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Abstract

This study explores the application of Casson fluid in cooling systems for rolling mills and heat exchangers, with an emphasis on reducing leading-edge accretion and optimizing heat transfer efficiency. By building on current optimization practices in metallurgical plants and power generation facilities, our research provides a structured approach to enhancing operational efficiency and reducing costs related to heat management and accretion. Our objective is to advance the understanding and use of non-Newtonian Casson fluids in managing leading-edge accretion, ultimately contributing to the development of more efficient and resilient materials for high-temperature applications. In this work, we analyze the unsteady flow of Casson fluid over an aligned flat plate with a moving slot, incorporating slip boundary conditions. The combined influence of these physical factors offers a comprehensive view of the complex interactions in such systems, which are relevant for industrial processes and advanced material manufacturing. The boundary layer equations are converted to nonlinear ordinary differential equations using the Blasius-Rayleigh-Stokes similarity variable. Solving these equations employs the shooting method in Matlab, where the Runge-Kutta-Fehlberg approach assesses convergence of the numerical solution. We present our findings with visualizations that illustrate the influence of various dynamic parameters on the flow field. This comparative study shows that non-Newtonian fluids improve heat retention, thermal stability, and energy efficiency. They enhance solutal mixing, reduce fouling, lower surface friction, and improve flow control, making them ideal for industrial applications. Non-Newtonian fluids offer better heat transfer, reduce particle deposition, and allow precise control over thermal and solutal properties, making them suitable for cooling systems, lubrication, and nanofluidic applications. Replacing water-based coolants with non-Newtonian Casson-type fluids boosts heat and mass transfer, operational efficiency, and system longevity. Future research should focus on optimizing these fluids under dynamic conditions to further enhance their performance in industry.

Keywords

Aligned flat plate leading edge accretion Casson fluid slip boundary conditions

Introduction

Global steel production is expected to increase dramatically in the coming years, though the globalization of the steel industry is still ongoing. Each year, new steel-making companies are established with production plants across all continents, a trend mirrored in the aluminum and non-ferrous metals industries. The rolling process, crucial for shaping metal sheets, generates substantial heat through friction and deformation in the roll bite. Effectively managing this heat is vital for maintaining product quality and preventing damage to both the workpiece and equipment. The efficiency of heat regulation directly influences the lifespan and reliability of rolling mills, with improper cooling leading to costly maintenance and operational downtime.

Colas et al.¹ and Guerrero et al.² conducted investigations on the hot rolling steel strip mill to examine the rolls’ surface and the wear mechanisms that lead to damage. They explored the heating and cooling cycles, which may cause defects in the rolled strips and potential damage. Their findings revealed that the oxidation of the work roll surface occurs due to diffusional processes, which can advance through cracks in the roll. In some cases, oxide formation can isolate healthy regions of the roll, causing them to break away and accelerating roll wear.

Sun et al.³ carried out an experimental study on the impact of scale thickness on rolling force and torque during hot rolling of low-carbon steel. They used nitrogen protection to regulate scale thickness and cooled the samples in a cooling box. The effects of the oxide scale on the work roll surface, as well as the prediction of thermal expansion and heat transfer efficiency between the rolls and strips, were also studied by Yukawa et al.⁴

In addition to rolling, heat exchangers and turbines in power plants also face challenges related to leading-edge accretion, where unwanted material builds up on heat transfer surfaces. This accretion can significantly reduce the efficiency of heat transfer systems, leading to increased energy consumption and decreased system lifespan. Cooper et al.,⁵ Müller-Steinhagen,⁶ and Sung et al.⁷ explored the heat exchanger fouling or accretion can be minimized through effective heat exchanger design, mechanical or chemical mitigation techniques, and by altering the coolant. Their study also highlighted that plate heat exchangers exhibit lower fouling rates than tubular equipment, thanks to the turbulence generated by the plate corrugations.

Regular cleaning and maintenance of the heat exchanger are essential for removing fouling-causing deposits, ensuring its continued efficiency. Surface treatments, such as coatings or nanoparticle layers, can enhance the surface’s resistance to fouling, thus extending the heat exchanger’s lifespan. Modifying fluid flow rate or pattern can help reduce the time the fluid spends in contact with surfaces, minimizing fouling. Additionally, pre-treating the fluid by adding chemicals or filtering impurities helps prevent debris buildup that can cause fouling. Designing the heat exchanger with reduced surface area contact and selecting fouling-resistant materials like stainless steel or titanium can further reduce the risk of fouling. Finally, applying fouling-resistant coatings, such as silicone or fluoropolymer, provides an added layer of protection, ensuring the system operates smoothly over time.

In industries ranging from power generation to chemical processing, advanced nozzle systems are often employed to mitigate these issues by continuously spraying coolant or water to regulate temperature and prevent overheating. However, high operational costs due to coolant consumption and system wear require innovative solutions. Raudensky et al.⁸ conducted an experimental study on the cooling of a steel plate using a set of flat nozzles with a non-rectangular spray angle. The study demonstrated that roll cooling should be enhanced by increasing water flow rather than water pressure.

Saha et al.⁹ examined the impact of the cooling system on roll performance and explored the thermal aspects of roll cooling on the roll surface at the hot strip mill at Tata Steel. Their model used to predict the roll cooling conditions under different thermal strains and stresses. Liu et al.¹⁰ studied how coiling temperature affects the surface quality and profile shape during the process cooling in hot-rolled steel strips. Their mathematical model, incorporating nonlinear structural results, demonstrated the stable performance of the cooling process, improving the steel product quality.

From this literature survey, the internal and external accretion or fouling of the particle at the surface reduces the heat transfer efficiency of the system. This accretion, by forming layers on the heat exchange surfaces, obstructs the smooth flow of fluid, leading to an increase in thermal resistance. In this case, the accretion formulates the boundary layers of the flow becomes unsteady and it oscillates due to thickness of the scale formation. The external flow model uses the nozzle system and internal flow model uses the coolant over the surfaces to improve the heat transfer efficiency in the system. By optimizing the nozzle configuration and ensuring adequate coolant distribution, both models aim to reduce the formation of fouling by keeping the surfaces cleaner and maintaining a stable flow, which ultimately enhances heat dissipation and system performance. Moreover, the use of advanced materials and coatings on the surfaces can mitigate fouling accumulation, allowing for sustained heat transfer efficiency over longer operational periods.

In 1997, Todd¹¹ investigated the boundary layers of the unsteady and oscillatory fluid flow. Todd introduced the Blasius-Rayleigh-Stokes similarity variable to characterize the boundary layer thickness under such conditions. Their study focused on the fluid flow over a flat surface with a moving slot. This moving slot controls the flow, causing oscillations that affect the thickness of the boundary layer, thereby influencing the overall heat transfer and flow dynamics. The boundary-layer behavior in these oscillating flows is crucial for understanding how unsteady conditions impact heat exchange and the potential for fouling or accretion. The similarity solutions for leading-edge ablation $(\frac{π}{2} < α < \frac{3 π}{4})$ and accretion $(0 < α < \frac{π}{2})$ on a semi-infinite flat plate, specifically addressing the laminar boundary-layer equations were presented in this study. It shows how the extreme situations of the similarity solution include both the Rayleigh Stokes shear layer and the classical Blasius boundary layer. This study revealed significant differences in boundary-layer solutions between ablation and accretion conditions. These findings are significant for the design of systems involving fluid flows over heated surfaces, where managing boundary-layer behavior under oscillatory or unsteady conditions is critical. Understanding the differences in flow characteristics between ablation and accretion conditions can help optimize system efficiency and reduce the negative effects of fouling or surface degradation.

The following literature review focuses on leading-edge accretion and ablation, phenomena that significantly impact the thermal and fluid dynamic performance of surfaces exposed to high-speed flows. Accretion refers to the accumulation of material or particles on a surface, while ablation involves the removal or erosion of material from the surface due to intense heat or fluid flow. When a surface experiences accretion, the boundary layer thickens, increasing resistance to heat transfer and potentially leading to flow instabilities. Conversely, under ablation conditions, the material from the surface is removed, causing a reduction in boundary layer thickness and possibly enhancing heat dissipation.

The unstable flow over a semi-infinite flat plate were studied by Fang.^12,13 Additionally, it explores flows from a slot on a moving surface, where the slot speed determines whether the flow behaves like a stretching or shrinking sheet. Dzulkifli et al.¹⁴ and Hussanan et al.¹⁵ explored the effects of edge accretion and ablation phenomena. Their study showed that the velocity and temperature fields are significantly impacted by the thinner thermal boundary layer in Blasius flat plate problems compared to Rayleigh-Stokes problems. A theoretical investigation on the heat transfer of magnetic nanofluids leading-edge accretion was carried out by Ilias et al.¹⁶ Their research demonstrates that MHD can effectively enhance the control of heat transfer and fluid flow in nanofluids. The study is significant as it investigates the alignment of magnetic fields with the flow, showing that the accretion process at the leading edge can substantially influence the thermal boundary layer behavior.

Ahmad et al.¹⁷ observed that the inclusion of magnetic particles enhanced heat transfer efficiency, attributed to increased thermal conductivity and the alignment of magnetic dipoles under an external magnetic field. Their results suggest that ferrofluids, which actively respond to magnetic fields, may offer improved thermal management in systems experiencing variable heat flux. Reddy et al.¹⁸ explored Blasius-Rayleigh-Stokes flow through a transient magnetic field, accounting for Joule heating effects. This study is significant for its use of a non-Fourier heat conduction model, incorporating thermal relaxation time to address limitations of the traditional Fourier law.

Basha et al.¹⁹ and Lu et al.²⁰ studied the effect of Prandtl number on unsteady boundary layer flow over a flat plate, specifically examining leading-edge accretion and ablation. Their numerical analysis sheds light on how variations in Prandtl number influence the heat transfer properties of nanofluids. Ali et al.²¹ explored the influence of Stefan blowing, thermal radiation, and Cattaneo Christov heat flux on nanofluid flow with microorganisms, considering conditions of leading-edge ablation and accretion. Using the finite element method, they simulated complex interactions among flow, heat transfer, and species concentration.

Kumar et al.²² and Mabood and Khan²³ explored laminar Rayleigh-Stokes and Blasius flow with ohmic heating, the process where electric current generates heat in a resistive material. The study concluded that the specified factors significantly influence flow and energy transport characteristics, with implications for improving thermal management in various applications. Blasius-Rayleigh-Stokes variables with slip effect were used to theoretically evaluate the unsteady convective flow of nanofluids by Adebayo et al.,²⁴ Ishaq and Ahmad,²⁵ and Al Nuwairan and Souayeh.²⁶

The forced convective boundary layer flow of gyrotactic microorganisms was investigated by Gangadhar et al.²⁷ To solve transformed boundary value problems, they employed a combination of the collocation and finite difference methods. Their results showed that temperature, velocity, volume fraction of nanoparticles, and microbe density were all significantly impacted by thermal radiation and chemical interactions. These findings, which have been supported by earlier research, have consequences for the creation of biomodified nanomaterials and microflow devices.

Leading-edge accretion can lead to convective and unsteady acceleration, significantly influencing flow dynamics and heat transfer efficiency in various applications. The study by Animasaun et al.²⁸ explores ternary-hybrid nanofluid flow on horizontal surfaces, revealing how convective and unsteady acceleration impacts thermal behavior, enhancing understanding of heat transfer in engineering applications. Wang et al.²⁹ explores concentration variations and thermal behavior in water-based nanofluids. It offers valuable insights into accretion phenomena and enhanced heat transfer in nanofluid applications. Vaidya et al.³⁰ utilizing Blasius-Rayleigh-Stokes variables to examined the impact of heat transport properties on the erratic flow across a moving plate. Their findings indicated that higher dimensionless reference temperatures and moving slot parameters reduced the conductivity of the nanofluid.

From this literature review, the water-based coolants can be prone to accretion or fouling due to the presence of minerals and other impurities in the water. Changing to a coolant that is less prone to fouling, it can help to reduce fouling. Changing to a coolant that is specifically designed to reduce fouling. Recent advancements in fluid dynamics have introduced Casson fluid as a potential coolant for improving heat transfer in such systems. Casson fluid, a type of non-Newtonian fluid, offers several advantages over traditional coolants due to its unique rheological properties, particularly in systems involving high thermal stresses.

Using Blasius-Rayleigh-Stokes variables, Oladapo et al.³¹ examined how radiation affected a Casson fluid flow over an inclined surface for solar aircraft applications. This work assessed the heat transfer efficiency of $Cu - Zr O_{2} / EG$ hybrid nanofluid parabolic trough solar collectors using the Galerkin-weighted residual method. The Casson nanoliquid flow over a stretching channel was investigated by Alammari³² and Nandi et al.³³ Their study emphasized the novelty of integrating PSO for numerical stability assessments, providing insights into optimizing the flow behavior in complex industrial applications. They analyzed the three-dimensional steady flow dynamics of the nanofluid, considering factors like nonlinear thermal radiation and binary chemical reactions.

The reviewed studies present a detailed view of advancements in non-Newtonian nanofluids, from early efforts to improve thermal conductivity using nanoparticles to more sophisticated models incorporating shear-dependent viscosity and heat transfer mechanisms. These insights are particularly valuable for optimizing the use of nanofluids in high-temperature environments and magnetic fields, with applications spanning advanced manufacturing, energy systems, and biomedical technologies. In this manuscript, we analyze the unsteady flow of a non-Newtonian Casson fluid over a horizontal flat plate with a moving slot, influenced by multiple slip conditions and oscillatory boundary layers. Buongiorno’s fluid model is considered due to concentration gradients. These physical phenomena provide critical insights into the fluid’s dynamic behavior. To solve this system, the boundary layer equations are transformed into nonlinear ordinary differential equations using the Blasius-Rayleigh-Stokes similarity variable. By employing the shooting method within Matlab, the Runge-Kutta-Fehlberg technique estimates the convergence of the numerical solution to these equations. To the best of our knowledge, this manuscript is the first to thoroughly examine the combined effects of Casson fluid with unsteady and oscillatory boundary layers combined with slip boundary conditions, Brownian motion, thermophoresis, Prandtl and Lewis numbers, thermal and solutal Grashof numbers for Casson fluid flow over a horizontally stretched sheet with a moving slot. This study delivers a comprehensive analysis of the momentum, temperature, and concentration profiles in the presence of moving slots and slip effects on a horizontal surface.

Mathematical model

Consider the unsteady $2 D$ laminar incompressible non-Newtonian Casson fluid flow over a semi-infinite flat plate $(x \geq 0)$ emerged with moving slot. Let $v (x, y, t)$ and $u (x, y, t)$ are the velocity components of $y$ and $x$ direction, $C (x, y, t)$ and $T (x, y, t)$ are concentration and temperature profiles, $C_{\infty}$ and $T_{\infty}$ are the ambient concentration and temperature. The physical model of this problem as depicted in Figure 1. Inspired by the studies of Song et al.³⁴ and Cao et al.,³⁵ Buongiorno’s nanofluid model is modified to incorporate the effects of thermophoresis and random nanoparticle motion resulting from concentration changes.

Figure 1.

Physical model.

The governing equations are characterized by Lu et al.²⁰

u_{x} + v_{y} = 0

(1)

u_{t} + u u_{x} + v u_{y} = ν (1 + \frac{1}{β}) u_{yy} + g β_{C} (C - C_{\infty}) + g β_{T} (T - T_{\infty})

(2)

T_{t} + u T_{x} + v T_{y} = σ T_{yy} + ε (D_{B} T_{y} C_{y} + \frac{D_{T}}{T_{\infty}} (T_{y})^{2})

(3)

C_{t} + u C_{x} + v C_{y} = D_{B} C_{yy} + \frac{D_{T}}{T_{\infty}} T_{yy}

(4)

where $ν$ kinematic viscosity, $β$ is Casson number modulates the influence of nonlinear effects, $g$ is the gravity, $β_{C}$ and $β_{T}$ are the solutal and thermal expansion coefficients, $σ$ is the thermal diffusivity, $ε$ is a ratio of heat capacities, $D_{T}$ and $D_{B}$ are thermophoretic and Brownian diffusion coefficients.

The relevant boundary conditions are characterized by

y = 0 \Rightarrow u (y) = L u_{y}, v (y) = 0, C (y) = C_{w}, T (y) = T_{w} + D T_{y}

(5)

y \to \infty \Rightarrow u (y) = 0, C (y) = C_{\infty}, T (y) = T_{\infty}

(6)

Introducing the Blasius-Rayleigh-Stocks similarity variable and stream functions is of the form

\begin{matrix} η = \frac{y}{\sqrt{ν t \cos α + \frac{ν x}{U_{w}} \sin α}}; \\ ψ = U_{w} \sqrt{ν t \cos α + \frac{ν x}{U_{w}} \sin α} f (η); \\ T (x, y, t) = T_{\infty} + θ (η) (T_{w} - T_{\infty}); \\ C (x, y, t) = C_{\infty} + φ (η) (C_{w} - C_{\infty}) \end{matrix}

(7)

Using (7), the governing equations with boundary conditions (1)–(6) are converted to nonlinear ordinary differential equations of the form:

(1 + β) f ″' (η) + \frac{β}{2} f ″ (η) (\cos α η + \sin α f (η)) - Gr θ - Gm φ = 0

(8)

\begin{matrix} θ ″ (η) + \frac{\Pr}{2} (\cos α η + \sin α f (η)) θ' (η) + (θ' (η))^{2} Nt \\ + Nb φ' (η) θ' (η) = 0 \end{matrix}

(9)

φ ″ (η) + \frac{Le}{2} (\cos α η + \sin α f (η)) φ' (η) + θ ″ (η) (\frac{Nt}{Nb}) = 0

(10)

where $Gr = \frac{g β_{T} (T_{w} - T_{\infty})}{ν U_{w} {(\cos α ν t + \sin α \frac{ν x}{U_{w}})}^{- 1}}$ and $Gm = \frac{g β_{C} (C_{w} - C_{\infty})}{ν U_{w} {(\cos α ν t + \sin α \frac{ν x}{U_{w}})}^{- 1}}$ are thermal and mass Grasof numbers, Le $= \frac{ν}{D_{B}}$ is a Lewis number, $\Pr = \frac{ν}{σ}$ is a Prandtl number, $Nt = \frac{ε D_{T} (T_{w} - T_{\infty})}{σ T_{\infty}}$ is thermophoresis diffusion parameter, and $Nb = \frac{ε D_{B} (C_{w} - C_{\infty})}{σ}$ is dimensional Brownian diffusion parameter.

Animasaun et al.³⁶ discusses the Brownian diffusion parameter as dimensionless, and Song et al.³⁴ and Cao et al.³⁵ discuss the energy and concentration equations to account for thermo-migration and haphazard motion. They normalize the concentration difference $▵ C$ by a reference concentration $C_{\infty}$ to eliminating the dimensions of concentration from $Nb$ . According to their study, to eliminate the dimensions of $Nb$ , we can normalize the concentration difference $▵ C = (C_{w} - C_{\infty})$ by ambient concentration $C_{\infty}$ . Which gives that the term $Nb = \frac{ε D_{B}}{σ} \times \frac{(C_{w} - C_{\infty})}{C_{\infty}}$ becomes normalized form of non-dimensional Brownian diffusion parameter.

The transformed boundary conditions are

\begin{matrix} η = 0 \Rightarrow f (η) = 0, f' (η) = γ f ″ (η), φ (η) = 1, θ (η) = 1 + τ θ' (η), \\ η \to \infty \Rightarrow f' (η) = 0, φ (η) = 0, θ (η) = 0 \end{matrix}

(11)

where $γ$ and $τ$ are momentum and thermal slip factors respectively.

Result and discussion

Todd¹¹ investigated the solutions of a new Blasius Rayleigh Stocks boundary layer for $0 < α < \frac{π}{2}$ . That is, accretion is occurring at the leading edge for $0 < α < \frac{π}{2}$ and the rate of accretion grows monotonically with $α$ decreasing. The non-dimensional governing equations (8)–(10) with corresponding boundary conditions (11) are solved by the Runge-Kutta-Fehlberg method with the shooting technique using Matlab. The solutions and their impacts are discussed in the following section with the comparison of Newtonian (Dash lines) and non-Newtonian fluids (solid lines).

Analysis of results

In this section, we discuss the analysis of the results from the comparative study of Newtonian and non-Newtonian fluids as the physical parameters increase. This study examines how variations in parameters such as the moving slot parameter, Casson fluid parameter, Brownian motion and thermophoresis parameters, Prandtl and Lewis numbers, thermal and solutal Grashof numbers, and slip parameters affect the behavior and performance of both fluid types. By analyzing these trends, we aim to highlight the differences in heat transfer, fluid flow, and solutal mixing, providing valuable insights into the practical applications of each fluid type.

Velocity profile

In Figure 2(a) the velocity of the non-Newtonian Casson fluid exhibits a higher peak compared to that of the Newtonian fluid, indicating greater resistance to flow deformation. An increase in moving slot parameter $α$ likely enhances fluid movement near the surface (e.g. due to increased suction or injection at the moving slot), leading to an increase in fluid velocity. The impact of the Casson fluid parameter $β$ is illustrated in Figure 2(b). The parameter $β$ characterizes the fluid’s non-Newtonian behavior, with higher values corresponding to stronger non-Newtonian effects, making the fluid more resistant to deformation. As $β$ increases, the fluid’s yield stress becomes more significant, leading to a reduction in velocity due to increased flow resistance.

Figure 2.

Comparison of Newtonian and non-Newtonian fluids in velocity while increasing moving slot and Casson fluid parameters: (a) impact of moving slot parameter $α$ and (b) impact of Casson fluid parameter $β$ .

Figure 3(a) shows the impact of $Gm$ , buoyancy-driven flow intensifies, causing an increase in fluid velocity. This occurs because the concentration gradient induces stronger natural convection, promoting fluid movement. In Figure 3(b) increase in $Gr$ indicates stronger thermal buoyancy effects because the temperature gradient causes the fluid to rise or move faster, driving natural convection. In Figure 4(a) a higher Prandtl number $\Pr$ indicates lower thermal diffusivity, meaning that heat spreads more slowly compared to momentum. As a result, convective motion slows down, and heat transfer becomes less efficient. From Figure 4(b) a higher $Le$ allows heat to spread more easily compared to mass, which can lead to a reduction in fluid motion as the driving forces for convective flow (such as concentration gradients) become less effective.

Figure 3.

Comparison of Newtonian and non-Newtonian fluids in velocity while increasing Grashof numbers: (a) impact of mass Grashof number $Gm$ and (b) impact of thermal Grashof number $Gr$ .

Figure 4.

Comparison of Newtonian and non-Newtonian fluids in velocity while increasing Pr and Le numbers: (a) impact of Prandtl number $\Pr$ and (b) impact of Lewis number $Le$ .

As $Nt$ increases in Figure 5(a) the thermophoretic effect becomes more pronounced, which can enhance particle movement within the fluid and result in an increase in fluid velocity, as thermophoretic forces actively drive particle motion. As $Nb$ increases in Figure 5(b) the random motion of particles becomes more dominant, causing greater momentum diffusion. This increased particle interaction creates more resistance to the bulk motion of the fluid, leading to a decrease in velocity.

Figure 5.

Comparison of Newtonian and non-Newtonian fluids in velocity while increasing diffusion parameters: (a) impact of thermal diffusion parameter $Nt$ and (b) impact of Brownian diffusion parameter $Nb$ .

As $γ$ increases in Figure 6(a) $6 a$ the fluid experiences less friction at the boundary, allowing it to move more freely. This reduction in boundary resistance results in an increase in fluid velocity. The thermal slip parameter $τ$ represents the extent to which the temperature at the boundary differs from the temperature in the adjacent fluid layers in Figure 6(b). Due to the slip condition $τ$ increases, the fluid becomes less heat transfer, and weakening the convective forces that drive fluid motion.

Figure 6.

Comparison of Newtonian and non-Newtonian fluids in velocity while increasing slip parameters: (a) impact of velocity slip parameter $γ$ and (b) impact of thermal slip parameter $τ$ .

Temperature profile

From Figure 7(a) the temperature distribution in the Casson fluid is more uniform across the boundary layer, indicating better thermal stability. As fluid velocity increases, there is greater mixing and more heat being transferred through convection. In Figure 7(b) as the Casson fluid parameter $β$ increases, the reduction in velocity causes convective heat transfer to diminish. Since the fluid moves more slowly, less heat is transported through convection, resulting in a decrease in overall temperature as thermal energy is less effectively distributed throughout the fluid.

Figure 7.

Comparison of Newtonian and non-Newtonian fluids in temperature while increasing moving slot and Casson fluid parameters: (a) impact of moving slot parameter $α$ and (b) impact of Casson fluid parameter $β$ .

In Figure 8(a) the increase in velocity due to enhanced buoyancy results in more vigorous convection, leading to greater heat transport. This causes a rise in temperature as the fluid more efficiently transfers heat from the heated surface or surrounding environment. Similarly, in Figure 8(b) as fluid velocity increases due to stronger buoyancy-driven convection.

Figure 8.

Comparison of Newtonian and non-Newtonian fluids in temperature while increasing Grashof numbers: (a) impact of mass Grashof number $Gm$ and (b) impact of thermal Grashof number $Gr$ .

In Figure 9(a) higher Prandtl number $\Pr$ values indicate lower thermal diffusivity. This results in a lower overall temperature distribution because heat is retained near the surface rather than being convected away efficiently. Increasing the Lewis number $Le$ in Figure 9(b) heat diffuses more rapidly throughout the fluid, leading to less heat retention near the surface or in localized regions. This faster diffusion of thermal energy results in a decrease in temperature, as the fluid doesn’t retain thermal energy long enough for effective convective heat transfer.

Figure 9.

Comparison of Newtonian and non-Newtonian fluids in temperature while increasing: (a) impact of Prandtl number $\Pr$ and (b) impact of Lewis number $Le$ .

An increase in $Nt$ in Figure 10(a) enhances the thermophoretic effect, intensifying the heat transfer process as particles are driven away from hotter regions, resulting in more dynamic interaction between particles and the fluid. In Figure 10(b) increased Brownian motion $(Nb)$ leads to the diffusion of thermal energy at the molecular level. Since particles are diffusing more randomly, convective heat transfer becomes less efficient. Consequently, the temperature decreases because heat is dissipated more diffusively rather than being convected away from the surface.

Figure 10.

Comparison of Newtonian and non-Newtonian fluids in temperature while increasing diffusion parameters: (a) impact of thermal diffusion parameter $Nt$ and (b) impact of Brownian diffusion parameter $Nb$ .

The momentum slip factor $(γ)$ affects the temperature boundary condition at the surface. In Figure 11(a) as $γ$ increases, the temperature near the surface also rises. The thermal slip factor $(τ),$ on the other hand, influences the thermal boundary condition at the surface. In Figure 11(b) higher values of $τ$ lead to a reduced temperature gradient near the surface.

Figure 11.

Comparison of Newtonian and non-Newtonian fluids in temperature while increasing slip parameters: (a) impact of velocity slip parameter $γ$ and (b) impact of thermal slip parameter $τ$ .

Concentration profile

The decrease in concentration with an increased moving slot parameter $α$ suggests enhanced diffusion of species or particles in the fluid due to higher velocity. The faster-moving fluid tends to thin out the concentration boundary layer, reducing the concentration of solute or particles near the surface in Figure 12(a). In Figure 12(b) the reduction in velocity weakens convective mass transfer, leading to a thicker concentration boundary layer.

Figure 12.

Comparison of Newtonian and non-Newtonian fluids in concentration while increasing moving slot and Casson fluid parameters: (a) impact of moving slot parameter $α$ and (b) impact of Casson fluid parameter $β$ .

In Figure 13(a) an increase in the mass Grashof number $Gm$ enhances convective flow, as solutes or particles are more effectively carried away. Similarly, in Figure 13(b) increased velocity due to stronger convective mass transport thins the concentration boundary layer near the surface.

Figure 13.

Comparison of Newtonian and non-Newtonian fluids in concentration while increasing Grashof numbers: (a) impact of mass Grashof number $Gm$ and (b) impact of thermal Grashof number $Gr$ .

In Figure 14(a) with decreased convective velocity and less efficient thermal transfer, mass diffusion of solutes or particles becomes more pronounced. The rise in concentration indicates that the slower-moving fluid causes greater accumulation of solutes or particles, which could result in issues like fouling or deposition. As shown in Figure 14(b) higher $Le$ leads to slower mass diffusion. This reduced mass diffusivity causes the concentration boundary layer to thin out more quickly, decreasing the concentration of solutes or particles near the surface. The decrease in concentration indicates that mass diffusion is slowing down, which could either be beneficial or detrimental depending on the application.

Figure 14.

Comparison of Newtonian and non-Newtonian fluids in concentration while increasing: (a) impact of Prandtl number $\Pr$ and (b) impact of Lewis number $Le$ .

In Figure 15(a) a stronger thermophoretic effect due to higher $Nt$ causes particles or solutes to be driven away from the heated surface toward cooler regions of the fluid. This movement leads to an increase in particle concentration in regions where they accumulate as a result of the thermophoretic force, causing an overall rise in concentration within the fluid. Similarly, in Figure 15(b) stronger Brownian motion $Nb$ leads to more effective particle dispersion throughout the fluid, resulting in a more uniform distribution.

Figure 15.

Comparison of Newtonian and non-Newtonian fluids in concentration while increasing diffusion parameters: (a) impact of thermal diffusion parameter $Nt$ and (b) impact of Brownian diffusion parameter $Nb$ .

As velocity increases, the convective mass transport of solutes or particles intensifies, leading to a thin concentration boundary layer. As a result the particles are more efficiently carried away due to the higher fluid velocity in Figure 16(a). Finally, Figure 16(b) provided that the higher values of $τ$ lead to a reduced temperature near the surface.

Figure 16.

Comparison of Newtonian and non-Newtonian fluids in concentration while increasing slip parameters: (a) impact of velocity slip parameter $γ$ and (b) impact of thermal slip parameter $τ$ .

Discussion of results

From Figures 2(a), 7(a), and 12(a) non-Newtonian fluids exhibit thicker boundary layers for velocity and temperature, while their concentration boundary layer remains thinner than that of Newtonian fluids. This behavior is attributed to the enhanced movement of the surface, which facilitates momentum transfer within the fluid. The moving slot has a significant effect on the boundary layers. By optimizing this parameter, cooling efficiency can be improved through the control of boundary layers. Non-Newtonian fluids show a more pronounced response to the moving slot, as evidenced by their thicker velocity and temperature boundary layers. This indicates a better retention of heat and momentum, which is advantageous in systems that require controlled thermal management. An increase in $α$ enhances the lubricant’s ability to cool and distribute momentum, thus reducing wear on rolling surfaces. Additionally, the influence on boundary layers improves heat transfer efficiency, ensuring a more uniform temperature distribution across surfaces.

Non-Newtonian fluids consistently maintain thicker boundary layers for velocity, temperature, and concentration than their Newtonian counterparts in Figures 2(b), 7(b), and 12(b). This behavior reflects the non-Newtonian characteristics of Casson fluids, where higher $β$ values correspond to stronger yield stress. This results in reduced momentum diffusion and slower mass transfer rates due to the structural properties of non-Newtonian fluids. Non-Newtonian Casson fluids with higher $β$ values are better at retaining heat, making them suitable for high-temperature industrial processes. Additionally, the increase in the concentration boundary layer slows down particle deposition, contributing to cleaner and more efficient operations in rolling mills and heat exchangers.

From Figures 3(a), 8(a), and 13(a) increasing $Gm$ leads to thicker velocity and temperature boundary layers in non-Newtonian fluids compared to Newtonian fluids. This reinforces their ability to retain heat and momentum more effectively. This behavior is attributed to buoyancy forces caused by solutal differences, which amplify convective heat transfer and drive the flow upwards. As $Gm$ increases, the solutal buoyancy force strengthens, accelerating the flow and improving both convective heat and mass transfer. Due to their higher viscosity and resistance to deformation, non-Newtonian fluids maintain thicker velocity and temperature boundary layers, ensuring better heat retention. The reduced concentration boundary layer suggests faster solutal mixing, which is beneficial for processes requiring efficient solutal transfer. The thicker velocity and temperature boundary layers in non-Newtonian fluids make them well-suited for applications where prolonged heat retention and uniform temperature distribution are critical, such as in rolling mills and heat exchangers. Moreover, the thinner concentration boundary layer promotes enhanced solutal mixing, which is advantageous in processes involving chemical reactions or nanoparticle dispersion.

Similarly, in Figures 3(b), 8(b), and 13(b) Newtonian fluids show a more pronounced increase in the velocity and temperature boundary layers with increasing $Gr$ , whereas non-Newtonian fluids exhibit comparatively thinner boundary layers due to their unique rheological properties. As $Gr$ increases, the velocity and temperature boundary layers become thicker due to thermal buoyancy effects that drive the fluid flow, while the concentration boundary layer decreases. This decrease in concentration is due to the faster solutal diffusion promoted by thermal buoyancy. Increasing $Gr$ highlights the influence of temperature gradients in driving fluid flow. For Newtonian fluids, the thinner boundary layers suggest faster thermal dissipation, while non-Newtonian fluids show enhanced heat retention due to their structural properties. Non-Newtonian fluids also maintain thinner concentration boundary layers compared to their velocity and thermal layers, indicating a controlled response to buoyancy forces that can be tailored for specific industrial applications. The increase in $Gr$ enhances buoyancy-driven cooling in systems such as natural convection heat exchangers. While Newtonian fluids may exhibit faster thermal dissipation, non-Newtonian fluids offer better control over the cooling process. The reduced concentration boundary layer associated with increasing $Gr$ also contributes to lower particle deposition on surfaces, helping to reduce fouling rates and maintenance costs.

From Figures 4(a), 9(a), and 14(a) it can be seen that Non-Newtonian fluids maintain thicker velocity and temperature boundary layers due to their inherent resistance to deformation, while their concentration boundary layers remain thinner compared to Newtonian fluids. Higher $\Pr$ values emphasize the dominance of thermal diffusivity over momentum diffusivity, which results in thinner velocity and thermal boundary layers. Non-Newtonian fluids retain thicker layers due to their unique flow properties, which balance heat retention and momentum diffusion. The increase in the concentration boundary layer with $\Pr$ reflects a reduction in mass transfer, which can be addressed by optimizing fluid properties for specific industrial applications. Thinner velocity and temperature boundary layers in Newtonian fluids suggest faster heat transfer rates. On the other hand, Non-Newtonian fluids, with their thicker boundary layers, provide more controlled thermal management, making them ideal for processes that require steady heat dissipation. The thicker concentration boundary layer with increasing $\Pr$ is beneficial for processes where slow solutal mixing is desired, such as in controlled chemical reactions.

Increasing $Le$ results in a reduction of all boundary layers, as higher mass diffusivity dampens momentum transfer. Non-Newtonian fluids exhibit thicker boundary layers across all parameters, highlighting their slower response to changes in diffusivity compared to Newtonian fluids, as shown in Figures 4(b), 9(b), and 14(b). Higher $Le$ values emphasize the dominance of mass diffusivity, which leads to thinner boundary layers for velocity, temperature, and concentration. However, Non-Newtonian fluids resist rapid changes due to their structural characteristics. The reduction in the concentration boundary layer with increasing $Le$ is advantageous for applications requiring efficient mass transport, such as in nanoparticle suspensions or chemical processes. The decrease in concentration boundary layer thickness with $Le$ enhances solutal transfer, which is beneficial for industrial processes like drying, distillation, and nanoparticle stabilization. The reduction in velocity and thermal boundary layers with increasing $Le$ improves heat transfer efficiency, which is crucial for heat exchangers and cooling systems. Additionally, enhanced mass diffusivity helps reduce particle accumulation on surfaces, aiding in the mitigation of fouling in rolling mills and heat exchangers.

Increasing $Nt$ enhances the velocity, temperature, and concentration boundary layers due to the increased thermophoretic forces that influence fluid motion, as shown in Figures 5(a), 10(a), and 15(a) Non-Newtonian fluids exhibit thicker boundary layers across all parameters compared to Newtonian fluids, reflecting their greater capacity to retain momentum, heat, and solutal particles under thermophoretic forces. The increase in boundary layer thickness for velocity, temperature, and concentration with increasing $Nt$ highlights the dominant role of thermophoresis in transporting heat and particles. Non-Newtonian fluids, with their inherent structural viscosity, amplify this effect, making them more suitable for applications requiring controlled heat and solutal transfer. Thicker boundary layers in non-Newtonian fluids suggest improved heat and momentum retention, which is particularly beneficial in systems requiring prolonged thermal stability. The increased velocity and temperature boundary layers in non-Newtonian fluids with high $Nt$ make them ideal for systems that need extended heat retention, such as in heat exchangers or thermal energy storage. Furthermore, the thicker concentration boundary layer with $Nt$ helps stabilize nanoparticle suspensions, reducing agglomeration and improving the efficiency of cooling or heating systems. The enhanced thermophoretic movement also reduces the accumulation of fouling particles, thus improving the operational efficiency of rolling mills and industrial heat exchangers.

In contrast, increasing $Nb$ reduces the velocity, temperature, and concentration boundary layers due to enhanced Brownian motion, as illustrated in Figures 5(a), 10(a), and 15(a). Non-Newtonian fluids, however, maintain thicker boundary layers even as $Nb$ increases, demonstrating a slower response to the enhanced diffusivity compared to Newtonian fluids. The reduction in all boundary layers with increasing $Nb$ reflects the intensified diffusion driven by Brownian motion. While Newtonian fluids exhibit sharper reductions in boundary layer thickness, non-Newtonian fluids show more gradual thinning, offering better control over diffusivity. The thinner concentration boundary layer facilitates faster mixing of nanoparticles or solutes, which is beneficial in chemical and biomedical applications. Additionally, the thinner velocity and temperature boundary layers with increasing $Nb$ improve heat transfer rates, which is critical for high-performance cooling systems. The enhanced solutal mixing resulting from thinner concentration boundary layers is advantageous in chemical processing, drug delivery, and nanofluidics. Non-Newtonian fluids with controlled $Nb$ can be tailored for applications that require specific nanoparticle behaviors, such as targeted drug delivery or advanced material synthesis.

From Figures 6(a), 11(a), and 16(a) Non-Newtonian fluids exhibit thicker velocity and temperature boundary layers due to their viscoelastic nature, while Newtonian fluids show minimal thickness. In contrast, Non-Newtonian fluids maintain thinner concentration boundary layers for the same $γ$ . The increase in velocity and temperature boundary layers with $γ$ results from reduced surface friction, which decreases flow resistance and promotes greater fluid motion. Non-Newtonian fluids amplify this effect due to their non-linear stress-strain behavior. The decrease in the concentration boundary layer indicates enhanced solutal mixing, making $γ$ a key parameter for optimizing mass transfer processes. By increasing velocity and temperature boundary layers, $γ$ provides better control over fluid motion and thermal transport, making it ideal for applications such as lubrication in rolling mills or cooling in heat exchangers. The reduced concentration boundary layer helps minimize particle deposition on surfaces, reducing fouling and improving operational efficiency. Enhanced slip flow reduces surface friction, leading to lower energy requirements for fluid transport in industrial systems.

In Figures 6(b), 11(b), and 16(b) Non-Newtonian fluids exhibit thicker boundary layers across all parameters compared to Newtonian fluids, even though $τ$ tends to induce thinner layers. The reduction in velocity and temperature boundary layers reflects the diminished effectiveness of heat transfer at the surface due to thermal slip. Non-Newtonian fluids show a more controlled reduction, maintaining better thermal stability. The thinner concentration boundary layer with $τ$ facilitates faster solutal transport, which is beneficial for applications involving nanoparticles or chemical species. The reduction in velocity and temperature boundary layers enhances heat dissipation, which is crucial for high-performance cooling systems like heat exchangers and turbine blades. The thinner concentration boundary layer with $τ$ also aids in stabilizing nanoparticles in nanofluids, preventing agglomeration and improving their thermal properties. Non-Newtonian fluids with thermal slip are advantageous for processes that require precise thermal and solutal control, such as biomedical cooling or advanced material manufacturing.

Conclusion

In this manuscript, we investigated the analysis heat transfer in Casson fluid flow due to aligned flat plate with leading edge accretion and slip boundary conditions. While using the Blasius Rayleigh-Stokes boundary layer, which reveals important distinctions between Newtonian and non-Newtonian Casson fluid behaviors. This analysis demonstrates the influence of leading-edge accretion rates on Casson fluid flow and thermal properties. The findings of the manuscript are listed as follows:

➢ From the comparison, the non-Newtonian Casson fluid flow gives better performance while increasing the leading edge accretion rate.

➢ Increasing the moving slot parameter $α$ , thermal Grashof number $α$ , mass Grashof number $Gm$ , and velocity slip parameter $γ$ all result in an increase in velocity and temperature, along with a decrease in concentration. These effects enhance fluid movement near the surface, leading to improved heat transfer due to higher velocity and more effective thermal convection, while reducing concentration near the plate. The increase in buoyancy-driven flow strengthens natural convection, improving heat transfer and leading to higher fluid temperatures, while reducing the accumulation of particles near the surface. The reduced friction at the boundary from higher $γ$ leads to increased velocity and temperature, boosting heat transfer efficiency and decreasing concentration due to enhanced convective mass transport.

➢ Conversely, increasing the Prandtl number $\Pr$ and the Casson fluid parameter $β$ results in a decrease in velocity, temperature, and concentration. This indicates that thermal diffusion is becoming more dominant than mass diffusion. A higher $\Pr$ leads to slower fluid motion and less effective heat transfer, reducing both temperature and concentration near the surface. As the fluid’s viscosity increases, resistance to flow grows, causing a drop in velocity and temperature, while concentration near the surface increases due to reduced convective transport.

➢ An increase in the thermophoresis diffusion parameter $Nt$ leads to increases in velocity, temperature, and concentration. This is due to enhanced particle movement driven by thermal gradients, which raises fluid velocity and temperature while also increasing particle concentration in certain regions due to thermophoretic forces.

➢ Increasing the Brownian diffusion parameter $Nb$ , Lewis number Le, and thermal slip parameter $τ$ results in decreases in velocity, temperature, and concentration. Stronger random motion of particles from increased Brownian diffusion leads to greater dispersion but lowers fluid velocity and temperature, reducing particle concentration near the surface as they are spread more uniformly. Similarly, weaker thermal interaction at the boundary due to increased $τ$ reduces heat transfer efficiency, leading to lower velocity and temperature, and resulting in decreased concentration due to weakened convective transport.

➢ This comparative study demonstrates that non-Newtonian fluids enhance heat retention and thermal stability, making them ideal for applications requiring sustained heat retention and efficient thermal management. These fluids ensure effective solutal mixing, reduce particle accumulation, mitigate fouling, lower surface friction, and improve flow control, all of which contribute to energy-efficient operations.

➢ The practical implications of these findings highlight the advantages of non-Newtonian fluids in industrial applications. They improve heat transfer efficiency, reduce fouling and particle deposition, and allow for precise control over thermal and solutal properties. These qualities make non-Newtonian fluids particularly well-suited for advanced cooling systems, lubrication processes, and nanofluidic applications in biomedicine and material processing.

➢ Finally, replacing water-based Newtonian coolants with non-Newtonian Casson-type fluids offers significant benefits, such as enhanced heat and mass transfer, improved operational efficiency, and increased system longevity. Future research could focus on optimizing these parameters under dynamic operating conditions to further improve the performance of non-Newtonian fluids in industrial applications.

The interplay of these parameters is crucial for determining the efficiency of heat transfer and mass transport in the system. Parameters that enhance fluid velocity and thermal convection generally improve heat transfer, while those that increase viscosity, slip, or random motion tend to reduce the overall efficiency of thermal and mass transport. Understanding these effects is vital for optimizing designs in applications like cooling systems, heat exchangers, and processes involving Casson fluid flow with accretion.

Future directions: Future research could explore the impact of external magnetic flux and thermal radiation could yield valuable insights for optimizing systems in specific applications.

Footnotes

Handling Editor: Chenhui Liang

Author contributions

The authors equally contributed in this manuscript. J Jayaprakash contributed to the methodology, validation, and prepared the original draft. Vediyappan Govindan was responsible for formal analysis, investigation, and supervision. Haewon Byeon secured funding for the project.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-RS-2023-00237287, NRF-2021S1A5A8062526) and local government-university cooperation-based regional innovation projects (2021RIS-003).

ORCID iD

Vediyappan Govindan

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Analysis of heat transfer in Casson fluid flow due to aligned flat plate with leading edge accretion and slip boundary conditions

Abstract

Keywords

Introduction

Mathematical model

Result and discussion

Analysis of results

Velocity profile

Temperature profile

Concentration profile

Discussion of results

Conclusion

Footnotes

Author contributions

Declaration of conflicting interests

Funding

ORCID iD

References