Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model

Nomenclature

a

constant ( s − 1 )

B 0

strength of magnetic field ( T )

c p

specific heat ( J kg K )

Cf x

coefficient of skin friction

Ec

coefficient of viscous dissipation (Eckert number)

f

nondimensional stream function

k 0

elastic parameter ( m 2 )

k *

absorption coefficient ( m − 1 )

M

magnetic parameter

Nu x

coefficient of heat transfer

Pr

Prandtl number

q r

radiative heat flux ( W s − 2 )

R

radiation parameter

Re

the local Reynolds number

T

temperature of the viscoelastic fluid (K)

T ∞

viscoelastic temperature away the sheet (K)

T w

temperature of the viscoelastic fluid beside the sheet (K)

U w

velocity of fluid due to stretching processes ( m s − 1 )

u

the x -component of the velocity vector ( m s − 1 )

v

the y -component of the velocity vector ( m s − 1 )

x , y

Cartesian coordinates (m)

Greek symbols

μ

viscosity of viscoelastic fluid ( kg m − 1 s − 1 )

η

similarity coefficient

ρ

density of viscoelastic fluid ( kg m − 3 )

κ

thermal conductivity of viscoelastic fluid ( W K − 1 m − 1 )

λ

the duration required for the heat flux to attain a state of equilibrium (s)

ν

kinematic viscosity ( m 2 s )

σ

electrical conductivity ( s m − 1 )

σ *

Stefan-Boltzmann constant ( J s − 1 m − 2 K − 4 )

Λ

viscoelastic parameter

γ

thermal relaxation parameter

θ

nondimensional temperature

Superscripts

w

condition along the sheet

′

differentiation with respect to η

∞

condition at the ambient

1 Introduction

In contemporary times, non-Newtonian fluids have become notably important in various industrial applications. Many synthetic fluids, as well as certain naturally occurring ones, display complex rheological properties, with viscoelasticity emerging as a significant fluid attribute. Over the last six decades, computational rheology, utilizing computational fluid dynamics to study fluids with non-Newtonian rheology, has matured into a well-established field. This area of study not only enhances our understanding of various physical phenomena but also provides valuable insights for engineering design purposes. Computational rheology involves conducting flow simulations using fluids characterized by non-Newtonian models, which are more intricate than the broadly defined Newtonian fluids. Given the complex nature of the constitutive equations, specialized techniques are necessary to tackle the inherent numerical challenges, even when the simulations are focused on fluid mechanics aspects [1]. Viscoelastic fluids represent a specific instance of the model proposed by Rivlin and Ericksen [2]. It possesses a diverse array of practical applications in various real-world contexts, encompassing fields such as computing, medicine, the polymer industry, engineering, and the automotive sector. Distinct from typical Newtonian fluids, these specialized substances referred to as viscoelastic fluids showcase an extra property of elasticity alongside their viscosity, enabling them to both retain and release shear energy. The previous studies [3–6] encompass a handful of studies concerning models for viscoelastic fluids with non-Newtonian behavior.

Heat transfer entails the exchange of thermal energy between two or more objects or systems. The process of heat transfer takes place when there is a temperature disparity either between different bodies or within distinct sections of a single body. This mechanism finds extensive use in technological and industrial contexts, such as cooling electronic devices, managing nuclear reactor temperatures, power generation, and various other applications [7]. Heat transfer occurs through various distinct mechanisms. The highly successful heat conduction model introduced by Fourier [8] and the mass diffusion model put forth by Fick [9] have garnered considerable interest due to their interchangeable characteristics and the omission of inconsistencies that arise in different scenarios within these theories. However, a significant drawback of these models is that they result in a parabolic energy equation, implying that any initial disturbance would be immediately felt by the system being analyzed. This phenomenon is commonly mentioned in the literature as the paradox of heat conduction. To address this constraint, Cattaneo [10] amended this principle by introducing an additional term involving relaxation time. Subsequently, Christov [11] enhanced the Cattaneo model by substituting the conventional derivative with Oldroyd’s upper-convected derivative. He formulated a unified energy equation for the governing issue, leading to the creation of the Cattaneo–Christov heat flux model. The importance of the Cattaneo–Christov heat flux model played a pivotal role in instigating more fundamental researches within this particular field of study [12,13].

Extensive research has been conducted on ordinary and partial differential equations (ODEs and PDEs, respectively) due to their extensive applicability across various domains, including fluid mechanics, viscoelasticity, biology, physics, and engineering. Because of the prevalence of ODEs and PDEs in these fields, a substantial amount of research has been dedicated to comprehending their properties, developing efficient solution methods, and unraveling their implications for diverse phenomena and systems. Consequently, considerable focus has been directed toward obtaining solutions for physically significant ODEs [14]. The utilization of the spectral collocation method (SCM) employing sixth-kind shifted Chebyshev polynomials ( CP 6 s ) [15] was applied to quantitatively address the nonlinear ODEs governing the physical scenario. This research stands out as the pioneering instance of numerically solving the provided model using the proposed novel numerical approach.

Building upon previous investigations, many researchers have effectively employed diverse numerical methodologies in this specific field. One such technique is the SCM, which stands as a versatile approximate analytical approach for deriving approximate solutions to differential equations. The SCM presents several merits in tackling such issues, as the Chebyshev coefficients of the solution can be easily computed using any accessible numerical software, resulting in superior computational efficiency. Chebyshev polynomials, recognized orthogonal polynomials defined within the interval [ − 1 , 1 ] , are commonly utilized due to their advantageous characteristics in approximating functions. Furthermore, this approach is distinguished by its remarkable precision, rendering it a dependable numerical method. With its capacity to ensure precise calculations and dependable results, the technique guarantees swift convergence rates and straightforward application in a variety of problem contexts, spanning finite and infinite domains, as demonstrated in previous works [16–19]. It exhibits rapid convergence, enabling accurate solutions to be attained with minimal terms, effectively conserving computational resources and time. Furthermore, its numerical stability mitigates the risk of errors, even when grappling with intricate or demanding issues. In addition, the method’s flexibility permits its utilization not only in differential equations but also in optimization problems, establishing its value across a wide array of domains. This method has been extensively utilized in different problems [20–23] and has played a significant role in conducting crucial research owing to these benefits.

The thermal radiation, viscous dissipation, and magnetic field have garnered considerable interest within the realm of thermal non-Newtonian fluids. Researchers and scholars have focused their attention on these phenomena, recognizing their importance in understanding the complex behavior of such fluids. The aim of incorporating these innovative aspects into the study of viscoelastic fluids is to enhance the precision in representing the transport properties of the fluid. The inclusion of the Cattaneo and Christov heat flux phenomenon becomes crucial in this context, especially when dealing with a surface that is subject to linear stretching. These considerations contribute to a more comprehensive understanding of the intricate thermal characteristics of viscoelastic fluids, thereby improving the accuracy of models used to describe their behavior. Furthermore, the application of the sixth-order shifted Chebyshev polynomials approximation and the utilization of the spectral collocation technique represent innovative elements in this investigation as an effective numerical approach. This numerical technique is utilized to visually demonstrate the influence of these characteristics using table and graphs.

2 Problem modeling

We examine the consistent two-dimensional movements of non-Newtonian viscoelastic fluids over a linearly extending surface. The Cartesian coordinate system is utilized in a manner where the x -axis aligns with the stretching surface, while the y -axis is perpendicular to it (Figure 1).

Figure 1

Description of the proposed viscoelastic model.

We are examining flow models that pertain to both second-grade fluids and elastico-viscous fluids. Let ( u , v ) denote the components of velocity in the ( x , y ) directions, respectively. Let’s represent the linear velocity along the x -direction as U w ( x ) = a x , where the constant a signifies the stretching velocity of the surface. The analysis incorporates the phenomenon of viscous dissipation, which holds considerable importance in fluid dynamics, thermal sciences, and various engineering domains. The significance of viscous dissipation lies in its ability to enhance precise and comprehensive assessments, thereby contributing to improved design, optimization, and innovation in diverse applications. In addition, the study delves into heat transfer mechanisms, exploring thermal radiation and employing the Cattaneo–Christov theory for heat flux. Given these assumptions, the governing equations relevant to the boundary layer can be articulated as follows [24]:

where ν = μ ρ represents the kinematic viscosity, μ stands for dynamic viscosity, ρ denotes the density of the viscoelastic fluid, c p represents specific heat, and k 0 is the elastic parameter. In this context, a positive value of k 0 indicates an elastico-viscous fluid, while a negative k 0 corresponds to a second-grade fluid. A k 0 value of zero is associated with a Newtonian fluid. T is the viscoelastic temperature, κ is the thermal conductivity coefficient, and λ stands for the time it takes for the heat flux to reach a state of relaxation. In addition, it is important to note that when λ equals zero, Eq. (3) simplifies to the conventional Fourier’s law of heat conduction. B 0 is the strength of the proposed magnetic field, and σ is the electric conductivity. Moreover, q r represents the radiative heat flux, and its magnitude depends on the temperature as described by the subsequent relationship [25]:

Clearly that the formula for q r involves constants: the Stefan-Boltzmann constant (symbolized as σ * ) and the absorption coefficient (denoted as k * ). These attributes hold considerable influence over the radiative heat flux and its interactions with the system. In addition, we make the assumption that the temperature disparity within the flow of the viscoelastic fluid enables us to express the term T 4 through a linear combination of temperatures. By employing Taylor’s series and focusing solely on terms of minimal order, we reach the subsequent formulation [26]:

The boundary conditions under consideration are as follows:

where T w is the sheet temperature and T ∞ is the ambient temperature. In this context, it is necessary to validate that the movement is solely a result of the wall being stretched and the absence of any velocity in the free stream, as indicated by the condition u → 0 as y → ∞ . The utilization of the following similarity transformations, expressed in relation to the functions f and θ , along with the introduction of the similarity variable η is expounded upon in the subsequent manner to solve the governing equations [27]:

It is evident that the expressions for u and v as presented in Eq. (8) fulfill the conditions set forth in Eq. (1). Now, Eqs. (2) and (3) are converted into the subsequent formats by using Eq. (8):

Likewise, the adjusted boundary conditions can be detailed in the following manner:

It is evident that our physical system is entirely controlled by the viscoelastic parameter Λ = a k 0 ρ μ , the magnetic number M = σ B 0 2 a ρ , the thermal relaxation parameter γ = a λ , the Eckert number Ec = U w 2 c p ( T w − T ∞ ) , the thermal radiation parameter R = 16 σ * T ∞ 3 3 κ k * and the Prandtl number Pr = μ c p κ . Once more, the prior framework adequately captures the actions of the non-Newtonian viscoelastic fluid Λ , defined by the existence of the viscoelastic parameter. Moreover, the omission of this parameter ( Λ = 0 ) allows us to streamline our proposed system into a Newtonian model.

3 Industrial and engineering measures

The crucial role of the local skin-friction coefficient C f x Re x 1 2 and the local Nusselt number Nu x Re x 1 2 lies in their ability to forecast fluid flow patterns and heat transfer rates, rendering them essential in diverse engineering and industrial contexts. Hence, it is crucial to finalize our investigation by examining how the governing parameters affect both of these measures. These measures can be presented in a dimensionless format in the subsequent manner:

where Re x = U w x ν is the local Reynolds number.

4 Procedure of solution using SCM

5 Discussion on results

Here, we discuss the results of the Eqs. (9) and (10) of a system of coupled nonlinear ODEs were computed numerically using the given method with the help of the Mathematica software version 11. This section pertains to the changes observed in velocity and temperature profiles as specific values of significant parameters are chosen. These parameters encompass the viscoelastic parameter Λ , magnetic number M , thermal relaxation parameter γ , thermal radiation parameter R , Eckert number Ec, and Prandtl number Pr. In this context, we are examining fluids with positive values of the elastico-viscous parameter ( Λ > 0 ). This objective is fulfilled by means of Figures 2–5. Figure 2 illustrates how the velocity f ′ ( η ) and temperature θ ( η ) fields are influenced by changes in the viscoelastic parameter Λ . Raising the viscoelastic parameter leads to heightened interfacial forces exerted on the fluid. Consequently, the fluid’s velocity profile might be impacted, potentially resulting in a decline as the viscoelastic parameter values increase. Furthermore, the temperature distribution and the thickness of the thermal boundary layer experience augmentation as this parameter is increased. The decrease in velocity as the viscoelastic parameter increases can be physically explained by the intensified intermolecular interactions and elevated internal friction within the fluid. These factors contribute to an enhanced resistance to the flow of the fluid.

Figure 2

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

Figure 3 demonstrates the influence of the magnetic parameter on both f ′ ( η ) and θ ( η ) , which can potentially cause a decrease in the velocity profile, particularly near the magnetic source. This occurs because the magnetic field has the ability to impede fluid movement, diminishing its kinetic energy and subsequently leading to a decline in the velocity field. Furthermore, the temperature profile displays an escalated trend as the magnetic parameter values increase. In a physical sense, when the magnetic parameter increases, the strength of the magnetic field becomes stronger, which in turn creates more opposition to the fluid’s movement. This heightened resistance contributes to a greater dissipation of energy as heat, ultimately leading to elevated temperatures within the fluid. Moreover, an in-depth understanding of the influence of the magnetic field on the heat transfer mechanism can be substantiated by comprehensively reviewing relevant research studies in this domain [33–37].

Figure 3

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

Figure 4(a) investigates the influence of variations in the thermal radiation parameter R on the temperature profile θ ( η ) of a viscoelastic fluid. This figure demonstrates that as the thermal radiation parameter increases, the temperature experiences a swift rise, causing the curve to become more pronounced. Consequently, due to this particular behavior of the thermal radiation parameter, the thermal boundary layer thickness expands. Figure 4(b) depicts the impact of the thermal relaxation parameter γ . A higher thermal relaxation parameter indicates that the fluid’s response to temperature changes is gradual. Consequently, as the thermal relaxation parameter increases, there is a decrease in the temperature profile. This effect also leads to a reduction in the thickness of the thermal boundary layer. It is important to note that when the thermal relaxation parameter γ equals zero, it results in the simplification of the heat flux expression to the conventional Fourier’s law. Physically, the reduction in temperature with an elevation in the thermal relaxation parameter can be ascribed to the extended duration required for the material to achieve thermal equilibrium. With an increase in the thermal relaxation parameter, the material demonstrates a sluggish response to temperature changes, resulting in an overall decrease in temperature.

Figure 4

(a) θ ( η ) for various R and (b) θ ( η ) for various γ .

Conversely, Figure 5(a) examines the impact of the Eckert number Ec on temperature expansion θ ( η ) . The observable trend is that the temperature profile rises with an increase in Eckert number values. This phenomenon arises from the conversion of the fluid’s kinetic energy into thermal energy through viscous dissipation, resulting in a steeper temperature gradient. In the examination of heat transfer, the Prandtl number holds significant importance, especially within the realm of convective heat transfer. Fluids with elevated Prandtl numbers, like honey or oil, possess greater momentum diffusivity compared to thermal diffusivity. This characteristic implies that these fluids are more resistant to the transfer of heat than to the transfer of momentum. Consequently, in Figure 5(b), the temperature profile θ ( η ) experiences a decrease as the Prandtl number Pr values increase. The physical explanation for the decrease in temperature with an increase in the Prandtl number lies in the intensified thermal diffusion relative to momentum diffusion. As the Prandtl number increases, heat is conducted more efficiently compared to the momentum transfer, leading to a decrease in overall temperature.

Figure 5

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

Table 1 presents a visual representation of how the skin friction coefficient and the local Nusselt number alter in response to changes in the governing parameters. This table showcases the relationship between these coefficients and the factors that exert control over them, providing a comprehensive overview of their variations. This table demonstrates that as the magnetic parameter values are heightened, they wield an influence over the fluid dynamics in the vicinity of the stretching sheet. Consequently, this alteration in magnetic parameter values leads to a reduction in the local Nusselt number, accompanied by an elevation in the skin friction coefficient. Furthermore, the augmentation of viscoelastic parameter values has a discernible impact on the behavior of the viscoelastic fluid adjacent to the sheet’s surface. This outcome manifests in a reduction of the skin friction coefficient and a concurrent escalation in the coefficient governing the rate of heat transport. Similarly, the tabulated data illustrates that the local Nusselt number exhibits higher values when considering the Prandtl number and the thermal relaxation parameter, whereas contrasting trends are observed for the thermal radiation parameter and the Eckert number.

6 Concluding remarks

To sum up, the investigation of viscoelastic fluids considering factors like viscous dissipation and the magnetohydrodynamics effects with the inclusion of Cattaneo and Christov heat flux presents a intricate and demanding challenge within the field of fluid mechanics. The Cattaneo and Christov heat flux model addresses the nonlocal nature of heat transfer, which becomes significant at diminutive length scales and elevated thermal gradients. Incorporating the effect of viscous dissipation into the heat equation allows for the conversion of mechanical energy into heat caused by the viscosity of the fluid. Frequently, viscous dissipation can manifest as a significant source of heat generation and may play a crucial role in the overall heat transfer process. The key findings from this research are outlined as follows:

1. The presence of magnetic fields can bring about increased organization in the fluid near the surface, resulting in a more streamlined flow pattern and a higher skin friction coefficient.

2. Enhancing the thermal relaxation parameter and the Prandtl number results in decreased temperature distribution and thinner thermal boundary layers. In contrast, raising the thermal radiation parameter and the Eckert number leads to increased temperature distribution and thicker thermal boundary layers.

3. Elevating both the Eckert number and the thermal radiation parameter can also result in a reduction of the heat transfer rate.

4. Raising the viscoelastic parameter value can lead to a higher fluid property density, resulting in a consequent decrease in the skin friction coefficient.

5. Potential future research directions emerging from this study could include the investigation of hybrid viscoelastic nanofluids under diverse thermal property conditions, representing a significant focal point. Also, we will try to present a more study of convergence of the proposed method, as well as control the efficiency and accuracy. Finally, the stochastic numerical computing approaches will use as promising alternative for solving such problems.