Effect of modulated boundary on heat and mass transport of Walter-B viscoelastic fluid saturated in porous medium

Anupama Singh; Atul Jakhar; Anand Kumar

doi:10.1515/nleng-2024-0014

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Effect of modulated boundary on heat and mass transport of Walter-B viscoelastic fluid saturated in porous medium

Anupama Singh , Atul Jakhar and Anand Kumar

Published/Copyright: September 5, 2024

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From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

This article depicts the heat and mass transport of the double-diffusive convective flow of Walter-B viscoelastic fluid in highly permeable porous media with an internal heat source. We used weakly nonlinear analysis to quantify the nature of heat and mass transport using the Ginzburg–Landau equation. The Ginzburg–Landau equation has been derived in terms of the amplitude of the stream function. The effect of physical parameters has been examined on Nusselt and Sherwood numbers, which has represented graphically. According to the boundary condition, we have discussed the four scenarios based on the phase angles. Our study has demonstrated that internal heat plays a significant role in heat transfer processes. Furthermore, the elastic parameter leads to a transient augmentation in the heat and mass transfer rate. The main output of the current study is that the highest transport was found when both the modulations were put in out-phase condition (Scenario 1).

Keywords: Walter-B viscoelastic fluid; internal heat source; double-diffusive convection; concentration modulation; porous medium; temperature modulation; Ginzburg–Landau equation

List of variables

A.: Latin symbols
q →: Velocity vector
Q: Internal heat
Le: Lewis number
Δ S: Solute difference beyond fluid layer
g →: Acceleration due to gravity
p →: Pressure
Nu: Nusselt number
S: Solute concentration
Ra T: Thermal Rayleigh number
Ra S: Solutal Rayleigh number
T: Temperature
Δ T: Temperature difference beyond fluid layer
Da: Darcy number
Ri: Internal Rayleigh number
Pr: Prandtl Number
t: Time
B.: Greek symbols
ϕ: Porosity
ε: Perturbation parameter
ϕ 1 , ϕ 2: Phase angle
κ T: Effective thermal diffusivity
κ S: Effective solutal diffusivity
ν: Kinematic viscosity
μ: Dynamic viscosity
β T: Thermal expansion coefficient
β S: Mass expansion coefficient
ρ: Density of fluid
ψ: Stream function
τ: Time scale parameter
μ ν: Viscoelastic constant
κ: Permeability
ω 1 , ω 2: Frequencies of modulation
δ 1: Amplitude of modulation
δ 2: Amplitude of modulation
Γ p: Elastic parameter
C.: Superscripts
′: Perturbed state
*: Non-dimensional value
D.: Subscripts
0: Critical state
b: Basic state

1 Introduction

Convection is a mechanism whereby fluid molecules move in bulk between locations of distinct temperatures, whereas diffusion occurs when molecules migrate randomly. Convective motions in a fluid occur when one aspect influences the density. It is a stabilizing effect in a fluid with a single solute, which can be in two or more components. The occurrence of double-diffusion requires two characteristics with different molecular diffusivity. Double-diffusive convection in porous media is a fluid dynamics phenomenon that explains a type of convection driven by two separate density gradients with differing rates of diffusion. Double-diffusive convection is feasible in the occurrence of two diffusing species in a bidisperse porous medium. Mendenhall and Mason [1] have discussed the double-diffusive process but have not presented the phenomenon physically. A detailed study on the double-diffusive convection was performed by Huppert and Turner [2]. They have shown significant developments in double-diffusive convection.

A porous medium is the most commonly described by its porosity or can be defined as a material containing pores or voids. There are promiscuous applications of natural convection through porous media in real-life problems. This implementation of porous media is in applied science and engineering, including geomechanics, oil extraction, and solidification of binary mixtures. The comprehensive studies of investigations on porous media and its applications are amassed admirably in the exemplary books of Ingham and Pop [3,4], Nield and Bejan [5], and Vafai [6]. Bég and Makinde [7] considered the convective Maxwell fluid flow to examine the viscoelastic flow and the species transfer in highly permeable porous media. They discovered that the Darcy and Deborah numbers had a negligible effect on the spread of the species.

Furthermore, so many researchers have scrutinized the flow through porous media. Shivkumara et al. [8] studied the onset of Darcy–Benard–Marangoni convection with high permeability. They have reported that effective viscosity and fluid viscosity are different and found that the onset convection is stabilized by a permeability constant. Recently, Torabi et al. [9] studied forced convection through porous media using pore-scale modeling. Basavaraja and Malashetty [10] investigated the thermal convection in a horizontal anisotropic porous layer under temperature and gravity modulation. They reported that the gravity modulation gives rise to the sub-critical motion, whereas temperature modulation leads to both super-critical and sub-critical motion. In addition, Wooding [11], Nield and Simmons [12], Postelnicue et al. [13], Shivkumara et al. [14], and Malashetty et al. [15] have published their articles on saturated porous medium.

Fluid convection caused by temperature and concentration gradient effects can result in complicated convection patterns. In many of the studies, a steady temperature and concentration gradient have been considered. There may be many situations in which the temperature or concentration gradient is a function of both time and space. This can be evaluated by solving the energy equation with appropriate time-dependent boundary conditions. Malashetty and Basavaraja [16,17], Bhadauria [18], Siddheshwar et al. [19], and Srivastava et al. [20] investigated the problem, which was followed by thermal convection in porous media with temperature modulation of the boundaries. Later on, another difficulty causes the emergence of a time-periodic gravity field perturbation, known in the literature as the g-gitter effect (Strong [21,22], Rees and Pop [23], Kuznetsov [24,25], Saravanan and Arunkumar [26], Saravanan and Sivakumar [27], Keshri et al. [28]). Siddheshwar et al. [19] have studied the onset of non-linear double-diffusive convection in porous media under temperature or gravity modulation. They discovered that the values of Darcy number show an opposite effect to the Nusselt number. Kumar et al. [29] have reported the effect of gravity modulation on double-diffusive convection of couple stress fluid. They discovered that increasing the Prandtl number and couple stress parameter leads to an increase in the rate of heat and mass transfer.

The studies on thermal instability under modulation presented earlier are for non-internal heating systems. However, it has been discovered that in many cases of enormous practical consequence, the material provides its own source of heat, resulting in a different approach for a convective flow to be established by local heat creation within the layer. Such a scenario can arise due to radioactive decay or rather modest exothermic and nuclear reactions that can occur within the substance. Internal heat, generated by nuclear fusion and radioactive decay, is the primary source of energy for celestial bodies, keeping them warm and active. The internal heating of the earth causes a temperature gradient between the inner and exterior of the earth’s crust, which aids convective flow and transfers thermal energy towards the planet’s surface. As a result, internal heat generation has become increasingly significant in various applications, including radioactive material storage, ignition and fire research studies, geophysical sciences, reactor safety analysis, and metal waste form development for spent nuclear fuel.

This study has various applications in various fields of science, engineering, and technology, motivating us to bring out the current work. However, few studies have looked into the effect of internal heating on convective flow in a fluid layer. The commencement of convection in a fluid saturating a horizontal layer of an anisotropic porous material with an internal heat source and an inclination temperature gradient is investigated by Parthiban and Palil [30]. Rao and Wang [31] analyzed the influence of internal heat generation on Rayleigh number in natural convection using different flow topologies in vertical porous cylinder. Rionero and Straughan [32] examined the study of convection in a fluid-saturated porous layer, which is heated internally with the gravitational field. Recently, a weakly nonlinear analysis by Bhadauria et al. [33] investigated natural convection in a rotating anisotropic porous layer with internal heat generation.

Nowadays, viscoelastic fluids are in colossal demand because of their wide range of essential applications in modern technology and industries, such as chemical engineering, reservoir engineering, polymer rheology, geophysics, and material production. Therefore, studies of instability in viscoelastic fluids are highly needed. Kumar et al. [34] presented the instability of Walter-B viscoelastic fluid under the rotational speed modulation. They have shown that modified Prandtl number decelerates heat transport due to the combined effect of the Taylor number and elastic parameter.

Herbert [35] reported the effect of elasticity on Couette flow. He considered a linear stability analysis and obtained that elasticity destabilizes the fluid flow. In addition, Bonn and Meunier [36] reported the theoretical and experimental literature for viscoelastic free boundary problems. Jakhar and Kumar [37,38] and Jakhar et al. [39] explored the effect of concentration modulation and internal heat on double- and triple-diffusive convection. Furthermore, many researchers presented their studies on the viscoelastic fluid in saturated porous media. Some of them are Slattery [40], Comissiong et al. [41], Bhadauria et al. [43], Comissiong et al. [44], Bhadauria and Kiran [45], Kumar et al. [46], and Shivkumara et al. [42].

The present study sketched the combined contribution of time-dependent temperature and concentration modulation on the onset of convection phenomena in the presence of an internal heat generator, which has a variety of applications in engineering, life sciences, and technology, such as complex mixtures, drug delivery systems with multiphase diffusion coefficients, coolant systems, materials science, and many more. From the aforementioned studies, researchers have studied heat and mass transport by considering the temperature modulation or concentration modulation separately. To the best of the author’s knowledge, no nonlinear study has investigated the influence of time-periodic boundary temperature and concentration modulation on heat and mass transfer of Walter-B viscoelastic fluid in the presence of internal heat. This article deals with the combined effect of temperature and concentration modulations on the non-Newtonian fluid. The current study has many applications, which encourages us to write this article.

2 Mathematical formulation of problem

Here, we have considered a Walter-B viscoelastic fluid saturated in porous medium between two infinite extent horizontal plates at z = 0 and z = d . The Cartesian coordinate system is established with the origin at the bottom plate and the z -axis pointing vertically upward. The gravity force acts in the downward direction. To maintain the temperature gradient, the fluid layer is maintained between temperate and concentration modulations at the plates (Figure 1). For double modulation, we adopted the time-dependent temperature field and solute concentration is modulated over the time. The internal heating is considered between the fluid flow, provided that the Joule heating and dissipation are ignored regarding heat transfer from the lower boundary (Davidson [47]). The flow equations are written as follows under the assumptions of the Boussinesq approximation (Kumar et al. [46], Shivkumara et al. [42]):

(1) ∇ . q → = 0 ,

(2) ρ 0 ϕ ∂ q → ∂ t + 1 ϕ ( q → . ∇ ) q → = − ∇ p + ρ g → − 1 k μ − μ ν ∂ ∂ t q → ,

(3) ∂ T ∂ t + 1 ϕ ( q → . ∇ ) T = κ T ∇ 2 T + Q ( T − T 0 ) ,

(4) ∂ S ∂ t + 1 ϕ ( q → . ∇ ) S = κ s ∇ 2 S ,

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

and g → = ( 0 , 0 , − g ) .

Figure 1

Geometry of problem.

The model under consideration takes the following boundary conditions

At z = 0 ,

(6) T = T 0 + Δ T 2 [ 1 + ε 2 δ 2 cos ( ω 1 t ) ] ,

(7) S = S 0 − Δ S 2 [ 1 − ε 2 δ 1 cos ( ω 2 t ) ] .

At z = d ,

(8) T = T 0 − Δ T 2 [ 1 − ε 2 δ 2 c o s ( ω 1 t + ϕ 2 ) ] ,

(9) S = S 0 + Δ S 2 [ 1 + ε 2 δ 1 cos ( ω 2 t + ϕ 1 ) ] .

In Eqs (1)–(9), q → is the velocity vector, ρ is the density, ϕ is the porosity, p is the pressure, κ is the permeability, μ is the dynamic viscosity, μ ν is the viscoelastic constant, T is the temperature, S is the concentration of fluid, κ T is the effective thermal diffusivity, κ S is the effective mass diffusivity, Q represents the internal heat, δ 1 , and δ 2 are small amplitudes of modulation, β S is the mass expansion coefficient, β T is the thermal expansion coefficient, ε is the perturbation parameter of the asymptotic analysis, Δ T is small temperature which is modulated upon, Δ S is the solutal difference beyond surface, ϕ 1 and ϕ 2 are the phase angles, respectively, δ 1 and δ 2 represent the amplitudes of modulation sequentially, and ω 1 and ω 2 are the frequencies of modulation.

Basic state: The basic state of the fluid is assumed to be quiescent, and the quantities at this state are as follows:

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

Now, substituting Eq. (10) into Eqs (1)–(4) along with the boundary condition defined earlier in Eqs (6)–(9), we found the following equations:

(11) ∂ p b ∂ z = ρ b g → ,

(12) ∂ T b ∂ t = κ T ∂ 2 T b ∂ z 2 + Q ( T b − T 0 ) ,

and

(13) ∂ S b ∂ t = κ S ∂ 2 S b ∂ z 2 .

The solutions for basic state with respect to the modulated boundary conditions are given by

(14) T b = T i ( z ) + ε 2 δ 2 T s ( z , t ) ,

(15) S b = S i ( z ) + ε 2 δ 1 S s ( z , t ) ,

where

(16) T i ( z ) = T 0 + Δ T 2 sin ( Ri ) sin Ri 1 − z d − sin Ri z d ,

(17) T s ( z , t ) = Δ T 2 Re B ( η ) e η z d + B ( − η ) e − η z d e − i ω 1 t ,

(18) S i ( z ) = S 0 − Δ S 2 1 − 2 z d ,

and

(19) S s ( z , t ) = Δ S 2 Re B ( λ 1 ) e λ 1 z d + B ( − λ 1 ) e − λ 1 z d e − i ω 2 t ,

where

B ( η ) = e − i ϕ 2 − e − η e η − e − η , η 2 = λ 2 − Ri , λ 2 = ( 1 − i ) 2 ω 1 2 , B ( λ 1 ) = e − i ϕ 1 − e − λ 1 e λ 1 − e − λ 1 , λ 1 2 = ( 1 − i ) 2 ω 2 Le 2 , and Ri = Q d 2 κ T .

Now, to accomplish the stability analysis, we consider the finite amplitude perturbation on basic state in the form:

(20) q → = q b → + q ′ → , p = p b + p ′ ( z ) , ρ = ρ b ( z ) + ρ ′ , T = T b ( z , t ) + T ′ , and S = S b ( z , t ) + S ′ ,

where ′ denotes the quantities at the perturb state.

On substituting the expressions for the variables from Eq. (20) to Eqs (1)–(5) and using the basic state solution, we obtained the equations as

(21) ∇ . q ′ → = 0 ,

(22) ρ 0 ϕ ∂ q ′ → ∂ t + 1 ϕ ( q ′ → . ∇ ) q ′ → = − ∇ p ′ + ρ ′ g → − 1 k μ − μ ν ∂ ∂ t q ′ → ,

(23) ∂ S ′ ∂ t + 1 ϕ ( q ′ → . ∇ ) S ′ + 1 ϕ ( q ′ → . ∇ ) S b = κ s ∇ 2 S ′ ,

(24) ∂ T ′ ∂ t + 1 ϕ ( q ′ → . ∇ ) T ′ + 1 ϕ ( q ′ → . ∇ ) T b = κ T ∇ 2 T ′ + Q T ′ ,

(25) and ρ ′ = − ρ 0 β T T ′ + ρ 0 β s S ′ .

For further evaluations, we have taken the curl of Eq. (22) to eliminate the pressure term. We have assumed all physical quantities to be independent of the y -coordinate, because of the study, which is barred form rolling with the y -axis. For two-dimensional convection, introduce the stream function ψ as u = ∂ ψ ∂ z and w = − ∂ ψ ∂ x . Now to non-dimensionalize Eqs (22)–(24), we have used the following scales:

( x , z ) = d ( x * , z * ) , T ′ = T * Δ T , S ′ = S * Δ S , t = d 2 κ T t * , ψ = ϕ κ T ψ * , ω 1 = κ T d 2 ω 1 * , ω 2 = κ T d 2 ω 2 * .

Hence, the non-dimensional form of governing equations is (asterisks are dropped for simplicity)

(26) − 1 Da ( ∇ 2 ψ ) + Ra S Le ∂ S ∂ x − Ra T ∂ T ∂ x = 1 Pr 1 − Γ p Da ∂ ∂ t ( ∇ 2 ψ ) − ∂ ( ψ , ∇ 2 ψ ) ∂ ( x , z ) ,

(27) ∂ S b ∂ z ∂ ψ ∂ x + 1 Le ∇ 2 S = ∂ S ∂ t − ∂ ( ψ , S ) ∂ ( x , z ) ,

(28) ∂ T b ∂ z ∂ ψ ∂ x + ( ∇ 2 + Ri ) T = ∂ T ∂ t − ∂ ( ψ , T ) ∂ ( x , z ) ,

where Da = k ϕ d 2 is the Darcy number, Ra S = g β s Δ S d 3 ν κ s , and Ra T = g β T Δ T d 3 ν κ T are the solutal and thermal Rayleigh number, respectively, Pr = ν κ T is the Prandtl number, Le = κ T κ s is the Lewis number, Γ p = μ ν ρ 0 d 2 is the elastic parameter, and Ri = Q d 2 κ T is the internal Rayleigh number. The expressions ∂ S b ∂ z and ∂ T b ∂ z , in Eqs (27)–(28), can be expressed as follows:

(29) ∂ T b ∂ z = f 1 ( z ) + ε 2 δ 2 f 2 ( z , t ) ,

(30) ∂ S b ∂ z = 1 + ε 2 δ 1 f ( z , t ) ,

where

f 1 ( z ) = − Ri 2 sin ( Ri ) [ cos ( Ri z ) + cos ( Ri ( 1 − z ) ) ] , f 2 ( z , t ) = Re [ B ( η ) e η z + B ( − η ) e − η z ] e − i ω 1 t , f ( z , t ) = Re [ B ( λ 1 ) e λ 1 z + B ( − λ 1 ) e − λ 1 z ] e − i ω 2 t ,

and Re denotes the real part, the other quantities having their usual mean described in the nomenclature section.

The stability analysis is performed for a short time period, for that we considered time parameter rescaled to slow time variation as τ = ε 2 t . The reduced form of Eqs. (26)–(28) is represented into a matrix form as:

(31) − 1 Da ∇ 2 Ra S Le ∂ ∂ x − Ra T ∂ ∂ x ( 1 + ε 2 δ 1 f ) ∂ ∂ x 1 Le ∇ 2 0 ( f 1 + ε 2 δ 2 f 2 ) ∂ ∂ x 0 ( ∇ 2 + Ri ) ψ S T = 1 Pr ε 2 1 − Γ p Da ∂ ∂ τ ( ∇ 2 ψ ) − ∂ ( ψ , ∇ 2 ψ ) ∂ ( x , z ) ε 2 ∂ S ∂ τ − ∂ ( ψ , S ) ∂ ( x , z ) ε 2 ∂ T ∂ τ − ∂ ( ψ , T ) ∂ ( x , z ) .

2.1 Finite amplitude equations (heat and mass transport)

In addition, the asymptotic expansion method has been used to perform weakly non-linear stability analysis. Accordingly, we have introduced a small perturbation parameter ε , which shows the deviation from the critical state. Thus, stream function ( ψ ), solute concentration ( S ), temperature ( T ), and solute Rayleigh number ( Ra S ) can be expanded as a power series of ε ≪ 1 as

(32) 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 + … , Ra S = Ra S 0 + ε 2 Ra S 2 + ε 4 Ra S 4 + … ,

where Ra S 0 is the critical value of the solute Rayleigh number at which starting convection takes place in the absence of any modulation.

2.1.1 First-order system (linear stability analysis)

In first-order system, comparing the coefficients of ε on both sides, we obtain the linear and homogeneous equations in the following:

(33) − 1 Da ∇ 2 Ra S 0 Le ∂ ∂ x − Ra T ∂ ∂ x ∂ ∂ x 1 Le ∇ 2 0 f 1 ∂ ∂ x 0 ( ∇ 2 + Ri ) ψ 1 S 1 T 1 = 0 0 0 .

The solution to the aforementioned system Eq. (33) can be accumulated in the following manner:

(34) ψ 1 = A ( τ ) sin ( κ c x ) sin ( π z ) , S 1 = κ c Le δ 2 A ( τ ) cos ( κ c x ) sin ( π z ) , T 1 = 4 π 2 κ c β γ A ( τ ) cos ( κ c x ) sin ( π z ) ,

where δ 2 = κ c 2 + π 2 is the total wave number. Also, the expression for the critical solute Rayleigh number is obtained as

(35) Ra S 0 = δ 4 κ c 2 Da + 4 π 2 δ 2 β γ Ra T ,

where the values for γ and β are denoted by the following expressions γ = Ri − δ 2 , β = 4 π 2 − Ri , and Ra T is the thermal Rayleigh number at the critical state.

2.1.2 Second-order system

On comparing the coefficient of ε 2 , the matrix representation of the second-order system can be acquired as;

(36) − 1 Da ∇ 2 Ra S 0 Le ∂ ∂ x − Ra T ∂ ∂ x ∂ ∂ x 1 Le ∇ 2 0 f 1 ∂ ∂ x 0 ( ∇ 2 + Ri ) ψ 2 S 2 T 2 = R 21 R 22 R 23 ,

where the expressions for R 21 , R 22 , and R 23 are as follows:

(37) R 21 = 0 , R 22 = − π κ c 2 A 2 ( τ ) Le 2 δ 2 sin ( 2 π z ) , R 23 = − 2 π 3 κ c 2 A 2 ( τ ) β γ sin ( 2 π z ) .

The solution for second order can be obtained as

(38) ψ 2 = 0 , S 2 = 1 8 π κ c A ( τ ) Le δ 2 sin ( 2 π z ) , T 2 = 2 π 3 γ κ c A ( τ ) β 2 sin ( 2 π z ) .

The horizontally averaged Nusselt number (Nu) and Sherwood number (Sh), for double-diffusive convection, are determined as follows:

(39) Nu = 1 + ∫ 0 2 π κ c ∂ T 2 ∂ z z = 0 d x ∫ 0 2 π κ c ∂ T b ∂ z z = 0 d x ,

(40) Sh = 1 + ∫ 0 2 π κ c ∂ S 2 ∂ z z = 0 d x ∫ 0 2 π κ c ∂ S b ∂ z z = 0 d x .

Now, by substituting the values from Eqs (29)–(30) and Eq. (38) into Eqs (39)–(40), we obtain

(41) Nu = 1 − 8 π 4 sin ( Ri ) γ Ri ( 1 + cos ( Ri ) ) κ c A ( τ ) β 2

and

(42) Sh = 1 + 1 4 A ( τ ) κ c Le δ 2 .

2.1.3 Third-order system

At the solution for the third order, we equate the coefficient of ε 3 from Eq. (31). Furthermore, putting the power series expansions of ψ , S , and T , the reduced form of equations can be obtained as

(43) − 1 Da ∇ 2 Ra S 0 Le ∂ ∂ x − Ra T ∂ ∂ x ∂ ∂ x 1 Le ∇ 2 0 f 1 ∂ ∂ x 0 ( ∇ 2 + Ri ) ψ 3 S 3 T 3 = R 31 R 32 R 33 ,

where

(44) R 31 = Ra S 2 κ c δ 2 A ( τ ) − δ 2 Pr 1 − Γ p Da ∂ A ( τ ) ∂ τ sin ( κ c x ) sin ( π z ) ,

(45) R 32 = L e δ 2 ∂ A ( τ ) ∂ τ − δ 1 f A ( τ ) − κ c Le 2 δ 2 A 3 ( τ ) cos ( 2 π z ) κ c cos ( κ c x ) sin ( π z ) ,

and

(46) R 33 = 4 π 2 β γ ∂ A ( τ ) ∂ τ − δ 2 f 2 A ( τ ) − 4 π 4 κ c 2 γ β 2 A 3 ( τ ) cos ( 2 π z ) κ c cos ( κ c x ) sin ( π z ) .

Using the Fredhölm solvability condition for the third-order solution, we obtain Ginzburg–Landau equation in the form:

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

where coefficients A 1 , A 2 , and A 3 are given by

A 1 = 4 ( π δ ) 2 β P r 1 − Γ p Da + 4 Le β π κ c δ 2 2 Ra S 0 + 16 π 4 κ c β γ 2 Ra T , A 2 = 4 β π κ c δ 2 Ra S 2 − 2 κ c δ 2 δ 1 Ra S 0 F 1 + 8 π 2 κ c 2 β γ δ 2 Ra T F 2 ,

and

A 3 = 1 β π κ c 2 Le δ 2 2 2 β + 3 Ri 4 β + 3 Ri Ra S 0 + 8 π 6 κ c 4 β 3 γ 2 Ra T ,

where

β = 4 π 2 − Ri , γ = Ri − δ 2 , δ 2 = π 2 + κ c 2 , F 1 = ∫ 0 1 f ( z ) f 1 ( z ) sin 2 ( π z ) d z , F 2 = ∫ 0 1 f 2 ( z ) sin 2 ( π z ) d z .

3 Results and discussion

This article delves into the concept of double-diffusive convection with a two-pronged modulation. External modulation of flows enables the regulation of heat and mass transport. We have made a weakly non-linear analysis for further investigations to quantify the heat and mass transport because linear stability analysis does not provide this information. Hence, it is quite essential to consider non-linear stability analysis.

The parameters that appeared in the present study are Pr, Le, Da, Ra T , Ri, Γ p , δ 1 , δ 2 , ω 1 , and ω 2 . The first six parameters are related to the fluid property, and the last four are concerned with the two external mechanisms, i.e., temperature and concentration modulation, that control the convective heat and mass transport. For all numerical evaluations, the common values of the arised parameters have been taken as Pr = 20 , Le = 3 , Da = 0.01 , Ri = 0.5 , Γ p = 0.6 , δ 1 = 0.5 , δ 2 = 0.4 , ω 1 = 3 , and ω 2 = 2 and positive values of Ra T are taken to obtain positive values of Ra S (Kumar et al. [34], and Siddheshwar et al. [19]). The combined effect of temperature and concentration modulation on heat and mass transport is delineated in the figures, which are the drawn graphs for Nusselt number (Nu) and Sherwood number (Sh) versus slow time ( τ ) .

The following scenarios are discussed in this article:

When both the modulation in out-phase scenario ( ϕ 1 = π , ϕ 2 = π ).
When concentration modulation in out-phase while temperature modulation at upper plate scenario ( ϕ 1 = π , ϕ 2 = − ι ∞ ).
When concentration modulation at upper plate while temperature modulation in out-phase scenario ( ϕ 1 = − ι ∞ , ϕ 2 = π ).
When both the modulation at upper plate scenario ( ϕ 1 = − ι ∞ , ϕ 2 = − ι ∞ ).

When both the plates at out-phase modulation ( ϕ 1 = π , ϕ 2 = π ).

Figures 2–10 depict the variation of Nusselt number(Nu) and Sherwood number (Sh) for slow time performed for out-phase modulation (OPM) at discrete values of chosen parameters. In Figure 2, the effect of Darcy number on Nusselt number and Sherwood number with the time ( τ ) is shown; as (Da) increases, the (Nu) and Sh show little negligible increment in their values, hence, systems get stabilized (considering rest parameters having their common values).

Figure 2

Effect of Da on Nu and Sh in Scenario 1.

Figure 3 exhibits the influence of δ 1 over Nu and Sh. As a result, both the heat and mass transport rate increases, destabilizing the system. Furthermore, the Nusselt number shows a slight decrement for the elastic parameter ( Γ p ) in Figure 4. Similarly, the Sherwood number also lessens as Γ p increases. This is due to the increase in viscoelasticity of the fluid. Figure 5 expresses the inversional effect of Lewis number (Le) over Nu. As Le increases, the Nusselt number decreases so fast, which is a sign of stability of the system due to a very fast decrement in the rate of heat transfer. The nature of Le has a reverse effect compared to the Prandtl number (Pr); enlarging the Pr increases the Nusselt number slowly, as shown in Figure 6.

$Figure 3 Effect of δ 1 {\delta }_{1} on Nu and Sh in Scenario 1.$

Figure 3

Effect of δ 1 on Nu and Sh in Scenario 1.

$Figure 4 Effect of Γ p \Gamma p on Nu and Sh in Scenario 1.$

Figure 4

Effect of Γ p on Nu and Sh in Scenario 1.

Figure 5

Effect of Le on Nu and Sh in Scenario 1.

Figure 6

Effect of Pr on Nu and Sh in Scenario 1.

The destabilizing effect is shown in Figures 5 and 6, in which the rate of mass transfer increases slightly as Le and Pr grow. This is because, when Le is increased, the solutal diffusivity decreases. Furthermore, the rate of heat transfer decreases with Le while increasing with Pr. This is due to an increment in the thermal diffusivity. On increasing the frequency of modulation ( ω 1 ) of the temperature, the rate of heat and mass transfer elevates (from Figure 7). Figure 8 shows that, as we increase the frequency of modulation ( ω 2 ) of solutal, the amplitude of modulation increases and inevitably lessens the heat and mass transfer. Therefore, the frequency of modulation ( ω 2 ) stabilizes the system (Kumar et al. [34]). For the thermal Rayleigh number ( Ra T ) , the Nusselt and Sherwood numbers both show an oscillatory effect. Hence, we can see the stabilizing impact on (Nu) and (Sh) in Figure 9. From Figure 10, we found that enlarging the internal Rayleigh number (Ri) makes the system destabilized as Nusselt number decreases. When concentration in out – phase modulation (OPM), while the temperature is at UPM (upper plate modulation) ( ϕ 1 = π , ϕ 2 = − i ∞ ):

$Figure 7 Effect of ω 1 {\omega }_{1} on Nu and Sh in Scenario 1.$

Figure 7

Effect of ω 1 on Nu and Sh in Scenario 1.

$Figure 8 Effect of ω 2 {\omega }_{2} on Nu and Sh in Scenario 1.$

Figure 8

Effect of ω 2 on Nu and Sh in Scenario 1.

$Figure 9 Effect of Ra T {{\rm{Ra}}}_{T} on Nu and Sh in Scenario 1.$

Figure 9

Effect of Ra T on Nu and Sh in Scenario 1.

Figure 10

Effect of Ri on Nu and Sh in Scenario 1.

A joint study of the temperature at UPM while the concentration in OPM (Scenario 2) is shown in Figures 11, 12, 13, 14 plotted for Nu and Sh versus τ . The effects of Da, δ 1 , Γ p , Le, Pr, ω 2 , Ra T , and Ri are quite similar to the previous results for Nusselt number (Nu) and Sherwood number (Sh) as Scenario 1 (Figures 2–10); therefore, we are not showing all graphical results here. The effect of ω 1 on Nu and Sh can be seen in Figure 11. As ω 1 increases, Nu and Sh fluctuate by considering the common values for remaining parameters. Figures 15, 16, 17, 18, 19, 20, 21, 22, 23 are manifested for the phase angles, ϕ 1 = − i ∞ , ϕ 2 = π (Scenario 3) and Figures 24, 25, 26, 27, 28, 29, 30, 31, 32 are for ϕ 1 = − i ∞ , ϕ 2 = − i ∞ (Scenario 4). In both cases, Lewis number (Le), elastic parameter ( Γ p ), internal Rayleigh number (Ri), ω 2 , and δ 1 show the same impact on Nusselt number and Sherwood number profiles as we discussed earlier in Scenario 2.

$Figure 11 Effect of ω 1 {\omega }_{1} on Nu and Sh in Scenario 2.$

Figure 11

Effect of ω 1 on Nu and Sh in Scenario 2.

Figure 12

Effect of Pr on Nu and Sh in Scenario 2.

$Figure 13 Effect of Γ p \Gamma p on Nu and Sh in Scenario 2.$

Figure 13

Effect of Γ p on Nu and Sh in Scenario 2.

Figure 14

Effect of Ri on Nu and Sh in Scenario 2.

Figure 15

Effect of Da on Nu and Sh in Scenario 3.

$Figure 16 Effect of δ 1 {\delta }_{1} on Nu and Sh in Scenario 3.$

Figure 16

Effect of δ 1 on Nu and Sh in Scenario 3.

$Figure 17 Effect of Γ p \Gamma p on Nu and Sh in Scenario 3.$

Figure 17

Effect of Γ p on Nu and Sh in Scenario 3.

Figure 18

Effect of Le on Nu and Sh in Scenario 3.

Figure 19

Effect of Pr on Nu and Sh in Scenario 3.

$Figure 20 Effect of ω 1 {\omega }_{1} on Nu and Sh in Scenario 3.$

Figure 20

Effect of ω 1 on Nu and Sh in Scenario 3.

$Figure 21 Effect of ω 2 {\omega }_{2} on Nu and Sh in Scenario 3.$

Figure 21

Effect of ω 2 on Nu and Sh in Scenario 3.

$Figure 22 Effect of Ra T {{\rm{Ra}}}_{T} on Nu and Sh in Scenario 3.$

Figure 22

Effect of Ra T on Nu and Sh in Scenario 3.

Figure 23

Effect of Ri on Nu and Sh in Scenario 3.

Figure 24

Effect of Da on Nu and Sh in Scenario 4.

$Figure 25 Effect of δ 1 {\delta }_{1} on Nu and Sh in Scenario 4.$

Figure 25

Effect of δ 1 on Nu and Sh in Scenario 4.

$Figure 26 Effect of Γ p \Gamma p on Nu and Sh in Scenario 4.$

Figure 26

Effect of Γ p on Nu and Sh in Scenario 4.

Figure 27

Effect of Le on Nu and Sh in Scenario 4.

$Figure 28 Effect of ω 1 {\omega }_{1} on Nu and Sh in Scenario 4.$

Figure 28

Effect of ω 1 on Nu and Sh in Scenario 4.

$Figure 29 Effect of ω 2 {\omega }_{2} on Nu and Sh in Scenario 4.$

Figure 29

Effect of ω 2 on Nu and Sh in Scenario 4.

Figure 30

Effect of Pr on Nu and Sh in Scenario 4.

$Figure 31 Effect of Ra T {{\rm{Ra}}}_{T} on Nu and Sh in Scenario 4.$

Figure 31

Effect of Ra T on Nu and Sh in Scenario 4.

Figure 32

Effect of Ri on Nu and Sh in Scenario 4.

On enlarging Darcy number, Nu and Sh decrease initially, then increase and repeat the same nature with slow time. Hence, we can find an oscillatory nature that stabilizes the system. Similarly, the Prandtl number also shows the exact stabilizing nature. Rate of heat and mass transport decreases with increasing value of Ra T , while for a small frequency of modulation ω 1 , the heat and mass transfer both get elevated. At a significant value of ω 1 , the heat and mass transfer are lesser. We conclude that while the frequency of modulation increases, the amplitude of modulation gets higher, consequently decreasing the value of heat and mass transfer. As a result, ω 1 stabilizes the system.

The presence of an internal heat source reduces the stationary convection. As the internal Rayleigh number (Ri) and Darcy number (Da) increas, the critical Rayleigh number ( Ra S 0 ) decreases, resulting in an earlier onset of convection (Table 1).

Table 1

Critical values of solutal Rayleigh number with different values of physical parameters

Ri	Da	kc	Ra S 0
0.1	0.01	3.09	3918
0.9	—	3.03	3680
2.9	—	2.87	3097
0.9	0.005	3.06	7360
—	0.02	3.02	1840
—	0.09	2.95	409

4 Conclusions

This article examines the consideration of the non-linear stability analysis for double-diffusive convection in a non-Newtonian Walter-B fluid that is heated and salted over time in the presence of an internal heat source. This study investigates the combined impact of temperature and concentration modulation on heat and mass transfer rates. A weakly non-linear theory employing an infinite asymptotic expansion method is utilized. By analyzing the interplay of phase angles, we identify distinct modulation scenarios. The influence of various dimensionless physical parameters on heat and mass transfer rates is meticulously examined, with the results presented in terms of Nusselt and Sherwood numbers. Drawing from results and discussion analysis of the convective system, we arrive at the following conclusions:

The rates of heat and mass transport exhibited a significant analysis in the OPM scenario compared to the other cases under consideration.
The effect of Lewis number (Le) is to decrease the rate of heat transfer in all cases; Le stabilizes the system.
The effect of the rising internal Rayleigh number (Ri) is to increase the value of Nu in all cases; hence, Ri destabilizes the system while exhibiting a decrease in Sh.
The elastic parameter ( Γ p ) destabilizes the system while the frequency of modulation shows a reverse effect.
The reduction in heat and mass transport rate that results from increasing the thermal Rayleigh number stabilizes the system.
The heat transfer diminishes with enlarging Darcy number (Da), which manifests similar outcomes for mass transfer.
Amplitude of modulation unsettles the system for all considering cases.
The rate of heat and mass transport for the considered scenarios can be arranged as follows:

Scenario 1 > Scenario 2 > Scenario 3 > Scenario 4 .

In light of the aforementioned phenomenon, it is evident that the temporal variations in temperature and concentration, coupled with the existence of an internal heat source, exert a pronounced influence on the dynamics of double-diffusive convection. Future research directions may involve examining the behavior using alternative non-Newtonian fluids with diverse rheological properties, shifting from Walter B fluid. Additionally, investigating the applicability of these principles in microfluidic devices, where heat and mass transfer are pivotal, presents a promising avenue for further exploration and innovation.

Acknowledgements

The first author wishes to express her gratitude toward UGC for funding under Grant JUNE18-419426.

Funding information: No funding is acquired in this research.
Author contributions: Anupama Singh: conceptualization; data curation; formal analysis; investigation; methodology; project administration; resources; software; validation; visualization; writing – original draft; writing – review and editing. Atul Jakhar: formal analysis, validation. Anand Kumar: formal analysis; supervision; writing – review and editing.
Conflict of interest: The authors declare that they have no conflict of interest.
Data availability statement: The data supporting the conclusions of the study are in the article.

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Received: 2023-11-24

Revised: 2024-05-18

Accepted: 2024-05-20

Published Online: 2024-09-05

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0014

Keywords for this article

Walter-B viscoelastic fluid; internal heat source; double-diffusive convection; concentration modulation; porous medium; temperature modulation; Ginzburg–Landau equation

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