Casson-Maxwell Nanofluid Dynamics

• Casson-Maxwell nanofluid is a blood-based ternary suspension that combines yield-stress and shear-thinning behavior with Maxwell viscoelastic relaxation.
• The model employs two-dimensional boundary-layer equations with magnetohydrodynamic forcing, thermal radiation, and a linear heat source, solved via similarity transformation and numerical methods.
• Practical insights reveal that viscoelastic relaxation and nanoparticle composition critically affect drag reduction and heat transfer efficiency.

A Casson-Maxwell nanofluid is a nanoparticle-laden non-Newtonian fluid whose rheology combines Casson yield-stress/shear-thinning behavior with Maxwell viscoelastic relaxation. In the cited arXiv literature, the explicit Casson-Maxwell formulation is a blood-based ternary nanofluid containing silver, copper, and aluminium oxide nanoparticles, studied under magnetohydrodynamic forcing, thermal radiation, and a linear heat source in a steady two-dimensional boundary-layer model that idealizes a stenosed artery as a stretching surface (Shivakumar et al., 8 Jul 2025). The term requires terminological care: several nearby nanofluid studies employ Casson rheology, Maxwell-type effective medium formulas, or electromagnetic effects, yet do not constitute Maxwell-fluid models in the viscoelastic sense (K et al., 2021).

1. Constitutive structure

The defining feature of a Casson-Maxwell nanofluid is the coexistence of two distinct non-Newtonian mechanisms. The Casson component introduces yield-stress and plastic-viscosity effects through a piecewise stress law involving the yield stress $p_y$, the plastic dynamic viscosity $\mu_b$, and a critical deformation measure. In the operational form used after transformation, Casson behavior enters through the factor

$\left(1+\frac{1}{\beta}\right),$

where $\beta$ is the Casson parameter. The Maxwell component introduces viscoelastic memory through the constitutive relation

$$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$

with relaxation time $\lambda$, viscosity coefficient $\eta$, and deformation-rate tensor $d_{ij}$. The formulation also invokes the upper-convected derivative, described as satisfying frame indifference or material objectivity (Shivakumar et al., 8 Jul 2025).

This combination means that the fluid is neither purely Casson nor purely Maxwell. The Casson term modulates viscous-yield response, whereas the Maxwell term contributes relaxation-dependent resistance to deformation. In the transformed momentum equation, these effects remain explicitly separated: the Casson contribution appears as a viscous multiplier and the Maxwell contribution appears as a nonlinear viscoelastic correction proportional to $\lambda$. This separation is central to interpreting reported trends, especially when comparing Casson-Maxwell flow against Casson-only behavior.

2. Governing transport model

The cited Casson-Maxwell formulation begins from the incompressible two-dimensional continuity, momentum, and energy equations. For velocity components $u(x,y)$ and $\mu_b$0, the continuity equation is

$\mu_b$1

The boundary-layer momentum equation includes Casson diffusion, Maxwell viscoelasticity, and MHD braking: $\mu_b$2 The energy equation contains thermal diffusion, radiative heat flux, Joule-heating or magnetic-dissipation effects, and a linear heat source: $\mu_b$3 Radiation is modeled with the Rosseland approximation,

$\mu_b$4

followed by the linearization

$\mu_b$5

This procedure augments the effective thermal diffusion term (Shivakumar et al., 8 Jul 2025).

The wall is stretched with velocity $\mu_b$6, and the thermal and kinematic boundary conditions are

$\mu_b$7

$\mu_b$8

After the similarity transformation

$\mu_b$9

the system reduces to coupled nonlinear ordinary differential equations: $\left(1+\frac{1}{\beta}\right),$0

$\left(1+\frac{1}{\beta}\right),$1

The transformed boundary conditions are

$\left(1+\frac{1}{\beta}\right),$2

$\left(1+\frac{1}{\beta}\right),$3

A notable modeling choice is that no separate nanoparticle concentration equation is solved; the nanoparticle fractions are fixed material parameters rather than dynamic species fields (Shivakumar et al., 8 Jul 2025).

3. Nanoparticle modeling and material systems

In the explicit Casson-Maxwell study, the suspension is a ternary or tri-hybrid nanofluid with blood as base fluid and $\left(1+\frac{1}{\beta}\right),$4, $\left(1+\frac{1}{\beta}\right),$5, and $\left(1+\frac{1}{\beta}\right),$6 as nanoparticle species. Its effective viscosity, density, heat capacitance, thermal conductivity, and electrical conductivity enter through nondimensional ratios $\left(1+\frac{1}{\beta}\right),$7, constructed from ternary nanofluid properties (Shivakumar et al., 8 Jul 2025). The accessible text makes clear that these ratios correspond to $\left(1+\frac{1}{\beta}\right),$8, $\left(1+\frac{1}{\beta}\right),$9, $\beta$0, $\beta$1, and $\beta$2, but the exact algebraic mixture rules are not given clearly.

Adjacent literature shows how nanoparticle property closures are often built in closely related Casson nanofluid models. One peristaltic CuO-water Casson study uses Hamilton-Crosser or Maxwell-Garnett-type conductivity relations to encode shape-dependent thermal transport, with needle particles characterized by sphericity $\beta$3 and $\beta$4, and lamina particles by sphericity $\beta$5 and $\beta$6; that study concludes that lamina-shaped nanoparticles have higher thermal conductivity than needle-shaped nanoparticles (K et al., 2021). A stenosed-artery Casson hybrid nanofluid study uses explicit effective-property formulas for $\beta$7- and $\beta$8-blood suspensions, including Maxwell-type thermal and electrical conductivity relations for nanofluid and hybrid nanofluid mixtures (Muqaddass et al., 16 Apr 2025).

These neighboring formulations are relevant because they establish the thermophysical modeling context in which Casson-Maxwell nanofluid studies operate. They do not, however, supply Maxwell viscoelastic rheology by themselves. In this literature, “Maxwell” may refer either to a constitutive law with relaxation time or to Maxwell-type effective medium formulas; only the former defines a Maxwell fluid.

4. Numerical treatment, surrogate prediction, and sensitivity analysis

The Casson-Maxwell boundary-value problem is solved numerically in MATLAB with bvp4c. The dependent variables are rewritten as

$\beta$9

leading to a first-order system with far-field conditions on $$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$0 and $$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$1. The numerical treatment emphasizes an initial guess satisfying the boundary conditions, adaptive mesh refinement, and the suitability of bvp4c for stiff nonlinear boundary-value problems. Validation is performed by comparing $$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$2 for several $$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$3 values under a parameter-free limit against earlier literature, and the reported agreement is close (Shivakumar et al., 8 Jul 2025).

A second computational layer is added through artificial neural network forecasting of the heat-transfer rate. The ANN input variables are the radiation parameter $$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$4, heat source $$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$5, Maxwell parameter $$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$6, Casson parameter $$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$7, magnetic parameter $$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$8, and the three nanoparticle volume fractions $$\left(1+\lambda \frac{d}{dt}\right)\tau_{ij}=2\eta d_{ij},$$9; the output is the Nusselt number $\lambda$0. Training uses the Levenberg-Marquardt backpropagation algorithm with a data partition of $\lambda$1 training, $\lambda$2 validation, and $\lambda$3 testing, corresponding in the reported run to 16, 4, and 4 target timesteps. The performance measures are mean squared error, coefficient of determination, and percentage error. The reported ANN performance is an overall $\lambda$4, best validation performance $\lambda$5 at the 4th epoch, and validation regression $\lambda$6 (Shivakumar et al., 8 Jul 2025).

The same study also applies Response Surface Methodology to the drag coefficient, identified with $\lambda$7. Using $\lambda$8, $\lambda$9, and $\eta$0 as factors, the fitted quadratic model attains $\eta$1 and adjusted $\eta$2. Sensitivity derivatives are then written explicitly, and the study concludes that the drag coefficient is most sensitive to variations in the Maxwell parameter. This result elevates viscoelastic relaxation from a constitutive detail to a quantitatively dominant control variable in the model’s optimization framework (Shivakumar et al., 8 Jul 2025).

5. Transport behavior and wall observables

The principal wall quantities are the skin-friction coefficient and the Nusselt number. With wall shear stress

$\eta$3

the nondimensional skin-friction expression is

$\eta$4

With wall heat flux

$\eta$5

the heat-transfer rate becomes

$\eta$6

These expressions establish $\eta$7 as the drag or wall-shear indicator and $\eta$8 as the heat-transfer indicator (Shivakumar et al., 8 Jul 2025).

The reported hydrodynamic behavior is unambiguous. The Casson-Maxwell nanofluid has a lower velocity profile than the Casson fluid, and velocity decreases as the magnetic parameter $\eta$9, the Maxwell parameter $d_{ij}$0, and the Casson parameter $d_{ij}$1 increase. The interpretation given is that Lorentz-force resistance, viscoelastic relaxation, and yield-stress-related non-Newtonian resistance all suppress motion. In biomedical terms, this slower motion is linked to increased residence time in the stenosed region (Shivakumar et al., 8 Jul 2025).

Thermally, temperature rises with the heat-source parameter $d_{ij}$2 and the radiation parameter $d_{ij}$3. Heat-transfer behavior is more differentiated than the temperature field itself. The local Nusselt number is reported to increase with larger $d_{ij}$4, larger $d_{ij}$5, and larger $d_{ij}$6, while decreasing with larger $d_{ij}$7, $d_{ij}$8, $d_{ij}$9, $\lambda$0, $\lambda$1, and $\lambda$2. The material conclusion drawn by the study is specific: increasing copper and aluminium oxide volume fractions increases heat-transfer rate, whereas increasing silver volume fraction decreases heat-transfer efficiency (Shivakumar et al., 8 Jul 2025).

The drag response is also quantified. The study reports that the skin friction or drag coefficient decreases by $\lambda$3 with a unit increase in the Maxwell parameter, increases by $\lambda$4 with a unit rise in the Casson parameter, and declines by $\lambda$5 with a unit increase in the magnetic parameter. It also lists linear slopes for $\lambda$6: approximately $\lambda$7 versus $\lambda$8, $\lambda$9 versus $u(x,y)$0, $u(x,y)$1 versus $u(x,y)$2, $u(x,y)$3 versus $u(x,y)$4, $u(x,y)$5 versus $u(x,y)$6, and $u(x,y)$7 versus $u(x,y)$8. The percentage wording is mathematically unconventional, but it is the form in which the paper states its regression-based trend summary (Shivakumar et al., 8 Jul 2025).

6. Biomedical interpretation, scope, and terminological boundaries

The biomedical interpretation centers on targeted drug delivery in stenosed arteries. Lower velocity is taken to imply greater residence time for nanoparticle carriers near the diseased site, improving interaction with the arterial wall and the stenotic region. Magnetic forcing is treated as a possible mechanism for externally guided transport, while radiative and internal heating effects are linked to hyperthermia-assisted release from temperature-sensitive nanoparticles (Shivakumar et al., 8 Jul 2025).

At the same time, the model has explicit scope limits. The stenosed artery is not represented by a full cylindrical wall-shape function but by a stretching-sheet idealization. No separate nanoparticle concentration transport equation is solved. The cited study also does not report entropy generation, pumping-power formulas, or pressure-drop relations. These limits matter when Casson-Maxwell nanofluid results are compared with lubrication-type arterial or peristaltic models that resolve geometry and wall transport differently (Shivakumar et al., 8 Jul 2025).

A recurring source of confusion is the use of “Maxwell” in neighboring nanofluid literature. The distinction is best summarized comparatively.

Study	Model status	Defining features
(Shivakumar et al., 8 Jul 2025)	True Casson-Maxwell nanofluid	Casson rheology plus Maxwell viscoelasticity; blood-based tri-hybrid nanofluid; MHD, radiation, heat source
(K et al., 2021)	Casson nanofluid, not Maxwell	CuO-water peristaltic transport in an asymmetric channel; Joule heating; velocity, thermal, and concentration slip
(Muqaddass et al., 16 Apr 2025)	Casson hybrid nanofluid, not Maxwell	Pulsatile magnetized $u(x,y)$9/Casson blood flow in a stenotic artery with Hall current

The peristaltic CuO-water study contains no viscoelastic relaxation-time constitutive law, no upper-convected derivative, and no Maxwell parameter; its non-Newtonian response is entirely Casson-type, with Joule heating increasing temperature and decreasing nanoparticle concentration, and with lamina-shaped nanoparticles exhibiting higher thermal conductivity than needle-shaped nanoparticles (K et al., 2021). The stenosed-artery hybrid nanofluid study includes Hall current, MHD effects, and Maxwell-type effective medium formulas for conductivity, but it likewise contains no Maxwell-fluid rheology; it models blood as a Casson fluid and reports that increasing the non-Newtonian parameter raises velocity and lowers temperature, whereas adding $\mu_b$00 and $\mu_b$01 nanoparticles lowers velocity and raises temperature (Muqaddass et al., 16 Apr 2025).

Accordingly, the encyclopedic meaning of Casson-Maxwell nanofluid should be reserved for models that combine Casson yield-stress behavior with Maxwell relaxation dynamics in the constitutive law. MHD terms, Joule heating, Maxwell-Garnett conductivity closures, or hybrid nanoparticle mixtures are adjacent features, not substitutes for Maxwell viscoelasticity.