow of a nanofluid over a stretching sheet in the presence of a magnetic field. They used a more advanced model called Cattaneo-Christov double-diffusion theory, instead of the more common Fourier’s and Fick’s laws, to describe how heat and particles move within the liquid. This means that the flow can be affected by external factors such as changes in temperature, concentration, rotation and even g-forces, allowing for more precise control. Researchers such as Rosenblatt and Tanaka [6] explored how tiny vibrations, or “g-jitter,” affect the stability of heat flow in fluids. Their work focused on a specific type of fluid movement called thermal convection. DDC under time-dependent temperature modulation is carried out by Bhadauria et al. [7], Siddheshwar et al. [8], and Bhadauria [9]. Later, Kumar et al. [10] also examined how the time-dependent gravity field in a couple-stress liquid affected DDC. It has been discovered that controlling the right tuning of modulation components may regulate the flow of mass and heat transport.

The production of magnetic fluids, fluid movement through ducts, oil reserves, and geophysical applications makes the convection–diffusion process in electrically conducting liquids very interesting. A detailed work on magneto-convection was well reported by Chandrasekhar [11]. Thompson [12] provided an illustration of the impact of a magnetic field on the modified Rayleigh–Jeffreys flow thermal convective theory. He used Jeffrey’s neutral stability approach to study oscillatory convection. In 1959, Nakagawa [13] conducted experiments to study the impact of magnetic field on a rotating fluid layer of mercury, verifying the theoretical findings of Chandrasekhar [11] that the transition between convection and overstability discontinued at a critical field strength. Additionally, Rudraiah [14] expanded the research of magneto-convection to include a double diffusing component. He engaged about the double diffusive magneto-convection neutral stability criteria. Srivastava et al. [15] examined how a magnetic field affected an anisotropic porous media under the influence of the Soret effect. According to their study, the Chandrasekhar number is in opposition to the heat transmission rate. Subsequently, researchers have focused on studying magneto-convection in various boundary circumstances and physical configurations. To investigate the impacts of time-periodic boundary conditions on magneto-convection, they have taken into account the g-jitter effect, temperature modulation, rotational modulation, and magnetic modulation. Several scholars have examined the effects of various boundary conditions on magneto-convection. These include Siddheshwar and Pranesh [16], Aniss et al. [17], Bhadauria and Srivastava [18], Kumar and Singh [19], Bhadauria and Kiran [20], Keshri et al. [21], Gupta et al. [22], Kushwaha et al. [23], and Manjula et al. [24].

Thermal buoyancy and solute buoyancy are the two main mechanisms that occur in DDC. Thermal buoyancy is created by density differences between warmer and cooler fluid areas, whereas solute buoyancy is caused by density changes induced by variations in solute concentration. Complex fluid flows can result from these buoyancy effects. Throughflow occurs when the interaction of these buoyant forces results in the formation of circulation patterns or flows within the fluid medium. This can include the upward movement of warm, less dense fluid and the downward flow of cool, denser fluid, as well as the movement of solute-rich and solute-poor fluid regions. These flows lead to the total mass and heat transmission within the system. Shivkumara and Khalili [25] investigated the instability analysis of DDC with throughflow and found that throghflow has a destabilizing effect on system even if the lower and upper boundaries are of the same type. Nield and Kuznetsov [26] worked on vertical cylinder with throughflow in DDC. They have investigated both oscillatory and non-oscillatory scenarios. Kumari and Murthy [27] investigated throughflow in vertical porous channel with DDC and found that the parameters Péclet number, buoyancy ratio, and Lewis number all have an effect on the neutral stability curves, as well as the critical wave number and Rayleigh number. Some of the researchers who had investigated the impact of different boundary conditions in the presence of throughflow are Barman et al. [28], Honnappa et al. [29], and Tripathi and Mahajan [30].

The preceding literature discusses two diffusing components in various physical circumstances such as fluid layers, porous media, Brinkman–Darcy porous medium and diverse boundary conditions. Each of them solely took spatial coordinates into account while determining the solutal boundary criteria. However, this is not always the case; in many physical events, space and time work together to determine the singular boundary conditions. We refer to this state as concentration modulation. It has pertinent important uses in food processing, ocean research, material science, and oil reservoirs. Gupta et al. [31] studied the effect of concentration/gravity modulation on mass transfer in a rotating fluid layer. Samah et al. [32] and Rudziva et al. [33] studied mass and heat transport rate numerically under rotation modulation. Furthermore, Gupta [34] extended the report of Gupta et al. [31] for couple stress liquid. He has solved the governing equations using the asymptotic analysis approach. The following are a few reports of studies on concentration modulation: Keshri et al. [35,36], Kumar et al. [37], Jakhar and Kumar [39], and Jakhar et al. [40,41]. Without taking the temperature profile into account, they examined the effects of concentration modification in fluid layers for several physical models. In several fields, including geophysics, material science, oil reservoir research, earth’s core study, and many more but the concentration modulation in the presence of temperature profiles have vast applications.

To the best of the author’s knowledge, no literature exists that clarifies how time-dependent concentration affects mass and heat transfers in double diffusive magneto-convection with throughflow. The present study will focus on Rayleigh–Bénard convection geometry because of its numerous applications in science, engineering, and industry. To enhance fluid mixing by causing convective flow – a highly valuable technique in heat exchangers, cooling systems, thermal management applications, thermal insulation systems and space exploration are some of them. Furthermore, natural convection has used in microgravity settings such as satellite applications and space laboratories. Scholars have recently discussed the distribution of mass and heat transfers under spatial coordinate-dependent concentration in double diffusive magneto-convection. The current work sketches the role of time-dependent concentration in the evolution of transport phenomena. Modulated concentration under magnetic field is used in magnetic resonance imaging to enhance contrast, separation of complex mixtures, drug delivery system, magnetic hyperthermia, magnetic sensors, magnetic data storage, and many other applications in life sciences and technology. We are motivated in carrying out this study because of the numerous applications of this research in various fields of science, engineering, and technology.

2 Mathematical formulation

A viscous incompressible and electrically conductive fluid is considered for the laminar flow between two infinitely horizontal parallel plates with a span of “d”. The fluid undergoes a heating process from the lower boundary, while cooled and salted from the above. The parallel plates are aligned with x -axis and the z -axis taken perpendicular to the plates in the upward direction. A constant magnetic field of strength H 0 is applied along the positive z -direction, while the gravity force is exerted in the downward z -direction. The vertical upward direction will serve as the reference frame for the throughflow, denoted as w 0 . The geometric representation of the problem is visually depicted in Figure 1. Using the Boussinesq approximation, the mathematical formulations of the complex dynamics of fluid motion, energy, solutal, and induction equations are expressed in the following manner (Siddheshwar et al. [8], Kumar et al. [10], Gupta et al. [31,34]):

(1) ∇ ⋅ q → = 0 ,

(2) ∇ ⋅ H → = 0 ,

(3) ρ 0 ∂ q → ∂ t + ( q → ⋅ ∇ ) q → = − ∇ p → + ρ g → + μ ∇ 2 q → + μ m ( H → ⋅ ∇ ) H → ,

(4) ∂ T ∂ t + ( q → ⋅ ∇ ) T = κ T ∇ 2 T ,

(5) ∂ S ∂ t + ( q → ⋅ ∇ ) S = κ S ∇ 2 S ,

(6) ∂ H → ∂ t + ( q → ⋅ ∇ ) H → − ( H → ⋅ ∇ ) q → = ν m ∇ 2 H → ,

(7) ρ = ρ 0 [ 1 − β T ( T − T 0 ) + β S ( S − S 0 ) ] ,

where g → = ( 0 , 0 , − g ) .

Figure 1

Geometrical representation of problem.

The following are the boundary conditions based on the model under consideration:

(8) T = T 0 + Δ T , S = S 0 − Δ S 2 [ 1 − ε 2 δ 1 cos ( Ω t ) ] , at z = 0 ,

(9) T = T 0 , S = S 0 + Δ S 2 [ 1 + ε 2 δ 1 cos ( Ω t + ϕ ) ] , at z = d .

This article demonstrates the effect of three types of solutal modulations:

Symmetry (in-phase modulation (IPM), ϕ = 0 ),
Asymmetry (out-phase modulation (OPM), ϕ = π ), and
Upper plate modulation (UPM), ϕ = − ι ∞ .

2.1 Basic state solution

The basic state solution can be found by taking

(10) q → = ( 0 , 0 , w 0 ) , ρ = ρ b ( z ) , p = p b ( z ) , T = T b ( z ) , and S = S b ( z , t ) .

Now, substituting Eqs. (10) in the system of Eqs. (1)–(7), we obtain:

(11) − ∇ p b + ρ b g → = 0 ,

(12) ∇ 2 T b − w 0 k T ∂ T b ∂ z = 0 ,

(13) κ s ∇ 2 S b − w 0 ∂ S b ∂ z = ∂ S b ∂ t .

Applying the boundary conditions Eqs. (8)–(9) in the above Eqs. (11)–(13), we obtain the basic state solutions

(14) T b = T 0 + Δ T e Pe z − e Pe 1 − e Pe ,

(15) S b = S s + ε 2 δ 1 f ,

where

S s = S 0 + Δ S 2 1 + e Pe − 2 e Pe z 1 − e Pe , f = Δ S 2 Re [ G e m z + J e − n z ] e − ι ω t , G = e − ι ϕ − e n 2 ( e m − e n ) , J = − e − ι ϕ + e m 2 ( e m − e n ) , λ 0 = P e 2 , λ 1 = ( 1 − ι ) ω 2 , m = λ 0 + λ 0 2 + 4 λ 1 2 2 , n = λ 0 − λ 0 2 + 4 λ 1 2 2 , and Pe = d w 0 k s .

Now, define the finite amplitude perturbation on the basic state of the form of

(16) q → = q b → + q ′ → , S = S b + S ′ , T = T b + T ′ , p → = p b → + p ′ → , ρ = ρ b + ρ ′ , and H → = H 0 + H ′ → .

By Eqs. (16) and (3)–(6), we obtain the following equations:

(17) ρ 0 ∂ q ′ → ∂ t + ( q ′ → ⋅ ∇ ) q ′ → = − ∇ p ′ → + ρ ′ g → + μ ∇ 2 q ′ → + μ m ( H ′ → ⋅ ∇ ) H ′ → + μ m H 0 ∂ H ′ → ∂ z ,

(18) ∂ T ′ ∂ t + ( q ′ → ⋅ ∇ ) T ′ + w 0 ∂ T ′ ∂ z + w ′ ∂ T b ∂ z = κ T ∇ 2 T ′ ,

(19) ∂ S ′ ∂ t + ( q ′ → ⋅ ∇ ) S ′ + w 0 ∂ T ′ ∂ z + w ′ ∂ S b ∂ z = κ S ∇ 2 S ′ ,

(20) ∂ H ′ → ∂ t + ( q ′ → ⋅ ∇ ) H ′ → − ( H ′ → ⋅ ∇ ) q ′ → − H 0 ∂ q ′ → ∂ z + w 0 ∂ H ′ ∂ z = ν m ∇ 2 H ′ → .

The following non-dimensional parameters have been defined:

Sc = ν κ S , Ra T = g β T Δ T d 3 ν κ T , Le = κ T κ S , T ′ = ( Δ T ) T * , Ω = ω * κ S d 2 , Ra S = g β S Δ S d 3 ν κ S , q ′ → = κ S q * d , t = d 2 t * κ S , S ′ = ( Δ S ) S * , ψ ′ = κ S ψ * , H ′ → = H 0 H * , ϕ ′ = d H 0 ϕ * , ( x ′ , z ′ ) = d ( x * , z * ) , Q = μ m H 0 2 d 2 ρ 0 ν κ S ,

where asterisks indicate the variables at dimensionless state.

Now, eliminating the pressure impact in Eq. (17) by considering the curl on both sides and introducing stream function u ′ = ∂ ψ ∂ z , w ′ = − ∂ ψ ∂ x and potential function H x = ∂ ϕ ∂ z , H z = − ∂ ϕ ∂ x . When the pressure term is removed from Eq. (17) and the aforementioned dimensionless parameters are used without the asterisks, we obtain the following set of equations:

(21) 1 Sc ∂ ∇ 2 ψ ∂ t + PeLe ∂ ( ∇ 2 ψ ) ∂ z − ∂ ( ψ , ∇ 2 ψ ) ∂ ( x , z ) = − Ra T Le ∂ T ∂ x + Ra S ∂ S ∂ x + ∇ 4 ψ − Q Pm ∂ ( ϕ , ∇ 2 ϕ ) ∂ ( x , z ) + Q Pm ∂ ∇ 2 ϕ ∂ z ,

(22) PeLe ∂ T ∂ z − Le ∇ 2 T = ∂ ( ψ , T ) ∂ ( x , z ) + ∂ ψ ∂ x ∂ T b ∂ z − ∂ T ∂ t ,

(23) PeLe ∂ S ∂ z − ∇ 2 S = ∂ ψ ∂ x ∂ S b ∂ z − ∂ S ∂ t + ∂ ( ψ , S ) ∂ ( x , z ) ,

(24) PeLe ∂ ϕ ∂ z − Pm ∇ 2 ϕ = ∂ ( ψ , ϕ ) ∂ ( x , z ) − ∂ ψ ∂ z .

The boundary conditions now become

(25) ψ = ∂ 2 ψ ∂ z 2 = T = S = 0 , z = 0 , 1 .

We used the slow timescale τ = ε 2 t in the aforementioned equations for the weakly nonlinear stability analysis, and we represented Eqs. (21)–(24) into matrix form as:

(26) − ∇ 4 + PeLe Sc ∂ ( ∇ 2 ) ∂ z Ra T Le ∂ ∂ x − Ra S ∂ ∂ x Q Pm ∂ ∂ z ∇ 2 ∂ T b ∂ z ∂ ∂ x − Le ∇ 2 + PeLe ∂ ∂ z 0 0 − ∂ S s ∂ z ∂ ∂ x 0 PeLe ∂ ∂ z − ∇ 2 0 − ∂ ∂ z 0 0 PeLe ∂ ∂ z − Pm ∇ 2 ψ T S ϕ = 1 Sc ∂ ( ψ , ∇ 2 ψ ) ∂ ( x , z ) − ε 2 Sc ∂ ( ∇ 2 ψ ) ∂ τ − Q Pm ∂ ( ϕ , ∇ 2 ϕ ) ∂ ( x , z ) ∂ ( ψ , T ) ∂ ( x , z ) − ε 2 ∂ T ∂ τ − ∂ ψ ∂ x ε 2 δ 1 f + ∂ ( ψ , S ) ∂ ( x , z ) − ε 2 ∂ S ∂ τ ∂ ( ψ , ϕ ) ∂ ( x , z ) − ε 2 ∂ ϕ ∂ τ .

We adopted a weakly nonlinear analysis using an asymptotic approach for the rate of mass and heat transport to investigate the instability. Therefore, the following enhanced power series of the perturbing parameter ( ε ≪ 1 ) uses the dependent variables ψ , ϕ , T , S , and Ra S (Siddheshwar and Pranesh [16], Bhadauria and Kiran [20], and Keshri et al. [21]):

(27) ψ = ε ψ 1 + ε 2 ψ 2 + ε 3 ψ 3 + ⋯ , S = ε S 1 + ε 2 S 2 + ε 3 S 3 + ⋯ , T = ε T 1 + ε 2 T 2 + ε 3 T 3 + ⋯ , ϕ = ε ϕ 1 + ε 2 ϕ 2 + ε 3 ϕ 3 + ⋯ , Ra S = Ra S 0 + ε 2 Ra S 2 + ε 4 Ra S 4 + ⋯ .

Substituting Eq. (27) into Eq. (26) and comparing the equal powers of ε on both sides, we will obtain distinct order cases.

2.2 First-order case

(28) − ∇ 4 + PeLe Sc ∂ ( ∇ 2 ) ∂ z Ra T Le ∂ ∂ x − Ra S ∂ ∂ x Q Pm ∂ ∂ z ∇ 2 ∂ T b ∂ z ∂ ∂ x − Le ∇ 2 + PeLe ∂ ∂ z 0 0 − ∂ S s ∂ z ∂ ∂ x 0 PeLe ∂ ∂ z − ∇ 2 0 − ∂ ∂ z 0 0 PeLe ∂ ∂ z − Pm ∇ 2 ψ 1 T 1 S 1 ϕ 1 = 0 0 0 0 .

The solution of Eq. (28) is obtained as follows:

(29) ψ 1 = A ( τ ) sin ( k c x ) sin ( π z ) , T 1 = − 4 π 2 A ( τ ) k c L e δ 2 β 1 2 cos ( k c x ) sin ( π z ) , S 1 = 4 π 2 A ( τ ) k c δ 2 β 2 2 cos ( k c x ) sin ( π z ) , ϕ 1 = A ( τ ) π P m δ 2 sin ( k c x ) cos ( π z ) ,

where the total wave number is δ 2 = k c 2 + π 2 , β 1 2 = Pe 2 + 4 π 2 , and β 2 2 = Le 2 Pe 2 + 4 π 2 . The critical solute Rayleigh number from Eqs. (28)–(29) is obtained as:

(30) Ra S 0 = δ 2 ( δ 4 + Q π 2 ) 4 π 2 k c 2 − Ra T β 1 2 β 2 2 .

At Pe = Ra T = Q = 0 , Eq. (30) yields a reduction in the critical value of the solute Rayleigh number

(31) Ra S 0 = δ 6 k c 2 .

The expression of Eq. (31) is in accordance with the results reported by Jakhar and Kumar [39] and Chandrasekhar [42]. The modulation frequency and amplitude terms are absent from the solute Rayleigh number formula mentioned earlier. This indicates that linear stability analysis or stationary convection are not significantly affected by time-dependent concentration variation.

2.3 Second-order case

(32) − ∇ 4 + PeLe Sc ∂ ( ∇ 2 ) ∂ z Ra T Le ∂ ∂ x − Ra S ∂ ∂ x Q Pm ∂ ∂ z ∇ 2 ∂ T b ∂ z ∂ ∂ x − Le ∇ 2 + PeLe ∂ ∂ z 0 0 − ∂ S s ∂ z ∂ ∂ x 0 PeLe ∂ ∂ z − ∇ 2 0 − ∂ ∂ z 0 0 PeLe ∂ ∂ z − Pm ∇ 2 ψ 2 T 2 S 2 ϕ 2 = R 21 R 22 R 23 R 24 ,

where

(33) R 21 = 0 , R 22 = − 2 A 2 ( τ ) k c 2 π 3 δ 2 Le β 1 2 sin ( 2 π z ) , R 23 = 2 A 2 ( τ ) k c 2 π 3 δ 2 β 2 2 sin ( 2 π z ) , R 24 = − A 2 ( τ ) k c π 2 2 δ 2 Pm sin ( 2 k c x ) .

The solution of the aforementioned system can be obtained as:

(34) ψ 2 = 0 , T 2 = A 2 ( τ ) π 2 k c 2 [ Pe ( cos ( 2 π z ) − 1 ) − 2 π sin ( 2 π z ) ] Le 2 δ 2 β 1 4 , S 2 = A 2 ( τ ) π 2 k c 2 [ PeLe ( 1 − cos ( 2 π z ) ) + 2 π sin ( 2 π z ) ] δ 2 β 2 4 , ϕ 2 = − A 2 ( τ ) π 2 8 k c Pm 2 δ 2 sin ( 2 k c x ) .

2.4 Third-order case

(35) − ∇ 4 + PeLe Sc ∂ ( ∇ 2 ) ∂ z Ra T Le ∂ ∂ x − Ra S ∂ ∂ x Q Pm ∂ ∂ z ∇ 2 ∂ T b ∂ z ∂ ∂ x − Le ∇ 2 + PeLe ∂ ∂ z 0 0 − ∂ S s ∂ z ∂ ∂ x 0 PeLe ∂ ∂ z − ∇ 2 0 − ∂ ∂ z 0 0 PeLe ∂ ∂ z − Pm ∇ 2 ψ 3 T 3 S 3 ϕ 3 = R 31 R 32 R 33 R 34 ,

where

(36) R 31 = 1 4 16 k c 2 π 2 Ra S 2 A ( τ ) δ 2 β 2 2 + 4 δ 2 A ′ ( τ ) Sc + π 4 ( 5 k c 2 + π 2 ) Q A 3 ( τ ) cos ( 2 k c x ) δ 4 Pm 2 sin ( k c x ) sin ( π z ) , R 32 = − 2 k c π 2 cos ( k c x ) sin ( π z ) [ k c 2 π A 3 ( τ ) ( 2 π cos ( 2 π z ) + Pe sin ( 2 π z ) ) − 2 Le β 1 2 A ′ ( τ ) ] δ 2 Le 2 β 1 2 , R 33 = k c 3 π 2 A 3 ( τ ) cos ( k c x ) sin ( π z ) [ 4 π 2 cos ( 2 π z ) + 2 LePe π sin ( 2 π z ) ] δ 2 β 2 2 − 4 k c π 2 cos ( k c x ) sin ( π z ) A ′ ( τ ) δ 2 β 2 2 − δ 1 f ( z ) A ( τ ) k c cos ( k c x ) sin ( π z ) , R 34 = π cos ( π z ) sin ( k c x ) [ π 2 A 3 ( τ ) cos ( k c x ) − 4 Pm A ′ ( τ ) ] 2 δ 2 Pm 2 .

We have used the solvability condition to build the Ginzburg–Landau equation and solve the aforementioned problem:

(37) ∂ A ( τ ) ∂ τ г 1 − A ( τ ) г 2 + A 3 ( τ ) г 3 = 0 ,

where

F 1 = 4 π 2 Le Pe 2 Q ( e ( Le + 1 ) Pe − 1 ) ( ( Le + 1 ) 2 Pe 2 + 2 π 2 ) δ 2 ( e Pe − 1 ) Pm ( e Le Pe − 1 ) ( β 1 2 + Pe 2 ( 2 Le 2 + 1 ) ) + 64 π 6 k c 2 Ra T δ 4 Le β 1 6 + 16 π 4 ( Pe 2 ( Le 2 + 16 π 2 β 2 2 ) + 64 π 2 ) ( δ 2 β 1 2 ( δ 4 + π 2 Q ) − 4 π 2 k c 2 Ra T ) δ 4 β 1 4 ( β 1 2 + 12 π 2 ) β 2 4 − π k c Le Pe 2 4 π k c ( δ 4 + π 2 ( δ 2 + π 2 ) ) + 4 π 7 k c ( e ( Le + 1 ) Pe − 1 ) δ 4 ( e Pe − 1 ) ( e Le Pe − 1 ) ( ( Le + 1 ) Pe ( Pe 2 ( Le + 1 ) + 4 π 2 ) ) , F 2 = − π k c Le Pe 2 ( e ( Le + 1 ) Pe − 1 ) ( − 16 π 3 k c Ra S 2 ( k c 2 + 1 ) ) β 2 2 δ 4 ( e Pe − 1 ) ( e Le Pe − 1 ) Pe ( Le + 1 ) ( ( Le + 1 ) 2 Pe + 4 π 2 ) + 2 I 1 δ 1 Pe ( 4 π 2 k c 2 Ra T − δ 2 β 1 2 ( δ 4 + π 2 Q ) ) δ 2 β 1 2 ( 1 − e Pe ) , F 3 = π 4 Le Pe Q ( e ( Le + 1 ) Pe − 1 ) ( ( Le + 1 ) 2 Pe 2 + 2 π 2 ) 2 δ 2 ( e Pe − 1 ) Pm 2 ( e Le Pe − 1 ) ( Le + 1 ) ( ( Le + 1 ) 2 Pe 2 + 4 π 2 ) − π k c Le Pe 2 ( e ( Le + 1 ) Pe − 1 ) ( 3 π 5 k c Q − π 7 Q ) 2 Pm 2 k c δ 4 ( e Pe − 1 ) ( e Le Pe − 1 ) Pe ( Le + 1 ) ( ( Le + 1 ) 2 Pe 2 + 4 π 2 ) + 64 π 6 k c 2 Ra T ( π 2 k c 2 ( Le 2 Pe 2 ( 1 − L e ) + 8 π 2 ) ) δ 2 Le 2 β 1 6 ( β 2 2 ( β 2 2 + 12 π 2 ) ) + 16 π 4 ( π 2 k c 2 ( Le 2 Pe 2 ( L e − 1 ) + 8 π 2 ) ) ( δ 2 β 1 2 ( δ 4 + π 2 Q ) − 4 π 2 k c 2 Ra T ) δ 4 β 1 4 ( β 1 2 + 12 π 2 ) β 2 4 ,

and

I 1 = ∫ 0 1 f ( z ) sin 2 ( π z ) d z .

Using the second-order solution, the Nusselt number Nu ( τ ) and Sherwood number Sh ( τ ) are obtained as follows:

(38) Nu ( τ ) = 1 − A ( τ ) 2 ( 4 π 4 k c 2 ( 1 − exp ( Pe ) ) ) β 1 4 Le 2 δ 2 Pe ,

(39) Sh ( τ ) = 1 − A ( τ ) 2 ( 4 π 4 k c 2 ( 1 − exp ( Le Pe ) ) ) β 2 4 Le δ 2 Pe .

3 Results and discussion

This article analyzes the rate of mass and heat transports caused by concentration modulation in the presence of throughflow of Newtonian fluid with induced magnetic field. The following factors influence how quickly mass and heat are transported: Sc, Le, Q, Pe, δ 1 , ω , and Pm. The critical solute Rayleigh number is expressed for stationary convection under the linear stability analysis. It does not provide any information, how the rate of mass and heat transport is influenced by the nonlinear terms. So, nonlinear stability analysis is required if the rate of mass and heat transports is to be determined. A weakly nonlinear stability analysis approach is used to simplify the system of differential equations, which leads to results in terms of the Ginzburg–Landau equation. The common values of physical parameters to computation of numerical results are as follows: Sc = 0.7 , Le = 0.7 , Q = 25 , Pe = 0.3 , δ 1 = 0.5 , ω = 2 , and Pm = 0.6 , respectively.

The rates of mass and heat transfers for IPM are depicted in Figures 2–6 for different values of physical parameter. From these figures, we see that the values of Nu and Sh begin at 1, indicating that mass and heat transports occur in conduction mode at the initial time. As time passed, fluid transports shifted from conduction to convection mode and it became steady state around τ = 0.4 (approximately). This means that the time-dependent concentration boundary conditions have little effect at this point.

Figure 2

Variation of Le on Nu and Sh.

Figure 2 depicts the effect of Le on the Nusselt number (Nu) and Sherwood number (Sh). Since the Lewis number is a ratio of thermal diffusivity to the mass diffusivity and when Lewis number increases, the thermal diffusivity increases the fluid or mass diffusivity decreases. In both cases, the rate of heat transfer slowed down, while the rate of mass transfer accelerated. Thus, Le increases Nu and decreases Sh. The relation between the Lewis number (Le), Nusselt number, and Sherwood number depends on both the specific problem being studied and the physical properties of the fluid or substance under consideration. Increasing the Lewis number enhances the thermal diffusivity or decreases the mass diffusivity.

In Figure 3, we examined the rate of mass and heat transport for Péclet number (Pe) on the system. Péclet number is increasing the rate of mass and heat transfer. Increasing the Pe number decreases the mass diffusivity or increases the flow velocity. The Nusselt number begins to deviate from 1 as the Péclet number increases and convective transport becomes more important. The Nusselt number increases with the Péclet number, indicating that convective heat transfer improves overall heat transfer rate. Péclet number is increasing the Sh profile by decreasing the mass diffusivity.

Figure 3

Variation of Pe on Nu and Sh.

Figure 4 shows the effect of magnetic Prandtl number (Pm) on Nusselt and Sherwood numbers. An increase in momentum diffusivity or decrease in magnetic diffusivity shows an increment in magnetic Prandtl number. Magnetic Prandtl number increases the Nusselt number profile, which indicates that either convective heat transfer increases or thermal conductivity decreases. Mass transfer rate increased on increment in Pm that shows a decrement in mass diffusivity or, increment in convective mass transfer rate. Overall, Pm destabliized the system on increasing its value. The effect of Chandrasekhar number ( Q ) on rate of mass and heat transport is studied in Figure 5. The effect of Q on Nu and Sh is the same as we obtained for Pm, which is a similar result reported by Rudraiah [14] and Bhadauria and Kiran [38]. Chandrasekhar number increases the rate of mass and heat transport, which is a destabliized effect on the system.

Figure 4

Variation of Pm on Nu and Sh.

Figure 5

Variation of Q on Nu and Sh.

The effect of Schmidt number (Sc) is observed in Figure 6 on Nusselt (Nu) and Sherwood number (Sh). The Schmidt number has negative posture on the mass and heat transport rate. This is due to the increment in Sc, indicating that the viscosity increases or mass diffusivity decreases. Hence, the Schmidt number stabilized the system.

Figure 6

Variation of Sc on Nu and Sh.

The mass and heat transport analysis for OPM is studied in Figures 7–11. In DDC, the OPM has been shown to cause a variety of interesting behaviours. The symmetry of the resulting flow patterns distinguishes IPM and OPM in DDC. OPM produces asymmetric flow patterns, whereas IPM produces symmetric flow patterns. Figure 7 depicts the effect of Lewis number (Le) on the Nusselt (Nu) and Sherwood (Sh) numbers profile. Enhancing the value of Le leads to a decrease in Nusselt number profile and an increase in Sherwood number profile. The aforementioned phenomenon happens due to either increase in thermal conductivity or decrement in mass diffusivity. The rate of heat and mass transfer is more in OPM than in the IPM.

Figure 7

Variation of Le on Nu and Sh.

The effect of Péclet number is inspected on Nusselt and Sherwood numbers in Figure 8. For small Péclet number, we could say convective transport is substantially smaller than diffusive transport (conduction dominates), the Nusselt number is close to 1. This suggests that convection does not considerably improve heat transport and the temperature gradient is predominantly driven by conduction. In OPM case, the Péclet number improves mass and heat transport. The rate is more than the IPM.

Figure 8

Variation of Pe on Nu and Sh.

Figure 9 shows the effect of magnetic Prandtl number on Nusselt and Sherwood numbers profile. The magnetic Prandtl number can help us understand the influence of magnetic forces on the fluid motion. A low magnetic Prandtl number suggests that the fluid is more susceptible to magnetic forces, whereas a high magnetic Prandtl number shows that momentum diffusion dominates magnetic diffusion in the fluid. The magnetic Prandtl number increases the Nu and Sh profiles. It shows a decrement in mass diffusivity or an increment in convective mass transport and a decrease in thermal conductivity or, increase in convective heat transfer. An increment in Pm resultant destabilize the system. The rate of mass and heat transport is seen more than IPM in this case.

Figure 9

Variation of Pm on Nu and Sh.

Figures 10 and 11 represent the effect of Chandrasekhar number ( Q ) and Schmidt number (Sc) on Nusselt number (Nu) and Sherwood number (Sh). The effect of Q on rate of mass and heat transport is similar to the previously discussed in the case of IPM. But the rate of transport is more than IPM. Figure 11 shows the oscillatory nature of rate of mass and heat transfer with an increment in value of Sc, which shows fluctuation in Nu and Sh profiles. For time scale of 2.6 to 3.5, rate of mass and heat transfer is increasing very slightly, but on the other hand for the time scale of 3.4 to 4, the rate of mass and heat transport is getting lower. For the large scale of time, it does not show any large impact on rate of mass and heat transport with Sc. Comparatively, it has more mass and heat transport than IPM.

Figure 10

Variation of Q on Nu and Sh.

Figure 11

Variation of Sc on Nu and Sh.

Figures 12–14 represent the instability analysis for the case of UPM. UPM in concentration modulation likely describes how changes or disturbances in the upper boundary or surface of the fluid (upper plate) influence or modulate variations in the concentrations of the two components (concentration modulation). This modulation has the potential to have a considerable impact on the overall DDC pattern, mixing processes, and mass and heat movement within the fluid.

Figure 12

Variation of Le on Nu and Sh.

Figure 12 shows the effect of Le, which is similar to other considered cases; but the rate of mass and heat transfer in UPM is higher than in IPM and lower than in OPM. Effect of Péclet number (Pe) is discussed in Figure 13. The Péclet number influences the Sherwood number in the same way as it influences the Nusselt number and increasing both the profiles. Figure 14 depicts the effect of magnetic Prandtl number on Nu and Sh profiles. Concomitantly, the Nusselt number (Nu) emerges as another pivotal parameter in the pursuit of understanding MHD phenomena. The Nusselt number characterizes the heat transfer enhancement due to the presence of a magnetic field. In MHD systems, magnetic fields can induce alterations in fluid flow patterns, causing fluctuations in temperature distribution and heat transfer rates. The Nusselt number increases illuminating the extent to which magnetic fields augment or impede heat transfer processes, thereby exerting a profound influence on the thermal behaviour of the system under study. Collectively, the magnetic Prandtl number on the Nusselt number has shown the emerging effect. Their intricate interplay embodies the intricate relation between fluid dynamics and magnetic fields, elucidating the profound changes in heat transfer rate. Figure 14 shows the similar effect as seen in IPM and OPM but the rate of mass and heat transport is higher than IPM but lower than in OPM. The effects of Sc and Q are the same as IPM and OPM; therefore, these figures are not included in this article.

Figure 13

Variation of Pe on Nu and Sh.

Figure 14

Variation of Pm on Nu and Sh.

4 Conclusion

This article shows how concentration modulation affects the magneto-convective flow of a Newtonian fluid trapped between two parallel, horizontal plates. We investigate time-varied concentration boundaries, which are composed of two components: a consistent element and a sinusoidal component that varies with time. The upper boundary is characterised by a higher solutal concentration compared to the lower boundaries. To address this model, we employ a series expansion technique known as the asymptotic analysis method. The solute Rayleigh equation is obtained for stationary convection. The Fredhölm solvability condition is used to derive the Ginzburg–Landau equation. The main conclusions of this study are as follows:

According to the solute Rayleigh number equation for stationary convection, the time-periodic limits have very little bearing on stationary convection.
The graphs of IPM reflect that after a certain time, the rate of mass and heat transports is steady. This happens because the concentration and temperature difference between upper and lower boundaries are constant in case of IPM.
In case of IPM, OPM, and UPM, Sc has a negligible effect on Nu and Sh profiles.
Pe, Q , and Pm amplify the pace of mass and heat transfer, which causes the system become unstable.
The Lewis number (Le) increases the mass transport rate while decreasing the heat transport rate.
Overall, we have noted that the rate of mass and heat transport order is as follows: OPM > UPM > IPM.

In comparison with previous situations, the examined cases indicate that mass and heat transfer rates are more in OPM. Thus, this study may be helpful at low heat transmission and large mass transfer, which has many applications in the industrial, food, glass industries among others.

The present study may be expanded to include non-Newtonian fluids because they are widely employed in industry, oil reservoirs, the food industry, material science, and engineering. This model is applicable whenever the mass diffusivity is more than compared to thermal diffusivity.

Acknowledgement

The authors are thankful to the editor and referees for their most valuable comments that helped us to modify this article to the present form.

Funding information: No funding is acquired in this research.
Author contributions: Atul Jakhar: conceptualization; data curation; formal analysis; investigation; methodology; project administration; resources; software; supervision; validation; visualization; writing – original draft; writing – review and editing. Anand Kumar: formal analysis; supervision; writing – review and editing. Priyanka Joshi: formal analysis, validation.
Conflict of interest: On behalf of all authors, the corresponding author states that there is no conflict of interest.
Data availability statement: The data that support the findings of this study are available within the article.

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Received: 2023-12-16

Revised: 2024-04-19

Accepted: 2024-04-22

Published Online: 2024-08-09

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0013

Keywords for this article

heat transfer; mass transfer; double diffusive convection; concentration modulation; MHD; horizontal plates; throughflow

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