Uncertainty Propagation for Power-Law, Bingham, and Casson Fluids: A Comparative Stochastic Analysis of a Class of Non-Newtonian Fluids in Rectangular Ducts

Abstract

This study presents a novel framework for uncertainty propagation in power-law, Bingham, and Casson fluids through rectangular ducts under stochastic viscosity (Case I) and pressure gradient conditions (Case II). Using the computationally efficient Stochastic Finite Difference Method with Homogeneous Chaos (SFDHC), validated via comparison with quasi-Monte Carlo simulations, we demonstrate significantly lower computational costs across varying Coefficients of Variation (COVs). For viscosity uncertainty (Case I), results show a 0.54–2.8% increase in mean maximum velocity with standard deviations reaching 75.3–82.5% of the COV, where the power-law model exhibits the greatest sensitivity (velocity variations spanning 71.2–177.3% of the mean at COV = 20%). Pressure gradient uncertainty (Case II) preserves mean velocities but produces narrower and symmetric distributions. We systematically evaluate the effects of aspect ratio, yield stress, and flow behavior index on the stochastic velocity response of each fluid. Moreover, our analysis pioneers a performance hierarchy: Herschel–Bulkley fluids show the highest mean and standard deviation of maximum velocity, followed by power-law, Robertson–Stiff, Bingham, and Casson models. A key finding is the extreme fluctuation of the Robertson–Stiff model, which exhibits the most drastic deviations, reaching up to 177% of the average velocity. The significance of fluid-specific stochastic analysis in duct system design is underscored by these results. This is especially critical for non-Newtonian flows, where system performance and reliability are greatly impacted by uncertainties in viscosity and pressure gradient, which reflect actual operational variations.

1. Introduction

Complex shear-dependent behavior in non-Newtonian fluids is essential for biological flows and polymer processing. Their variable viscosity necessitates the use of specific models, such as power-law (shear-thinning/thickening) [1,2], Bingham (yield stress fluids) [3], and Casson (e.g., for modeling blood flow or food pastes) [4], in contrast to Newtonian fluids. Complex geometries can now be modeled thanks to developments in rheology and computational techniques [5,6]. These models are essential because, for instance, the shear-thinning behavior enhances flow efficiency in ducts by producing flatter velocity profiles than Newtonian fluids [7]. These nonlinearities are well captured by finite difference/element methods [8]. Moreover, a yield-stress-dependent plug region is revealed by analytical solutions for Bingham fluids [9], and precise cement and slurry flow predictions are made possible by contemporary regularization techniques [10]. Due to their nonlinear viscosity, Casson fluids have smaller plug regions; consequently, finite volume techniques are often used to model their blood flow behavior, particularly in biomedical applications [11]. Recent studies have shown the necessity of these rheological models for evaluating large-scale geophysical risks like landslides and tsunamis [12] and have advanced our understanding of Casson fluids under complex conditions like magneto-convective rotation [13].

Unlike the deterministic approaches described above, stochastic methods deal with real-world uncertainties in viscosity, yield stress, and boundary conditions, while deterministic approaches assume fixed parameter values [14]. Stochastic analysis of fluid dynamics has become a crucial tool for managing uncertainty in complex fluid systems found in industrial operations and geophysical phenomena. While Amirdjanova [15] and Zirbel [16] introduced novel vorticity-based and Lagrangian approaches, respectively, Mikulevicius and Rozovskii [17] laid the groundwork for stochastic fluid mechanics by providing rigorous mathematical frameworks. Advanced numerical methods, such as Breuer and Petruccione’s [18] method for stochastic computational fluid dynamics and Benth and Gjerde’s [19] solver for the stochastic pressure equation, were made possible by these early studies. Since then, the field has advanced to use sophisticated uncertainty quantification techniques. For example, Xiu and Karniadakis [20] demonstrated the effectiveness of generalized polynomial chaos expansions in modeling fluid uncertainties, and Prieto [21] used stochastic particle methods to investigate non-Newtonian droplet behavior.

Computational efficiency for complex flows is given priority in recent advances in stochastic fluid modeling. In his comparison of stochastic and deterministic duct flow algorithms, Sochi [22] emphasized the advantages of energy minimization. De et al. [23] modified these for viscoelastic porous media, while Bedrossian et al. [24] created Lagrangian models for turbulence. Guadagnini et al.’s fracture studies [25], Galal’s non-Newtonian velocity models [26,27], and Herschel–Bulkley duct flow analyses [28] all demonstrate an increasing emphasis on uncertainty quantification. Rinkens et al. [29] and Rezaeiravesh et al. [30] developed frameworks for generalized Newtonian fluids built on Knio and Le Maître’s chaos methods [31] and Gunzburger’s collocation techniques [32]. Terasawa and Yoshida [33] laid the theoretical groundwork (existence/uniqueness for stochastic power-law fluids), which Breit [34] expanded upon. The vibrational dynamics of a nonlinear Bingham model under stochastic conditions was investigated by Kan et al. [35]. For stochastic unsteady mixed convection flow of Casson nanofluids, Nawaz et al. [36] introduced an effective two-stage computational approach that increased numerical precision. Ali et al. [37] examined mixed convective Casson fluid flows while accounting for Hall and ion slip effects by employing artificial neural networks to analyze stochastic behaviors via non-Fourier double diffusion theories. Complex thermal and flow interactions become easier to understand as a result.

Despite their extensive use in many different domains, prior studies have shown a notable deficiency in stochastic modeling and solutions for non-Newtonian fluids in rectangular ducts, including power-law, Bingham, and Casson fluids. This study addresses a critical gap by pioneering stochastic models for these fluids, examining their behavior under uncertainties in viscosity and pressure gradients across two distinct cases. We also assess the sensitivity of each fluid’s stochastic response to variations in aspect ratio, yield stress, and flow behavior index. Furthermore, a comparative analysis against established Herschel–Bulkley [28] and Robertson–Stiff [27] models under identical conditions provides novel, quantitative evidence for key performance differences. The analysis employs the Stochastic Finite Difference Method with Homogeneous Chaos (SFDHC) [38], a computationally efficient approach that provides statistical metrics and the Probability Density Function (PDF) of the solution process, with strong agreement confirmed against Quasi-Monte Carlo Simulation (QMCS). A key advantage of our approach is its computational performance, being over 200 times faster than the QMCS benchmark [39]. This study is organized to first establish the stochastic modeling foundation (Section 2), then derive the new stochastic formulations (Section 3), outline the numerical methodology (Section 4), present and discuss the resulting data (Section 5), and finally, conclude with the key insights (Section 6).

2. Random Fields Representation and Solution

Uncertain parameters are frequently represented as random fields, which are discretized into random variables (RVs) or stochastic processes (SPs). The Karhunen–Loève Expansion (KLE) approximates an SP as a truncated infinite series based on a known covariance kernel, $K\left(x_1,y_1;x_2,y_2\right)$ expressed as [40].

$$\mathcal{S}\left(x,y;\theta \right)=\overline{\mathcal{S}}\left(x,y\right)+{\displaystyle {\sum }_{i=1}^{N_{KL}}}\sqrt{{\lambda }_i}{h}_i\left(x,y\right){\xi }_i\left(\theta \right),$$

(1)

where $\overline{\mathcal{S}}\left(x,y\right)$ is the mean, ${\left\{{\xi }_i\left(\mathsf{\theta }\right)\right\}}_{\mathrm{i}=1}^{N_{KL}}$ are uncorrelated RVs, and ${\lambda }_i,h_i\left(x\right)$ are eigen-pairs derived from the Fredholm integral equation.

$${\int }_{\Omega }K\left(x_1,y_{1 };x_2,y_2\right) h_i\left(x_2,y_2\right) d{x}_2 d{y}_2={\lambda }_i h_i\left(x_1,y_1\right),$$

(2a)

with $\Omega$ as the spatial domain. For second-order SPs, an exponential covariance kernel

$$K\left(x_1,y_1;x_2,y_2\right)={\mathcal{k}}^2{e}^{-{\displaystyle \frac{\left|x_1-x_2\right|}{{lc}_x}}-{\displaystyle \frac{\left|y_1-y_2\right|}{{lc}_y}}},$$

(2b)

is often used, where $\mathcal{k}$ is the Coefficient of Variation (COV), and ${lc}_x$, ${lc}_y$ are correlation lengths. A second-order SP can be fully described by its covariance kernel since all its variables have finite means and variances. The main advantage is that it makes it possible to use strong analytical techniques like the KLE, which simplifies the complex random fields into a number of uncorrelated RVs for effective modeling and uncertainty quantification. If separable, the kernel yields eigen-pairs in one dimension as $\left\{{\acute{\lambda }}_i,{\acute{h}}_i\left(x\right)\right\}$ and $\left\{{\acute{\stackrel{´}{\lambda }}}_j,{\acute{\stackrel{´}{h}}}_j\left(y\right)\right\}$, combined as ${\lambda }_k={\acute{\lambda }}_i{\acute{\stackrel{´}{\lambda }}}_j$ and $h_k\left(x,y\right)={\acute{h}}_i\left(x\right) {\acute{\stackrel{´}{h}}}_j \left(y\right)$, sorted in descending order. For the used exponential covariance kernel, the eigen-pairs can be computed analytically or using the numerical solution of Equation (2a), as detailed in [40]:

$$h_i^{\left(e\right)}\left(x\right)={\displaystyle \frac{cos\left({\beta }_i^{\left(e\right)}x\right)}{\sqrt{\mathcal{b}+\frac{sin\left(2{\mathcal{b}\beta }_i^{\left(e\right)}\right)}{{2\beta }_i^{\left(e\right)}}}}}, \mathrm{and} h_i^{\left(o\right)}\left(x\right)={\displaystyle \frac{sin\left({2\beta }_i^{\left(e\right)}x\right)}{\sqrt{\mathcal{b}-\frac{sin\left(2{\mathcal{b}\beta }_i^{\left(o\right)}\right)}{{2\beta }_i^{\left(o\right)}}}}},$$

(3a)

$${\lambda }_i^{\left(e\right)}={\displaystyle \frac{2\upsilon }{{\left( {\beta }_i^{\left(e\right)}\right)}^2+{\upsilon }^2}},\mathrm{and} {\lambda }_i^{\left(o\right)}={\displaystyle \frac{2\upsilon }{{\left( {\beta }_i^{\left(o\right)}\right)}^2+{\upsilon }^2}},$$

(3b)

In the above description, $h_i^{\left(e\right)}\left(x\right)$ and $h_i^{\left(o\right)}\left(x\right)$ are the even and odd eigenfunctions, respectively. The corresponding eigenvalues are ${\lambda }_i^{\left(e\right)}$ and ${\lambda }_i^{\left(o\right)}$, where $\mathcal{b}$ is the half-domain of the SP, and υ is the reciprocal of the correlation length (i.e., υ = 1/$lc$). To obtain the eigen-pairs in Equation (3a,c), $\beta$ must first be calculated by solving the two transcendental equations in (3c).

$$\upsilon -\beta tan\left(\beta \mathcal{b}\right)=0,\mathrm{and} \beta -\upsilon tan\left(\beta \mathcal{b}\right)=0.$$

(3c)

When the response’s covariance is unknown, Polynomial Chaos Expansion (PCE) represents it as a series of RVs ${\left\{{\xi }_i\left(\mathsf{\theta }\right)\right\}}_{\mathrm{i}=1}^{N_{KL}}$ with deterministic coefficients, truncated to order $P$ as follows:

$${\left\{{\Psi }_i\left(\xi \right)\right\}}_{i=0}^{\mathrm{\infty }}=a_o{\Gamma }_o+\sum _{k_1=1}^{\mathrm{\infty }}{a}_{k_1}{\Gamma }_1\left({\xi }_{k_1}\left(\theta \right)\right)+\sum _{k_1=1}^{\mathrm{\infty }}\sum _{k_2=1}^{\mathrm{\infty }}{a}_{{k_{1}k}_2}{\Gamma }_{{k_{1}k}_2}\left({\xi }_{k_1}\left(\theta \right),{\xi }_{k_2}\left(\theta \right)\right)+\dots ,$$

(4)

The truncated form of the response function, up to order $P$, is

$$w\left(x,y,{\xi }_{N_{KL}}\left(\theta \right)\right)\cong {\sum }_{i=0}^P{\mathcal{C}}_i\left(x,y\right){\Psi }_i\left[\left\{{\xi }_{N_{KL}}\left(\theta \right)\right\}\right],$$

(5)

or simply,

$$w\left(x,y,\xi \right)={\sum }_{i=0}^P{\mathcal{C}}_i\left(x,y\right){\Psi }_i\left(\xi \right),$$

(6)

with a deterministic coefficient vector as follows:

$$\mathbb{V}\left(x,y\right)={[{\mathcal{C}}_o\left(x,y\right) {\mathcal{C}}_1\left(x,y\right).....{\mathcal{C}}_P\left(x,y\right)]}^T,$$

(7)

and ${\left\{{\Psi }_i\right\}}_{i=0}^P$ are orthogonal basis functions with total terms $N_{PC}$ given by the following equation:

$$N_{PC}=P+1={\displaystyle \frac{\left(N_{KL}+p\right)!}{p!N_{KL}!}}.$$

(8)

For a second-order expansion $(p=2)$, this results in 6 polynomials for $N_{KL}=2$ (denoted as SFD2) and 15 polynomials for $N_{KL}=4$ (denoted as SFD4). The specific polynomials for the SFD2 and SFD4 are ${{\{\Psi }_i\}}_{i=1}^6=\{1,{\xi }_1,{\xi }_2,{\xi }_1{\xi }_2,{\xi }_1^2-1,{\xi }_2^2-1\}$ and ${{\{\Psi }_i\}}_{i=1}^{15}={ \left\{1,{ \xi }_1,{ \xi }_2,{ \xi }_3, { \xi }_4, { \xi }_1{ \xi }_2,\right.{ \xi }_1\xi }_3,{{\xi }_1\xi }_4,{{\xi }_2\xi }_4,{{{\xi }_3\xi }_4,\xi }_1^2-1,{\xi }_2^2-1,\left.{\xi }_3^2-1,{\xi }_4^2-1\right\}$, respectively. Consequently, the SFDHC method [38] uses the KLE to represent parameters with dimensionality $N_{KL}$ and the PCE of order $p$ to model the response. Galerkin projection transforms the stochastic system into a deterministic one, which is solved for the solution field $\mathbb{C}\left(x,y\right)$ using finite difference methods (FDM). The PDF is then derived using Equations (6) and (7), with the mean and variance provided in Equations (9) and (10), respectively.

$$〈w\left(x,y,\xi \right)〉={\mathcal{C}}_o\left(x,y\right),$$

(9)

$$Var\left(w\left(x,y,\xi \right)\right)=\sum _{i=1}^P〈{\Psi }_i^2〉{\mathcal{C}}_i^2\left(x,y\right)$$

(10)

3. Stochastic Modeling and Solution of Power-Law, Bingham, and Casson Fluids in Rectangular Ducts

3.1. The Stochastic Modeling Framework

The deterministic foundation of this problem considers a steady, laminar, and fully developed flow of an incompressible, isothermal, purely viscous non-Newtonian fluid within a rectangular duct. The duct has dimensions $\left(L\right)$ in the ($x)$-direction and $\left(\alpha L\right)$ in the $\left(y\right)$-direction. As shown in Figure 1, the duct walls are aligned with the ($x)$- and ($y)$-directions, and the fluid flows in the ($z)$-direction, driven by a constant pressure gradient $\left(dp/dz\right)$ [8].

However, uncertainty in fluid dynamic systems occurs from a variety of sources. Temperature changes or impurities impacting viscosity and density cause variability in material properties, while boundary conditions present obstacles in the form of thermal fluctuations, geometric imperfections, and deviations in pressure gradient. Other sources include initial conditions and external factors, such as turbulence or fluctuating heat inputs, as do measurement errors caused by sensor limits or equipment degradation over time. Modeling limitations, such as simplified constitutive relations or discretization errors. Operational challenges like manufacturing tolerances and fluctuating flow rates further contribute to system complexity. Furthermore, flow disruptions due to trapped air, inadequate thermal insulation, and the inherent non-uniformity of non-Newtonian fluids affecting viscosity highlight the need for stochastic approaches to better represent real-world behaviors in applications [14,15]. These complexities highlight the significance of using stochastic methods to depict real-world behavior more accurately.

Here, the continuity, momentum, and energy equations [8], which have been specially adapted for the study of stochastic non-Newtonian fluid flow, are the governing equations for this investigation. The stochastic version of these equations is as follows:

$${\displaystyle \frac{\partial }{\partial x}}\left(\mu \left(x,y;\theta \right){\displaystyle \frac{\partial w\left(x,y;\theta \right)}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(\mu \left(x,y;\theta \right){\displaystyle \frac{\partial w\left(x,y;\theta \right)}{\partial y}}\right)-{\displaystyle \frac{dp\left(\theta \right)}{dz}} =0,$$

(11)

$$\rho c_{p}w\left(x,y;\theta \right){\displaystyle \frac{\partial T\left(x,y;\theta \right)}{\partial z}}=k_c \left({\displaystyle \frac{{\partial }^{2}T\left(x,y;\theta \right)}{\partial x^2 }}+{\displaystyle \frac{{\partial }^{2}T\left(x,y;\theta \right)}{\partial y^2 }} \right).$$

(12)

In these equations, $T$ represents the fluid temperature, $k_c$ its thermal conductivity, $\rho$ its density, $c_p$ the specific heat capacity, and $w\left(x,y;\theta \right)$ the stochastic velocity component along the z-axis. The terms ${\mu }_{PL }\left(x,y;\theta \right)$, ${\mu }_{BH }\left(x,y;\theta \right)$ and ${\mu }_{CS }\left(x,y;\theta \right)$ denote the fluid’s uncertain apparent viscosity for power-law, Bingham and Casson fluids, respectively, where an overbar $\left(e.g., \overline{.} \right)$ indicates the mean value of a parameter. This yield

$${\overline{\mu }}_{PL}\left(x,y\right)=m{\left(\sqrt{{\left({\displaystyle \frac{\partial \overline{w}}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial \overline{w}}{\partial y}}\right)}^2+e_o}\right)}^{n-1},$$

(13)

$${\overline{\mu }}_{BH}\left(x,y\right)={\tau }_o/\left(\sqrt{{\left({\displaystyle \frac{\partial \overline{w}}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial \overline{w}}{\partial y}}\right)}^2+e_o}\right)+{\overline{\mu }}_o$$

(14)

$${\overline{\mu }}_{CS}\left(x,y\right)={\left({\left({\tau }_o/\sqrt{{\left({\displaystyle \frac{\partial \overline{w}}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial \overline{w}}{\partial y}}\right)}^2+e_o}\right)}^{1/2}+K_c\right)}^2$$

(15)

Here, the parameters $m$ and $n$ denote the fluid’s consistency index and flow behavior index, respectively, while ${K_c}^2$ denotes the Casson viscosity coefficient, ${\tau }_o$ is the yield stress, and $e_o$ is a small constant serving as a regularization parameter to ensure numerical stability in regions of vanishing shear rate. Leveraging the geometrical symmetry of the duct, the computational domain is reduced to one-quarter of the full cross-section. This simplification yields the following deterministic boundary conditions (B.C.s) for the momentum equation, which is the primary subject of this investigation:

$$\overline{w}\left(0,y\right)= \overline{w}\left(x,0\right)=0 , {\displaystyle \frac{\partial \overline{w}\left(L/2,y\right)}{\partial x}}={\displaystyle \frac{\partial \overline{w}\left(x,\alpha L /2\right)}{\partial y}}= 0.$$

(16)

A non-dimensional representation of the governing Equation (11) is derived for generality, employing the variable definitions given below [8]:

$\mathcal{x}={\displaystyle \frac{x}{L}}$, $\mathcal{y}={\displaystyle \frac{y }{L}}$, $\mathcal{z}={\displaystyle \frac{z }{L}}$, $\overline{\eta }=\frac{\overline{\mu }}{{\overline{\mu }}_r}$, and $\overline{\mathcal{w}}={\displaystyle \frac{\overline{w}}{{\overline{w}}_m}}$, given that ${\overline{\mu }}_r=m {\left(D_h\right)}^{1-n}/{\left({\overline{w}}_{av}\right)}^{1-n}$, where $D_h$ represents the hydraulic diameter and ${\overline{w}}_m$ is the average velocity, calculated as ${\overline{w}}_m={\displaystyle \frac{1}{{\alpha L}^2}}{\int }_0^{\alpha L}{\int }_0^{L}w dx dy$. This transformation results in the non-dimensional form of the equation:

$${\displaystyle \frac{\partial }{\partial \mathcal{x}}}\left(\eta \left(\mathcal{x},\mathcal{y};\theta \right){\displaystyle \frac{\partial \mathcal{w}\left(\mathcal{x},\mathcal{y};\theta \right)}{\partial \mathcal{x}}}\right)+{\displaystyle \frac{\partial }{\partial \mathcal{y}}}\left(\eta \left(\mathcal{x},\mathcal{y};\theta \right){\displaystyle \frac{\partial \mathcal{w}\left(\mathcal{x},\mathcal{y};\theta \right)}{\partial \mathcal{y}}}\right)+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{1+\alpha }{\alpha }}\right)}^{2}f\left(\theta \right) Re=0.$$

(17)

In this context, $f\left(\theta \right)$ denotes the friction factor, defined as $f\left(\theta \right)=L(-dp(\theta )/dz)/2\rho {\overline{w}}_m^2$, where $Re$ is the dimensionless Reynolds number, given by $Re=\rho L{ \overline{w}}_m/{\overline{\mu }}_r$, and $\rho$ represents the fluid’s density. As a result, the boundary conditions and the non-dimensional viscosity $\overline{\eta } \left(\mathcal{x},\mathcal{y}\right)$ are expressed as follows:

$$\overline{\mathcal{w}}\left(0,\mathcal{y}\right)= \overline{\mathcal{w}}\left(\mathcal{x},0\right)=0 , {\displaystyle \frac{\partial \overline{\mathcal{w}}\left(1/2,\mathcal{y}\right)}{\partial \mathcal{x}}}={\displaystyle \frac{\partial \overline{\mathcal{w}}\left(\mathcal{x},\alpha /2\right)}{\partial \mathcal{y}}}= 0.$$

(18)

The dimensionless viscosity for the three non-Newtonian fluids is defined as follows:

$${\overline{\eta }}_{PL}\left(\mathcal{x},\mathcal{y}\right)={\left(\sqrt{{\left({\displaystyle \frac{\partial \overline{\mathcal{w}}}{\partial \mathcal{x}}}\right)}^2+{\left({\displaystyle \frac{\partial \overline{\mathcal{w}}}{\partial \mathcal{y}}}\right)}^2+e_o}\right)}^{n-1},$$

(19)

$${\overline{\eta }}_{BH}\left(\mathcal{x},\mathcal{y}\right)={\tau }_D/\left(\sqrt{{\left({\displaystyle \frac{\partial \overline{\mathcal{w}}}{\partial \mathcal{x}}}\right)}^2+{\left({\displaystyle \frac{\partial \overline{\mathcal{w}}}{\partial \mathcal{y}}}\right)}^2+e_o}\right)+1,$$

(20)

$${\overline{\eta }}_{CS}\left(\mathcal{x},\mathcal{y}\right)={\left({\left({\tau }_D/\sqrt{{\left({\displaystyle \frac{\partial \overline{\mathcal{w}}}{\partial \mathcal{x}}}\right)}^2+{\left({\displaystyle \frac{\partial \overline{\mathcal{w}}}{\partial \mathcal{y}}}\right)}^2+e_o}\right)}^{1/2}+1\right)}^2.$$

(21)

Given the definition of the dimensionless yield stress ${\tau }_D={\tau }_o/m {\left( {\overline{w}}_m/L\right)}^n$ and the non-dimensional average velocity expressed as ${\overline{\mathcal{w}}}_m={\displaystyle \frac{4}{\alpha }}{\int }_0^{0.5 \alpha }{\int }_0^{0.5}\overline{\mathcal{w}} d\mathcal{x}d\mathcal{y}$. Numerical integration is performed via Simpson’s 1/3 rule, as described in [41]:

$$\begin{gathered} {\sum }_{q=\mathrm{1,3},5}^{\mathcal{Q}-1} {\sum }_{r=\mathrm{1,3},5}^{\mathcal{R}-1} \left(\left({\overline{\mathcal{w}}}_{q,r}+{\overline{\mathcal{w}}}_{q+2,r}+{\overline{\mathcal{w}}}_{q,r+2}+{\overline{\mathcal{w}}}_{q+2,r+2}\right)\right.+4\left( \right.{\overline{\mathcal{w}}}_{q,r+1}+{\overline{\mathcal{w}}}_{q+1,r}+\hspace{1em} \\ {\overline{\mathcal{w}}}_{q+2,r+1}+\left.{\overline{\mathcal{w}}}_{q+1,r+2}\right)+\left.16 {\overline{\mathcal{w}}}_{q+1,r+1}\right)={\displaystyle \frac{9}{4∆x∆y}} , 2\le q\le \mathcal{Q}+1, 2\le r\le \mathcal{R}+1, \end{gathered}$$

(22)

with $\mathcal{Q}$ and $\mathcal{R}$ defining the number of discrete intervals in the $\mathcal{x}$ and $\mathcal{y}$ spatial directions.

Using the KLE in Equation (1) to discretize the random fields $\eta \left(x,y;\theta \right)$ and $f\left(\mathcal{x},\mathcal{y};\theta \right)$, and the PCE in Equation (6) to represent the velocity field $\mathcal{w}$, yields

$$\eta \left(\mathcal{x},\mathcal{y};\theta \right)=\overline{\eta }\left(\mathcal{x},\mathcal{y}\right)\left(1+{\epsilon }_{\eta }\sum _{i=1}^{N_{KL}}{\sqrt{{\lambda }_i }g}_i\left(\mathcal{x},\mathcal{y}\right){ \xi }_i\right),$$

(23)

$$f\left(\mathcal{x},\mathcal{y};\theta \right)=\overline{f}\left(1+{\epsilon }_{dp}\sum _{i=1}^{N_{KL}}{\sqrt{{\gamma }_i}h}_i\left(\mathcal{x},\mathcal{y}\right){ \xi }_i\right),$$

(24)

$$\mathcal{w}\left(\mathcal{x},\mathcal{y};\theta \right)=\sum _{i=0}^P{\mathcal{C}}_i\left(x,y\right) {\Psi }_i\left(\xi \right),$$

(25)

where ${\lambda }_i$,$f_i\left(\mathcal{x},\mathcal{y}\right)$ and ${\gamma }_i$, $g_i\left(\mathcal{x},\mathcal{y}\right)$ represent the eigen-pairs of $\eta \left(\mathcal{x},\mathcal{y}\right)$ and $f\left(\mathcal{x},\mathcal{y};\theta \right)$, respectively, with ${\epsilon }_{\eta }$ and ${\epsilon }_{dp}$ serving as their control factors. Inserting Equations (23)–(25) into the non-dimensional momentum Equation (17) results in:

$$\begin{gathered} {\displaystyle \frac{\partial }{\partial \mathcal{x}}}\left(\overline{\eta }\left(\mathcal{x},\mathcal{y}\right) \left(1+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}}{\sqrt{{\lambda }_i }g}_i\left(\mathcal{x},\mathcal{y}\right){ \xi }_i\right) {\sum }_{j=0}^P{\displaystyle \frac{\partial {\mathcal{C}}_j\left(\mathcal{x},\mathcal{y}\right)}{\partial \mathcal{x}}} { \Psi }_j\left(\xi \right)\right)+{\displaystyle \frac{\partial }{\partial \mathcal{y}}}\left(\overline{\eta }\left(\mathcal{x},\mathcal{y}\right) (1+ \\ {\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}}{g}_i\left(\mathcal{x},\mathcal{y}\right){ \xi }_i) {\sum }_{j=0}^P{\displaystyle \frac{\partial {\mathcal{C}}_j\left(\mathcal{x},\mathcal{y}\right)}{\partial \mathcal{y}}} { \Psi }_j\left(\xi \right)\right)+\overline{f}Re \left(1+{\epsilon }_{dp}{\sum }_{i=1}^{N_{KL}}{\sqrt{{\gamma }_i}h}_i\left(\mathcal{x},\mathcal{y}\right){ \xi }_i\right)=0. \end{gathered}$$

(26)

The concluding Equation (26) provides a stochastic model for the behavior of non-Newtonian fluids within rectangular ducts.

3.2. Problem Solution Using SFDHC

One-quarter of the duct domain is divided into $\mathcal{Q}$ and $\mathcal{R}$ equal intervals along the $\mathcal{x}$- and $\mathcal{y}$ -directions, respectively. The non-dimensional momentum Equation (17) is then formulated at each grid node $\left(q,r\right)$. Since the dimensionless viscosity $\overline{\eta }$ is not known in advance, it is initially approximated using Equations (19)–(21) with, $\overline{\mathcal{w}}=2\mathit{sin}\pi \mathcal{x}\mathit{sin}{\displaystyle \frac{\pi \mathcal{y}}{\alpha }}$ for the first iteration $\left(\mathcal{l}=1\right)$. Subsequently, the central finite difference scheme is applied as follows [8]:

$$\begin{array}{ll}{\left(∆\mathcal{x}\right)}^{-2}[({\eta }_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)}& +{\eta }_{\left(\mathcal{l}\right)}^{\left(q,r\right)}){\mathcal{w}}_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)}-\left({\eta }_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)}+2{\eta }_{\left(\mathcal{l}\right)}^{\left(q,r\right)}+{\eta }_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)}\right){\mathcal{w}}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}\\ & +\left({\eta }_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)}+{\eta }_{\left(\mathcal{l}\right)}^{\left(q,r\right)}\right){\mathcal{w}}_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)}]\hfill \\ & +{\left(∆\mathcal{y}\right)}^{-2}[\left({\eta }_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)}+{\eta }_{\left(\mathcal{l}\right)}^{\left(q,r\right)}\right){\mathcal{w}}_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)}\hfill \\ & -\left({\eta }_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)}+2{\eta }_{\left(\mathcal{l}\right)}^{\left(q,r\right)}+{\eta }_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)}\right){\mathcal{w}}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}\hfill \\ & +\left({\eta }_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)}+{\eta }_{\left(\mathcal{l}\right)}^{\left(q,r\right)}\right){\mathcal{w}}_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)}]+{\overline{f}Re|}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}=0\hfill \end{array}$$

(27)

the last equation is expanded as follows:

$$\begin{array}{c}{\left(∆\mathcal{x}\right)}^{-2}(-{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q-1,r\right)}{\Psi }_i- \hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q-1,r\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q-1,r\right)}-{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q-1,r\right)}{\Psi }_i-\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q,r\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q-1,r\right)}+{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q,r\right)}{\Psi }_i+\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q-1,r\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q,r\right)}+{2\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q,r\right)}{\Psi }_i+\hfill \\ 2{\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q,r\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q,r\right)}+{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q,r\right)}{\Psi }_i+\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q+1,r\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q,r\right)}-{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q+1,r\right)}{\Psi }_i-\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q+1,r\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q+1,r\right)}-{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q+1,r\right)}{\Psi }_i-\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q,r\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q+1,r\right)})\hfill \\ {+\left(∆\mathcal{y}\right)}^{-2}(-{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q,r-1\right)}{\Psi }_i-\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q,r-1\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q,r-1\right)}-{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q,r-1\right)}{\Psi }_i-\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q,r\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q,r-1\right)}+{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q,r\right)}{\Psi }_i+\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q,r-1\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q,r\right)}+{2\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q,r\right)}{\Psi }_i+\hfill \\ 2{\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i} g_i^{\left(q,r\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q,r\right)}+{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q,r\right)}{\Psi }_i+\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q,r+1\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q,r\right)}-{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q,r+1\right)}{\Psi }_i-\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q,r+1\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q,r+1\right)}-{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=0}^P{\mathcal{C}}_{i,\left(\mathcal{l}\right)}^{\left(q,r+1\right)}{\Psi }_i-\hfill \\ {\epsilon }_{\eta }{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}{\sum }_{i=1}^{N_{KL}}{\sum }_{j=0}^P\sqrt{{\lambda }_i}{g}_i^{\left(q,r\right)}{\xi }_i{\Psi }_j{\mathcal{C}}_{j,\left(\mathcal{l}\right)}^{\left(q,r+1\right)})={\left({\displaystyle \frac{1+\alpha }{\alpha }}\right)}^2{\overline{f}Re|}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}(1+\hfill \\ {\epsilon }_{dp}{\sum }_{i=1}^{N_{KL}}\sqrt{{\gamma }_i}{h}_i^{\left(q,r\right)}{\xi }_i)\hfill \end{array}$$

(28)

The random elements of Equation (28) are transformed into a deterministic form using a Galerkin projection technique [40]. The procedure requires multiplication by ${\Psi }_k$ followed by evaluation of the ensemble averages, producing:

$$\begin{gathered} {\left(∆\mathcal{x}\right)}^{-2}\left[-\left( {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}} {\mathbb{D}}_i^{\left(q-1,r\right)}\right)+ {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}} {\mathbb{D}}_i^{\left(q,r\right)}\right)\right){\mathbb{V}}_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)}+ \\ \left( {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}} {\mathbb{D}}_i^{\left(q-1,r\right)}\right)+2 {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q ,r\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}} {\mathbb{D}}_i^{\left(q,r\right)}\right)+{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)} \left(\mathbb{C}+ \\ {\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}} {\mathbb{D}}_i^{\left(q+1,r\right)}\right)\right){\mathbb{V}}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}-\left( {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}} {\mathbb{D}}_i^{\left(q+1,r\right)}\right)+ {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}} {\mathbb{D}}_i^{\left(q,r\right)}\right)\right){\mathbb{V}}_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)}\right]+ \\ {\left(∆\mathcal{y}\right)}^{-2}\left[-\left( {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}} {\mathbb{D}}_i^{\left(q,r-1\right)}\right)+ {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}} {\mathbb{D}}_i^{\left(q,r\right)}\right)\right){\mathbb{V}}_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)}+ \\ \left( {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}} {\mathbb{D}}_i^{\left(q,r-1\right)}\right)+2 {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}}{\mathbb{D}}_i^{\left(q ,r\right)} \right)+{\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)} \left(\mathbb{C}+ \\ {\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}}{\mathbb{D}}_i^{\left(q,r+1\right)} \right)\right){\mathbb{V}}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}-\left( {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{\mathcal{N}{N}_{KL}}{\mathbb{D}}_i^{\left(q,r+1\right)} \right)+ {\overline{\eta }}_{\left(\mathcal{l}\right)}^{\left(q,r\right)} \left(\mathbb{C}+{\epsilon }_{\eta }{\sum }_{i=1}^{N_{KL}}{\mathbb{D}}_i^{\left(q ,r\right)}\right)\right){\mathbb{V}}_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)}\right]= \\ {\left({\displaystyle \frac{1+\alpha }{\alpha }}\right)}^2{\overline{f}Re|}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}\left(\mathbb{A}+{\epsilon }_{dp}{\sum }_{j=1}^{N_{KL}}{\mathbb{B}}_j^{\left(q,r\right)}\right), \end{gathered}$$

(29)

where $\mathbb{A}$ is $N_{PC}\times 1$ vector with entries $a_i=〈{\Psi }_i〉$, ${\mathbb{B}}_{\left(q,r\right)}$ is $N_{PC}\times N_{KL}$ vectors with entries $b_{ij}=\sqrt{{\gamma }_j}{g}_j^{\left(q,r\right)}〈{\Psi }_i{\xi }_j〉$, $\mathbb{C}$ is $N_{PC}\times N_{PC}$ matrix with elements, $c_{ij}=〈{\Psi }_i{\Psi }_j〉$ and ${\mathbb{D}}_i^{\left(q,r\right)}$ is an $N_{PC}\times N_{PC}\times N_{KL}$ array with elements, and $d_{ijk}=\sqrt{{\lambda }_k}{g}_k^{\left(q,r\right)}〈{\Psi }_i{\Psi }_j{\xi }_k〉$.

Following the initialization of $\overline{\eta }\left(\mathcal{x},\mathcal{y}\right)$, the values of ${\mathbb{V}}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}$ are computed at all mesh points using Equation (29) during iteration $\left(\mathcal{l}\right)$. These values are then used to determine the stochastic velocity via Equation (25), with the mean value calculated using Equation (9). This mean value at each grid point is subsequently employed to update ${\overline{\eta }}_{\left(\mathcal{l}+1\right)}^{\left(q,r\right)}$ in the next iteration through the viscosity Equations (19)–(21) for the three studied fluids, utilizing the finite difference approximations: ${\displaystyle \frac{\partial \overline{\mathcal{w}}}{\partial \mathcal{x}}}={\displaystyle \frac{{\overline{\mathcal{w}}}_{\left(\mathcal{l}\right)}^{\left(q+1,r\right)}-{\overline{\mathcal{w}}}_{\left(\mathcal{l}\right)}^{\left(q-1,r\right)}}{2∆\mathcal{x}}}$ and ${\displaystyle \frac{\partial \overline{\mathcal{w}}}{\partial \mathcal{y}}}={\displaystyle \frac{{\overline{\mathcal{w}}}_{\left(\mathcal{l}\right)}^{\left(q,r+1\right)}-{\overline{\mathcal{w}}}_{\left(\mathcal{l}\right)}^{\left(q,r-1\right)}}{2∆\mathcal{y}}}$. This process continues until the required precision is achieved, satisfying the convergence criterion:

$$max\left|\left({\mathbb{V}}_{\left(\mathcal{l}+1\right)}^{\left(q,r\right)}-{\mathbb{V}}_{\left(\mathcal{l}\right)}^{\left(q,r\right)}\right)/{\mathbb{V}}_{\left(\mathcal{l}+1\right)}^{\left(q,r\right)}\right|\le Er.$$

(30)

Equation (29) is configured to analyze two separate uncertainty cases. The first case (Case I) isolates the impact of variability in the apparent viscosity. This is simulated by activating the viscosity uncertainty parameter (${\epsilon }_{\eta }=1$) while holding the pressure gradient uncertainty constant (${\epsilon }_{dp}=0$). Conversely, the second case (Case II) examines the exclusive effect of an uncertain pressure gradient, which is modeled by setting ${\epsilon }_{dp}=1$ and ${\epsilon }_{\eta }=0$. While Equation (29) admits a direct solution for forced flow, the free convection case necessitates a coupled approach. This involves the simultaneous numerical solution of Equations (29) and (22), where the latter is evaluated using Simpson’s one-third rule, to resolve the velocity and the $fRe$ product concurrently.

4. Numerical Implementation

The outcomes obtained from Equations (19)–(21) and (29) were coded in MATLAB, Version 9.13.0 (R2022b) [42]. The spatial domain corresponding to a quarter-section of the duct was discretized using ${∆}_{\mathcal{x}}{=∆}_{\mathcal{y}}=1/50$, meeting the accuracy criterion in Equation (30) with $Er=1\times {10}^{-4}$. The regularization parameter $e_o$ was chosen via sensitivity analysis to ensure numerical stability without affecting physical results. It must be small enough to have a negligible impact on the velocity field outside the infinitesimal plug region. Gao and Hartnett [8] recommend $e_o<1\times {10}^{-4}$; we used $e_o=1\times {10}^{-8}$, to ensure its effectiveness. For applications with very low yield stresses or different numerical schemes, $e_o$ may need adjustment to balance accuracy and convergence.

To validate the deterministic segment of the solution, both ${\epsilon }_{\eta }$ and ${\epsilon }_{dp}$ were initialized to zero, allowing for comparison with benchmark results from the literature. For the power-law fluid, the aspect ratios $\alpha =0.5$ and $1$, along with flow indices $n=0.5,0.8$, and 1.0, were adopted. The results were validated against solutions from Gao and Hartnett [8] and Syrjäla [43], as shown in

Table 1. Comparison of the $fRe$ product: current vs. published results for power-law fluids at different $\alpha$ and $n$.

α	1			0.50
n	1.00	0.80	0.50	1.00	0.80	0.50
Gao and Hartnett [8]	14.229	9.918	5.723	15.551	10.658	6.002
Syrjala [43]	14.22708	9.91546	5.72140	15.54806	10.65524	5.99867
Present study	14.22894	9.94518	5.75781	15.55180	10.71018	6.05171
|Difference| (%)	0.013073	0.299734	0.636383	0.024077	0.515601	0.88416

, exhibiting excellent agreement with prior studies and a maximum difference of 0.88416%. Unlike power-law fluids, no direct prior solutions exist for Bingham and Casson fluids for the specific problem addressed here, precluding similar comparisons.

The SFDHC was applied using a second-order PCE for two stochastic dimensions: $N_{KL}$ = 2 and 4. These configurations are referred to as SFD2 and SFD4, respectively. To validate the SFDHC outputs, a Quasi-Monte Carlo Simulation (QMCS) [39] was utilized to verify the SFD2 and SFD4 results in the initial test cases. Furthermore, the uncertainty quantification was assessed for three coefficients of variation (COV), 10%, 15%, and 20%, of the uncertain parameter’s mean value. This evaluation generated three separate solution sets for each technique, denoted as (SFD2-10, SFD2-15, SFD2-20), (SFD4-10, SFD4-15, SFD4-20), and (MCS-10, MCS-15, MCS-20).

The assigned COVs are derived from empirical studies. The 10–15% variability in viscosity corresponds to the dispersion observed in hematite-based drilling muds due to particle size distribution, as experimentally documented by Tehrani et al. [44]. Similarly, the uncertainty in pressure gradient is consistent with the ±11.7% variations measured by Fadl et al. [45] in field studies of iron ore slurry pipelines exhibiting flow-induced stratification. These ranges were conservatively extended to 20% to encompass potentially greater variability in other non-Newtonian fluids due to the multiple sources of uncertainty previously discussed. Further experimental study is needed to identify the most suitable probability distributions for fluid parameter uncertainties and their COVs. In both the PCE and QMCS analyses, a sample size of $N_{\mathrm{R}}=2\times {10}^3$ was used to generate the random variables. While a larger sample size would be preferable, computational limitations dictated this choice.

5. Numerical Results and Discussion

5.1. Stochastic Analysis of Power-Law, Bingham, and Casson Fluids

The PDFs were calculated at each mesh point using SFD2, SFD4, and QMCS. For the three fluids, power-law, Bingham, and Casson, the mean velocity, the Standard Deviation (S.D.), approximate minimum, approximate maximum, range, and deterministic velocity were determined and labeled as ${\mu }_{\mathcal{w}}$, ${\sigma }_{\mathcal{w}}$, $\widetilde{Min}$, $\widetilde{Max}$, $\tilde{R}$, and ${\mathcal{w}}_D$, respectively. Key statistical metrics for the maximum centerline velocity (${\mathcal{w}}_{max}$) are summarized in

Table 2. Power-law fluid: maximum velocity statistics for the two cases.

Case	Method of Solution	${\mathit{\mu }}_{\mathcal{w}}$	${\mathit{\mu }}_{\mathcal{w}}/{\mathcal{w}}_{\mathit{D}}$	${\mathit{\sigma }}_{\mathcal{w}}$	${\mathit{\sigma }}_{\mathcal{w}}/{\mathit{\mu }}_{\mathcal{w}}$	$\widetilde{\mathit{M}\mathit{i}\mathit{n}}$	$\widetilde{\mathit{M}\mathit{a}\mathit{x}}$	$\tilde{\mathit{R}}$
Case I (Uncertainty Viscosity)	SFD2-10	2.2680	1.0061	0.1743	0.0778	1.8392	2.9547	1.1155
	SFD4-10	2.2691	1.0066	0.1747	0.0779	1.8405	2.9784	1.1379
	MCS-10	2.2695	1.0068	0.1750	0.0780	1.8250	3.0176	1.1925
	SFD2-15	2.2860	1.0141	0.2696	0.1193	1.7171	3.4363	1.7193
	SFD4-15	2.2885	1.0152	0.2706	0.1197	1.7193	3.4824	1.7632
	MCS-15	2.2898	1.0158	0.2721	0.1203	1.6680	3.6517	1.9837
	SFD2-20	2.3126	1.0259	0.3754	0.1643	1.6516	4.0313	2.3797
	SFD4-20	2.3173	1.0280	0.3778	0.1650	1.6502	4.1104	2.4602
	MCS-20	2.3206	1.0294	0.3844	0.1677	1.5365	4.6661	3.1296
Case II (Uncertainty Pressure gradient)	SFD2-10	2.2543	1.0000	0.1744	0.0783	1.7074	2.8052	1.0978
	SFD4-10	2.2543	1.0000	0.1763	0.0792	1.6544	2.8111	1.1566
	MCS-10	2.2559	1.0007	0.1835	0.0823	1.6150	2.8186	1.2036
	SFD2-15	2.2543	1.0000	0.2616	0.1174	1.4340	3.0807	1.6467
	SFD4-15	2.2543	1.0000	0.2645	0.1188	1.3545	3.0895	1.7350
	MCS-15	2.2568	1.0011	0.2753	0.1235	1.2954	3.1008	1.8054
	SFD2-20	2.2543	1.0000	0.3488	0.1566	1.1606	3.3562	2.1955
	SFD4-20	2.2543	1.0000	0.3527	0.1584	1.0546	3.3679	2.3133
	MCS-20	2.2576	1.0015	0.3671	0.1646	0.9758	3.3830	2.4072

,

Table 3. Bingham fluid: maximum velocity statistics for the two cases.

Case	Method of Solution	${\mathit{\mu }}_{\mathcal{w}}$	${\mathit{\mu }}_{\mathcal{w}}/{\mathcal{w}}_{\mathit{D}}$	${\mathit{\sigma }}_{\mathcal{w}}$	${\mathit{\sigma }}_{\mathcal{w}}/{\mathit{\mu }}_{\mathcal{w}}$	$\widetilde{\mathit{M}\mathit{i}\mathit{n}}$	$\widetilde{\mathit{M}\mathit{a}\mathit{x}}$	$\tilde{\mathit{R}}$
Case I (Uncertainty Viscosity)	SFD2-10	2.0280	1.0059	0.1536	0.0757	1.6486	2.6317	0.9831
	SFD4-10	2.0287	1.0063	0.1538	0.0757	1.6493	2.6377	0.9884
	MCS-10	2.0292	1.0065	0.1538	0.0757	1.6358	2.6686	1.0328
	SFD2-15	2.0434	1.0128	0.2373	0.1160	1.5392	3.0530	1.5137
	SFD4-15	2.0453	1.0137	0.2379	0.1162	1.5407	3.0665	1.5259
	MCS-15	2.0466	1.0143	0.2387	0.1166	1.4960	3.1979	1.7018
	SFD2-20	2.0663	1.0241	0.3300	0.1596	1.4790	3.5716	2.0925
	SFD4-20	2.0698	1.0267	0.3315	0.1600	1.4778	3.5975	2.1197
	MCS-20	2.0726	1.0272	0.3359	0.1619	1.3788	4.0195	2.6407
Case II (Uncertainty Pressure gradient)	SFD2-10	2.0161	1.0000	0.1548	0.0767	1.5301	2.5051	0.9750
	SFD4-10	2.0161	1.0000	0.1562	0.0774	1.4876	2.5060	1.0183
	MCS-10	2.0174	1.0006	0.1625	0.0805	1.4538	2.5182	1.0644
	SFD2-15	2.0161	1.0000	0.2322	0.1151	1.2871	2.7496	1.4625
	SFD4-15	2.0161	1.0000	0.2343	0.1161	1.2234	2.7509	1.5275
	MCS-15	2.0181	1.0010	0.2437	0.1207	1.1727	2.7693	1.5966
	SFD2-20	2.0161	1.0000	0.3096	0.1534	1.0441	2.9941	1.9500
	SFD4-20	2.0161	1.0000	0.3123	0.1548	0.9592	2.9959	2.0367
	MCS-20	2.0187	1.0007	0.3250	0.1609	0.8916	3.0204	2.1288

and

Table 4. Casson fluid: maximum velocity statistics for the two cases.

Case	Method of Solution	${\mathit{\mu }}_{\mathcal{w}}$	${\mathit{\mu }}_{\mathcal{w}}/{\mathcal{w}}_{\mathit{D}}$	${\mathit{\sigma }}_{\mathcal{w}}$	${\mathit{\sigma }}_{\mathcal{w}}/{\mathit{\mu }}_{\mathcal{w}}$	$\widetilde{\mathit{M}\mathit{i}\mathit{n}}$	$\widetilde{\mathit{M}\mathit{a}\mathit{x}}$	$\tilde{\mathit{R}}$
Case I (Uncertainty Viscosity)	SFD2-10	1.9003	1.0050	0.1468	0.0752	1.5871	2.5269	0.9398
	SFD4-10	1.9510	1.0054	0.1470	0.0753	1.5875	2.5270	0.9395
	MCS-10	1.9515	1.0056	0.1469	0.0752	1.5745	2.5551	0.9806
	SFD2-15	1.9557	1.0134	0.2257	0.1148	1.4749	2.9147	1.4398
	SFD4-15	1.9573	1.0142	0.2262	0.1149	1.4759	2.9172	1.4413
	MCS-15	1.9585	1.0148	0.2267	0.1151	1.4335	3.0357	1.6021
	SFD2-20	1.9866	1.0237	0.3151	0.1585	1.4233	3.4220	1.9987
	SFD4-20	1.9896	1.0253	0.3164	0.1589	1.4221	3.4300	2.0079
	MCS-20	1.9922	1.0266	0.3200	0.1605	1.3276	3.8088	2.4811
Case II (Uncertainty Pressure gradient)	SFD2-10	1.9299	1.0000	0.1484	0.0765	1.4726	2.4080	0.9354
	SFD4-10	1.9299	1.0000	0.1497	0.0771	1.4337	2.4073	0.9736
	MCS-10	1.9311	1.0006	0.1556	0.0802	1.4019	2.4212	1.0193
	SFD2-15	1.9299	1.0000	0.2216	0.1142	1.2336	2.6301	1.3964
	SFD4-15	1.9299	1.0000	0.2234	0.1151	1.1754	2.6290	1.4536
	MCS-15	1.9317	1.0009	0.2324	0.1196	1.1281	2.6498	1.5217
	SFD2-20	1.9299	1.0000	0.2969	0.1530	1.0062	2.8770	1.8707
	SFD4-20	1.9299	1.0000	0.2993	0.1542	0.9283	2.8756	1.9472
	MCS-20	1.9323	1.0013	0.3113	0.1602	0.8648	2.9034	2.0386

for the two uncertainty cases. These results, generated for the power-law, Bingham, and Casson models, respectively, correspond to three coefficients of variation ($\mathcal{k}$) with fixed parameters $\alpha =1$, ${\tau }_D=0.5$, and $n=1.5$. The results for SFD4 were presented at $\mathcal{k}$ = 10% and 20%, while results for SFD2, SFD4, and QMCS were provided at $\mathcal{k}$ = 15% to assess the convergence of the applied solution methods. Furthermore,

Table 5. Comparative maximum velocity statistics for five non-Newtonian fluids (Cases I and II).

Case	Model	${\mathit{\mu }}_{\mathcal{w}}$	${\mathit{\mu }}_{\mathcal{w}}/{\mathcal{w}}_{\mathit{D}}$	${\mathit{\sigma }}_{\mathcal{w}}$	${\mathit{\sigma }}_{\mathcal{w}}/{\mathit{\mu }}_{\mathcal{w}}$	$\widetilde{\mathit{M}\mathit{i}\mathit{n}}$	$\widetilde{\mathit{M}\mathit{a}\mathit{x}}$	$\tilde{\mathit{R}}$
Case I (Uncertainty Viscosity)	Herschel–Bulkley [28]	2.3054	1.0152	0.2729	0.1199	1.7115	3.4200	1.7085
	Power-law	2.2885	1.0152	0.2706	0.1197	1.7193	3.4824	1.7632
	Robertson–Stiff [27]	2.2627	1.0151	0.2671	0.1194	1.7017	3.6421	1.9404
	Bingham	2.0453	1.0145	0.2379	0.1162	1.5407	3.0665	1.5259
	Casson	1.9573	1.0142	0.2262	0.1149	1.4759	2.9172	1.4413
Case II (Uncertainty Pressure gradient)	Herschel–Bulkley [28]	2.2709	1.0000	0.2667	0.1190	1.3892	3.1893	1.8001
	Power-law	2.2543	1.0000	0.2645	0.1188	1.3545	3.0895	1.7350
	Robertson–Stiff [27]	2.2291	1.0000	0.2613	0.1185	1.2493	3.1102	1.8608
	Bingham	2.0161	1.0000	0.2343	0.1161	1.2234	2.7509	1.5275
	Casson	1.9299	1.0000	0.2234	0.1151	1.1754	2.6290	1.4536

presents the key statistics for the three investigated fluids compared with Herschel–Bulkley (HB) [28] and Robertson–Stiff (RS) [27] fluids under the same two uncertainty scenarios at $\mathcal{k}$ = 15%.

A series of figures illustrate the results further. For the power-law fluid, Figure 2a illustrates the mean and the S.D. along the duct’s centerline, specifically along the $\mathcal{x}$-axis at the midpoint of the $\mathcal{y}$-axis (i.e., $\mathcal{y}=\alpha /2$), for $\alpha =1$ and $n=1.5$ in Case I, across $\mathcal{k}$ = 10%, 15%, and 20%, for all solution methods. Figure 2b presents the PDFs for maximum velocity using all solution methods. Figure 3a,b display contour plots of the mean and S.D., respectively, across all points in the duct’s quarter section using SFD4 with $\mathcal{k}$ = 15%. The same metrics for the power-law fluid in Case II are depicted in Figure 4 and Figure 5. Similarly, Figure 6 and Figure 7 illustrate these metrics for the Bingham fluid, while Figure 8 and Figure 9 present the same metrics for the Casson fluid.

The figures and tables above summarize the key findings. For the power-law fluid, uncertain viscosity in Case I had a minimal impact on the maximum velocity mean, which increased by only 0.66%, 1.52%, and 2.8% of ${\mathcal{w}}_D$ for $\mathcal{k}$ = 10%, 15%, and 20%, respectively. However, the maximum velocity S.D. was substantially affected, with the S.D. ranging from 77.9% to 82.5% of the COV across these $\mathcal{k}$ values. Additionally, Case I exhibited a significant variation range in ${\mathcal{w}}_{max}$, with values of 1.1379, 1.7632, and 2.4602 for the respective $\mathcal{k}$ values. The ($\widetilde{Min}$, $\widetilde{Max}$) range reached (71.2%, 177.3%) of the velocity mean at $\mathcal{k}$ = 20%. In contrast, Case II, which accounted for an uncertain pressure gradient, showed no effect on the mean of ${\mathcal{w}}_{max}$. Furthermore, it demonstrated slightly lower S.D.s and narrower ranges across all three COV values, with ($\widetilde{Min}$, $\widetilde{Max}$) reaching (46.7%, 149.4%) at $\mathcal{k}$ = 20%.

For the Bingham fluid, Case I also showed minor increases in the mean ${\mathcal{w}}_{max}$ as 0.63%, 1.37%, and 2.67% relative to ${\mathcal{w}}_D$ for $\mathcal{k}$ = 10%, 15%, and 20%, respectively. The S.D.s were 75.7%, 77.5%, and 80% of the COV across these $\mathcal{k}$ values. The ${\mathcal{w}}_{max}$ ranges were 0.9884, 1.5259, and 2.1197, with the ($\widetilde{Min}$, $\widetilde{Max}$) range reaching (71.4%, 173.8%) of the velocity mean at $\mathcal{k}$ = 20%. In contrast, Case II had no effect on the mean ${\mathcal{w}}_{max}$, exhibiting slightly lower S.D.s and a narrower range. Here, the ($\widetilde{Min}$, $\widetilde{Max}$) range was (47.6%, 148.6%) of the mean value at $\mathcal{k}$ = 20%.

Casson fluid exhibited similar behavior in both uncertainty cases. In Case I, the mean increases in ${\mathcal{w}}_{max}$ relative to the ${\mathcal{w}}_D$ were 0.54%, 1.42%, and 2.53% for uncertainty levels $\mathcal{k}$ = 10%, 15%, and 20%, respectively. The S.D.s corresponded to 75.3%, 76.6%, and 79.5% of the COV across these $\mathcal{k}$ values. The ${\mathcal{w}}_{max}$ ranges were 0.9395, 1.4413, and 2.0079, with the ranges ($\widetilde{Min}$, $\widetilde{Max}$) spanning 71.5% to 172.4% of the velocity mean at $\mathcal{k}$ = 20%. In contrast, Case II showed no effect on the mean ${\mathcal{w}}_{max}$, with slightly lower S.D.s and a narrower range of 48.1% to 149.0% of the mean velocity at $\mathcal{k}$ = 20%.

In conclusion, all three fluids exhibited minor increases in the mean value of ${\mathcal{w}}_{max}$ for Case I, while Case II showed no increase. The S.D.s ranged from 75.3% to 82.5% of the COV of the investigated uncertain parameters. Case I displayed a longer right tail, with maximum values deviating farther from the mean than the minimum values. In contrast, Case II produced nearly symmetrical PDF curves. Given the substantial ranges of ${\mathcal{w}}_{max}$ observed, these variations should be carefully accounted for in both the analysis and design of such duct systems.

5.2. Investigation of Key Fluid Parameters’ Impact

To further investigate the influence of key parameters on fluid behavior, Figure 10, Figure 11 and Figure 12 present the effects of aspect ratio; $\alpha$ = 1, 0.75, and 0.5, dimensionless yield stress; ${\tau }_D$= 1, 0.5, and 0, and flow behavior index; $n$ = 1.5, 1, and 0.5 on both the mean velocity and its S.D. at the duct’s centerline. The investigation considers Case I using SFD4 with $\mathcal{k}$ = 15%. Figure 10 examines the power-law fluid, with panels (a) and (b) showing the effects of $\alpha$ and $n$, respectively. Figure 11 presents results for the Bingham fluid, where panels (a) and (b) demonstrate the influence of $\alpha$ and ${\tau }_D$. Similarly, Figure 12 analyzes the Casson fluid, with panels (a) and (b) illustrating the impact of $\alpha$ and ${\tau }_D$.

This analysis revealed that decreasing aspect ratio ($\alpha$) increases mean velocity near the walls ($0\lesssim x\lesssim 0.35$ for power-law, $0\lesssim x\lesssim 0.36$ for Bingham, and $0\lesssim x\lesssim 0.38$ for Casson fluids) before decreasing toward the centerline, with lower $\alpha$ values exhibiting higher S.D. values. For power-law fluids, both mean velocity and S.D. increase with decreasing $n$ within $0\lesssim x\lesssim 0.22$ but decrease beyond this interval. For Bingham and Casson fluids, lower ${\tau }_D$ values slightly reduce both velocity metrics within $0\lesssim x\lesssim 0.25$ while increasing them in outer regions.

Changes in aspect ratio ($\alpha$), flow index ($n$), and yield stress (${\tau }_D$) have a substantial effect on duct design, as the stochastic analysis shows. While a lower α can improve heat transfer, it also accelerates wall erosion and necessitates stronger duct materials because it increases near-wall velocity with a flatter velocity profile. A lower $n$ (strong shear-thinning) for power-law fluids results in a flatter velocity profile too, but it also increases flow unpredictability and increases the possibility of core flow stagnation. A higher ${\tau }_D$ for yield-stress fluid requires stronger pumps to start the flow and produces a bigger, more stable central plug, which is better for moving suspensions but renders the system extremely vulnerable to changes in material properties. Therefore, to account for these large velocity variations and guarantee system reliability, designers need to employ stochastic models to add conservative safety factors.

5.3. Comparative Stochastic Analysis of Five Non-Newtonian Fluid Models

The present study’s solutions, along with published results for Herschel–Bulkley [28] and Robertson–Stiff fluids [27], allow for a comprehensive comparison of a class of five non-Newtonian fluid behaviors in rectangular ducts. Table 5 summarizes the key statistics of ${\mathcal{w}}_{max}$ for both uncertainty cases examined. Figure 13a displays the mean velocity and its S.D. across the duct centerline for these fluids using SD4 at $\mathcal{k}$ = 0.15, while Figure 13b presents the PDFs of ${\mathcal{w}}_{max}$ for Case I. Corresponding results for Case II are shown in Figure 14a,b.

The results presented in Table 5 and Figure 13 and Figure 14 reveal several key findings. For both uncertainty cases, the highest values of mean velocity (${\mu }_{\mathcal{w}}$), velocity ratio (${\mu }_{\mathcal{w}}/{\mathcal{w}}_D$), and S.D. (${\sigma }_{\mathcal{w}}$) at the duct center were observed for Herschel–Bulkley fluid, followed by power-law, Robertson–Stiff, Bingham, and Casson fluids. Notably, ${\mu }_{\mathcal{w}}/{\mathcal{w}}_D$ reached unity in Case II for all fluids. This velocity hierarchy remained consistent within the interval $0.21\lesssim x\lesssim 0.50$ but reversed outside this region. The Robertson–Stiff fluid exhibited the broadest velocity range, with $\tilde{R}$ = 1.9404 (Case I) and $\tilde{R}$ = 1.8608 (Case II), corresponding to ($\widetilde{Min,}\widetilde{Max}$) ranges of (75.21%, 160.96%) and (56.05%, 139.53%) of the mean value, respectively. The PDFs showed distinct tail behavior: Case I displayed longer right tails for all fluids, indicating a higher probability of elevated velocities, while Case II exhibited nearly symmetric distributions. These results provide valuable insights into the behavior of these five non-Newtonian fluid models in rectangular duct flows.

5.4. Analysis of the Applied Solution Methods

Deterministically, the problem under investigation was discretized into a $\left(\mathcal{Q}+1\right)\times \left(\mathcal{R}+1\right)$ grid, totaling 2601 mesh nodes and thus generating an equivalent number of linear equations. This system was iterated 11 times to meet the convergence criterion outlined in Equation (30), leveraging an initial velocity estimate as detailed in Section 3.2. The QMCS method achieves faster convergence than traditional MCS by employing deterministic, low-discrepancy sequences for more uniform sampling, thereby improving accuracy with fewer simulations. It utilized $2\times {10}^3$ random inputs ($N_{\mathrm{R}}$), generating $\left[\left(\mathcal{Q}+1\right)\times \left(\mathcal{R}+1\right)\times N_{\mathrm{R}}\right]$ linear equations, totaling $5202\times {10}^3$, which underscores the significant computational demand. For robust MCS results, $N_{\mathrm{R}}$ is typically set around $1\times {10}^5$, though often lower for QMCS [38]. Due to limited computational resources, $N_{\mathrm{R}}$ was restricted, with all calculations performed serially on a computer equipped with an 11th Gen Intel^® Core™ i7-1195G7 processor @ 2.90 GHz and 16 GB RAM. This constraint underscores the enhanced reliability of SFDHC’s results in this study. Conversely, the SFDHC addresses $\left[\left(\mathcal{Q}+1\right)\times \left(\mathcal{R}+1\right)\times N_{PC}\right]$ linear equations, yielding 15,606 and 39,015 for SFD2 and SFD4, respectively. This demonstrates that SFDHC is about $\left(N_{\mathrm{R}}/N_{PC}\right)$-fold faster than QMCS.

For higher-dimensional or complex geometries, increasing node count to capture response variability escalates computational cost. The value of $N_{PC}$ depends on both the number of dimensions ($N_{KL}$) and the polynomial chaos order ($p$). Typically, lower values are used initially, with $N_{KL}$ and $p$ adjusted until convergence is achieved, as demonstrated here with the close agreement between SFD2 and SFD4 and further verified against QMCS results. More intricate uncertainties require higher $N_{PC}$, raising computational expense. Nevertheless, this study demonstrates SFDHC’s lower computational cost compared to QMCS, with a cost ratio of 243, 207, 274, 203, and 221 times faster for HB, PL, RS, BH, and CA fluids, respectively [27,28].

Additionally, SFDHC surpasses collocation by offering exponential convergence and a complete response function solution, not mere point estimates, at an optimized computational cost. Moreover, SFDHC can be viewed as an extension of the classical Finite Difference Method (FDM) for stochastic systems. If a problem is solvable with FDM, it can be addressed with SFDHC, regardless of complexity, nonlinearity, geometry, dimensions, or other factors [38].

6. Conclusions

This study pioneers the analysis of power-law (PL), Bingham (BH), and Casson (CA) fluid behavior under conditions of uncertain viscosity and stochastic pressure gradients, a novel approach examined through two distinct benchmark scenarios (Case I and Case II). While all three fluids showed slight increases in the mean of maximum stochastic velocity (${\mathcal{w}}_{max}$) for Case I, no such increase was observed in Case II. The standard deviations (S.D.) ranged from 75.3% to 82.5% of the coefficient of variation (COV) of the uncertain parameters. In Case I, the probability density function (PDF) of maximum velocity exhibited a longer right tail, whereas Case II produced nearly symmetrical PDF curves. Given the wide variations in ${\mathcal{w}}_{max}$, these findings should be carefully considered in the analysis and design of duct systems. Additionally, the study explored the effects of aspect ratio ($\alpha$), yield stress (${\tau }_D$), and flow behavior index ($n$) on each fluid’s stochastic response. Reducing $\alpha$ was found to increase mean velocity near the walls before a subsequent decline toward the centerline. For power-law fluids, both mean velocity and S.D. rose with decreasing $n$ within the interval $0\lesssim x\lesssim 0.22$, but decreased beyond this range. In contrast, Bingham and Casson fluids showed slightly reduced velocities and S.D. at lower ${\tau }_D$ values within $0\lesssim x\lesssim 0.25$, while exhibiting increases in outer regions. A comparison with published data on Herschel–Bulkley (HB) and Robertson–Stiff (RS) fluids revealed that the HB fluid had the highest mean velocity and S.D., followed by PL, RS, BH, and CA fluids. This velocity hierarchy remained consistent within $0.21\lesssim x\lesssim 0.50$ but reversed outside this range. The RS fluid displayed the broadest velocity range, with normalized ranges ($\tilde{R}$) of 1.9404 (Case I) and 1.8608 (Case II), corresponding to intervals of (75.21%, 160.96%) and (56.05%, 139.53%) of the mean value, respectively. Future study may include implementing different fluid models in varying geometries. The SFDHC method extends the classical Finite Difference Method (FDM) to stochastic systems, effectively handling any problem solvable by FDM, regardless of complexity, nonlinearity, geometry, or dimensions. It demonstrates high efficiency, performing over 200 times faster than the applied QMCS. The findings of this study hold significant implications for real-world fluid systems, particularly in industries where precise flow control is critical, such as chemical processing, petroleum engineering, biomedical devices, and food manufacturing. The observed high variations in maximum velocity across all fluid models underscore the necessity for robust design strategies to accommodate the significant variations revealed in this study. Future studies may extend this approach to higher dimensions and more complicated geometries for different fluid models. This would entail analyzing heat transfer and connecting the model with uncertain thermal conditions, as well as examining multiple simultaneous uncertainty effects, which would provide a more comprehensive and accurate risk assessment and design for such engineering systems.