Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation

Mohamed M. Khader; Hijaz Ahmad; Mohamed Adel; Ahmed M. Megahed

doi:10.1515/phys-2024-0016

Article Open Access

Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation

Mohamed M. Khader , Hijaz Ahmad , Mohamed Adel and Ahmed M. Megahed

Published/Copyright: May 4, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

In the current study, we delve into examining the movement of a nanofluid within a Williamson boundary layer, focusing on the analysis of heat and mass transfer (HMT) processes. This particular flow occurs over a sheet that undergoes nonlinear stretching. A significant facet of this investigation involves the incorporation of both the magnetic field and the influence of viscous dissipation within the model. The sheet is situated within a porous medium, and this medium conforms to the Darcy model. Since more precise outcomes are still required, the model assumes that both fluid conductivity and viscosity change with temperature. In this research, we encounter a system of extremely nonlinear ordinary differential equations that are treated through a numerical technique, specifically by employing the spectral collocation method. Graphical representations are used to illustrate how the relevant parameters impact the nanoparticle volume fraction, velocity, and temperature profiles. The study involves the computation and analysis of the effect of physical parameters on the local Sherwood number, skin friction coefficient, and local Nusselt number. Specific significant findings emerging from the present study highlight that the rate of mass transfer is particularly influenced by the thermophoresis factor, porous parameter, and Williamson parameter, showing heightened effects, while conversely, the Brownian motion parameter demonstrates an opposing pattern. The results were computed and subjected to a comparison with earlier research, indicating a notable degree of conformity and accord.

Keywords: Williamson nanofluid; magnetic field; porous medium; shifted airfoil polynomials; spectral collocation method; convergence analysis

Nomenclature

A , B: constants
B 0: magnitude of magnetic field
c p: the heat capacity for a nanofluid
c: a constant indicating the rate of stretching
C: concentration
C w: nanofluid concentration beside sheet
D B: diffusion rate constant
D T: thermophoresis coefficient
Ec: Eckert number
f: non-dimensional stream function
k: the porous medium’s permeability characteristic
M: magnetic factor
n: positive constant related to nonlinear stretching
Nu x: Nusselt number
Pr: Prandtl number
r , s: constants
Re: Reynolds number
Sc: Schmidt number
T: temperature symbol
T w: temperature beside the sheet
T ∞: temperature away the sheet
U w: fluid velocity induced by stretching processes
u , v: the velocity’s components in the x - and y -directions, respectively
W e: Williamson parameter
: Greek symbols
η: similarity coefficient
μ: viscosity at arbitrary point
μ ∞: ambient viscosity
κ: thermal conductivity
κ ∞: ambient conductivity
ρ: density
ρ ∞: ambient density
Γ: time coefficient of Williamson nanofluid
ψ: stream function
θ: dimensionless temperature
δ: porous parameter
ϕ: dimensionless concentration
σ: electrical conductivity
α: viscosity parameter
ε: conductivity parameter
λ b: parameter governing Brownian motion
λ t: parameter governing thermophoresis phenomenon

1 Introduction

Non-Newtonian fluids, characterized by their departure from the typical behavior seen in Newtonian fluids, demonstrate a broad range of significance and suitability in various engineering and industrial applications. They are essential components in a variety of industrial situations, including manufacturing, chemical engineering, materials research, and more. Their flexibility and ability to adapt to diverse requirements make them indispensable in a variety of applications. Non-Newtonian fluids have unique rheological properties that are characterized by shear-thinning or shear-thickening behavior. These characteristics provide advantages in areas including fluid transport, mixing, and heat transfer, which improves the efficiency and performance of processes. As a result, these fluids are crucial in defining and improving the field of contemporary engineering and industrial practices. Lately, there has been an increasing amount of research concentrating on various non-Newtonian models, including but not limited to the viscoelastic system [1], Maxwell system [2], Casson system [3], power-law system [4], Carreau system [5], and Eyring-Powell system [6].

The Williamson model, which is an important component in the production of emulsion sheets, is an example of one type of such model. This concept is used in many different contexts, such as the flow of plasma, the creation of photographic films, and the comprehension of blood dynamics. This model that was introduced by Williamson [7] is recognized for its shear-thinning behavior within the realm of non-Newtonian fluids. He emphasized the notable significance of this model in differentiating plastic flow from the viscous flow. Furthermore, he found that this model holds equal importance in the field of biological engineering, such as quantifying mass and heat transfer within blood vessels and in hemodialysis applications. Given its paramount importance, a plethora of scholars have committed their time and resources to thoroughly exploring the dynamics of Williamson fluid flow under a wide spectrum of conditions, reflecting the profound interest and dedication within the research community to comprehensively understand this type of fluid. Due to the significance of such fluids, they play a fundamental role in various critical engineering and industrial sectors. They can be analyzed numerically using the homotopy analysis method [8], applied in stagnation point flow [9], influenced by chemical reactions over a stretching cylinder [10], studied within porous mediums [11], examined under the influences of viscous dissipation and slip velocity [12], subjected to thermal radiation and chemical reaction effects [13], explored under the influence of multiple slip boundaries [14], and investigated within nanofluid contexts with magnetic field effects [15].

As suggested by its name, nanofluid consists of the primary fluid mixed with exceptionally small nanoparticles [16]. The defining trait of nanofluid lies in its remarkable thermal conductivity. In this context, the nanoscale layer serves as an effective thermal connector, connecting the solid nanoparticles to the primary fluid [17]. Nanofluids offer significant benefits in various biomedical fields, including enhancing surgical safety through cooling, improving cancer therapy, enhancing X-ray generator performance, boosting high-power laser applications, and facilitating magnetic cell separation [18]. Moreover, the application of magnetic nanofluid systems holds critical significance in the realms of biomedical technology and drug delivery [19]. In addition, the viscosity of the primary fluid in nanofluids plays a role in influencing the Brownian motion of nanoparticles, consequently impacting the thermal conductivity of the nanofluid. Several studies exploring nanofluids, particularly focusing on radiation and slip velocity phenomena [20], prescribed heat flux and viscous dissipation [21], and the combination of the heat and mass transfer (HMT) with a magnetic field [22] pose significant challenges and offer practical applications in various engineering fields. These investigations highlight the profound relevance and potential impact of nanofluids in diverse real-world scenarios.

It is worth noting that obtaining analytical solutions for most ordinary differential equations (ODEs) is a challenging task. Consequently, many researchers focus on obtaining numerical solutions for ODEs. In our case, we place particular emphasis on the numerical solution of the given system. With the objective in sight, we have developed a hybrid spectral collocation method (SCM) that incorporates the shifted airfoil polynomials of second kind (SAP2s). It is essential to highlight that the SCM has been successfully employed for the computational treatment of diverse and intriguing mathematical models. This has been accomplished by utilizing a variety of basis functions, including Bernstein [23], Legendre [24], Chebyshev [25], and Vieta-Lucas polynomials [26]. In the method we propose, employing these basis functions leads to the emergence of a set of nonlinear algebraic equations. These equations are utilized to derive the polynomial form of the required solution. The key advantage of our method, distinguishing it from the previously established SCM in studies by Khader and Sharma [27] and Delkhosh and Cheraghian [28], lies in its unique characteristics. Another advantage is found in the convergence rate, signifying the greater precision of the method in contrast to alternative approaches. Furthermore, we perform an error assessment of the airfoil basis functions applied in our method, making a comparison with the Chebyshev and Legendre polynomials mentioned in studies by Khader and Adel [29] and Adel et al. [30].

The previous research in the existing literature has neglected to present the combined influences of variable fluid properties and viscous dissipation in the scenario of Williamson nanofluid flow over a nonlinearly stretching sheet (SS) through a porous medium, potentially creating a gap in the literature. Hence, the novelty of this current study lies in investigating the flow of dissipative Williamson nanofluid over a nonlinearly SS within a Darcy porous medium, while accounting for variable fluid properties and the presence of a magnetic field. Another distinctive aspect of the current study is the numerical treatment of the problem using the SCM with the SAP2s.

2 Flow analysis

We will elucidate the physical issue at hand by assuming that the flowing liquid behaves as a Williamson nanofluid with a time parameter denoted as Γ . The x -axis is aligned along the sheet’s length, while the y -axis is oriented in the perpendicular direction relative to the sheet’s surface, as depicted in Figure 1.

Figure 1

Illustration of nanofluid flow over a nonlinearly stretching sheet.

The fluid’s motion results from the nonlinear stretching of a sheet in the presence of a magnetic field and the occurrence of viscous dissipation effects. As a consequence of the stretching operation, it may lead to the generation of a fluid velocity profile expressed as U w = c x n , where the parameter c remains constant, and the exponent n characterizes the behavior. The sheet is situated within a porous medium characterized by Darcy’s law. In the present work, we assume that the thermal conductivity κ and viscosity μ of the nanofluid vary with temperature, as previously described by Megahed’s work [31]. It is worth noting that all other properties of the fluid remain constant. In our analysis, we delve into the aspects of the HMT while considering the impact of both Brownian motion characteristics with coefficient D B and thermophoresis phenomenon with coefficient D T . Following the preliminary elucidation and the application of approximations for the specific problem we intend to investigate, it is now imperative to introduce the mathematical equations that accurately represent the prescribed conditions. These equations can be presented subsequently as indicated in studies by Megahed [32] and Abbas et al. [32]:

(1) ∇ · ( u , v ) = 0 ,

(2) ∇ u ⋅ ( u , v ) = 1 ρ ∞ ∂ ∂ y μ ∂ u ∂ y + Γ 2 ∂ u ∂ y 2 − σ B 0 2 ρ ∞ u − μ ρ ∞ k u ,

(3) ∇ T ⋅ ( u , v ) = 1 ρ ∞ c p ∂ ∂ y κ ∂ T ∂ y + τ D B ∂ C ∂ y ∂ T ∂ y + D T T ∞ ∂ T ∂ y 2 + μ ρ ∞ c p ∂ u ∂ y 2 + Γ 2 ∂ u ∂ y 3 ,

(4) ∇ C ⋅ ( u , v ) = D B ∂ 2 C ∂ y 2 + D T T ∞ ∂ 2 T ∂ y 2 .

Here, ∇ = ( ∂ ∂ x , ∂ ∂ y ) . It is important to note that the first equation signifies the continuity equation, the second equation delineates the momentum equation, the third equation represents the energy equation, and the last equation corresponds to the concentration equation. In this research, we adopt the same approach for the magnetic field as previously described by Abbas et al. [33] and Jain et al. [34]. Furthermore, it is important to see that the porous medium proposed in our model is presumed to adhere to the Darcy model, as previously elucidated by Mehta et al. [35]. The term T represents the temperature for the Williamson nanofluid, C is the nanofluid concentration, ρ ∞ denotes the fluid density under ambient conditions, τ is the proportion of the heat capacity of the nanomaterial in relation to that of the fluid, and the property denoted as c p corresponds to the specific heat at constant pressure. Within the same framework, the nonlinear stretching of the impermeable sheet, along with the variations in concentration C w and temperature T w at the sheet’s surface as well as in the ambient C ∞ , T ∞ , can be described by the ensuing set of circumstances:

(5) u = c x n , v = 0 , T w = T ∞ + A x r , C w = C ∞ + B x s , at y = 0 ,

(6) u → 0 , T → T ∞ , C → C ∞ , as y → ∞ .

At this stage, we will present the dimensionless functions, denoted as f , θ , and ϕ , which are dependent on the variable η , and they can be expressed subsequently [33]:

(7) η = c ρ ∞ x n − 1 μ ∞ 1 2 y , ψ ( x , y ) = c x n + 1 μ ∞ ρ ∞ 1 2 f ( η ) ,

(8) θ ( η ) = T − T ∞ T w − T ∞ , ϕ ( η ) = C − C ∞ C w − C ∞ .

Here, μ ∞ represents the dynamic viscosity at the ambient conditions, and ψ ( x , y ) is the stream function that satisfies Eq. (1) as per the following relationships:

(9) u = ∂ ψ ∂ y , v = − ∂ ψ ∂ x .

On the contrary, several key assumptions in this study involve the viscosity changing exponentially in response to temperature, while the thermal conductivity exhibits linear variations with temperature, as specified by the following relationships [31]:

(10) μ = μ ∞ e − α θ , κ = κ ∞ ( 1 + ε θ ) .

Furthermore, derived from these expressions, the parameter α pertains to viscosity, while the parameter ε corresponds to thermal conductivity. By applying Eqs. (7) and (8) to the main Eqs. (1)–(4), we have confidence that Eq. (1) is rigorously fulfilled, while the rest of the equations undergo simplification to the following form:

(11) e − α θ ( 1 + W e f ″ ) f ‴ − α θ ′ f ″ 1 + W e 2 f ″ + n + 1 2 f f ″ − n f ′ 2 − M f ′ − δ e − α θ f ′ = 0 ,

(12) 1 Pr [ ε θ ′ 2 + ( 1 + ε θ ) θ ″ ] + n + 1 2 f θ ′ − r f ′ θ + λ b θ ′ ϕ ′ + λ t θ ′ 2 + Ec f ″ 2 + W e 2 f ″ 3 e − α θ = 0 ,

(13) 1 S c ϕ ″ + n + 1 2 f ϕ ′ − s f ′ ϕ + λ t S c λ b θ ″ = 0 .

Also, the corresponding boundary conditions (B.Cs) can be defined as follows:

(14) f ( 0 ) = 0 , f ′ ( 0 ) = 1 , θ ( 0 ) = 1 , ϕ ( 0 ) = 1 ,

(15) f ′ → 0 , θ → 0 , ϕ → 0 , at η → ∞ .

Here, it is important to examine the structure of the Eqs. (11)–(13) resulting from the proposed physical model. These equations exhibit significant nonlinearity, rendering exact solutions unattainable. Consequently, numerical methods are necessary to address this challenge. In this case, the SCM has been employed for its effectiveness and precision in numerical treatment. The variables mentioned below can be utilized to determine the parameters that dictate the properties of nanofluid velocity, nanofluid concentration, and nanofluid temperature, as illustrated in the previous system (11)–(15):

(16) W e = 2 c 3 2 x 3 m − 1 2 ν ∞ Γ , δ = μ ∞ x 1 − n c ρ ∞ k , Ec = U w 2 c p ( T w − T ∞ ) , Pr = μ ∞ c p κ ∞ ,

(17) λ t = τ ρ ∞ D T ( T w − T ∞ ) μ ∞ T ∞ , M = σ B 0 2 c ρ ∞ , Sc = μ ∞ ρ ∞ D B , λ b = τ ρ ∞ D B ( C w − C ∞ ) μ ∞ ,

where W e is the local Williamson fluid parameter, δ is the porous parameter, Ec is the Eckert number, λ t is the thermophoresis parameter, and λ b is the Brownian motion parameter. Upon concluding this analysis, we observe that both the Williamson fluid parameter and the Eckert number exhibit variations dependent on the spatial coordinate x . To address this challenge, the solution is to set the values of r and s as equal to 2/3 and n as equal to 1/3.

2.1 Quantities crucial for practical engineering purposes

The comprehensive characterization of our problem can be achieved by thoroughly examining the local skin friction coefficient (LSFC) denoted as C f x , the local Sherwood number (LSN) Sh x , and the local Nusselt number (LNN) represented as Nu x , which can be expressed as follows:

(18) C f x = − 2 Re x − 1 2 1 + δ 2 f ″ ( 0 ) f ″ ( 0 ) e − α θ ( 0 ) , Nu x = − Re x 1 2 θ ′ ( 0 ) , Sh x = − Re x 1 2 ϕ ′ ( 0 ) ,

where Re x = U w ρ ∞ x μ ∞ is the local Reynolds number.

3 Approximation functions and numerical application

3.1 Airfoil polynomials and their convergence analysis

We present some concepts and properties of the standard airfoil polynomials of the second-kind (AP2s), as well as, the shifted airfoil polynomials. In addition, we mentioned some theorems related to the convergence analysis of approximation by using this kind of polynomials.

We will achieve this aim through the following points:

The AP2s are obtained from the following formula:
A k ( z ) = sin ( ( 0.5 + k ) ξ ) sin ( 0.5 ξ ) , z = cos ( ξ ) , − 1 ≤ z ≤ 1 .
The airfoil polynomials can be generated from the recurrence identity as follows:
A k ( z ) = 2 z A k − 1 ( z ) − A k − 2 ( z ) , k = 2 , 3 , … .
The first four polynomials are obtained as follows:
A 0 ( z ) = 1 , A 1 ( z ) = 2 z + 1 , A 2 ( z ) = 4 z 2 + 2 z − 1 , A 3 ( z ) = 8 z 3 + 4 z 2 − 4 z − 1 .
Based on the previous formulae, the values of these polynomials at z = 0 , 1 , − 1 are obtained as follows:
A k ( − 1 ) = ( − 1 ) k , A k ( 0 ) = ± 1 , A k ( 1 ) = 2 k + 1 .
The family of the AP2s is orthogonal on ( − 1 , 1 ) :
∫ − 1 1 ( 1 − z ) ( 1 + z ) − 1 A i ( z ) A j ( z ) d z = π , i = j ; 0 , i ≠ j .
The explicit expansion form of the AP2s is given by:
(19) A k ( z ) = ∑ i = 0 k ( − 1 ) i ( 1 + 2 k ) ! 2 k ( 1 + 2 i ) ! ( 2 k − 2 i ) ! ( 1 − z ) i ( 1 + z ) k − i , k = 0 , 1 , 2 , … .
The distinct and real roots (or zeros) of A k ( z ) of degree k on ( − 1 , 1 ) and obtained by solving the equation A k ( z ) = 0 are as follows:
(20) z i = − cos ( π ( 2 i − 1 ) ( 2 k + 1 ) − 1 ) , i = 1 , 2 , … , k .
The SAP2s of degree k on the interval [ 0 , 1 ] is given by implemented the change in variable z = 2 η − 1 , as A k s ( η ) = A k ( 2 η − 1 ) . So, by using the formula in (19), we obtain the polynomials in the following explicit form:
(21) A k s ( η ) = ∑ i = 0 k ( − 1 ) i ( 1 + 2 k ) ! ( 1 + 2 i ) ! ( 2 k − 2 i ) ! ( 1 − η ) i η k − i , k = 0 , 1 , 2 , … .
The orthogonality relation for the SAP2s on ( 0 , 1 ) is given as follows:
∫ 0 1 ( 1 − η ) ∕ η A i s ( η ) A j s ( η ) d η = 0.5 π , i = j ; 0 , i ≠ j .
The zeros of the SAP2 A k s ( η ) of order k are within ( 0 , 1 ) and given by:
(22) η i = 0.5 ( 1 + z i ) , z i = − cos ( ( 2 i − 1 ) ( 2 k + 1 ) − 1 π ) , i = 1 , 2 , … , k .
We can obtain a new form of the explicit formula (21) as a power of η , if we employ the following binomial expansion:
( 1 − η ) i = ∑ j = 0 i ( − 1 ) i − j i ! j ! ( i − j ) ! η i − j .
After substituting the previous binomial expansion into (21) and some modifications, we can obtain a new analytic form for SAP2s ( k = 0 , 1 , 2 , … ):
(23) A k s ( η ) = ∑ i = 0 k ∑ j = k − i i ( − 1 ) 2 j − k + i ( 1 + 2 k ) ! j ! ( 1 + 2 j ) ! ( 2 k − 2 j ) ! ( k − i ) ! ( j − k + i ) ! η i .
The set of SAP2s satisfies the following inequality (for all k ≥ 0 ):
(24) ∣ A k s ( η ) ∣ ≤ 2 k + 1 , ∀ η ∈ [ 0 , 1 ] .

The function Ω ( η ) ∈ L 2 * [ 0 , 1 ] may be expressed in a linear combination of SAP2s as follows [36]:

(25) Ω ( η ) = ∑ i = 0 ∞ c i A i s ( η ) .

With the help of the orthogonality relation, we can obtain the unknown coefficients c i in a closed form.

Now, to prove that (25) is uniformly convergent, we need to estimate the coefficients c i , i = 0 , 1 , 2 , … in (25) through the following theorem.

Theorem 1

[37] For the function Ω ( η ) ∈ L 2 * [ 0 , 1 ] ∩ C ( 2 ) [ 0 , 1 ] be written as (25). Then, the upper bound for the coefficients c i , i = 2 , 3 , 4 , … defined in (25) will estimate as follows:

(26) ∣ c i ∣ < 4 ϒ π i 4 , ϒ = max η ∈ [ 0 , 1 ] ∣ Ω ″ ( η ) ∣ , i = 2 , 3 , … .

If the infinite series solution (25) is truncated up to its first ( m + 1 ) -terms to approximate the function Ω ( η ) , then it may be written as follows:

(27) Ω m ( η ) = ∑ i = 0 m c i A i s ( η ) .

With the help of the weighted L 2 norm on [ 0 , 1 ] related to the weight function ω ˆ ( η ) = ( 1 − η ) ∕ η , we can estimate the approximate error e m ( η ) , and the global error E m ( η ) , by computing the upper bound for each of them through the following theorem.

Theorem 2

For the function Ω ( η ) ∈ L 2 * [ 0 , 1 ] ∩ C ( 2 ) [ 0 , 1 ] be given as in (25) and satisfied that ϒ = max η ∈ [ 0 , 1 ] ∣ Ω ″ ( η ) ∣ . Let Ω m ( η ) and Ω m + 1 ( η ) be two consecutive approximations of Ω ( η ) . Then we have the following two different estimations:

The upper bound for the approximate error e m ( η ) = ∣ Ω m + 1 ( η ) − Ω m ( η ) ∣ holds:
(28) ‖ e m ( η ) ‖ 2 < 8 ϒ π m 4 .
The upper bound for the global error E m ( η ) = ∣ Ω ( η ) − Ω m ( η ) ∣ satisfies:
(29) ‖ E m ( η ) ‖ 2 < ϒ m 3.5 8 7 π .

Proof

The details of the proof of the two estimations (28) and (29) can be found in the study by Hari and Mohammad [37] through Theorems 2 and 3, respectively.□

Through the following theorem, we present an estimation of the n -derivative for the approximation function Ω m ( η ) .

Theorem 3

Let Ω ( η ) be approximated by SAP2s as (27) and also suppose n > 0 , then:

(30) Ω m ( n ) ( η ) ≃ ∑ i = n m c i χ j , ℓ i , n η j − n ,

where χ j , ℓ i , n is given by

(31) χ j , ℓ i , n = ∑ j = 0 i ∑ ℓ = i − j j × ( − 1 ) 2 ℓ − i + j ( 1 + 2 i ) ! ℓ ! j ! ( 1 + 2 ℓ ) ! ( 2 i − 2 ℓ ) ! ( i − j ) ! ( ℓ − i + j ) ! ( j − n ) ! .

Proof

Since differentiation operator D n is a linear we have:

(32) Ω m ( n ) ( η ) = ∑ i = 0 m c i D n ( A i s ( η ) ) .

By employing the basic rules of the differentiation, we can find:

(33) D n A i s ( η ) = 0 , i = 0 , 1 , … , n − 1 , n > 0 .

Also, for i = n , n + 1 , … , m , by using the basic rules of differentiation, we obtain:

(34) D n ( A i s ( η ) ) = ∑ i = n m ∑ j = 0 i ∑ ℓ = i − i j c i × ( − 1 ) 2 ℓ − i + j ( 1 + 2 i ) ! ℓ ! ( 1 + 2 ℓ ) ! ( 2 i − 2 ℓ ) ! ( i − j ) ! ( ℓ − i + j ) ! D n η j = ∑ i = n m ∑ j = 0 i ∑ ℓ = i − i j c i × ( − 1 ) 2 ℓ − i + j ( 1 + 2 i ) ! ℓ ! j ! ( 1 + 2 ℓ ) ! ( 2 i − 2 ℓ ) ! ( i − j ) ! ( ℓ − i + j ) ! ( j − n ) ! η j − n .

A combination of Eqs. (31)–(34) gives us the required formula (30).□

3.2 Procedure of solution using SCM-SAP2s

Now as an extension of the previous work, we use the SAP2s to approximate the solution of the proposed problem under study (11)–(15), where these polynomials used as a set of basis functions with the SCM as a numerical technique [37].

Now, we apply SCM to obtaining the approximate solution of the model (11)–(13). We approximate f ( η ) , θ ( η ) , and ϕ ( η ) by f m ( η ) , θ m ( η ) , and ϕ m ( η ) , respectively, as follows:

(35) f m ( η ) = ∑ i = 0 m a i A i s ( η ) , θ m ( η ) = ∑ i = 0 m b i A i s ( η ) , ϕ m ( η ) = ∑ i = 0 m c i A i s ( η ) .

With using Eqs. (11)–(13), (35) and the formula (30), we obtain:

(36) ( 1 + W e ( f m ( 2 ) ( η ) ) ) ( f m ( 3 ) ( η ) ) − α ( θ m ( 1 ) ( η ) ) ( f m ( 2 ) ( η ) ) × ( 1 + 0.5 W e ( f m ( 2 ) ( η ) ) ) + [ 0.5 ( n + 1 ) ( f m ( η ) ) ( f m ( 2 ) ( η ) ) − n ( f m ( 1 ) ( η ) ) 2 − M ( f m ( 1 ) ( η ) ) ] e α θ m ( η ) − δ ( f m ( 1 ) ( η ) ) = 0 ,

(37) ( 1 + ε ( θ m ( η ) ) ) ( θ m ( 2 ) ( η ) ) + ε ( θ m ( 1 ) ( η ) ) 2 + Pr [ 0.5 ( n + 1 ) ( f m ( η ) ) ( θ m ( 1 ) ( η ) ) − r ( f m ( 1 ) ( η ) ) ( θ m ( η ) ) + λ b ( θ m ( 1 ) ( η ) ) ( ϕ m ( 1 ) ( η ) ) + λ t ( θ m ( 1 ) ( η ) ) 2 ] + Pr Ec [ ( f m ( 2 ) ( η ) ) 2 + 0.5 W e ( f m ( 2 ) ( η ) ) 3 ] e − α θ m ( η ) = 0 ,

(38) ( ϕ m ( 2 ) ( η ) ) + Sc [ 0.5 ( n + 1 ) ( f m ( η ) ) ( ϕ m ( 1 ) ( η ) ) − s ( f m ( 1 ) ( η ) ) ( ϕ m ( η ) ) ] + ( λ t ∕ λ b ) ( θ m ( 2 ) ( η ) ) = 0 .

The previous system (36)–(38) will be collocated at m − 2 of nodes η p , which defined in (22) as follows:

(39) ( 1 + W e ( f m ( 2 ) ( η p ) ) ) ( f m ( 3 ) ( η p ) ) − α ( θ m ( 1 ) ( η p ) ) × ( f m ( 2 ) ( η p ) ) ( 1 + 0.5 W e ( f m ( 2 ) ( η p ) ) ) + [ 0.5 ( n + 1 ) ( f m ( η p ) ) ( f m ( 2 ) ( η p ) ) − n ( f m ( 1 ) ( η p ) ) 2 − M ( f m ( 1 ) ( η p ) ) ] e α θ m ( η p ) − δ ( f m ( 1 ) ( η p ) ) = 0 ,

(40) ( 1 + ε ( θ m ( η p ) ) ) ( θ m ( 2 ) ( η p ) ) + ε ( θ m ( 1 ) ( η p ) ) 2 + Pr [ 0.5 ( n + 1 ) ( f m ( η p ) ) ( θ m ( 1 ) ( η p ) ) − r ( f m ( 1 ) ( η p ) ) ( θ m ( η p ) ) + λ b ( θ m ( 1 ) ( η p ) ) ( ϕ m ( 1 ) ( η p ) ) + λ t ( θ m ( 1 ) ( η p ) ) 2 ] + Pr Ec [ ( f m ( 2 ) ( η p ) ) 2 + 0.5 W e ( f m ( 2 ) ( η p ) ) 3 ] e − α θ m ( η p ) = 0 ,

(41) ( ϕ m ( 2 ) ( η p ) ) + Sc [ 0.5 ( n + 1 ) ( f m ( η p ) ) ( ϕ m ( 1 ) ( η p ) ) − s ( f m ( 1 ) ( η p ) ) ( ϕ m ( η p ) ) ] + ( λ t ∕ λ b ) ( θ m ( 2 ) ( η p ) ) = 0 .

Also, the BCs (14)–(15) may be approximated by substituting from Eq. (35) in (14)–(15) as follows:

(42) ∑ i = 0 m ± a i = 0 , ∑ i = 0 m a i f i ( 1 ) ( 0 ) = 1 , ∑ i = 0 m ± b i = 0 , ∑ i = 0 m ± c i = 1 ,

(43) ∑ i = 0 m a i f i ( 1 ) ( η ∞ ) = 0 , ∑ i = 0 m b i θ i ( η ∞ ) = 0 , ∑ i = 0 m c i ϕ i ( η ∞ ) = 0 .

Eqs. (39)–(43) give a system of 3 ( m + 1 ) algebraic equations. We will apply the Newton iteration method to solving this nonlinear system for the coefficients a i , b i , c i , i = 0 , 1 , … , m .

4 Model validation

To assess the accuracy and reliability of the current code developed through the SCM, we have computed the values of − θ ′ ( 0 ) for the case, where δ = α = M = W e = Ec = ε = r = 0 , as well as for many values of the parameter n . This computation was performed using MATHEMATICA software, and the results are given in Table 1. Analysis of the data given in Table 1 reveals a remarkable concurrence between the results generated by the current code and those obtained by Cortell [38]. This strong agreement between the two sets of data substantiates the validity and reliability of the numerical code we have employed in this study.

Table 1

Comparison of the values of ‒ θ ′ ( 0 ) for various values of n with the results of Cortell [38] when δ = α = M = W e = Ec = ε = r = 0 and Pr = 1.0

n	Cortell [38]	Present work
0.2	0.610262	0.61026199520
0.5	0.595277	0.59527660029
1.5	0.574537	0.57453690125

Through Table 2, we give the evaluated values of the residual error function (REF) to verify the validity of the approximate method applied, where the values of the parameters are taken δ = α = W e = Ec = ε = 0.2 , M = 0.5 , λ t = 0.1 , λ b = 0.8 , n = 1 3 , r = s = 2 3 , and Pr = Sc = 2.0 . In light of these values, we can confirm the accuracy of the proposed method.

Table 2

The values of the REF obtained by the presented scheme

η	REF of f ( η )	REF of θ ( η )	REF of ϕ ( η )
0.0	4.951456 × 1 0 ‒ 7	4.963025 × 1 0 ‒ 7	1.635504 × 1 0 ‒ 7
1.0	3.654023 × 1 0 ‒ 7	3.952145 × 1 0 ‒ 6	3.963025 × 1 0 ‒ 6
2.0	2.015975 × 1 0 ‒ 6	2.965412 × 1 0 ‒ 5	4.875421 × 1 0 ‒ 6
3.0	3.654187 × 1 0 ‒ 6	0.963258 × 1 0 ‒ 6	6.320054 × 1 0 ‒ 7
4.0	0.012450 × 1 0 ‒ 6	1.021458 × 1 0 ‒ 6	9.951024 × 1 0 ‒ 7
5.0	3.963258 × 1 0 ‒ 6	6.963258 × 1 0 ‒ 7	0.758945 × 1 0 ‒ 6
6.0	0.654120 × 1 0 ‒ 7	3.963258 × 1 0 ‒ 6	2.852240 × 1 0 ‒ 5
7.0	9.951753 × 1 0 ‒ 5	2.650210 × 1 0 ‒ 5	2.689423 × 1 0 ‒ 5
8.0	3.602405 × 1 0 ‒ 6	9.002546 × 1 0 ‒ 5	1.753210 × 1 0 ‒ 7

5 Interpretation of numerical results

In our research study, we delved into the intricate dynamics of a nanofluid as it traversed a boundary layer (BL) over a nonlinear SS capable of extension in two dimensions. The application of the Williamson model facilitated our exploration of this phenomenon. To confine this surface, a porous medium was employed, introducing variability in the fluid’s properties. In addition, our investigation encompassed an examination of the effect of both M and α on the overall system. We conducted a thorough examination of how these factors exerted their influence on key parameters, including velocity, the distribution of the HMT. Our investigation aimed to understand the intricate interplay between these elements and their collective impact on the dynamic behavior of the system. To depict the fluctuations in the computed outcomes, we utilized the SCM. This method relies on the utilization of shifted airfoil polynomials, and we employed the Mathematica software to create visual representations of the data. This approach allowed us to effectively visualize and communicate the variations in the calculated results through graphical representations.

Figure 2 illustrates the velocity profiles denoted as f ′ ( η ) , temperature profiles represented by θ ( η ) , and concentration profiles indicated by ϕ ( η ) . These profiles are associated with different viscosity parameters, denoted as α . The figure serves to visually communicate the variations in f ′ ( η ) , θ ( η ) , and ϕ ( η ) across the range of α considered in our analysis. The depicted figure reveals a coherent trend: with the escalation of viscosity parameters, a consistent rise is evident in the concentration within the free stream and the temperature of the fluid. In contrast, the velocity distribution and the thickness of the BL exhibit a divergent pattern, diminishing as the viscosity parameters increase. This observed consistency and contrast offer valuable insights into the interplay between viscosity parameters and the dynamic characteristics of the system.

$Figure 2 (a) f ′ ( η ) f^{\prime} \left(\eta ) for various α \alpha and (b) ϕ ( η ) \phi \left(\eta ) and θ ( η ) \theta \left(\eta ) for various α \alpha .$

Figure 2

(a) f ′ ( η ) for various α and (b) ϕ ( η ) and θ ( η ) for various α .

Figure 3 illustrates the impact of variations in δ on the distribution of f ′ ( η ) , θ ( η ) , and ϕ ( η ) . These visual representations provide insight into how alterations in the porous parameter contribute to the changes observed in f ′ ( η ) , θ ( η ) , and ϕ ( η ) profiles within the system under consideration. Clearly discernible from the data is the fact that elevating the porous parameter results in an augmentation of the concentration distribution, the thickness of the temperature BL, and the temperature distribution. Conversely, a contrasting trend is observed in the velocity distribution, which decreases as the porous parameter is increased. These findings highlight the intricate relationship between the porous parameter and the dynamic characteristics of the system, shedding light on the nuanced effects on concentration, temperature, and velocity distributions. Physically, better fluid flow across the medium, which permits the transportation of more nanoparticles and raises dispersion rates and concentrations, is the reason why increased porosity parameter results in higher nanofluid concentrations.

$Figure 3 (a) f ′ ( η ) f^{\prime} \left(\eta ) for various δ \delta and (b) ϕ ( η ) \phi \left(\eta ) and θ ( η ) \theta \left(\eta ) for various δ \delta .$

Figure 3

(a) f ′ ( η ) for various δ and (b) ϕ ( η ) and θ ( η ) for various δ .

Figure 4 illustrates how variations in M impact the f ′ ( η ) , θ ( η ) , and ϕ ( η ) . These changes have the potential to lead to an augmentation in both temperature and concentration fields. Simultaneously, there is an observable decrease in f ′ ( η ) , especially in the vicinity of the magnetic source. This phenomenon arises due to the capacity of M to hinder the flow of the fluid, resulting in a reduction of its kinetic energy and consequent decrease in the velocity field. In addition, an increase in the magnetic parameter values correlates with an elevated trend in both the θ ( η ) and ϕ ( η ) . Physically, as the magnetic parameter rises, the magnetic field strength intensifies, creating increased resistance against the fluid’s motion. This heightened resistance results in more energy dissipation as heat, ultimately causing a rise in fluid temperatures.

$Figure 4 (a) f ′ ( η ) f^{\prime} \left(\eta ) for various M M and (b) ϕ ( η ) \phi \left(\eta ) and θ ( η ) \theta \left(\eta ) for various M M .$

Figure 4

(a) f ′ ( η ) for various M and (b) ϕ ( η ) and θ ( η ) for various M .

Figure 5 depicts the influence of changes in W e on the f ′ ( η ) , θ ( η ) , and ϕ ( η ) . The diagram visually represents how alterations in the parameter W e affect the scaled velocity, concentration, and temperature values. As the local Williamson nanofluid parameter rises, there is a simultaneous elevation in both θ ( η ) and ϕ ( η ) , tending to converge toward a limit as the parameter approaches infinity. Conversely, the velocity undergoes an inverse trend as the same parameter increases. In a physical sense, as the local Williamson fluid parameter rises, the velocity of the nanofluid decreases. This is due to the fluid exhibiting greater non-Newtonian characteristics and increased viscosity, ultimately leading to a reduction in flow speed.

$Figure 5 (a) f ′ ( η ) f^{\prime} \left(\eta ) for various W e {W}_{e} and (b) ϕ ( η ) \phi \left(\eta ) and θ ( η ) \theta \left(\eta ) for various W e {W}_{e} .$

Figure 5

(a) f ′ ( η ) for various W e and (b) ϕ ( η ) and θ ( η ) for various W e .

Figure 6(a) explores how changes in Ec affect the θ ( η ) , in a Williamson nanofluid. This analysis aims to understand the impact of variations in Ec on the thermal characteristics of the nanofluid. This illustration reveals that with an increase in the Eckert parameter, there is a rapid elevation in temperature, resulting in a more pronounced curve. Consequently, owing to this distinctive response of the Eckert parameter, the thickness of the thermal BL expands. Figure 6(b) illustrates the influence of variations in the thermal conductivity parameter, denoted as ε . A heightened thermal conductivity parameter ε signifies a direct and more responsive reaction of the fluid to temperature changes. Therefore, as ε increases, there is a corresponding elevation in the temperature profile. This phenomenon additionally results in an increase in the thickness of the thermal BL.

$Figure 6 (a) θ ( η ) \theta \left(\eta ) for various Ec and (b) θ ( η ) \theta \left(\eta ) for various ε \varepsilon .$

Figure 6

(a) θ ( η ) for various Ec and (b) θ ( η ) for various ε .

Figure 7 shows the changes in both the ϕ ( η ) and θ ( η ) by manipulating the values of λ b . Clearly, an escalation in λ b induces greater random motion among nanofluid particles, causing an expansion in both the temperature profile and the thermal thickness. Conversely, the concentration field exhibits an opposite trend. Physically, this phenomenon arises because the Brownian motion of particles is essentially a result of the cumulative effects of collisions between fluid molecules and the surfaces of the particles.

$Figure 7 (a) θ ( η ) \theta \left(\eta ) for various λ b {\lambda }_{b} (b) ϕ ( η ) \phi \left(\eta ) for various λ b {\lambda }_{b} .$

Figure 7

(a) θ ( η ) for various λ b (b) ϕ ( η ) for various λ b .

Figure 8 depicts the impact of variations in λ t on the distribution of ϕ ( η ) and θ ( η ) within the specified domain. In the BL, thermophoresis phenomenon serves a dual purpose as it facilitates the diffusion of both various species and thermal energy, including nanoparticles. The figure unmistakably indicates that elevating the thermophoresis parameter results in an augmentation of the thicknesses for both the thermal and concentration BLs. Physically, the thermophoresis phenomenon contributes to the conveyance of both temperature and particles within the BL. As the system undergoes heating, a positive outcome arises with the improved dispersion of nanoparticles. This enhancement in nanoparticle distribution is directly linked to the increase in λ t .

$Figure 8 (a) θ ( η ) \theta \left(\eta ) for various λ t {\lambda }_{t} and (b) ϕ ( η ) \phi \left(\eta ) for various λ t {\lambda }_{t} .$

Figure 8

(a) θ ( η ) for various λ t and (b) ϕ ( η ) for various λ t .

Table 3 exhibits the numerical values corresponding to the skin friction coefficient Re x 1 2 C f x , local Sherwood number Re x − 1 2 Sh x , and local Nusselt number Re x − 1 2 Nu x . This table illustrates that as the values of the magnetic parameter increase, they exert an impact on the fluid dynamics in the vicinity of the stretching sheet. Hence, the adjustment in magnetic parameter values results in a decrease in both the LSN and the LNN, coupled with an increase in the skin friction coefficient. Moreover, it is noteworthy that the LSFC experiences an elevation with an increase in the local Williamson nanofluid parameter. Conversely, it demonstrates a declining pattern with higher values of both the viscosity parameter and the porous parameter. Furthermore, with the elevation of the thermophoresis parameter, thermal conductivity parameter, and Brownian motion parameter, there is a demonstrated decrease in both the LNN and the LSFC. In addition, it is evident that an elevation in the nanofluid Eckert number, Brownian motion parameter, and thermal conductivity parameter leads to an increase in the LSN. Conversely, the LSN exhibits a significantly lower value with both the thermophoresis parameter and the local Williamson nanofluid parameter.

Table 3

Values of Re x − 1 2 Nu x , Re x 1 2 C f x , and Re x − 1 2 Sh x for varying the governing parameters with r = s = 2 3 , n = 1 3 , Pr = 2.0 , and Sc = 2.0

α	δ	M	W e	Ec	ε	λ b	λ t	Re x 1 2 C f x	Re x − 1 2 Nu x	Re x − 1 2 Sh x
0.0	0.2	0.5	0.2	0.2	0.2	0.8	0.1	1.028510	0.475591	1.12629
0.2	0.2	0.5	0.2	0.2	0.2	0.8	0.1	0.930432	0.467730	1.09556
0.4	0.2	0.5	0.2	0.2	0.2	0.8	0.1	0.839264	0.458904	1.06236
0.2	0.0	0.5	0.2	0.2	0.2	0.8	0.1	0.978946	0.489121	1.12303
0.2	0.2	0.5	0.2	0.2	0.2	0.8	0.1	0.930432	0.467730	1.09556
0.2	0.5	0.5	0.2	0.2	0.2	0.8	0.1	0.759543	0.437799	1.05885
0.2	0.2	0.0	0.2	0.2	0.2	0.8	0.1	0.708824	0.532651	1.17971
0.2	0.2	0.5	0.2	0.2	0.2	0.8	0.1	0.930432	0.467730	1.09556
0.2	0.2	1.0	0.2	0.2	0.2	0.8	0.1	1.102403	0.412270	1.02896
0.2	0.2	0.5	0.0	0.2	0.2	0.8	0.1	0.859314	0.474861	1.11499
0.2	0.2	0.5	0.2	0.2	0.2	0.8	0.1	0.930432	0.467730	1.09556
0.2	0.2	0.5	0.4	0.2	0.2	0.8	0.1	1.066561	0.456185	1.06779
0.2	0.2	0.5	0.2	0.0	0.2	0.8	0.1	0.932297	0.566191	1.08608
0.2	0.2	0.5	0.2	0.2	0.2	0.8	0.1	0.930432	0.467730	1.09556
0.2	0.2	0.5	0.2	0.5	0.2	0.8	0.1	0.927625	0.319948	1.10977
0.2	0.2	0.5	0.2	0.2	0.0	0.8	0.1	0.931568	0.497637	1.09356
0.2	0.2	0.5	0.2	0.2	0.2	0.8	0.1	0.930432	0.467730	1.09556
0.2	0.2	0.5	0.2	0.2	0.5	0.8	0.1	0.928977	0.430564	1.09796
0.2	0.2	0.5	0.2	0.2	0.2	0.2	0.1	0.936043	0.712043	0.92535
0.2	0.2	0.5	0.2	0.2	0.2	0.8	0.1	0.930432	0.467730	1.09556
0.2	0.2	0.5	0.2	0.2	0.2	1.5	0.1	0.925614	0.297561	1.11521
0.2	0.2	0.5	0.2	0.2	0.2	0.8	0.0	0.930966	0.482198	1.12225
0.2	0.2	0.5	0.2	0.2	0.2	0.8	0.2	0.929903	0.454015	1.07069
0.2	0.2	0.5	0.2	0.2	0.2	0.8	0.5	0.928492	0.418783	1.00451

6 Conclusions

This research introduced a new approach by employing the SCM to numerically analyze the magnetohydrodynamic dissipative Williamson nanofluid flow over a nonlinearly SS within a Darcy porous medium. In addition, the study extensively examines mass and heat transfer efficiency, while also addressing the combined effects of changing fluid properties. The numerical findings are visually presented through graphs and tables and are extensively discussed. The primary findings that have emerged from this investigation can be summarized as follows:

To enhance the energy distribution of the Williamson nanofluid flow, it is imperative to employ a non-Newtonian fluid characterized by a high Williamson parameter and to augment the viscosity parameter.
The presence of a porous medium has the capability to bring about increased regularity in the fluid near the surface, resulting in a smoother flow pattern and a reduced rate of the HMT, along with a diminished skin friction coefficient.
An increase in both α and M resulted in higher temperatures of the nanofluid, an augmentation in thermal thickness, and an elevated concentration field.
Elevating the Williamson parameter led to an increased rise in the LSFC across the BL, accompanied by a decrease in both the LNN and LSN.
An augmentation in the magnetic number results in an upsurge in the magnitude of the LSFC. Nevertheless, the incorporation of variable fluid viscosity and the existence of the porous parameter contribute to its subsequent decrease.
Possible avenues for future research stemming from this study may involve expanding upon the current findings by incorporating variable heat flux utilizing various types of polynomials or fractional derivatives.

Acknowledgments

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RP23092).

Funding information: This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIURP23092).
Author contributions: Conceptualization: M.M.K. and H.A.; methodology: M.M.K. and M.A.; software: M.M.K., A.M.M. and H.A.; validation: M.M.K. and H.A.; formal analysis: M.M.K., H.A., and M.A.; investigation: M.M.K., M.A., and A.M.M.; resources: M.M.K. and M.A.; data curation: M.M.K., H.A., and A.M.M.; writing–original draft preparation: M.M.K., M.A. and H.A.; writing–review and editing: M.M.K., M.A., and A.M.M.; visualization: M.M.K. and H.A.; supervision: M.M.K. and H.A.; project administration: M.M.K. and H.A. 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.

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Received: 2023-11-28

Revised: 2024-02-19

Accepted: 2024-04-02

Published Online: 2024-05-04

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

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0016

Keywords for this article

Williamson nanofluid; magnetic field; porous medium; shifted airfoil polynomials; spectral collocation method; convergence analysis

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