Analysis of heat mass transfer in a squeezed Carreau nanofluid flow due to a sensor surface with variable thermal conductivity

Nourhan I. Ghoneim; A. M. Amer; Khalid S. M. AL-Saidi; Ahmed M. Megahed

doi:10.1515/nleng-2024-0021

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Analysis of heat mass transfer in a squeezed Carreau nanofluid flow due to a sensor surface with variable thermal conductivity

Nourhan I. Ghoneim , A. M. Amer , Khalid S. M. AL-Saidi and Ahmed M. Megahed

Published/Copyright: September 13, 2024

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From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

This research offers an advanced investigation into the examination of squeezed flow and heat mass transfer mechanisms of non-Newtonian Carreau dissipative nanofluids across a sensor surface. This analysis takes into consideration both variable thermal conductivity and variable viscosity aspects. It is widely accepted that the phenomenon of viscous dissipation has a significant impact on both the temperature distribution and heat transfer characteristics within nanofluids. Hence, it is being considered here. The governing equations of the problem are formulated using the Carreau model for the non-Newtonian fluid for the nanofluid. The thermal conductivity of the sensor surface is assumed to vary linearly with the temperature. The resulting nonlinear ordinary differential equations are solved numerically using the shooting method. The effects of various parameters such as suction parameter and magnetic parameter on the flow, the solutal characteristics, and thermal characteristics are analyzed. The results show that the slip parameter, the magnetic parameter, and the suction parameter have a significant effect on the flow and thermal fields. The heat transfer rate is improved by the squeezed flow index parameter and the Weissenberg number, but reduced by the power law index parameter and the Eckert number. Ultimately, the precision and reliability of the proposed approach are confirmed by benchmarking our data against previous findings. Understanding how variable viscosity impacts flow characteristics, heat transfer efficiency, and the performance of heat exchangers and cooling systems optimizes the design of nanofluids for efficient thermal systems in practical applications.

Keywords: Carreau nanofluid; two-dimensional squeezed flow; sensor surface; variable fluid viscosity

1 Introduction

The combination of a base fluid with suspended nanoparticles results in a non-Newtonian nanofluid [1]. Non-Newtonian fluids exhibit varying viscosity based on the rate of deformation, while nanofluids contain suspended nanoparticles. The resultant combination yields a complex fluid with distinctive rheological properties [2,3]. Numerous research studies have been carried out to explore the performance of non-Newtonian nanofluids in various fields including but not limited to heat transfer, energy, and medicine. For instance, in their research study [4], Bowers et al. analyzed the heat transfer properties of non-Newtonian nanofluids in a microchannel heat sink. Mohsenian et al. [5] conducted a study examining how the non-Newtonian behavior affects the flow and heat transfer of nanofluids in a converging–diverging channel. These research investigations offer significant contributions to the understanding of non-Newtonian nanofluid behavior and its possible applications.

The impact of the viscous dissipation phenomenon on engineering applications such as microfluidics, heat exchangers, and lubrication systems can be substantial. In microfluidics, for instance, viscous dissipation can lead to a temperature increase that can compromise the accuracy and performance of microdevices. In heat exchangers, viscous dissipation can affect both the heat transfer rate and the pressure drop. Similarly, in lubrication systems, it can influence the lubricant temperature, viscosity, and overall performance. As a result, it is crucial to comprehend and account for the effects of viscous dissipation in the design and optimization of these engineering applications. Considering the importance of viscous dissipation phenomenon, several researchers [6–9] have employed different geometrical models to investigate the flow of nanofluids in the presence of viscous dissipation.

The flow of squeezed nanofluid is a crucial phenomenon in various engineering applications such as microfluidics, chemical processing, and biomedical devices. The exceptional properties exhibited by squeezed nanofluid make it highly suitable for use in various biomedical applications, such as drug delivery, cancer treatment, and bio-imaging. Its unique features, including improved thermal and optical properties, make it a highly promising option for these applications. The high thermal conductivity of squeezed nanofluid makes it a suitable option for heat transfer applications. It has the potential to be used in cooling systems for electronic devices, heat exchangers, and solar collectors. Numerous research studies have been carried out to explore the flow and thermal properties of squeezed nanofluid over different geometries [10–12]. The results of these research investigations offer essential knowledge regarding the behavior of squeezed nanofluid flow, which can be instrumental in optimizing the design and enhancing the performance of devices that utilize this flow. Moreover, certain notable studies, for instance, the work by Rafeek et al. [13], investigated the behavior of a non-Newtonian Carreau magnetic fluid subjected to changes in thermal conductivity and dissipation within the confines of a micro-cantilever sensor, specifically in a squeezing regime. The literature’s limitations encompass a failure to discuss the combination of variable viscosity with both thermophoresis and Brownian motion, within the framework of non-Newtonian Carreau dissipative nanofluids flowing across a sensor surface. Our research builds on Rafeek et al.’s work [13] by incorporating the nanofluid model, which accounts for thermophoresis and Brownian phenomena, to examine how variable thermal properties interact with nanofluid behavior in the Carreau model framework, characterized by variable thermal viscosity. While Rafeek et al. focused on variable thermal conductivity in the Carreau model, our study extends this by introducing nanofluid dynamics, offering new insights into these interactions. By integrating nanoparticle suspensions into the Carreau nanofluid model with variable thermal viscosity, we explore how nanoparticles influence the variable thermal properties, deepening our understanding of the interplay between nanofluid characteristics and thermal conductivity.

Our study extends upon the research conducted by Rafeek et al. [13] by introducing an innovative analysis that considers the influence of temperature-dependent viscosity and the effects of both thermophoresis and Brownian motion in conjunction with Carreau nanofluid flow dynamics. While phenomena such as magnetic fields, viscous dissipation, and variable thermal conductivity have been commonly studied in both Rafeek et al.’s work [13] and ours, our novel investigation into variable thermal viscosity, alongside thermophoresis and Brownian motion, offers potential enhancements to the heat and mass transfer processes. We employed the shooting method to numerically manage our proposed model. These elements together represent a unique and inventive contribution to advancing our comprehension of Carreau nanofluid flow dynamics and heat mass transfer mechanisms.

2 Mathematical modeling

In this section, we will formulate the mathematical model of the two-dimensional squeezed non-Newtonian Carreau nanofluid flow under the impact of magnetic field and viscous dissipation. The squeezed flow of nanofluids refers to the flow of nanofluids in a narrow channel or conduit. The exploration of mathematical models for squeezing flows has been prompted by their wide-ranging and valuable uses in the fields of biological, mechanical, and medical engineering. These flows are commonly encountered in scenarios like sensors, presenting significant prospects for effectively controlling vibrations and regulating lubrication. Therefore, we will assume the flow of a squeezed Carreau nanofluid between two parallel plates with height h ( t ) . Assume that u ( x , y ) and v ( x , y ) represent the velocity components in the x and y directions, respectively (Figure 1).

Figure 1

Physical model of the problem.

Furthermore, it is assumed that the free stream undergoing squeezing occurs from the tip to the surface. In addition, it is assumed that the lower plate remains stationary, while the upper plate undergoes squeezing. The micro-cantilever sensor, which has a length of L , is positioned within the channel. In a squeezing fluid, the behavior of the fluid near the sensor surface is essential because the squeezing motion affects the pressure distribution on the surface. When the upper plate moves toward the lower plate, a high-pressure region is created in the fluid between the plates, resulting in deformation of the sensor surface. As a consequence, the output signal of the sensor, which is utilized to measure physical quantities such as fluid velocity and viscosity, is influenced by this deformation. The external free stream velocity U drives the flow, while a magnetic field of strength B 0 is applied perpendicular to the channel and in the y -direction. Subsequently, with the boundary-layer approximation employed, the continuity, momentum, energy, and concentration equations are expressed as follows [3,14,15]:

(1) ∂ u ∂ x + ∂ v ∂ y = 0 ,

(2) ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = − 1 ρ ∞ ∂ p ∂ x + 1 ρ ∞ ∂ ∂ y μ ( T ) ∂ u ∂ y 1 + Γ 2 ∂ u ∂ y 2 n − 1 2 − σ B 0 2 ρ ∞ u ,

(3) ∂ U ∂ t + U ∂ U ∂ x = − 1 ρ ∞ ∂ p ∂ x − σ B 0 2 ρ ∞ U ,

(4) ∂ T ∂ t + u ∂ T ∂ x + v ∂ T ∂ y = 1 ρ ∞ c p ∂ ∂ y κ ∂ T ∂ y + μ ( T ) ρ ∞ c p ∂ u ∂ y 2 × 1 + Γ 2 ∂ u ∂ y 2 n − 1 2 + τ D B ∂ C ∂ y ∂ T ∂ y + D T T ∞ ∂ T ∂ y 2 ,

(5) ∂ C ∂ t + u ∂ C ∂ x + v ∂ C ∂ y = D B ∂ 2 C ∂ y 2 + D T T ∞ ∂ 2 T ∂ y 2 ,

where the density of the nanofluid is represented by ρ ∞ , while the dynamic viscosity of the nanofluid is denoted by μ ( T ) , which is a function of the temperature T . Additionally, the pressure is denoted by p , the time constant is represented by Γ , and the thermal conductivity is denoted by κ . Likewise, the concentration of nanoparticles is represented by C , and the Brownian diffusion coefficient is denoted by D B . In addition, the thermophoretic phenomenon is characterized by a thermophoretic diffusion coefficient, which is represented by D T . Furthermore, τ represents the quotient obtained by dividing the heat capacity of the microparticles by the heat capacity of the base fluid. Consequently, the boundary conditions associated with the geometry of the problem are presented as follows [15]:

(6) u ( x , y , t ) = 0 , v ( x , y , t ) = v 0 ( t ) , C ( x , y , t ) = C w = C ∞ + q 2 x μ ∞ a ρ ∞ , T ( x , y , t ) = T w = T ∞ + q 1 x μ ∞ a ρ ∞ , at y = 0 ,

(7) u ( x , y , t ) = a x , T ( x , y , t ) = T ∞ , C ( x , y , t ) = C ∞ , as y → ∞ ,

where the symbols used in the equation are as follows: q 1 represents the heat flux at the wall, q 2 represents the mass flux at the wall, and μ ∞ represents the viscosity of the surrounding environment. Also, v 0 ( t ) is the permeable velocity, which can be assumed as v 0 ( t ) = v i . As it is a conventional method in dealing with systems of partial differential equations, the utilization of similarity transformations involves the introduction of new variables that are linked to the initial variables through a scaling factor [16]. Thus, the following transformations are often utilized [15]:

(8) η = y a ρ ∞ μ ∞ , ψ = x a μ ∞ ρ ∞ , u = a x f ′ ( η ) , a = 1 S + b t , v = − a μ ∞ ρ ∞ f ( η ) ,

(9) θ ( η ) = T − T ∞ T w − T ∞ , U = a x , and ϕ ( η ) = C − C ∞ C w − C ∞ .

In the given context, the non-dimensional variables are used to simplify the mathematical description of the problem. Specifically, η represents the dimensionless similarity variable, S is a dimensionless constant, ϕ corresponds to the dimensionless concentration, b is a squeezed flow index, θ refers to the non-dimensional fluid temperature, and f is the non-dimensional stream function. Furthermore, in fluid mechanics, the stream function ψ is a function that is utilized to analyze and visualize fluid flow patterns. It is characterized by having a gradient that is perpendicular to the direction of fluid flow, and it is often employed to satisfy the continuity equation (1) in two-dimensional, incompressible flows. In certain situations, it may be desirable to express a physical problem in terms of a system of ordinary differential equations. Additionally, here, we make the assumption that the nanofluid thermal conductivity κ and the viscosity μ are dependent on the dimensionless temperature θ based on these relationships [17]:

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

After applying Eqs (8)–(10) and satisfying the continuity Eq. (1), the momentum Eqs (2) and (3), the energy Eq. (4), and concentration Eq. (5) can be expressed as follows:

(11) e − α θ ( 1 + W e 2 f ″ 2 ) n − 3 2 [ − α θ ′ f ″ ( 1 + W e 2 f ″ 2 ) + ( 1 + n W e 2 f ″ 2 ) f ‴ ] + b 2 η + f f ″ + ( b − M − f ′ ) f ′ + ( M − b + 1 ) = 0 ,

(12) 1 Pr ( ε θ ′ 2 + ( 1 + ε θ ) θ ″ ) + ( f θ ′ − f ′ θ ) + Ec f ″ 2 e − α θ × ( 1 + W e 2 f ″ 2 ) n − 1 2 − b 2 ( θ − η θ ′ ) + Λ 1 θ ′ 2 + Λ 2 θ ′ ϕ ′ = 0 ,

(13) ϕ ″ + Pr Le f ϕ ′ − f ′ ϕ − b 2 ( ϕ − η ϕ ′ ) + Λ 1 Λ 2 θ ″ = 0 .

Furthermore, the corresponding boundary conditions can be expressed as follows:

(14) f = − f 0 , f ′ = 0 , θ = 1 , ϕ = 1 , at η = 0 ,

(15) f ′ = 1 , θ = 0 , ϕ = 0 , as η → ∞ .

Furthermore, it is apparent that the aforementioned system is predominantly governed by the factors listed as follows:

(16) W e = a 3 x 2 Γ 2 ν ∞ 1 2 , M = σ B 0 2 a ρ ∞ , Pr = μ ∞ c p κ ∞ , Ec = U 2 c p ( T w − T ∞ ) , and f 0 = v i a ν ∞ .

The aforementioned factors are, respectively, denoted as the Weissenberg number, the magnetic parameter, the Prandtl number, the Eckert number, and the surface permeable velocity parameter. Moreover, the remaining controlling parameters can be expressed as follows:

(17) Λ 1 = τ D T ( T w − T ∞ ) ν ∞ T ∞ , Λ 2 = τ D B ( C w − C ∞ ) ν ∞ , and Le = κ ∞ ρ ∞ c p D B .

These parameters are denoted, respectively, as the thermophoresis parameter, the Brownian motion parameter, and the Lewis number. Additionally, there are precise physical quantities that can mathematically describe the flow behavior, heat transfer, and mass transfer in an intriguing manner. The quantities known as the local skin-friction coefficient Cf x , the local Nusselt number Nu x , and the local Sherwood number Sh x are defined as follows:

(18) Re x 1 2 Cf x = − e − α θ ( 0 ) [ 1 + W e 2 ( f ′ ′ ( 0 ) ) 2 ] n − 1 2 f ′ ′ ( 0 ) ,

(19) Re x − 1 2 Nu x = − θ ′ ( 0 ) ,

(20) Re x − 1 2 Sh x = − ϕ ′ ( 0 ) ,

where Re x = a x 2 ν ∞ is the local Reynolds number.

3 Validating the code

After formulating the mathematical model for squeezed Carreau nanofluid flow that considers the impact of viscous dissipation, the next step involves developing a numerical shooting approach to solve the equations. The shooting method provides numerous advantages. It is adaptable for diverse problems [3], capable of managing linear and nonlinear equations, and apt for intricate boundary conditions. This technique systematically converts boundary value problems into initial value problems, solvable through common numerical integration methods. It proves particularly valuable when analytical solutions are elusive, rendering it a valuable asset across multiple scientific and engineering domains. However, it is crucial to validate the code to ensure precise outcomes before performing any simulations. This section aims to outline the validation process and compare the results with literature data. Therefore, to ensure the accuracy and reliability of the current numerical scheme, it is validated by comparing its solutions with previously published results from studies conducted by Khaled and Vafai [18] and Shankar et al. [19]. For different values of Prandtl number Pr and squeezed flow index parameter b where W e = M = f 0 = ε = Ec = Λ 1 = Λ 2 = 0 , the comparison was conducted under the condition θ ′ ( 0 ) = − 1 instead of θ ( 0 ) = 1 . Table 1 has been presented by the authors to demonstrate the validation process. However, the authenticity of the current numerical solutions is endorsed through the following table, which confirms their accuracy and reliability.

Table 1

Comparison of θ ( 0 ) for different values of Pr and b with the results of Khaled and Vafai [18] and Shankar et al. [19] when W e = M = f 0 = ε = Ec = Λ 1 = Λ 2 = 0

Pr	b	Khaled and Vafai [18]	Shankar et al. [19]	Present work
2.0	1.0	0.65412	0.654123423120187	0.654123423120069
5.0	1.0	0.43561	0.435614607270683	0.435614607271201
6.7	1.0	0.38182	0.381823375689146	0.381823375688510
6.7	1.0	0.38182	0.381823375689146	0.381823375687892
6.7	1.5	0.33084	0.330840498714310	0.330840498713120
6.7	2.0	0.29544	0.295440261684154	0.295440261685121

4 Graphical and tabular findings with discussion

In this section, the outcomes derived through the aforementioned method have been presented and explained. The shooting method, which is a proficient numerical technique, has been employed to compute the solutions of the governing equations that define the current physical model. Figure 2 illustrates how the power law index parameter n in the velocity f ′ ( η ) , concentration ϕ ( η ) , and temperature θ ( η ) plot performs as the similarity malleable changes. As the power law index parameter increased, the velocity profile of Carreau nanofluids decreased, whereas for both the temperature and concentration fields, the opposite trend was observed. Physically, the power law index for a Carreau fluid is a metric that characterizes the fluid’s propensity to exhibit either shear-thinning or shear-thickening behavior. This index explains the connection between the shear stress and the shear rate of the fluid. This parameter has practical implications in predicting the behavior of Carreau fluids in different applications, including industrial and biomedical settings. Although the influence of the power law index parameter on flow properties resembles previous research conducted by Rafeek et al. [13], our innovative approach involving nanofluids, characterized by variable viscosity, has enhanced the effect of the power law index parameter, particularly on heat and mass transfer rates.

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

Figure 2

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

Figure 3 displays the changes in the viscosity parameter α and the distribution of temperature θ ( η ) , nanoparticles concentration ϕ ( η ) , and nanofluid velocity f ′ ( η ) . It is evident that an increase in the viscosity parameter leads to an increase in the velocity profile within the boundary-layer region. In physical terms, viscosity can be influenced by temperature. When the temperature of a substance rises, its molecules tend to move at a faster pace, resulting in a decrease in viscosity. Conversely, if the temperature decreases, molecular motion slows down, leading to an increase in viscosity. Similarly, the same graph indicates a trend of decreasing temperature and concentration profiles, as well as boundary-layer thickness, as the viscosity parameter increases. Variable viscosity in nanofluid flow is essential for its behavior and performance. Nanofluids alter properties such as thermal conductivity and viscosity by suspending nanoparticles in a base fluid. Temperature-dependent viscosity affects flow characteristics, heat transfer efficiency, and performance in heat exchangers and cooling systems. This understanding optimizes nanofluid design for efficient thermal systems.

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

Figure 3

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

Figure 4 shows the smooth plotting of unique profiles f ′ ( η ) , ϕ ( η ) , and θ ( η ) as the squeezed flow index parameter b increases. To provide further clarity, it is worth mentioning that the squeezed flow index is a minor parameter that is depicted to range from 0.1 to 1.2. Physically, the degree of compression in a flow can be evaluated using the squeezed flow index, and it has the potential to modify the flow’s behavior, thereby impacting the velocity distribution. With this fact in consideration, it is evident from Figure 4 that the concentration and temperature profiles exhibit a decreasing trend with respect to the squeezed flow index parameter. Moreover, the velocity field of the nanofluid flow exhibits a similar decreasing trend, albeit with a lesser impact as compared to the temperature and concentration fields. Physically, when the squeezed flow index parameter rises, there is a possibility of altering how the nanofluid flows. This alteration can impact how the nanofluid spreads out, gathers into groups, and engages with the sensor surface. Consequently, these alterations could subsequently have an effect on the concentration of nanoparticles close to the sensor surface, potentially causing a decrease. Furthermore, squeezing fluid flows have diverse applications in mechanical, biological, and medical engineering fields. Sensors incorporate these flows and can be efficiently employed to manage vibrations and oversee lubrication [13]. While the impact of the squeezed flow parameter on flow properties aligns with prior studies, affirming the validity of our approach, our novel utilization of nanofluids, distinguished by variable viscosity, has improved the influence of the squeezed flow parameter, particularly on boundary-layer thickness.

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

Figure 4

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

The concentration profile ϕ ( η ) , temperature profile θ ( η ) , and velocity profile f ′ ( η ) are all influenced by the magnetic parameter M , as shown in Figure 5. As depicted in Figure 5, when the squeezing mechanism is in operation, the nanofluid motion experiences opposing Lorentz forces due to the presence of a magnetic field. A dynamic effect is observed for increasing values of the magnetic number. Therefore, when the magnetic parameter reaches its maximum value, the concentration and temperature curves display a decreasing trend, in contrast to the opposite trend observed in the velocity field. Physically, raising the magnetic field parameter can amplify the magnetic forces that prompt nanoparticles to come together, accelerate sedimentation, and perhaps induce movement toward distinct zones. When these influences amalgamate, they culminate in a reduction of the apparent nanoparticle concentration within the nanofluid. Additionally, modern sensor technologies place a high value on understanding how a magnetic field affects non-Newtonian nanofluids. Micro-cantilevers are used in biomedical applications as sensors for biological, physical, or chemical parameters. These sensors track changes in vibrational frequency or cantilever bending to identify the factors at play [13]. Moreover, the integration of nanoparticles into the base fluid, along with the consideration of variable fluid viscosity, exerts a notable influence in augmenting the thermal characteristics of the Carreau nanofluid, particularly when subjected to a magnetic field.

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

Figure 5

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

Figure 6 demonstrates the effects of the Weissenberg number We on the velocity distribution f ′ ( η ) of Carreau nanofluid, as well as its temperature θ ( η ) and concentration ϕ ( η ) characteristics. Here, we must mention that the Weissenberg number is determined by the ratio of the product of the characteristic time scale of flow and the relaxation time of the fluid, to the characteristic length scale of the flow. An increase in the Weissenberg number leads to enhancements in both the velocity distribution f ′ ( η ) and the boundary-layer thickness. Furthermore, the concentration and temperature profiles exhibit a slight decreasing trend throughout the boundary-layer as the cumulative Weissenberg number increases.

$Figure 6 (a) f ′ ( η ) f^{\prime} \left(\eta ) and θ ( η ) \theta \left(\eta ) for various W e {W}_{e} and (b) ϕ ( η ) \phi \left(\eta ) for various W e {W}_{e} .$

Figure 6

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

Figure 7 illustrates the impact of the surface permeable velocity parameter f 0 on the velocity fields f ′ ( η ) , as well as the thermal θ ( η ) and solutal ϕ ( η ) behaviors. It has been observed that the velocity profile increases as the absolute value of the surface permeable velocity parameter increases. Physically, the increased attachment of the fluid to the sensor surface resulting from the absolute value of the surface permeable velocity parameter leads to an enhancement in the velocity profile. Furthermore, it has been observed that as the absolute value of the surface permeable velocity parameter increases, both the concentration and temperature of the nanofluid decrease. Although the effect of the permeable velocity parameter on flow properties corresponds to previous research, validating our methodology, our innovative utilization of nanofluids characterized by variable viscosity has enhanced the impact of the permeable velocity parameter. This enhancement is particularly evident in its influence on the rates of heat and mass transfer, as well as the thickness of the boundary-layer.

$Figure 7 (a) f ′ ( η ) f^{\prime} \left(\eta ) and θ ( η ) \theta \left(\eta ) for various f 0 {f}_{0} and (b) ϕ ( η ) \phi \left(\eta ) for various f 0 {f}_{0} .$

Figure 7

(a) f ′ ( η ) and θ ( η ) for various f 0 and (b) ϕ ( η ) for various f 0 .

Figure 8(a) illustrates the impact of altering the thermal conductivity factor ε on the temperature fields θ ( η ) of the Carreau nanofluid flow. First, we must refer that a material characteristic that specifies how well a substance transmits heat is called the thermal conductivity parameter. Also, it shows how quickly heat moves through a material per unit temperature difference. There is no doubt that an increase in the thermal conductivity factor results in improvements to the temperature distribution and the accompanying thermal thickness. Furthermore, Figure 8(b) illustrates how the temperature profile θ ( η ) of the Carreau nanofluid is influenced by changes in the Eckert number Ec. To begin with, it is important to note that the Eckert number is calculated by squaring the fluid velocity and dividing it by the product of the specific heat at constant pressure and the temperature difference between the fluid and its surroundings. Figure 8(b) shows that as the Eckert number increases, both the temperature of the nanofluid and its corresponding thermal boundary thickness increase as well. Furthermore, the influence of the Eckert number, which characterizes the phenomenon of viscous dissipation, on the temperature distribution can be validated through the significant study conducted by Khan et al. [20].

$Figure 8 (a) θ ( η ) \theta \left(\eta ) for various ε \varepsilon and (b) θ ( η ) \theta \left(\eta ) for various Ec.$

Figure 8

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

The impact of the thermophoresis parameter Λ 1 on the distribution of concentration ϕ ( η ) and temperature θ ( η ) is illustrated in Figure 9. It is important to note that the thermophoresis parameter measures the intensity of the thermophoretic force exerted on particles due to temperature differences between the fluid and the surface of the particles. Due to the characteristics of the thermophoresis parameter, an increase in its value results in a less viscous flow field and greater distribution of concentration and temperature. Physically, the mobility of nanoparticles in the nanofluid is enhanced as the thermophoresis parameter rises because of temperature differences. Due to their tendency to congregate in cooler areas, the concentration of nanoparticles increases noticeably. The effectiveness of the impact depends on elements such as fluid flow dynamics, temperature gradient, and nanoparticle characteristics.

$Figure 9 (a) θ ( η ) \theta \left(\eta ) for various Λ 1 {\Lambda }_{1} and (b) ϕ ( η ) \phi \left(\eta ) for various Λ 1 {\Lambda }_{1} .$

Figure 9

(a) θ ( η ) for various Λ 1 and (b) ϕ ( η ) for various Λ 1 .

Figure 10 presents the results of the temperature θ ( η ) and concentration fields ϕ ( η ) , along with the Brownian motion factor Λ 2 . To begin with, it is important to note that the behavior of nanoparticles and their impact on the fluid flow and heat transfer characteristics is significantly influenced by the Brownian motion factor. It can be observed that higher values of the Brownian motion factor lead to an increase in the temperature of the nanofluid θ ( η ) and its corresponding thermal thickness, while only a slight decrease is seen in the concentration distribution ϕ ( η ) . In physical terms, the Brownian motion factor is a nondimensional parameter that quantifies the erratic movement of particles that are suspended in a fluid as a result of collisions with the fluid molecules. This factor is determined by the ratio of the thermal energy of the fluid to the binding energy of the particle and is influenced by factors such as the size of the particles, temperature, and viscosity of the fluid.

$Figure 10 (a) θ ( η ) \theta \left(\eta ) for various Λ 2 {\Lambda }_{2} and (b) ϕ ( η ) \phi \left(\eta ) for various Λ 2 {\Lambda }_{2} .$

Figure 10

(a) θ ( η ) for various Λ 2 and (b) ϕ ( η ) for various Λ 2 .

At this point, the calculation of the skin friction coefficient Cf x Re x 1 2 , Nusselt number Nu x Re x , and Sherwood number Sh x Re x has become imperative, as they are critical in predicting and analyzing heat transfer and fluid flow characteristics in a range of engineering applications. Additionally, examining the heat transfer rate within heated fluid flows enhances effectiveness, fosters innovation, and exerts an impact on a diverse range of technological and scientific progressions that shape the contemporary world [21]. Table 2 clearly indicates that the skin-friction coefficient is amplified by increasing values of the power law index n , viscosity parameter α , and magnetic number M . Conversely, increasing the squeezed flow index parameter b , Weissenberg number We , and Eckert number Ec diminishes the skin-friction coefficient. Furthermore, it is observed that the Sherwood number rises significantly for high values of the viscosity parameter, squeezed flow index parameter, magnetic number, and Weissenberg number. In addition, the local Nusselt number is seen to increase with higher values of the thermal conductivity parameter, squeezed flow index parameter, and Weissenberg number. Conversely, an opposite trend is observed for the power law index, magnetic parameter, and Eckert number. Finally, it should be noted that the findings of this study are particularly useful in applications related to heat transfer in solar collectors and oil recovery.

Table 2

Skin friction coefficient Cf x Re x 1 2 , local Nusselt number Nu x Re x , and local Sherwood number Sh x Re x against values of n , α , b , M , We , f 0 , ε , Ec , Λ 1 , and Λ 2 with Pr = 2.0 and Le = 1.0

n	α	b	M	We	f 0	ε	Ec	Λ 1	Λ 2	Cf x Re x 1 2	Nu x Re x	Sh x Re x
0.5	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.23320	0.511371	1.474281
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.58401	0.442785	1.360510
1.0	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.78676	0.411065	1.306991
0.8	0.0	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.29233	0.421792	1.327304
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.58401	0.442785	1.360510
0.8	0.5	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	2.17450	0.475219	1.408951
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.58401	0.442785	1.360510
0.8	0.2	0.6	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.50351	0.576392	1.586131
0.8	0.2	1.2	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.37907	0.716881	1.824401
0.8	0.2	0.1	0.0	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.40331	0.445604	1.324960
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.58401	0.442785	1.360510
0.8	0.2	0.1	2.0	3.0	‒ 0.1	0.2	0.2	0.1	0.8	2.00696	0.428415	1.433411
0.8	0.2	0.1	0.5	0.1	‒ 0.1	0.2	0.2	0.1	0.8	1.78568	0.411232	1.307340
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.58401	0.442785	1.360510
0.8	0.2	0.1	0.5	5.0	‒ 0.1	0.2	0.2	0.1	0.8	1.51009	0.454703	1.376711
0.8	0.2	0.1	0.5	3.0	‒ 0.9	0.2	0.2	0.1	0.8	2.41375	0.754805	2.508692
0.8	0.2	0.1	0.5	3.0	‒ 0.5	0.2	0.2	0.1	0.8	1.96332	0.588067	1.902991
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.58401	0.442785	1.360510
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.0	0.2	0.1	0.8	1.58226	0.434245	1.363221
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.5	0.2	0.1	0.8	1.58395	0.446643	1.357770
0.8	0.2	0.1	0.5	3.0	‒ 0.1	1.0	0.2	0.1	0.8	1.58585	0.447587	1.354981
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.0	0.1	0.8	1.63227	0.584202	1.344901
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.5	0.1	0.8	1.51448	0.231213	1.383862
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.8	0.1	0.8	1.44822	0.020301	1.407161
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.0	0.8	1.59433	0.471671	1.342781
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.1	0.8	1.58401	0.442785	1.360510
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.2	0.3	0.8	1.56578	0.391512	1.407432
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.0	0.1	0.8	1.63903	0.598612	1.347403
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.5	0.1	0.8	1.58401	0.442785	1.360510
0.8	0.2	0.1	0.5	3.0	‒ 0.1	0.2	0.8	0.1	0.8	1.50239	0.205752	1.362962

5 Concluding remarks

The study investigates the two-dimensional squeezed flow past a sensor surface of a dissipative Carreau nanofluid under the influence of varying viscosity, magnetic field and thermal conductivity. Although certain fluid properties may not be temperature dependent, the viscosity and thermal conductivity of the nanofluid are considered to vary with temperature. The unsteady boundary-layer equations for concentration, energy, and momentum were converted to dimensionless equations based on certain controlling parameters and were then numerically solved using the shooting method. Graphs and tables are utilized to investigate the impact and sensitivity of various parameters on the flow. The key takeaways can be outlined as follows:

The thermal profile is observed to rise as the power law index, thermal conductivity parameter, and Eckert number increase.
The temperature profile decreases significantly as the absolute values of the surface permeable velocity parameter, squeezed flow index parameter, Weissenberg number, and magnetic parameter increase.
Incorporating temperature-dependent viscosity and the presence of a magnetic field amplifies the boundary-layer thickness and velocity field of Carreau nanofluid.
The skin friction coefficient at the surface is augmented as the viscosity parameter, power law index, and magnetic number increase, while the opposite trend is observed for the squeezed flow index parameter, Weissenberg number, and Eckert number.
Increasing the magnetic number, squeezed flow index parameter, and viscosity parameter leads to a reduction in the thickness of the concentration boundary-layer.

Acknowledgement

We would like to thank International Maritime College Oman, National University of Science and Technology and The Research Council (TRC) of the Sultanate of Oman, MoHERI, for supporting project number (BFP/GRG/EI/22/080), Oman.

Funding information: This research was funded by The Research Council (TRC) of the Sultanate of Oman, MoHERI, Project number (BFP/GRG/EI/22/080), Oman.
Author contributions: All authors have acknowledged responsibility for the full content of this manuscript and have approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: No data was used for the research described in the article.

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Received: 2023-05-22

Revised: 2024-04-13

Accepted: 2024-06-12

Published Online: 2024-09-13

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0021

Keywords for this article

Carreau nanofluid; two-dimensional squeezed flow; sensor surface; variable fluid viscosity

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