This study explores the steady flow and heat transfer of an Ostwald–de Waele (power-law) fluid across a cylinder with nonlinear stretching, encompassing convective heating, nonlinear radiation, Joule heating, and a variable magnetic field. The governing boundary-value problem is worked out using MATLAB’s BVP4C collocation scheme and justified against existing literature. Three fluid types pseudoplastic, Newtonian, and dilatant fluids are observed to assess the influence of shear-dependent viscosity. The findings show that stretching nonlinearity plays a primary role in transport phenomena: nonlinear stretching (m = 2) consistently leads to higher skin-friction coefficients and larger Nusselt numbers than linear stretching, indicating strengthened near-wall gradients. Thermal responses depend on the controlling factor Ec, M, κ, ϕ, and R produce higher temperatures under linear stretching, whereas Pr reduces temperature more effectively under nonlinear stretching. Curvature enhances heat removal for all fluids, while magnetic damping suppresses heat transfer. These findings offer guidance for polymer extrusion and thermal processing of cylindrical materials.

We consider the axisymmetric, radial extrusion of Newtonian and shear-thinning, power-law fluids from a cylindrical source, which displace an ambient inviscid fluid of equal density. In unconfined geometries, the upper and lower fluid interfaces are stress free, and the flow is dominated by extensional stresses everywhere. In a layer of extruded shear-thinning fluid, a radially growing viscosity field, associated with a radially decaying velocity field, causes the current to bulge near the cylindrical source, with the thickness of the layer growing without bound over time. In contrast, with a Newtonian fluid, the thickness of the fluid layer never exceeds the height of the cylindrical source. We compute numerical solutions to this system, and find similarity solutions describing its late-time behaviour for values of the rheological power-law exponent $1\leqslant n\leqslant 3/2$ . We also consider extrusion between parallel plates, in which the shear-thinning fluid displaces the inviscid fluid and fills the cell completely up to a grounding line, beyond which it separates from the boundaries to extend freely. In this case, we find similarity solutions for values of the power-law exponent $n \geqslant 1$ .

ABSTRACT In order to reveal the complexity of the internal flow of bubble plume in power law fluid, the flow characteristics and chaotic characteristics of plume are studied by experiment and theory. The chaotic characteristic parameters (correlation dimension D , K entropy, and Lyapunov exponent λ ) of gas velocity under different superficial gas velocities and liquid concentrations are analyzed. With the increase of superficial gas velocity and concentration of carboxymethyl cellulose sodium (CMC) aqueous solution, the values of chaotic characteristic parameters all increase. The rheological characteristics of the liquid phase have a significant effect on the flow field and chaotic characteristics. According to the power spectrum of velocity signal, the chaotic characteristics of the velocity are significant in power law fluid, and the energy of the flow field is high. The disorder of the gas phase becomes intense with the motion of the plume, and the energy of velocity signal increases. The complexity of gas–liquid two‐phase flow field was revealed, laying the foundation for the study of the flow mechanism of gas–liquid two‐phase flow.

A phenomenological model for the heat transfer coefficient in turbulent pipe flow of shear-thinning power-law fluids

During the drilling stage of oil wells, the flow of drilling fluids through the annular space is influenced by discontinuities and high-permeability regions in the geological formation. This phenomenon, known as lost circulation, reduces operational efficiency and compromises well integrity. One of the preventive control methods for this issue is the selection of a drilling fluid whose formulation is best suited to the drilling phase and well characteristics. This process can be assisted through numerical simulations using CFD (Computational Fluid Dynamics), where the annular space is simplified as a vertical channel, fractures are modeled as transverse channels with constant thickness through which leakage flow occurs, and the high-permeability geological formation is treated as a porous medium. This characterization refers to a partially porous and fractured channel (PPFC), modeled as a bidisperse porous medium, where both the free-flow and porous regions are considered homogeneous. This study aims to analyze lost circulation in a PPFC with multiple discrete fractures. The flow is assumed to be steady, incompressible, and the fluid is modeled as a power-law fluid. The porous medium is considered isotropic. Flow in the free channel and fractures is governed by the Navier-Stokes equations, while flow in the high-permeability region is described by the generalized porous media equation proposed by Vafai and Tien (1981). Numerical simulations are performed using the finite volume method in OpenFOAM version 24.06, employing an adapted version of the porousSimpleFoam solver to represent non-Newtonian fluid flow in porous media. The simulations are set up with a porosity of ε = 0.7 and permeability of 10⁻⁶ m². The drilling fluid is idealized with a consistency index k = 0.205 Pa·sⁿ, a power-law index n = 0.568, a density ρ = 1012.60 kg/m³, and a Reynolds number Re = 1000, calculated based on the annular space. The results are obtained from combinations of 2 or 3 fractures with thicknesses eₓ = 4, 16, and 40 mm, spaced at distances h = 100, 200, and 400 mm. The analysis focuses on evaluating the velocity, pressure, and viscosity fields, the leakage rate, and the pressure drop along the channel.

This study introduces a computationally efficient, second-order precise algorithm for investigating thermal power-law fluids in stick-slip flows under various thermal boundary conditions. The study includes two main contributions. First, it introduces a novel numerical algorithm based on a modified artificial compressibility formulation that employs the Taylor-Galerkin finite element method. To demonstrate its efficacy, we compare this algorithm with an equivalently accurate scheme, the Taylor Galerkin/pressure correction algorithm, highlighting the superior convergence characteristics of the proposed algorithm, which requires nearly half the number of time steps. Second, investigation highlights the critical influence of disregarding the temperature dependence of viscous fluid on key parameters, such as viscous dissipation and heat transfer rate. Ignoring this dependence can lead to substantial underestimations of these parameters, thus affecting the simulation's accuracy. Additionally, we examine the influence of dimensionless parameters on average and local Nusselt numbers, finding remarkable agreement with previous findings. Highlights Proposed a high-accuracy algorithm for non-Newtonian thermal flows based on the extension of artificial compressibility equations. Combined of two-step Lax-Wendroff scheme and Galerkin finite element methods in the proposed algorithm. Compared the proposed algorithm with the Taylor-Galerkin pressure correction algorithm. Results demonstrate the accuracy and computational efficiency of the proposed algorithm at low Mach numbers. Investigated the consequences of ignoring the temperature dependence of a fluid's viscosity on pressure drop, viscous dissipation, and heat transfer under different thermal boundary conditions and stick-slip scenarios.

Abstract This paper conducts a numerical analysis of the sedimentation dynamics of two-dimensional rigid circular particles suspended in a power-law fluid flow within a computational channel. The fluid dynamics are represented by the power-law model, encompassing three specific flow conditions: shear-thinning (n = 0.9), Newtonian (n = 1), and shear-thickening (n = 1.1). The finite element technique (FEM) is integrated with the fictitious boundary method (FBM) to address the fluid-particle interaction issue. The particles, with radii R1 = 0.2 and R2 = 0.125, display intricate behaviors such as drafting, kissing, and tumbling (DKT) during sedimentation. The research investigates the influence of particle size and initial particle placement on their behavior in the fluid. Principal findings reveal that shear-thinning fluids (n < 1) precipitate faster particle interactions, heightened instability, and expedited transitions into the tumbling phase, whereas shear-thickening fluids (n > 1) enhance wake stability, resulting in delayed interactions and more gradual settling. The research underscores the importance of the power-law index in ascertaining the overall particle dynamics and fluid-particle interaction.
This research enhances the comprehension of particle movement in non-Newtonian fluids and has ramifications for diverse engineering applications, including sedimentation processes, fluid transport, and industrial processing. The simulations are performed using the FEATFLOW Software package which utilizes the finite element method (FEM) for the numerical solutions.

While anisotropic extrudate swell in polymer processing is fundamentally driven by physical viscoelastic recovery, this paper proposes a theoretical framework to explicitly isolate and map the purely geometric and kinematic components of this phenomenon. Serving as a mathematical proof-of-concept, a multi-velocity-centre hypothesis is proposed. By introducing a semi-empirical, lumped material-flow calibration parameter, the macroscopic diameter swell ratio is mathematically extended to the discrete local flow field of a rectangular slit die. To evaluate its validity, the analytical framework is subjected to a numerical test for kinematic consistency utilizing isothermal, inelastic power-law fluid CFD simulations, thereby separating geometric mapping from complex viscoelastic stress relaxation. Results indicate that analytical predictions show good agreement with CFD data (error < 5%) strictly within the core zone of high-aspect-ratio dies. However, due to the infinite-slit assumption, 3D flow kinematics near die edges induce velocity decay, leading to local deviations that require future empirical corrections. Although comprehensive physical extrusion experiments and non-isothermal viscoelastic coupling are required for industrial deployment, this semi-empirical kinematic mapping provides a foundational mathematical basis that could potentially inform future inverse die-profile design and shape distortion compensation.

Experimental study on three-dimensional atomization characteristics and droplet size-velocity correlation of power-law fluid in liquid-liquid pintle injectors

Heat transfer and flow dynamics of power-law fluids in a lid-driven cavity with a flexible flow modulator

• A refined model for power-law fluid flow in eccentric annuli is proposed. • The hybrid method precisely locates the zero-shear-stress surface. • Directly links flow rate to pressure drop, solving a key engineering problem. • Validated against CFD and experimental data with excellent accuracy. • Quantifies the non-linear impact of eccentricity and flow index on pressure loss. Predicting laminar power-law fluid flow in eccentric annuli is crucial for managing wellbore pressure in complex drilling operations. This study presents a refined semi-analytical model that rigorously implements the concentric annular superposition principle. The core novelty lies in a hybrid numerical-analytical technique that employs hypergeometric functions and the bisection method to precisely locate the zero-shear-stress position within each discretized element, thereby overcoming the inaccuracies of previous approximation-based methods. This framework enables the direct and accurate calculation of key hydraulic parameters—including asymmetric velocity profiles, shear distributions, and frictional pressure drops—from given engineering flow rates. The model's high precision is demonstrated by reducing velocity continuity errors at the zero-shear interface to below 0.0007 m/s. Key findings confirm that peak velocity and shear occur on the wide-gap side, while increasing eccentricity significantly reduces frictional pressure gradients. Notably, at high eccentricities (ε > 0.4), the influence of the flow behavior index (n) becomes more pronounced, with fluids exhibiting higher n values showing lower frictional pressure gradients.

This article examines the thermohaline stability of a power-law fluid saturating a porous layer in the presence of a magnetic field. The system stability is analyzed using both linear and weakly nonlinear instability theories. Within the linear framework, the Galerkin method is employed to derive analytical expressions for the Rayleigh number corresponding to steady and oscillatory modes of instability. Takens–Bogdanov and Hopf bifurcation points are identified, highlighting the transition mechanisms between different instability regimes. An increase in the Hartmann number delays the onset of convection. The critical Rayleigh number is a monotonic increasing function of the solute Rayleigh number, whereas it is a non-monotonic function of the Peclet number. To investigate heat and mass transport characteristics, an amplitude equation is derived in the weakly nonlinear regime. The results reveal that increasing the Hartmann, Lewis, and Peclet numbers enhances both heat and mass transport, whereas an opposite trend is observed with increasing the solute Rayleigh number.

Optical coherence tomography velocimetry (OCTV) was demonstrated with in-line processing of biologics for the first time. OCTV allowed the velocity of concentrated monoclonal antibodies (mAbs at 39.5-84.7 mg mL-1) to be probed in 3.4 pL volumes over distances 0-5 mm from the pipe walls. The large penetration depth is facilitated by the relatively low turbidity of mAbs at near-infrared wavelengths (1300 nm). The mAb solutions could be concentrated in situ and the changes to the viscoelasticity measured. Higher concentration mAb solutions became shear thinning (following the power law fluid model) and the amplitude of their velocity fluctuations decreased. Furthermore, dropping the pH of the mAb solutions induced a gelation phase transition and complex changes to the mAb rheology could be observed with OCTV e.g. thixotropy and the formation of a stationary boundary layer. Thus, in situ formulation of mAbs could be explored with OCTV under industrially relevant conditions.

This study investigates mixed forced convection heat transfer of a non-Newtonian power-law fluid flowing in a thick-walled rectangular channel subjected to a sinusoidal wall temperature with a non-zero mean value. Viscous dissipation is fully taken into account, whereas axial conduction in the fluid is neglected under the assumption of high Peclet numbers. An asymptotic analytical solution for the temperature field is developed, from which the local Nusselt number is obtained, while the average Nusselt number is computed numerically. The influence of the power-law index n, Brinkman number Br, Peclet number Pe, and wall thickness ratio on heat transfer and temperature distribution is systematically analyzed. The results show that decreasing n (shear-thinning behavior) increases the velocity, reduces the bulk temperature, and significantly enhances the average Nusselt number, especially at higher dimensionless frequencies of the imposed sinusoidal temperature. The originality of this work lies in the simultaneous consideration of non-Newtonian rheology, viscous dissipation and conjugate heat transfer in a thick channel under sinusoidal thermal excitation, providing useful guidelines for the design and optimization of systems involving polymers, lubrication, compact heat exchangers and biomedical flows.

The Numerical Study of flow behaviour in a two- dimensional square lid- driven cavity containing power- law fluids. The work aims to investigate the influence of the variation of the moving length of the top lid on the fluid flow behaviour, velocity profiles and vortex creation. The power- law fluid model is employed to simulate non- Newtonian fluids with power-law indices varying from shear- thinning (n<1) to shear- thickening (n>1) fluids. The finite volume method with a staggered grid arrangement and the simple algorithm for the pressure- velocity coupling technique is employed to solve the governing equations. The results show that a decrease in the moving length of the top lid results in localized flow perturbations with substantial influences on the primary vortex intensity and secondary vortex formation. The flow penetration inside the cavity is deeper for shear- thinning fluids with a smaller moving length, while resistance to flow is higher for shear- thickening fluids. The Reynolds number is held constant at Re=100 for all computations to demonstrate the dominance of viscous forces.

Abstract The dynamics of solid particles and fluid solid interaction in particulate flows with a power law model are examined. An Eulerian approach is employed to simulate this flow across a fixed computational grid, utilizing the fictitious boundary method to efficiently handle complex geometries. The multigrid finite element solver FEATFLOW is extended to account for non-Newtonian fluids. Three types of rheological behavior are considered: (i) shear-thinning (n = 0.9), (ii) Newtonian (n = 1), and (iii) shear-thickening (n = 1.1). Numerical experiments are conducted with two falling particles, focusing on the drafting, kissing, and tumbling (DKT) phenomena in power-law fluids. Results from these experiments are presented and compared across shear-thinning, Newtonian, and shear-thickening fluids by varying the initial particle positions and the particle radii. The effects of particle-particle interactions on particle motion and the overall behavior of the fluid-particle system are analyzed. This study validates the Fictitious Boundary Method (FBM) by comparing it with benchmark results and offers valuable insights into particle-fluid interactions.

A simple approach for effective CFD simulation of turbulent pipe transport of shear-thinning, power-law fluids

Abstract The red fruit oil (RFO) ( Pandanus conoideus Lam) emulsion has potential application as a natural colorant in food products. The characteristics of the emulsion can be influenced by the oil, emulsifier, and stabilizer used. This research aimed to determine the effects of different RFO brands and pH emulsion on the characteristics of RFO emulsion. The factors used in this research were oil brands (A, B, C, D, E) and pH levels (3.5, 7, 8). The formulation of O/W emulsion consists of 20% oil (w/v), 1% of polysorbate 80, and 0.3% of xanthan gum. The results show that different oil brands influence the total carotenoid content. The emulsions have an orange color, with total carotenoid content around 644.31 – 1,213.48 ppm. The result showed that the pH level only influences the carotenoid content. There is no difference in rheological properties of emulsions with different pH levels. The emulsions in this study exhibited non-Newtonian fluidity, specifically shear-thinning behavior, and were characterized by the Power Law fluid model. The emulsions remain visually stable during one month of storage. These findings support sustainability by utilizing locally sourced natural oil as natural colorant emulsion based, that contributes to sustainable food system.

Development and testing of a new pore network algorithm for modeling flows of power law fluids in porous media

Multiphase lattice Boltzmann flux solver for non-Newtonian power-law fluid flows with high efficiency and stability