EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6958 ISSN 1307-5543 – ejpam.com Published by New York Business Global Numerical Study of Nano-encapsulated Phase Change Materials on Non-Newtonian Nanofluid Flow over a Stretching Sheet Leli Deswita1,∗, Syamsudhuha Syamsudhuha1, Rustam Effendi1, Asral Asral2 1 Department of Mathematics, Faculty of Mathematics and Science, Universitas Riau, 28293 Pekanbaru, Riau, Indonesia 2 Department of Mechanical Engineering, Engineering Faculty, Universitas Riau, 28293 Pekanbaru, Riau, Indonesia Abstract. This study analyzes the hydrodynamic and thermal behavior of a power-law nanofluid containing nano-encapsulated phase change materials (NEPCMs) over a non-isothermal stretching sheet. The combined effects of non-Newtonian rheology and latent heat transport due to NEPCM inclusions are formulated through boundary-layer theory and the power-law constitutive relation. The resulting nonlinear similarity equations are solved numerically using the shooting technique with Runge–Kutta integration. Results indicate that embedding NEPCM particles within the power-law nanofluid significantly enhances heat transfer through latent heat absorption and re- lease, though it also increases wall shear stress. Higher power-law indices suppress velocity and temperature distributions, mitigating this shear enhancement. These findings reveal a competing interaction between fluid rheology and the thermal storage capacity of NEPCMs. The study offers new insights into the coupled flow and heat transfer mechanisms of NEPCM-suspended power-law nanofluids, with relevance to advanced cooling, polymer processing, and energy storage applica- tions. 2020 Mathematics Subject Classifications: 41A30, 76F12, 65M60, 65N12 Key Words and Phrases: Runge–Kutta method, NEPCM, phase change heat transfer, nanoflu- ids, boundary layer 1. Introduction The behavior of non-Newtonian fluids is integral to numerous natural processes and industrial applications due to their shear rate- and shear stress-dependent viscosity, which is a distinctive property of these fluids. This distinctive property of non-Newtonian fluids ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6958 Email addresses: leli.deswita@lecturer.unri.ac.id (L. Deswita), syamsudhuha@lecturer.unri.ac.id (S. Syamsudhuha), rustam@lecturer.unri.ac.id (R. Effendi), asral@lecturer.unri.ac.id (A. Asral) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 2 of 18 has enabled their use in a variety of industrial and engineering contexts, including the manufacture of plastics, mining, transportation, aerodynamics, lubrication, and biological and natural systems such as bodily fluids and wet sand. Non-Newtonian fluids display a range of rheological behaviours, including shear thinning and shear thickening, which depend on the shear rate applied. Due to their intricate nature, these fluids cannot be described by a single constitutive equation. Consequently, numerous mathematical models have been proposed to characterise their behavior, with the power-law model being the most widely adopted for describing pseudoplastic and dilatant fluids. Extensive research has been conducted on power-law fluid flows, highlighting their multifaceted properties and applications. Cobble [1] analyzed the flow of power-law fluids over a flat plate under a magnetic field and pressure gradient, considering fluid injection and ejection. Hassanien et al. [2] and Tashtoush et al. [3] examined the heat transfer behavior of power-law fluids over a stretching surface with variable surface temperature, incorporating suction or injection effects. Sahu et al. [4] investigated the momentum and heat transfer in power-law fluids near a continuously moving plate, while Xu and Liao [5] examined heat transfer over a stretching flat sheet. Prasad and Vajravelu [6] analyzed magnetohydrodynamic (MHD) flow over a non-isothermal stretching sheet, extending the problem to include variable thermal conductivity and non-uniform heat generation [7]. Further studies have incorporated additional complexities, such as radiation and heat generation [8], slip velocity effects [9], non-linear stretching surfaces [10], and non-uniform heat generation [11]. Recent investigations have expanded into the effects of porosity on inclined plates [12] and unsteady electro-magnetohydrodynamic flows involving active and passive nanoparticle fluxes [13]. Alabdulaal et al. [14] reported that the power-law exponent exerts a favorable effect on heat transfer, but its influence on skin friction remains complex. A recent innovation in heat transfer research involves the development of nanofluids containing nanoparticles made from highly conductive materials such as carbon, metals, and metal oxides. A more advanced class of nanofluids, known as nano-encapsulated phase change material (NEPCM) suspensions, incorporates phase change materials encapsulated within nanoparticles, thus providing an additional latent heat storage mechanism. Studies on NEPCM-based Newtonian nanofluids include Ghalambaz et al. [15], who investigated convection dynamics in NEPCM-water systems within enclosures, and Hajjar et al. [16], who analyzed NEPCM suspensions in periodically heated cavities. Other works, such as Alhashash and Saleh [17], examined heat transfer enhancement via enclosure rotation, while Wang et al. [18] explored entropy generation in localized NEPCM systems. Recently, Sheikholeslami et al. [19] proposed an enhancement methodology for photovoltaic systems, encompassing the integration of a nanofluid filter and a paraffin-based storage apparatus. Despite these advances, the interaction of NEPCMs with power-law fluids remains largely unexplored, particularly under boundary-layer flow conditions. Only a few stud- ies, such as Ghalambaz et al. [20] and Ali et al. [21], have considered NEPCM suspensions in non-Newtonian enclosures. However, to the best of the authors’ knowledge, no inves- tigation has addressed the hydrodynamic and thermal behavior of NEPCM-suspended power-law nanofluids over a stretching sheet with variable surface temperature. L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 3 of 18 Nano Encapsulated Phase Change Materials y x Figure 1: Schematic model of the flow over a stretching sheet contains NEPCMs suspended in the non- Newtonian base fluid. The novelty of the present work lies in formulating and analyzing, for the first time, the boundary-layer flow and heat transfer of a power-law nanofluid embedded with NEPCMs over a non-isothermal stretching surface. The study provides a detailed parametric analysis of the effects of power-law index, NEPCM concentration, fusion temperature, and heating parameter on velocity, temperature, skin friction, and heat transfer rate. By coupling the rheological behavior of non-Newtonian fluids with the latent heat characteristics of NEPCMs, this research establishes a new framework for enhancing thermal transport in shear-dependent suspensions, with implications for advanced cooling, polymer processing, and energy storage applications. 2. Mathematical Formulation Consider a two-dimensional, steady-state, incompressible boundary layer flow of a power-law nanofluid over a linearly stretching sheet. The x-axis is aligned with the flow direction, and the y-axis is normal to the surface. The sheet is stretched linearly, generating the flow as illustrated in Fig. 1. The fluid contains nano-encapsulated phase change materials (NEPCMs) suspended in a non-Newtonian base fluid. These NEPCMs absorb and release latent heat during phase transitions as they experience temperature variations within the boundary layer. The wall temperature distribution follows T = bxγ + Tw, (1) where Tw is the reference wall temperature, and γ is the power-law exponent describing the surface temperature variation. Using the boundary layer approximations, the governing equations are given as follows [22, 23]: L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 4 of 18 ∂u ∂x + ∂v ∂y = 0, (2) u ∂u ∂x + v ∂u ∂y = νnf ∂ ∂y (∣∣∣∣∂u∂y ∣∣∣∣n−1 ∂u ∂y ) , (3) u ∂T ∂x + v ∂T ∂y = αnf ∂2T ∂y2 . (4) Equation (2) ensures mass conservation, while Eq. (3) represents the momentum bal- ance in the boundary layer for a power-law fluid. Equation (4) governs the thermal energy transport, accounting for conduction and convection. The corresponding boundary conditions are: u = ax, v = vw(x), T = bxγ + Tw at y = 0, u = 0, T = T∞ as y → ∞. (5) Here, u and v are the velocity components, a is the stretching rate, and vw(x) denotes the wall mass flux. The nanofluid’s kinematic viscosity and effective thermal diffusivity are given by νnf = µnf/ρnf and αnf = knf/(ρCp)nf , respectively. The effective density of the nanofluid is expressed as: ρnf = (1− ϕ)ρbf + ϕρns, (6) where ϕ is the nanoparticle volume fraction, and subscripts bf and ns denote the base fluid and NEPCM solid particles, respectively. The NEPCM particle density is calculated as: ρNm = (1 + ι)ρNcρNs ρNs + ιρNc , (7) where ρNc and ρNs are the core and shell densities, and ι is the core-to-capsule mass ratio (ι ≈ 0.447 [24]). The nanofluid’s specific heat capacity is expressed as: (Cp)nf = (1− ϕ)(ρCp)bf + ϕ(ρCp)Nm ρNm . (8) This relation considers both the base fluid and the encapsulated nanoparticles. To capture the phase change behavior, the specific heat of NEPCMs varies with tem- perature as: Cp,Nc = Cp,Nc,l + [ 90◦ ( hsf TMr − Cp,Nc,l ) sin ( 180◦ T − Tf + TMr/2 TMr )] Γ, (9) L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 5 of 18 where TMr is the melting temperature range and Γ defines the active phase change region. This expression ensures smooth numerical convergence by representing latent heat effects through a sinusoidal model. The effective thermal conductivity and viscosity properties of the suspension are: keff = 1 + kNmϕ, (10) µeff = 1 + µNmϕ. (11) Next, the similarity transformation is applied to reduce the governing PDEs into ODEs: ψ = (νnfa 2n−1x2n)1/(n+1)f(η), Θ(η) = T − T∞ Tw − T∞ , η = ( a2−nx1−n νnf )1/(n+1) y. (12) Here, ψ is the stream function defined by u = ∂ψ ∂y and v = −∂ψ ∂x . Substituting these transformations yields the following dimensionless equations: (1 + µNmϕ) [(1− ϕ) + ϕρs/ρbf ] f ′′′(f ′′)n−1 + 2n n+ 1 ff ′′ − (f ′)2 = 0, (13) (1 + kNmϕ) CrPr Θ′′ + 2n n+ 1 fΘ′ − γΘf ′ = 0. (14) Equation (13) governs the momentum transport of the non-Newtonian nanofluid, while Eq. (14) describes the corresponding energy transfer. The boundary conditions are rewritten as: f = 0, f ′ = 1, Θ = 1 at η = 0, f ′ = 0, Θ = 0 as η → ∞. (15) Finally, the key dimensionless performance measures are given as follows: 2CfRe 1/2 x = (1 + µNmϕ)[f ′′(0)]n, (16) NuRe−1/2 x = −(1 + kNmϕ)Θ ′(0). (17) Equations (16) and (17) represent the skin friction and local Nusselt number, respec- tively, providing insights into the shear and heat transfer performance along the stretching surface. 3. Computational Methodology The coupled nonlinear boundary value problem (BVP) given by Eqs. (13)–(14), sub- ject to the boundary conditions (15), was solved using the single–shooting method. This approach transforms the BVP into an equivalent initial value problem (IVP) whose un- known initial slopes are determined iteratively such that the far–field boundary conditions are satisfied [25]. L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 6 of 18 The first step is to reduce the system to a first-order system. We then introduce the dependent variables: y1 = f, y2 = f ′, y3 = f ′′, y4 = Θ, y5 = Θ′, (18) such that f ′′′ and Θ′′ can be obtained directly from the governing equations. Let A = (1 + µNmϕ)[ (1− ϕ) + ϕ ρs ρbf ] , B = CrPr (1 + kNmϕ) . (19) From Eq. (13), the third derivative of f is expressed as y′3 = f ′′′ = y22 − 2n n+1y1y3 An (y3)n−1 . (20) Similarly, from Eq. (14), the second derivative of Θ is given by y′5 = Θ′′ = B ( γ y4 y2 − 2n n+ 1 y1y5 ) . (21) The complete first–order system becomes y′1 = y2, (22) y′2 = y3, (23) y′3 = y22 − 2n n+1y1y3 An (y3)n−1 , (24) y′4 = y5, (25) y′5 = B ( γ y4 y2 − 2n n+ 1 y1y5 ) . (26) The boundary conditions at η = 0 are y1(0) = 0, y2(0) = 1, y4(0) = 1, (27) while y3(0) = α and y5(0) = β are unknown initial slopes to be determined. At the far–field (η → ∞), the requirements are y2(∞) → 0, y4(∞) → 0. (28) The shooting method treats α and β as free parameters and integrates Eqs. (22)–(26) from η = 0 to a sufficiently large ηmax, where the “infinity” conditions are approximated. The residual vector is defined as R(α, β) = [ y2(ηmax;α, β) y4(ηmax;α, β) ] , (29) L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 7 of 18 and the goal is to find (α, β) such that R ≈ 0. The IVP above was integrated using the Runge–Kutta method. The far–field was truncated to ηmax in the range [4, 12] to ensure that f ′(ηmax) and Θ(ηmax) were suffi- ciently close to zero. The nonlinear system for (α, β) was solved using a Newton-type root-finding algorithm with Jacobian approximation. Convergence was declared when the infinity-norm of the residual vector was less than 0.000001. To avoid division by zero in Eq. (24) when y3 → 0, a small regularisation parameter ε was introduced, replacing (y3) n−1 by (max(|y3|, ε))n−1 in the denominator. The choice ε ≈ 10−12 was found sufficient to maintain numerical stability without affecting accuracy. Additionally, parameter con- tinuation was employed for challenging parameter regimes: solutions obtained for simpler cases (e.g., n = 1, γ = 0) were used as initial guesses for nearby parameter values. To verify the accuracy of the numerical algorithm, we compared the present results with the benchmark solutions of Hassanien et al. [2] for the case of zero capsule volume fraction (ϕ = 0.0). Table 1 reports the values of f ′′(0) from the literature and the present computations together with the absolute and relative deviations. The absolute differences between the present results and Hassanien et al. are 3.0×10−5, 2.0×10−5, and 1.0×10−5 for n = 0.3, n = 0.5 and n = 0.8, respectively, which correspond to relative deviations of approximately 0.0020%, 0.0017% and 0.0010%. Such discrepancies are negligible and lie well within typical numerical round-off and truncation errors for boundary-layer similarity solutions. The small deviations can be attributed to minor differences in numerical im- plementation: (i) the finite truncation of the computational domain used to approximate the far-field boundary, (ii) tolerance criteria for the shooting/Newton iterations, and (iii) interpolation/rounding conventions when reporting tabulated results. Table 1: The values f ′′(0) in comparison to results from the literature. n Present work [2] 0.3 1.48680 1.48677 0.5 1.16526 1.16524 0.8 1.02884 1.02883 Currently, no experimental data are available for the specific configuration of a power- law fluid containing NEPCM capsules over a non-isothermal stretching sheet; hence, direct experimental validation is not feasible. Nevertheless, the benchmark comparison confirms the accuracy of the hydrodynamic solver for the limiting case ϕ = 0. The NEPCM- augmented model adopts constitutive and latent-heat formulations consistent with recent literature, though its underlying assumptions (e.g., single-phase approximation, capsule homogeneity) require future experimental verification. Accordingly, experimental valida- tion is recommended as a priority for subsequent investigations. L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 8 of 18 4. Results and Discussion Subsequent numerical analyses are conducted using dimensionless variables across the defined parameter ranges: NEPCM volume fraction (0.0 ≤ ϕ ≤ 0.05), power-law index (0 ≤ n ≤ 0.9), fusion temperature (0.1 ≤ Θf ≤ 0.3), and heating parameter (−1 ≤ γ ≤ 1). The Prandtl and Stefan numbers are fixed at Pr = 6.2 and Ste = 0.2, respectively. (a) (b) f' Θ (c) Cr Figure 2: Impact of the power-law index on velocity (a), temperature (b) and heat capacity profile (c) at ϕ = 0.05, Θf = 0.3, Ste = 0.2 and γ = −1. Figure 2 shows the evolution of the velocity (a), temperature (b) and heat capacity profile (c) by varying the power-law index at ϕ = 0.05, Θf = 0.3, and γ = −1. The veloc- ity decreases markedly as n increases, indicating enhanced viscous resistance within the boundary layer. This trend reflects the dominance of pseudoplastic (shear-thinning) be- havior for n < 1, where effective viscosity increases with reduced shear rate. For reference, the opposite trend would occur in a shear-thickening fluid (n > 1). The observed sup- pression of velocity with increasing n demonstrates the rheological control of momentum transport. L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 9 of 18 The temperature field shows a reduction with higher n, due to weakened convec- tive motion and diminished thermal diffusion. At lower n values, thermal penetration is stronger, occasionally producing temperature overshoots near the plate—an outcome of more efficient energy redistribution in shear-thinning fluids. The corresponding shift of the heat capacity ratio profile toward the surface at larger n implies greater local energy storage in the near-wall region, consistent with enhanced viscous heating resistance. In practical terms, this behavior is crucial for non-Newtonian materials processing such as polymer extrusion and coating flows, where optimizing the rheological index allows regu- lation of both heat distribution and wall shear, thereby improving product uniformity and preventing overheating in viscous materials. (a) (b) f' Θ (c) Cr Figure 3: Impact of the nano encapsulated PCM concentration on velocity (a), temperature (b) and heat capacity profile (c) at n = 0.4, Θf = 0.3, and γ = −1. Figure 3 shows the evolution of the velocity (a), temperature (b) and heat capacity profile (c) by varying NEPCMs concentration at n = 0.4, Θf = 0.3, and γ = −1. In- creasing ϕ slightly elevates the fluid velocity, attributed to buoyancy-assisted momentum transfer from encapsulated particles. More notably, the temperature profile rises substan- L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 10 of 18 tially with higher NEPCM content, demonstrating the ability of the capsules to absorb and release latent heat during phase transition. The thermal boundary layer expands, enhancing heat storage and delaying thermal decay away from the wall. The observed behavior aligns with experimental and numerical studies such as Adel et al. [26], where NEPCM addition improved energy buffering capacity. The broaden- ing of the heat capacity ratio profile reflects more distributed heat absorption across the boundary layer. From an engineering perspective, increasing NEPCM concentration is beneficial for systems requiring temperature stabilization, such as in electronics cooling, thermal regulation of solar collectors, or battery thermal management where latent heat buffering mitigates temperature fluctuations under variable loads as reported by Sheik- holeslami [27]. (a) (b) f' Θ (c) Cr Figure 4: Impact of the fusion temperature on velocity (a), temperature (b) and heat capacity profile (c) at n = 0.4, ϕ = 0.05, and γ = −1. Figure 4 shows the evolution of the velocity (a), temperature (b) and heat capacity profile (c) by varying fusion temperature at n = 0.4, ϕ = 0.05, and γ = −1. The velocity remains essentially unchanged with variations in Θf , confirming that hydrodynamics are L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 11 of 18 dominated by rheology rather than thermal phase behavior. However, temperature and heat capacity ratio increase significantly with higher fusion temperature, as higher Θf delays melting and promotes greater latent heat release near the wall region. The shift of the heat capacity ratio curve toward the surface indicates localized energy storage, while its compression suggests a narrower effective melting zone. These behaviors are critical in applications such as heat exchangers or thermal energy storage tanks, where tuning the PCM fusion temperature can optimize the thermal response for specific operating ranges. (a) (b) f' Θ (c) Cr Figure 5: Impact of the temperature parameter on velocity (a), temperature (b) and heat capacity ratio (c) at n = 0.4, ϕ = 0.05, and Θf = 0.3. Figure 5 shows the evolution of the velocity (a), temperature (b) and heat capacity ratio (c) by varying temperature parameter at n = 0.4, ϕ = 0.05, and Θf = 0.3. The velocity profile remains almost unaffected by γ, underscoring the hydrodynamic robustness of power-law fluids under thermal perturbations. However, the temperature decreases under positive γ (heating) due to enhanced heat removal, while negative γ (cooling) leads to greater thermal accumulation. The proximity of the heat capacity ratio profile to the plate at higher γ indicates L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 12 of 18 that energy storage becomes concentrated near the surface during heating. The decou- pling between velocity and temperature responses implies that, in practical devices such as polymer-film stretchers or cooling channels, thermal performance can be controlled via surface heating conditions without disturbing flow stability—a desirable feature for precision manufacturing and thermal regulation systems. (a) (b) � � � -1 / (n + 1 ) N U x 2 � � � 1 / (n + 1 ) C f Figure 6: Impacts of the NEPCMs volume fraction on the Nusselt number (a) and the skin friction coefficient (b) versus the power law index for Θf = 0.3, γ = 0.0. Figures 6–8 present the combined influence of NEPCM concentration, power-law in- dex, fusion temperature, and heating parameter on the Nusselt number and skin friction coefficient. In Figure 6, increasing NEPCM concentration enhances the Nusselt number due to the latent heat contribution of encapsulated particles, while higher n values (shear- thinning behavior) promote a modest additional increase by improving thermal boundary layer structure. Conversely, skin friction rises with NEPCM concentration but decreases L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 13 of 18 with n, illustrating a competition between viscous thickening by particle addition and viscosity reduction through shear-thinning. The decline in skin friction from n = 0.4 to n = 0.7 suggests that the rheological effect dominates over the particle-induced drag enhancement. (a) (b) � � � -1 / (n + 1 ) N U x 2 � � � 1 / (n + 1 ) C f Figure 7: Impacts of the heating parameter on the Nusselt number (a) and the skin friction coefficient (b) versus the power law index for Θf = 0.3, n = 0.4. Figure 7 shows the effect of heating parameter on the local Nusselt number (a) and the skin friction coefficient (b) with respect to the power law index Θf = 0.3, n = 0.4. It shows that heating parameter γ primarily affects heat transfer while leaving skin fric- tion nearly unchanged, highlighting that thermal boundary manipulation alters energy transport without destabilizing flow, a feature desirable for thermally controlled surface processes. Figure 8 shows how fusion temperature affects (a) the local Nusselt number and (b) the L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 14 of 18 (a) (b) � � � -1 / (n + 1 ) N U x 2 � � � 1 / (n + 1 ) C f Figure 8: Impacts of the fusion temperature on the Nusselt number (a) and the skin friction coefficient (b) versus the heating parameter for ϕ = 0.05, n = 0.4. skin friction coefficient, when the power-law index is held constant at n = 0.4 and ϕ = 0.05. It reveals that higher fusion temperatures reduce the Nusselt number because phase change occurs less readily, limiting latent heat utilization. The interplay between γ and Θf also shows that excessive heating can counteract PCM storage benefits, underscoring the need for optimized material selection. This insight is valuable for the design of NEPCM- enhanced heat exchangers or microchannel coolers, where selecting the proper PCM fusion temperature is crucial to achieving efficient heat dissipation under varying operational loads. L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 15 of 18 5. Conclusions This study investigated the hydrodynamic and thermal behavior of power-law nanoflu- ids embedded with nano-encapsulated phase change materials (NEPCMs) over a non- isothermal stretching sheet. The governing boundary-layer equations were transformed into similarity form and solved numerically using the shooting method combined with Runge–Kutta integration and Newton iteration. The results revealed that increasing the power-law index markedly reduces the fluid velocity and suppresses the temperature dis- tribution, indicating that shear-thickening behavior enhances viscous resistance and sig- nificantly influences the flow dynamics. Furthermore, higher NEPCM concentrations were found to thicken the thermal boundary layer and elevate the heat capacity ratio, thereby improving the overall heat transfer performance, as evidenced by the increase in Nusselt numbers. The study also demonstrated that the fusion temperature and the heating parameter strongly affect the thermal field and heat storage capacity, while exerting a negligible impact on velocity profiles. This finding suggests a decoupling between the hydrodynamic and thermodynamic behaviors of NEPCM-based power-law fluids. Additionally, it was ob- served that the inclusion of NEPCMs enhances heat transfer but simultaneously increases wall shear stress. However, this effect is mitigated at higher power-law indices, indicating a competitive interaction between the rheological characteristics of the base fluid and the concentration of NEPCMs. Overall, this work provides new insights into the coupled rheological and thermal mechanisms governing NEPCM-laden non-Newtonian fluids. The findings contribute to the understanding of complex heat transfer processes in shear-dependent suspensions and are expected to inform the design and optimization of advanced thermal management sys- tems. Potential applications include energy storage units, polymer extrusion processes, and high-performance cooling systems, where improved heat transfer is required without compromising flow stability. Future research could extend the present analysis to un- steady, three-dimensional, and magnetohydrodynamic configurations to broaden the ap- plicability of NEPCM-based non-Newtonian suspensions in industrial and energy-related technologies. Data Availability The data that support the result of this study are available upon request from the authors. Conflict of Interests The author declares that there is no competing of interests in publishing this article. L. Deswita et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6958 16 of 18 Acknowledgements This research was funded by Daftar Isian Pelaksanaan Anggaran (DIPA) Universitas Riau Nomor SP DIPA-139.03.2.693425/2025 Date 02 December 2024. References [1] MH Cobble. Magnetohydrodynamic flow for a non-newtonian power-law fluid having a pressure gradient and fluid injection. Journal of Engineering Mathematics, 14(1):47– 55, 1980. [2] I.A. Hassanien, A.A. Abdullah, and R.S.R. Gorla. 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