Dynamic-behavior of Casson-type hybrid nanofluids due to a stretching sheet under the coupled impacts of boundary slip and reaction-diffusion processes

2.1 Dimensional analysis framework

The system’s dynamics are mathematically represented through four coupled conservation principles, the mass balance (continuity), the momentum transport incorporating non-Newtonian impacts, the energy transfer with radiative heat flux and velocity slip boundary conditions, and the species transport with reaction kinetics. This formulation rigorously captures the interplay between hydrodynamic, thermal, and chemical processes under the defined physical constraints. The complete system of coupled transport equations is presented below [30]:

where Γ is the Casson parameter, t is the time, T is the dimensional temperature, D _B is the diffusion coefficient and σ is the electrical conductivity. Also, the energy conservation analysis (11) reveals a temperature-dependent radiative transfer mechanism q _r , where thermal radiation effects are quantified through the Rosseland diffusion approximation. This approach models radiative flux as a function of the local temperature gradient ∂ T ∂ y , expressed mathematically as [31], 32]:

where k* signifies the mean absorption coefficient and σ* denotes the Stefan–Boltzmann constant. Further, the chemical reaction phenomenon that studied here in our model is represented using a homogeneous first-order kinetic model, a common simplification in fluid dynamic analyses of reactive transport. This formulation is applicable to both consumptive reaction types, such as those occurring on catalytic surfaces. The first-order approach offers a physically justified and mathematically manageable framework, aligning with established methodologies in nanofluid reactivity research [33].

2.2 Physical system boundary definitions

The domain’s boundary constraints are formally defined to capture essential interfacial phenomena governing fluid-wall interactions. These mathematically prescribed conditions incorporate three critical physical effects, the first is the hydrodynamic slip, the second is the thermal slip and the concentration slip is the third phenomenon at the boundary surface. Their rigorous implementation ensures the governing equation solutions remain physically consistent with real-world system behavior. These phenomena can be summarized mathematically as [29]:

where v ₀ is the suction velocity, L ₁, L ₂ and L ₃ are the slip coefficients of the velocity, the temperature and the concentration respectively.

2.3 Canonical dimensionless quantities

The dimensional analysis approach employs scaling transformations to reduce system complexity through two key mechanisms, the first is consolidation of independent variables into dimensionless groups, and the second is conversion of governing partial differential equations into a simplified ordinary differential system. This methodology yields three significant advantages: enhanced physical insight through characteristic parameter ratios, improved numerical stability by eliminating unit dependence, and clearer pattern recognition in solution behavior. The subsequent section details the complete nondimensionalization procedure systematically deriving each transformed equation from its dimensional counterpart. This procedure based on the following form [29]:

2.4 Similarity-transformed equation set

The transformation to dimensionless form serves two critical analytical purposes, the first is the reduction of system complexity through characteristic scaling, and the second is the universalization of results for broader physical interpretation. By introducing normalized variables, the governing equations distill to their essential dynamics, revealing the dominant dimensionless groups that control flow and thermal transport mechanisms. The transformed system, comprising the ordinary differential equations (Eq. (17)) and their associated boundary constraints, represents a generalized formulation applicable across multiple scales and operating conditions. Below we present this refined mathematical framework that forms the basis for our subsequent analysis:

The boundaries edge conditions are formulated through these expressions:

The parametric analysis reveals eight dominant control variables governing the coupled transport phenomena, the magnetic field factor M modifying Lorentz forces, the unsteadiness parameter γ dictating flow instability, the velocity, thermal and concentration slip parameters (λ ₁, λ ₂, λ ₃) which regulating wall–fluid interaction, the chemical reaction factor C _r controlling species conversion, the radiation parameter R, the Prandtl number Pr and Schmidt number Sc. These parameters collectively determine the system’s hydrodynamic behavior, thermal transport efficiency, and mass diffusion characteristics through their interconnected influences on boundary layer development. Also, these factors can be defined as:

Further, we now examine three key transport metrics, the wall shear stress (Cf _x ), the species flux (Sh _x ), and the heat flux (Nu _x , with their definitive expressions below:

where R e = u w ρ i x μ i is the local Reynolds number.

4.1 Graphical representation of parameter impacts on the model

This section provides a comprehensive numerical investigation after employing shooting technique of the hybrid nanofluid’s flow behavior under multiple interacting parameters, presenting the results through detailed graphs that analyze their collective effects on velocity profiles, temperature distributions, concentration patterns, and related transport properties. Additionally, the range of parameters that controlling the model are 0.5 ≤ Γ ≤ 2.0, 0.0 ≤ λ ₁ ≤ 0.4, 0.0 ≤ M ≤ 5.0, 0.5 ≤ γ ≤ 2.0, 0.0 ≤ R ≤ 5.0, 0.0 ≤ λ ₂ ≤ 0.3, 0.0 ≤ λ ₃ ≤ 0.3 and 0.0 ≤ C _r  ≤ 4.0. Figure 2 illustrates the influence of the Casson parameter Γ on the profiles of velocity f′(η), concentration H(η), and temperature θ(η). This figure demonstrates that elevated Casson parameter Γ values substantially reduce the dimensionless velocity gradient f′(η), consistent with the rheological behavior of yield-stress fluids where increased Γ enhances viscoplastic resistance and flow opposition. This velocity suppression consequently diminishes convective transport, leading to elevated temperature θ(η) and concentration H(η) profiles as thermal energy and solute species accumulate near the surface. The resulting boundary layer thickening reflects a shift from convective to diffusion-dominated transport regimes under high yield stress conditions. Physically, an increase in the Casson parameter corresponds to a rise in the fluid’s yield stress, enhancing its resistance to deformation. This effect restricts fluid motion, resulting in the accumulation of thermal energy and solute species adjacent to the boundary. Consequently, both the thermal and concentration boundary layers experience noticeable thickening.

Figure 2:

Effect of the Casson parameter γ on the transport behavior of the hybrid nanofluid. (a) Velocity profile f′(η) for different values of Γ. (b) Temperature θ(η) and concentration H(η) distributions for varying Γ.

Figure 3 presents the effects of the velocity slip parameter λ ₁ on the flow and transport characteristics of a hybrid nanofluid composed of copper (Cu) and aluminum oxide (Al₂O₃) nanoparticles dispersed in a Casson base fluid. Increasing the slip parameter λ ₁ reduces the velocity gradient f′(η), particularly at the boundary (f′(0)), but it enhances both thermal θ(η) and concentration H(η) fields and the corresponding boundary thickness, due to weakened fluid-wall adhesion. This slip-induced deceleration contrasts with no-slip conditions where peak velocities occur near the wall. While hybrid nanofluids inherently improve transport, velocity slip disrupts wall-to-fluid momentum transfer, diminishing overall flow intensity despite the nanoparticles’ thermal advantages. Physically, rising slip factor diminish frictional resistance at the boundary, thereby impairing momentum diffusion and convective transport. This reduction in fluid mixing promotes the accumulation of heat and solute particles near the surface, leading to an expansion of both thermal and solutal boundary layer thicknesses.

Figure 3:

Influence of the velocity slip parameter λ ₁ on hybrid nanofluid flow characteristics. (a) Velocity distribution f′(η) at different slip values. (b) Temperature θ(η) and concentration H(η) responses to varying λ ₁.

Figure 4 illustrates the flow patterns of a hybrid nanofluid with and without velocity slip at the surface (λ ₁ = 0.4 and λ ₁ = 0.0, respectively). When there is no slip (λ ₁ = 0.0), the streamlines are closely spaced near the boundary, indicating a strong interaction between the fluid and the wall and the development of typical parabolic velocity profiles. With slip phenomenon (λ ₁ = 0.4) significantly alters the flow topology, this means that streamlines become more dispersed adjacent to the wall, indicating reduced momentum transfer due to partial fluid detachment. This slip-induced deceleration decreases wall shear stress by approximately 30 % while simultaneously thickening the hydrodynamic boundary layer. The change from even to uneven flow patterns highlights how slip disrupts viscous forces, making inertia more influential, this is very important for designing microfluidic devices or lubricants where wall slip is significant.

Figure 4:

Streamline patterns of the hybrid nanofluid under different velocity slip conditions. (a) Streamlines for the no-slip case λ ₁ = 0.0. (b) Streamlines for the slip case λ ₁ = 0.4.

Figure 5 demonstrates that higher magnetic field intensity M suppresses flow velocity f′(η) while simultaneously elevating thermal θ(η) and concentration H(η) profiles, due to Lorentz-force-induced flow resistance and enhanced energy/mass retention near the boundary. Physically, the observed trends can be attributed to the magnetohydrodynamic effects, when a transverse magnetic field interacts with the conductive nanofluid, it generates Lorentz forces that oppose flow direction. This electromagnetic resistance slows bulk fluid motion, increasing momentum boundary layer thickness. Reduced velocity weakens convective heat/mass transfer, causing energy and solute accumulation near the surface. Consequently, both θ(η) and H(η) profiles elevate, while their boundary layers expand-demonstrating the tradeoff between magnetic flow control and transport efficiency.

Figure 5:

Effect of the magnetic parameter M on hybrid nanofluid momentum, thermal, and concentration fields. (a) Velocity profile f′(η) for various values of M. (b) Temperature H(η) and concentration H(η) profiles for different magnetic field strengths.

Figure 6 compares the flow patterns of a hybrid nanofluid under the absence (M = 0.0) and presence (M = 5.0) of a magnetic field. When magnetic forces are absent (M = 0.0), the streamlines are closely packed and aligned in a parallel manner, which is characteristic of viscous flow that encounters no opposing forces. Clearly that, with a strong magnetic field (M = 5.0), Lorentz forces alter the flow patterns, leading to wider spacing between streamlines (slower flow), further lead to curved paths, and small areas of separated flow near the surface. This change shows how magnetic fields shift the flow from being mainly controlled by viscosity to being governed by magnetohydrodynamic effects-important for applications like electromagnetic pumps or cooling systems in fusion reactors where flow regulation is key.

Figure 6:

Streamline configurations of the hybrid nanofluid with and without magnetic field effects. (a) Streamlines for M = 0.0. (b) Streamlines for M = 5.0.

Figure 7 illustrates that a rise in the unsteadiness parameter γ causes a notable change in the hybrid nanofluid’s behavior. Larger γ values lead to a decrease in flow velocity f′(η) because of time-dependent momentum spreading, while at the same time, they increase the thermal θ(η) and concentration H(η) profiles through two processes, the first is the cyclical speeding up and slowing down of the flow enhances the trapping of energy and solute near the boundary, and the second is the changes over time disrupt stable convective flow patterns, making diffusion the more dominant transport mechanism. This leads to thicker boundary layers for both temperature and concentration (increasing by as much as 25 % when γ = 1.8 compared to γ = 0.8) but less steep changes in velocity-demonstrating the opposing influences of γ on momentum transfer versus heat and mass transfer. These observations are especially important for systems with pulsating flow, where the unsteadiness parameter significantly affects how the system performs.

Figure 7:

Influence of the unsteadiness parameter γ on hybrid nanofluid flow and transport. (a) Velocity variations f′(η) for different γ values. (b) Temperature θ(η) and concentration H(η)profiles for varying γ.

Figure 8 demonstrates how thermal radiation R and thermal slip λ ₂ collectively alter heat transfer in a Cu–Al₂O₃/Casson hybrid nanofluid. An increase in R leads to a greater temperature distribution θ(η) because the radiative heat flux adds to the heat transfer occurring through conduction and convection. This effect is especially pronounced at higher radiation values (R = 5.0), where the thermal boundary layer becomes approximately 20 % thicker compared to non-radiative cases (R = 0.0). On the other hand, enhancing λ ₂ lessens the temperature changes, especially near the wall by reducing the thermal interaction between the fluid and the surface, leading to a noticed decrease in the local Nusselt number in terms of (θ′(0)) by as much as 15 % when λ ₂ equals 0.3. This interplay reveals a critical tradeoff, while radiation augments heat transfer, thermal slip diminishes it by disrupting energy exchange at the boundary. The physical and fundamental reason stems from the dual role of thermal radiation and slip effects is that the radiation contributes supplementary energy flux, raising the bulk temperature of the fluid, while thermal slip reduces energy exchange at the wall-fluid interface, thereby suppressing conductive heat transfer at the boundary.

Figure 8:

Effects of thermal radiation R and thermal slip λ ₂ on heat transfer characteristics. (a) Temperature distribution θ(η) under different radiation levels. (b) Temperature response θ(η) for varying thermal slip values λ ₂.

Figure 9 demonstrates how both the concentration slip parameter λ ₃ and chemical reaction rate C _r impact the hybrid nanofluid’s behavior. Increasing λ ₃ reduces species concentration, especially near the wall H(0) due to weakened fluid-surface adhesion, suppressing diffusive mass transfer and reduces the corresponding boundary layer. Further, higher C _r enhances near-wall consumption of chemical species, sharpening concentration gradients and reduces the concentration distribution. In the Cu–Al₂O₃/Casson nanofluid, these factors demonstrate a balancing act: increasing slip at the surface λ ₃ reduces the buildup of mass near the wall, while intensified chemical reactions C _r actively consume the species, modifying the behavior of the boundary layer. The findings underscore the competing roles of surface slip and reaction rates in governing mass transport within non-Newtonian nanofluids is a crucial consideration for optimizing applications such as catalytic reactors or localized drug delivery systems. The observed trend is physically attributed to the enhanced consumption of reactant species near the surface under elevated reaction rates. This localized depletion establishes a steeper concentration gradient, which promotes diffusive transport toward the wall and ultimately lowers the overall solute concentration throughout the domain.

Figure 9:

Effects of concentration slip λ ₃ and chemical reaction parameter C _r on mass transfer in the hybrid nanofluid. (a) Concentration profile H(η) for different values of λ ₃. (b) Concentration field H(η) for varying chemical reaction rates C _r .

4.2 Numerical analysis of parameter impacts on Cf _x , Nu _x and Sh _x

Table 2 quantifies how key dimensionless groups affect surface transport coefficients ( 1 2 C f x R e 1 2 , N u x R e − 1 2 , S h x R e − 1 2 ) for fixed Sc = 1.5 and Pr = 4.5, revealing parametric trends in wall shear stress, thermal exchange, and species transfer. The Casson parameter Γ exhibits an inverse relationship with all transport metrics, which implies higher G values reduce flow velocity as well as the wall shear stress, heat transfer, and mass transfer due to enhanced viscoplastic resistance within the fluid. Further, the velocity slip coefficient reduces wall shear stress and both transfer rates ( N u x R e − 1 2 , S h x R e − 1 2 ) by decreasing fluid-surface adhesion. Conversely, increasing the magnetic parameter elevates 1 2 C f x R e 1 2 through Lorentz-force resistance while lowering N u x R e − 1 2 and S h x R e − 1 2 via flow suppression. The unsteadiness coefficient exhibits dual effects, slightly increasing wall shear stress, while decreasing thermal and mass transfer performance ( N u x R e − 1 2 , S h x R e − 1 2 ), attributed to time-dependent flow disruptions. In contrast, thermal radiation parameter significantly boosts heat transfer rates N u x R e − 1 2 without materially affecting friction or concentration distributions, demonstrating selective energy enhancement. Finally, thermal slip (λ ₂) diminishes heat transfer N u x R e − 1 2 by weakening boundary thermal coupling, while concentration slip (λ ₃) analogously reduces mass transfer S h x R e − 1 2 ). Additionally, higher reaction rates (C _r ) progressively decrease S h x R e − 1 2 ) values through wall-adjacent species consumption.

4.3 Research contribution in context

The current investigation advances the ongoing research in non-Newtonian unsteady Casson hybrid nanofluid dynamics by introducing a comprehensive model that concurrently incorporates slip boundary mechanisms, magnetohydrodynamic forces, radiative heat transfer, and chemically reactive processes. While prior studies have extensively explored individual physical phenomena, including slip flow [1], magnetohydrodynamic (MHD) effects [8], thermal radiation [9], Casson rheology [32], and chemical reaction kinetics [33], their interconnected behavior in Casson hybrid nanofluids remains largely unexamined. Most existing literature addresses these mechanisms in isolation, leaving a critical gap in understanding their synergistic impact on heat and mass transfer in multi-physics flow environments. The findings reveal that higher Casson parameter values amplify viscoplastic resistance, which in turn restrains momentum diffusion while promoting the growth of thermal and solutal boundary layers. This trend consistent with established non-Newtonian fluid studies. Furthermore, the slip and radiation impacts documented in this work build upon prior research by quantitatively evaluating their combined impact on transport efficacy and boundary layer development. Thus, this study not only validates and complements prior literature but also advances it by integrating multiple complex mechanisms, providing new insights with potential industrial relevance.