Enhanced Thermal Performance of Variable-Density Maxwell Nanofluid Flow over a Stretching Sheet Under Viscous Dissipation: A Maritime Technology Perspective

Abstract

This scientific research examines the intricate dynamics of Maxwell nanofluid flow across a stretching surface with Stefan blowing impacts, with a particular focus on maritime thermal management applications. The analysis integrates multiple physical phenomena including magnetohydrodynamic forces, the energy dissipation phenomenon, and thermal density variations within Darcy porous media. Special attention is given to Stefan blowing’s role in modifying thermal and mass transfer boundary layers. We derive an enhanced mathematical formulation that couples Maxwell fluid properties with nanoparticle transport under combined magnetic and density-gradient conditions. Computational results demonstrate the crucial influence of viscous heating and blowing intensity on thermal performance, with direct implications for naval cooling applications. The reduced governing equations form a nonlinear system that requires robust numerical treatment. We implemented the shooting technique to solve this system, verifying its precision through systematic comparison with established benchmark solutions. The close correspondence between results confirms both the method’s reliability and our implementation’s accuracy. The primary results of this study indicate that raising the Stefan blowing and density parameters causes notable changes in the temperature and concentration fields. The Stefan blowing parameter enhances both temperature and concentration near the wall by affecting thermal diffusion and nanoparticle distribution. In contrast, the density parameter reduces these values because of increased fluid resistance.

1. Introduction

Maxwell nanofluids represent an innovative class of working fluids that integrate the viscoelastic characteristics of Maxwell materials with nanoparticle-enhanced thermal transport properties. These hybrid fluids show particular promise for maritime engineering applications [1], where they significantly enhance thermal management in vessel-cooling circuits, propulsion components, and offshore thermal exchange systems [2]. Their unique combination of shear resistance and thermal stability renders them exceptionally suitable for demanding marine operating conditions. As a distinct category of viscoelastic non-Newtonian fluids, Maxwell fluids have attracted significant research interest for their capacity to represent stress relaxation behavior in polymer processing and industrial applications. Their unique combination of elastic and viscous properties makes them particularly relevant for extrusion systems, lubricant formulations, and advanced thermal management solutions. Recent studies have advanced understanding of Maxwell fluid dynamics under various physical conditions. For instance, Afify and Elgazery [3] examined nanoparticle-enhanced Maxwell fluid flow past an elongating surface, analyzing coupled magnetohydrodynamic and chemical reaction effects on boundary layer development. Building upon previous work, Kumar and Reddy [4] investigated radiative heat transfer effects in chemically reactive Maxwell nanofluid flows, providing computational solutions for enhanced energy transport applications. Their subsequent research [5] systematically analyzed coupled thermal radiation and reaction kinetics in magnetohydrodynamic (MHD) flows over deformable surfaces, demonstrating electromagnetic regulation of thermal processes. Meanwhile, Al Rashdi et al. [6] implemented the Cattaneo-Christov heat flux formulation to study the MHD slip flow of Maxwell nanofluids through porous media along vertical substrates, overcoming the transient heat transfer constraints inherent in classical Fourier conduction models. Complementing these investigations, Khan et al. [7] numerically examined heat generation in porous media flows of Maxwell fluids using Cattaneo-Christov heat flux theory, offering critical insights for high-heat industrial applications. Their subsequent work (Khan et al. [8]) advanced the field by analyzing dual-diffusion mechanisms in exponentially stretching surface flows, with important implications for atmospheric and geological transport phenomena. Arif et al. [9] developed a stochastic framework for transient Maxwell nanofluid dynamics between moving boundaries, establishing new computational approaches for modeling uncertainty in adaptive fluid systems. Collectively, these contributions have substantially enhanced both fundamental knowledge and applied solutions for Maxwell fluid systems across multiple disciplines.

Stefan blowing describes mass transfer phenomena induced by surface phase transitions (evaporation/condensation), significantly affecting adjacent fluid dynamics and thermal transport. In nanofluid systems, this process critically modifies boundary layer development and can either intensify or diminish thermal and species transfer based on flow orientation. Practical implementations span diverse engineering fields, including industrial drying processes, advanced thermal barriers, aerospace thermal protection systems, and optimized surface coating techniques. Contemporary research by Manjunatha et al. [10] demonstrated these effects through analysis of chemically reactive nanofluid flow along curved deformable surfaces, revealing substantial impacts on coupled heat and mass transfer mechanisms. Recent investigations have demonstrated the significant role of Stefan blowing in various nanofluid flow configurations. Haider et al. [11] analyzed transient magnetohydrodynamic nanofluid flow subject to combined electrical and thermal fields, quantifying its effects on both thermal and momentum boundary layer development. Parallel work by Jyothi et al. [12] investigated Casson nanofluid dynamics around moving slender geometries, showing how Stefan blowing mechanisms can simultaneously decrease thermal gradients while improving flow regulation. More recently, Saleem and Hussain [13] studied nonlinear radiative transfer in Williamson-type nanofluids with magnetic effects, establishing new insights into thermal boundary layer manipulation through Stefan blowing phenomena. Konai et al. [14] examined transient Casson nanofluid dynamics with Stefan blowing, demonstrating its stabilizing effect on unsteady flows. Zhang et al. [15] further advanced this field by analyzing Jeffery nanofluids with volumetric heating, revealing how Stefan blowing, when coupled with radiative and convective heat transfer, substantially modifies thermofluidic characteristics. Together, these works highlight Stefan blowing’s critical importance in optimizing nanofluid thermal technologies.

Accounting for density variations, especially those influenced by temperature, is essential for precise modeling of non-Newtonian nanofluid dynamics under thermal gradients. This consideration is particularly crucial in marine applications, where rapid temperature fluctuations near submerged structures alter fluid characteristics, directly impacting buoyancy forces, flow patterns, and thermal efficiency in critical systems such as vessel-cooling circuits, hull coatings, and subsea propulsion units. Early work by Siddiqa et al. [16] established these effects through analysis of natural convection around circular geometries, demonstrating how thermal–density coupling modifies flow recirculation patterns. Recent advances have demonstrated the significance of coupled thermofluidic phenomena in nanofluid applications. Ullah et al. [17] investigated chemically reactive and viscous dissipative nanofluid flows under magnetic fields, with practical implications for precision machining and advanced lubrication systems. Boukholda et al. [18] further developed this framework by integrating temperature-dependent density effects with heat sink configurations in magnetized nanofluids, specifically addressing thermal management challenges in microelectronics and marine cooling applications. Cutting-edge work by Haider et al. [19] examined the synergistic interaction between density variations and viscous heating in Ree-Eyring magneto-nanofluids, demonstrating improved thermal and mass transport performance under slip boundary conditions, findings that are particularly applicable to next-generation marine heat exchangers and adaptive thermal coatings.

Despite significant advances in nanofluid research, the coupled influence of Stefan blowing, thermal-density variations, and viscous dissipation phenomenon on MHD transport of non-Newtonian Maxwell nanofluids through porous media remains underexplored. Our work addresses this critical knowledge gap by establishing an integrated computational framework that resolves the complex coupling between viscoelastic fluid behavior, heat transfer mechanisms, and mass diffusion processes in challenging marine and industrial applications.

2. Mathematical Modeling

The mathematical model that appears below in Figure 1, which accurately describes the two-dimensional, time-independent flow of an incompressible Maxwell nanofluid with density variations and non-Newtonian characteristics due to a stretching sheet surface, is governed by the following relations. The mass conservation principle is maintained through the continuity equation, formulated as follows:

$${\displaystyle \frac{\partial }{\partial x}}\left(\overline{\rho }u\right)+{\displaystyle \frac{\partial }{\partial y}}\left(\overline{\rho }v\right)=0,$$

(1)

where u and v are the velocity vector components in the x- and y-directions, respectively, and $\overline{\rho }$ represents the temperature-dependent fluid density. The momentum equation subsequently accounts for both non-Newtonian effects characteristic of Maxwell fluids and externally applied forces, including porous media drag impacts and the magnetic field interactions within the model [20]:

$$\rho \left(u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}\right)+\rho {{\rm Y}}_0\left(v^2{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}+2uv{\displaystyle \frac{{\partial }^{2}u}{\partial x\partial y}}+u^2{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}\right)+\left(\sigma B_0^2+{\displaystyle \frac{\mu }{k}}\right)u=\mu \left({\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}\right),$$

(2)

where $\rho$ is the fluid density, which is assumed to be dependent on temperature in this model, $\mu$ is the dynamic viscosity, ${{\rm Y}}_0$ denotes the fluid relaxation time parameter, which has a unit of time that is typically expressed in seconds s, $\sigma$ is the electrical conductivity, k represents the permeability of the porous medium and $B_0$ is the applied magnetic field strength. Likewise, the energy equation that accounts for the heat conduction mechanism, Brownian motion properties, thermophoresis phenomenon and viscous dissipation takes the following form:

$$\begin{array}{cc}\hfill v{\displaystyle \frac{\partial T}{\partial y}}-\tau {\displaystyle \frac{\partial T}{\partial y}}\left({\displaystyle \frac{D_T}{T_{\infty }}}{\displaystyle \frac{\partial T}{\partial y}}+D_B{\displaystyle \frac{\partial C}{\partial y}}\right)& ={\displaystyle \frac{\kappa }{\rho c_p}}\left({\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}\right)-u{\displaystyle \frac{\partial T}{\partial x}}+\hfill \\ & {\displaystyle \frac{1}{\rho c_p}}\left\{\mu {\left({\displaystyle \frac{\partial u}{\partial y}}\right)}^2-\rho {{\rm Y}}_0\left[2uv{\displaystyle \frac{\partial u}{\partial x}}{\displaystyle \frac{\partial u}{\partial y}}+v^2{\left({\displaystyle \frac{\partial u}{\partial y}}\right)}^2\right]\right\},\hfill \end{array}$$

(3)

where T is the temperature of the Maxwell nanofluid, $D_B$ and $D_T$ are the thermophoretic and Brownian diffusion coefficients, respectively, $\kappa$ is the thermal conductivity of the fluid, $\tau$ represents the ratio of nanoparticle heat capacity to that of the base fluid and $c_p$ is the specific heat at constant pressure. Finally, the concentration equation incorporating nanoparticle diffusion and thermophoretic effects takes the following form:

$$\begin{array}{c}\hfill u{\displaystyle \frac{\partial C}{\partial x}}+v{\displaystyle \frac{\partial C}{\partial y}}=D_B{\displaystyle \frac{{\partial }^{2}C}{\partial y^2}}+{\displaystyle \frac{D_T}{T_{\infty }}}{\displaystyle \frac{{\partial }^{2}T}{\partial y^2}},\end{array}$$

(4)

where C is the nanoparticle concentration. This study introduces a significant advancement by systematically incorporating temperature-sensitive thermophysical properties, particularly density ($\rho \left(T\right)$), into the Maxwell nanofluid framework, as expressed in the following mathematical formulation [21]:

$$\overline{\rho }=e^{-\lambda \left(\frac{T-T_{\infty }}{T_w-T_{\infty }}\right)},\rho \left(T\right)={\rho }_c\overline{\rho },$$

(5)

where $\lambda$ is the density factor, $T_{\infty }$ is the constant ambient temperature, $T_w$ is the temperature of the nanofluid along the sheet and ${\rho }_c$ is a constant density. To accurately characterize the status of the system and ensure that the governing Equations (1)–(4) are properly implemented within the model framework, it is crucial to specify appropriate physical boundary conditions. The relevant conditions apply to both the stretching sheet surface and the far-field fluid region. Notably, Stefan blowing/suction [22], which is taken into consideration through this model, is a phase-change-induced mass transfer mechanism that significantly enhances thermal and mass transport efficiency in applications like thermal insulation, maritime applications, membrane separation processes, and aerospace thermal management systems.

$$\begin{array}{c}\hfill u=cx,T=T_w,v=-D_B\left({\displaystyle \frac{1}{1-C_w}}\right)\left({\displaystyle \frac{\partial C}{\partial y}}\right),C=C_w,aty=0,\end{array}$$

(6)

$$C\to C_{\infty },T\to T_{\infty },u\to 0,aty\to \infty ,$$

(7)

where c is a positive constant, $C_{\infty }$ is the nanofluid concentration far away from the sheet and $C_w$ is the nanofluid concentration along the sheet.

Non-Dimensionalized Governing Equations

To enhance computational efficiency, key system variables are identified and governing equations are reduced to dimensionless form. This approach simplifies analysis while preserving essential physics, clarifies the influence of dimensionless control parameters, and provides deeper insight into the system’s dynamics and its heat and mass mechanism [23]:

$$v=-{\displaystyle \frac{1}{\overline{\rho }}}{\displaystyle \frac{\partial \Psi }{\partial x}},u={\displaystyle \frac{1}{\overline{\rho }}}{\displaystyle \frac{\partial \Psi }{\partial y}},\Psi =\sqrt{c\nu }xf\left(\eta \right),$$

(8)

$$\theta \left(\eta \right)={\displaystyle \frac{T_{\infty }-T}{T_{\infty }-T_w}},\varphi \left(\eta \right)={\displaystyle \frac{C_{\infty }-C}{C_{\infty }-C_w}},\eta =\sqrt{{\displaystyle \frac{c}{\nu }}}{\int }_0^y\overline{\rho }dy.$$

(9)

The converted forms of Equations (1)–(4), which were generated through substitution of relations (8) and (9), inherently satisfy mass conservation (1). The following analytical approach transforms these into coupled nonlinear ordinary differential equations governing momentum, species transport, and thermal energy, producing the following key results:

$$e^{-\lambda \theta }\left(f^{‴}-\lambda f^{″}{\theta }^{\prime }\right)+f{f}^{″}-f^{\prime 2}+{\rm Y}f\left[\left(2f^{\prime }+\lambda f{\theta }^{\prime }\right)f^{″}-f{f}^{‴}\right]-e^{\lambda \theta }\left(\Lambda +M\right)f^{\prime }=0,$$

(10)

$${\displaystyle \frac{e^{-\lambda \theta }}{Pr}}\left({\theta }^{″}-\lambda {\theta }^{\prime 2}\right)+{\theta }^{\prime }f+\left({\displaystyle \frac{{\Delta }_t{\theta }^{\prime 2}+{\Delta }_b{\varphi }^{\prime }{\theta }^{\prime }}{e^{2\lambda \theta }}}\right)+Ec\left\{e^{-\lambda \theta }{f}^{″}+{\rm Y}\left(2f^{\prime 2}-f{f}^{″}\right)f\right\}{f}^{″}=0,$$

(11)

$${\varphi }^{″}+{\displaystyle \frac{{\Delta }_t}{{\Delta }_b}}{\theta }^{″}+S_{c}f{\varphi }^{\prime }{e}^{2\lambda \theta }-\lambda \left({\theta }^{\prime }{\varphi }^{\prime }+{\displaystyle \frac{{\Delta }_t}{{\Delta }_b}}{\theta }^{″}\right)=0.$$

(12)

To comply with the established specifications, the relevant boundary conditions are adjusted accordingly. This adaptation ensures methodological precision and consistency by verifying full compliance with all requirements. The following specific criteria must be satisfied:

$$f\left(0\right)=\left({\displaystyle \frac{\Omega }{S_c}}\right)e^{-2\lambda }{\varphi }^{\prime }\left(0\right),f^{\prime }\left(0\right)=1\varphi \left(0\right)=1,\theta \left(0\right)=1,$$

(13)

$$f^{\prime }\to 0,\theta \to 0,\varphi \to 0,\mathrm{as}\eta \to \infty .$$

(14)

Clearly, the governing parameters are defined as follows. Each plays a critical role in determining the system’s performance and outcomes.

$$\Omega =\left({\displaystyle \frac{C_w-C_{\infty }}{1-C_w}}\right),S_c={\displaystyle \frac{\nu }{D_B}},Ec={\displaystyle \frac{u_w^2}{c_p\left(T_w-T_{\infty }\right)}},{\Delta }_b={\displaystyle \frac{\tau \left(C_w-C_{\infty }\right)D_B}{\nu }},$$

(15)

$$M={\displaystyle \frac{\sigma B_0^2}{{\rho }_{c}a}},{\Delta }_t=={\displaystyle \frac{D_T\left(T_w-T_{\infty }\right)\tau }{T_{\infty }\nu }},{\rm Y}=c{{\rm Y}}_0,\Lambda ={\displaystyle \frac{\mu }{kc{\rho }_c}},Pr={\displaystyle \frac{\mu c_p}{\kappa }},$$

(16)

The previously introduced dimensionless parameters that characterize the system’s physical behavior are the Stefan blowing coefficient $\Omega$, the Schmidt number $S_c$, the Eckert number $Ec$, the Brownian motion parameter ${\Delta }_b$, the magnetic field parameter M, the thermophoresis parameter ${\Delta }_t$, the Maxwell parameter ${\rm Y}$, the porous parameter $\Lambda$ and the Prandtl number Pr. The proposed physical model incorporates three key dimensionless parameters with broad industrial applications. These are the local skin friction coefficient $C{f}_x$, Sherwood number $S{h}_x$, and Nusselt number $N{u}_x$. Their physical significance and practical relevance will be elucidated through the following analysis:

$$C{f}_{x}R{e}^{\frac{1}{2}}=-\left[f^{″}\left(0\right)-{\rm Y}{e}^{\lambda }\left(-2f\left(0\right)f^{\prime 2}\left(0\right)+f^2\left(0\right)f^{″}\left(0\right)\right)\right],$$

(17)

$$N{u}_{x}R{e}^{\frac{-1}{2}}=-{\theta }^{\prime }\left(0\right),S{h}_{x}R{e}^{\frac{-1}{2}}=-{\varphi }^{\prime }\left(0\right).$$

(18)

3. Limitations and Future Work

While this study provides foundational insights through a simplified model, assuming steady laminar flow, ideal nanoparticle distribution, and Darcy-type porous media, we recognize that real-world maritime systems operate under more complex conditions. Phenomena such as turbulence, particle aggregation, thermal radiation, and chemical reactivity were beyond the scope of the current work but represent critical factors for future investigation. To bridge the gap between theory and practice, subsequent research should integrate these multifaceted physics and emphasize experimental validation using seawater-based nanofluids. A particularly promising direction is the optimization of Stefan blowing intensity to actively control boundary layer dynamics and achieve breakthrough thermal performance in marine heat management systems.

4. Validating the Code

Previous research by Abel et al. [24] examined the problem of magnetohydrodynamic flow and the thermal transport mechanism in Maxwell fluids under specialized boundary conditions. Their work represents a special case of the more comprehensive theoretical framework developed in our current investigation. When simplifying our governing equations (by setting $\Lambda =\Omega =\lambda =0$) and parameters to align with their study, we obtain identical numerical results (see

Table 1. $-f^{″}\left(0\right)$ values in varying both M and ${\rm Y}$ with $\Lambda =\Omega =\lambda =0$.

${\rm Y}$	M	Abel et al. [24]	Present Work
0.0	0.0	0.999962	0.99999988998
0.2	0.0	1.051948	1.05194770096
0.4	0.0	1.101850	1.10184899072
0.6	0.0	1.150163	1.15016285014
0.0	0.2	1.095445	1.09544488902
0.2	0.2	1.188270	1.18826890258
0.4	0.2	1.275878	1.27587777031
0.6	0.2	1.358733	1.35873288095

). In addition, a validation exercise was performed against the findings of Nadeem et al. [25], as reported in

Table 2. $-{\theta }^{\prime }\left(0\right)$ values in varying both ${\Delta }_t$ with $Pr=10$, ${\Delta }_b=0.1$ and $\gamma =\Lambda =M=\Omega =Ec=\lambda =0$.

${\Delta }_{\mathit{t}}$	Nadeem et al. [25]	Present Work
0.1	0.9524	0.952377098
0.2	0.6932	0.693189907
0.3	0.5201	0.520055911
0.4	0.4026	0.402588093
0.5	0.3211	0.321088907

, by evaluating the Nusselt number $-{\theta }^{\prime }\left(0\right)$ for various values of ${\Delta }_t$ under conditions $Pr=10$, ${\Delta }_b=0.1$ and $\gamma =\Lambda =M=\Omega =Ec=\lambda =0$. This precise agreement fulfills two key objectives, confirms the reliability of our model, and illustrates its ability to extend and encompass previously established solutions in the literature.

5. Interpretation of Results

This section explores the Maxwell nanofluid model, examining how changes in its governing parameters affect the flow behavior, heat transfer, and mass diffusion. The findings demonstrate that parameters like Stefan blowing intensity, magnetic field strength, porosity of the medium, fluid density, and viscous dissipation phenomenon significantly influence the velocity $f^{\prime }\left(\eta \right)$, temperature $\theta \left(\eta \right)$, and concentration $\varphi \left(\eta \right)$ profiles within the boundary layer. The parameter ranges selected in this study are grounded in the thermophysical properties of seawater-based nanofluids, ensuring direct relevance to maritime thermal management. The Prandtl number ($Pr=5.0$) aligns with typical seawater values (5–8) at moderate operating temperatures. The Schmidt number ($Sc=2.0$) reflects realistic diffusion dynamics for dissolved salts and engineered nanoparticles (commonly ≈1–3). Similarly, the Eckert number ($Ec=0.2$) captures moderate viscous dissipation effects consistent with practical maritime flow conditions (often ≪1, with 0.1–0.3 being widely adopted). These deliberate, physically justified choices bridge theoretical modeling and real-world oceanic cooling applications, enhancing the practical significance of the results. Therefore, the selected parameters are intentionally aligned with the thermophysical characteristics of seawater nanofluids, ensuring that our model accurately reflects real-world maritime thermal management challenges. By anchoring our simulation in empirically grounded values, we bridge theoretical fluid dynamics with practical ocean engineering, enhancing the framework’s immediate applicability to advanced cooling technologies in marine environments. This section commences with an investigation into the Stefan blowing parameter’s $\Omega$ effect on flow behavior, as shown in Figure 2. To clarify, the Stefan blowing parameter $\Omega$ measures the intensity of fluid injection from the surface into the boundary layer, acting as a mechanism for mass addition or “blowing” from the wall. From a physical perspective, this parameter quantifies the fluid injection rate normal to the elongating surface. When $\Omega$ rises, a modest velocity profile $f^{\prime }\left(\eta \right)$ augmentation becomes apparent. This minor enhancement stems from the injected fluid imparting additional momentum near the boundary, though the stretching mechanism remains the primary flow driver. In contrast, the temperature and concentration distributions exhibit more pronounced changes compared to velocity. The introduced fluid traps thermal energy and nanoparticles near the surface, expanding both the thermal and concentration boundary layers. Consequently, this impedes heat and mass dissipation from the wall region, elevating localized temperature and particle concentration levels.

Figure 3 makes it clear that the porous parameter $\Lambda$ distinctly affects the velocity $f^{\prime }\left(\eta \right)$, temperature $\theta \left(\eta \right)$, and concentration $\varphi \left(\eta \right)$ profiles. An increase in $\Lambda$ physically translates to more flow resistance within the porous medium. This results in greater frictional drag and, consequently, a decreased velocity profile. Further, the same resistance that hinders flow also keeps heat and nanoparticles from escaping the sheet surface. This confinement makes the thermal and solutal layers thicker, which in turn elevates temperature and concentration levels close to the surface. The reason for this dual impact is that porous structures hinder fluid flow while also confining the spread of energy and particles. Consequently, the porous parameter $\Lambda$ is a vital factor in systems involving nanofluids within a porous medium.

As Figure 4 shows, the Maxwell parameter ${\rm Y}$ affects $\theta \left(\eta \right)$, $\varphi \left(\eta \right)$ and $f^{\prime }\left(\eta \right)$ profiles in distinct ways because of its viscoelastic nature. A rise in ${\rm Y}$ enhances the fluid’s elastic memory, which creates more internal resistance. This, in turn, slows down momentum transfer and reduces the velocity distribution. The fluid’s viscoelastic nature has a dual effect, in which it marginally improves the thermal boundary layer while notably expanding the solutal boundary layer. By restricting energy dispersal and nanoparticle diffusion, it traps heat and concentrates particles, which results in a slight increase in temperature and a more pronounced rise in concentration near the surface.

Figure 5 illustrates that the magnetic field parameter M uniquely impacts the flow field in terms of $\theta \left(\eta \right)$, $\varphi \left(\eta \right)$ and $f^{\prime }\left(\eta \right)$. As M rises, the magnetic field generates a Lorentz force that opposes fluid motion. This opposition boosts hydrodynamic resistance, causing a suppressor in the velocity profile. On the other hand, the magnetic field’s inhibition of fluid motion causes the thermal and solutal boundary layers to thicken. By preventing fluid from moving freely, it confines heat energy and nanoparticles to the area near the surface, leading to higher temperatures and concentration levels. The physical reason for this phenomenon is that magnetic effects impede momentum transfer while simultaneously limiting heat and mass transport by restricting fluid circulation. This makes this parameter a key factor, especially in magnetohydrodynamic nanofluid systems where the goal is to electromagnetically control heat and mass transfer.

In Figure 6, we notice that the density parameter $\lambda$ can significantly influence the flow dynamics in the form of $f^{\prime }\left(\eta \right)$ and transport properties in the form of both $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ for the Maxwell nanofluid model. From a physical standpoint, a higher density factor corresponds to a more compact fluid structure, which increases viscous resistance and helps to hinder fluid movement. This increased inertia results in a distinct decline in velocity distribution, since the denser medium responds more sluggishly to the stretching boundary. Simultaneously, this heightened resistance restricts nanoparticle movement, reducing their concentration near the sheet surface as mass transfer becomes less efficient. Additionally, the temperature distribution $\theta \left(\eta \right)$ also diminishes as the same parameter $\lambda$ increases, because denser fluid layers demonstrate lower thermal reactivity and slower heat diffusion, resulting in less efficient thermal energy transfer.

The graph in Figure 7 shows the influence of the Brownian motion parameter ${\Delta }_b$ on the dimensionless temperature $\theta \left(\eta \right)$ and concentration $\varphi \left(\eta \right)$ profiles. When $\Delta$ increases (specifically from $0.2$ to $0.5$), we observe a rise in the temperature profile and a corresponding decrease in the concentration profile. Physically, Brownian motion improves thermal conductivity by increasing nanoparticle movement. This agitation disrupts the thermal boundary layer, enhancing heat transfer. A higher intensity of random particle motion leads to greater energy exchange and, consequently, higher temperatures near the surface. Further, enhancing the same parameter can promote nanoparticle dispersion away from the surface, leading to a lower concentration, especially near the sheet, due to increased randomness.

As Figure 8 illustrates, the thermophoresis parameter ${\Delta }_t$ notably expands both the thermal $\theta \left(\eta \right)$ and concentration $\varphi \left(\eta \right)$ boundary layers. However, its impact is more substantial on mass transfer than on heat transfer. Therefore, as ${\Delta }_t$ increases, the temperature distribution $\theta \left(\eta \right)$ shows a moderate rise due to the migration of nanoparticles from hotter to cooler regions, which redistributes thermal energy. However, the impact on concentration $\varphi \left(\eta \right)$ is stronger, as thermophoresis directly drives nanoparticle movement, leading to a more noticeable accumulation in the boundary layer. This trend is very evident in the figure, where adjustments made with constant parameters cause a slight increase in heat but exert a stronger impact on concentration profiles. Finally, the computed velocity, temperature, and concentration profiles are directly correlated with critical maritime performance metrics, including thermal enhancement efficiency and hydrodynamic resistance, demonstrating the model’s immediate relevance to shipboard cooling systems. By translating numerical results into tangible engineering insights, this work provides a practical framework for optimizing thermal management in marine vessels, from naval ships to commercial carriers.

At this step and from

Table 3. $\frac{S{h}_x}{\sqrt{R{e}_x}}$, $\frac{N{u}_x}{\sqrt{R{e}_x}}$ and $C{f}_{x}R{e}_x^{\frac{1}{2}}$ values under impact of controlling factors with $S_c=2.0,Pr=5.0$ and $Ec=0.2$.

$\Omega$	$\Lambda$	${\rm Y}$	M	$\mathit{\lambda }$	${\Delta }_{\mathit{b}}$	${\Delta }_{\mathit{t}}$	${\mathit{Cf}}_{\mathit{x}}{\mathit{Re}}_{\mathit{x}}^{\frac{1}{2}}$	$\frac{{\mathit{Nu}}_{\mathit{x}}}{\sqrt{{\mathit{Re}}_{\mathit{x}}}}$	$\frac{{\mathit{Sh}}_{\mathit{x}}}{\sqrt{{\mathit{Re}}_{\mathit{x}}}}$
0.0	0.5	0.3	0.5	0.3	0.3	0.1	1.85486	0.361815	0.962961
0.4	0.5	0.3	0.5	0.3	0.3	0.1	1.82521	0.164096	0.825509
0.6	0.5	0.3	0.5	0.3	0.3	0.1	1.81293	0.092607	0.770876
0.4	0.0	0.3	0.5	0.3	0.3	0.1	1.56683	0.307311	0.859341
0.4	0.5	0.3	0.5	0.3	0.3	0.1	1.82521	0.164096	0.825509
0.4	1.0	0.3	0.5	0.3	0.3	0.1	2.05324	−0.02453	0.807554
0.4	0.5	0.0	0.5	0.3	0.3	0.1	1.73356	0.181274	0.834812
0.4	0.5	0.8	0.5	0.3	0.3	0.1	1.97355	0.134959	0.811285
0.4	0.5	1.3	0.5	0.3	0.3	0.1	2.11693	0.104591	0.800538
0.4	0.5	0.3	0.0	0.3	0.3	0.1	1.56683	0.307295	0.859341
0.4	0.5	0.3	0.5	0.3	0.3	0.1	1.82521	0.164096	0.825509
0.4	0.5	0.3	1.0	0.3	0.3	0.1	2.05325	0.024545	0.807498
0.4	0.5	0.3	0.5	0.0	0.3	0.1	1.43928	0.103147	0.675692
0.4	0.5	0.3	0.5	0.4	0.3	0.1	1.97875	0.180201	0.891299
0.4	0.5	0.3	0.5	0.7	0.3	0.1	2.53277	0.209633	1.152970
0.4	0.5	0.3	0.5	0.3	0.2	0.1	1.82901	0.291815	0.788805
0.4	0.5	0.3	0.5	0.3	0.3	0.1	1.82521	0.164096	0.825509
0.4	0.5	0.3	0.5	0.3	0.5	0.1	1.82012	0.004423	0.849231
0.4	0.5	0.3	0.5	0.3	0.3	0.0	1.82682	0.244082	0.816055
0.4	0.5	0.3	0.5	0.3	0.3	0.1	1.82521	0.164096	0.825509
0.4	0.5	0.3	0.5	0.3	0.3	0.2	1.82289	0.088690	0.840912

, we observe that the obtained numerical data clearly highlight how different physical and operational parameters significantly impact key engineering quantities, namely the skin friction coefficient $C{f}_{x}R{e}_x^{\frac{1}{2}}$, the Nusselt number $\frac{N{u}_x}{\sqrt{R{e}_x}}$, and the Sherwood number $\frac{S{h}_x}{\sqrt{R{e}_x}}$. Variations in these governing dimensionless quantities offer valuable insights into the mechanism of velocity gradients, heat transfer rates, and mass diffusion along the sheet surface. Elevating the porous medium parameter $\Lambda$ intensifies the skin friction coefficient while weakening the thermal and concentration boundary layers. Conversely, an increase in the Stefan blowing parameter $\Omega$ leads to a noticeable reduction in skin friction force but causes a noticeable decline in both heat and mass transfer rates, as reflected by the lowered Nusselt and Sherwood numbers, highlighting the damping influence of the blowing mechanism. Likewise, increasing the Maxwell parameter ${\rm Y}$ leads to more resistance, which boosts the value of skin friction while simultaneously lowering both heat and mass transfer. At the same time, the magnetic field parameter M has the opposite effect on transfer rates, intensifying skin friction but diminishing thermal and solutal transport due to the Lorentz force. A substantial rise in the density parameter $\lambda$ significantly improves momentum, heat, and the mass transfer mechanism, as indicated by the increased skin friction and Nusselt and Sherwood numbers. Also, a rise in the thermophoresis parameter ${\Delta }_t$ diminishes heat transfer but enhances mass transfer. This implies that particle movement driven by temperature gradients promotes greater species diffusion. Additionally, the Brownian motion parameter ${\Delta }_b$ only slightly lowers skin friction; it leads to a reduction in the Nusselt number and a small increase in the Sherwood number.

6. Concluding Remarks

This work provides a thorough theoretical study of dissipative Maxwell nanofluid dynamics across a stretching surface, accounting for Stefan blowing, magnetic forces, and porous medium impacts. The formulated model simultaneously considers density-dependent thermal variations. The nonlinear governing equations were successfully resolved through a shooting technique, yielding solutions that closely match established reference data. This validation underscores the reliability and effectiveness of the methodology for marine thermal regulation systems. The study yielded the following core conclusions:

• Raising the Stefan blowing parameter reduces the wall shear stress and substantially reduces heat and mass transfer, highlighting its suppressive impact on momentum, thermal, and concentration exchange along the sheet.
• The porous medium and Maxwell and magnetic field parameters each boost flow resistance, which in turn increases skin friction. However, they also hinder heat and mass transfer by thickening the thermal and concentration boundary layers and through the influence of fluid elasticity and the Lorentz force.
• When Brownian motion intensifies, it leads to better heat transfer and higher surface temperatures. This also results in a lower nanoparticle concentration because of enhanced diffusion.
• A higher density parameter diminishes velocity, temperature, and concentration profiles. This occurs because increased fluid inertia weakens the transport of momentum, heat, and mass close to the stretching surface.
• A rise in the Stefan blowing parameter subtly boosts velocity but considerably increases temperature and concentration profiles. This happens because it traps heat and nanoparticles closer to the surface, making the thermal and solutal boundary layers thicker.
• Increasing the thermophoresis parameter significantly thickens the concentration boundary layer and moderately enhances the temperature profile.