Journal of Non-Newtonian Fluid Mechanics最新文献_第4页

The influence of viscoelasticity on the dynamics of encapsulated microbubbles near a rigid surface forced by ultrasound 粘弹性对超声作用下刚性表面包封微泡动力学的影响

IF 2.8 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics

Pub Date : 2026-01-01 Epub Date: 2025-11-10 DOI: 10.1016/j.jnnfm.2025.105518

H. Furukawa , S. Iwata , T.N. Phillips , S.J. Lind , M.J. Walters

The dynamics of thin-shell encapsulated microbubbles (EMBs) in viscoelastic fluids forced by ultrasound are investigated in this paper. EMBs, which are gas-filled microbubbles encased in a stiff albumin or flexible lipid shell, have been shown to improve the performance of biomedical procedures such as ultrasound contrast imaging and sonoporation. To gain computationally efficient initial insights, the flow is assumed irrotational and axisymmetric, and is solved via the boundary element method. The viscoelastic fluid is modelled using the Oldroyd B model with both the fluid and the properties of the shell accounted for through the dynamic boundary condition at the bubble surface. A large bubble shell thickness is found to have a significant stabilising effect on the bubble, markedly reducing bubble deformation and response to the ultrasound pulse. For realistic ultrasound and biological fluid parameters, shell properties appear to dominate over fluid rheology. Although at lower shell thicknesses the dynamics are governed by a competition between viscous, elastic and inertial forces. A larger response is observed for lower frequency ultrasound and for pressure amplitudes typical to sonoporation, large translational movement in the direction of the pulse is predicted as well as deformation and the potential for bubble fragmentation. The model and quantitative insights herein have the potential to form the basis of a low-cost computational tool useful for EMB design, fabrication and characterisation in the near future.

研究了超声作用下粘弹性流体中薄壳包封微泡的动力学特性。EMBs是包裹在硬白蛋白或柔性脂质外壳中的充满气体的微泡，已被证明可以改善超声对比成像和超声成像等生物医学程序的性能。为了获得计算效率高的初始见解，假设流动是无旋转和轴对称的，并通过边界元方法求解。粘弹性流体模型采用Oldroyd B模型，通过气泡表面的动态边界条件考虑了流体和壳的性质。研究发现，大的泡壳厚度对泡具有显著的稳定作用，显著降低泡的变形和对超声脉冲的响应。对于真实的超声和生物流体参数，壳性质似乎比流体流变学更重要。尽管在较低的壳体厚度下，动力学是由粘性、弹性和惯性力之间的竞争所控制的。对于低频超声和典型的声穿孔压力幅值，可以观察到更大的响应，预测脉冲方向的大平移运动以及变形和气泡破碎的可能性。本文的模型和定量见解有可能在不久的将来成为EMB设计、制造和表征的低成本计算工具的基础。

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引用次数: 0

Accurate correlations for turbulent pipe flow of shear-thinning fluids 剪切变稀流体湍流管道流动的精确关联

IF 2.8 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics

Pub Date : 2026-01-01 Epub Date: 2025-10-25 DOI: 10.1016/j.jnnfm.2025.105517

Noman Yousuf , Daniel Lester , Andrew Chryss , Murray Rudman , Nicky Eshtiaghi

Turbulent pipe flow of generalised Newtonian (GN) fluids is prevalent in many industries. However, outstanding challenges include the accurate prediction of both the pressure gradient (or friction factor C_f) and the Reynolds number Re_turb which marks the onset of predominantly turbulent pipe flow. Although validated over a limited range of conditions, conventional methods such as empirical correlations and numerical simulations with unvalidated turbulence closures yield large errors when applied outside this range. This study addresses these challenges by using a validated direct numerical simulation (DNS) method for shear-thinning GN turbulent pipe flow, which resolves all spatiotemporal scales of the flow and so does not require turbulence closure. DNS simulations of shear-thinning turbulent pipe flow are used to develop simple and accurate correlations for Re_turb and C_f for both Herschel-Bulkley (HB) and Sisko rheological models (which capture the behaviour of most shear-thinning GN fluids) over a wide range of flow and rheology parameters. The DNS-based correlations for C_f are found to exhibit superior accuracy (L₂ error ∼6 %) compared to most conventional empirical correlations (L₂ error ∼16 %). We also demonstrate that these correlations can be used to estimate rheological parameters from combined laminar and turbulent pipe flow data. Hence, these correlations offer a simple, robust and accurate method for prediction of turbulent pipe flow of shear-thinning fluids and estimation of their rheological parameters.

广义牛顿（GN）流体的紊流在许多工业中都很普遍。然而，突出的挑战包括准确预测压力梯度（或摩擦系数Cf）和雷诺数return，后者标志着主要湍流管道流动的开始。尽管在有限的条件范围内得到了验证，但传统的方法，如经验关联和未经验证的湍流闭包的数值模拟，在此范围外应用时会产生很大的误差。本研究通过使用一种经过验证的直接数值模拟（DNS）方法来解决这些挑战，该方法用于剪切减薄GN湍流管道流动，该方法解决了流动的所有时空尺度，因此不需要湍流关闭。剪切变薄湍流管道流动的DNS模拟用于开发Herschel-Bulkley （HB）和Sisko流变模型（捕获大多数剪切变薄GN流体的行为）在广泛的流动和流变参数范围内的Returb和Cf的简单而准确的相关性。与大多数传统的经验相关性（L2误差~ 16%）相比，发现基于dns的Cf相关性具有更高的准确性（L2误差~ 6%）。我们还证明了这些相关性可以用于从层流和湍流管道流动数据中估计流变参数。因此，这些相关性为预测剪切变稀流体的湍流管道流动和估计其流变参数提供了一种简单、可靠和准确的方法。

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引用次数: 0

Small-amplitude forced oscillation of encapsulated gas bubbles in elastic microchannels: effect of blood shear-thinning 弹性微通道中封装气泡的小振幅强迫振荡：血液剪切变薄的影响

IF 2.8 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics

Pub Date : 2026-01-01 Epub Date: 2025-10-22 DOI: 10.1016/j.jnnfm.2025.105514

M. Pourjafar-Chelikdani , G. Biglari , P. Zakeri , K. Sadeghy

Encapsulated spherical gas bubbles are increasingly used as contrast agents in ultrasound imaging. The main restriction is the shear stress that is experienced by the vessel wall during bubble oscillations as it may damage its endothelial cells. Previous attempts to estimate the wall shear stress treated blood as a Newtonian fluid. In the present work, using the Carreau rheological model, we numerically investigate the effect of blood shear-thinning behavior on the wall shear stress. To ensure that the bubble remains nearly spherical during oscillations, simulation is restricted to small amplitudes for the acoustic pulse. To calculate the bubble’s instantaneous radius, we relied on the de Jong’s model for a linearly viscoelastic shell made of the Kelvin–Voigt material. By assuming that the micro-vessel obeys the Hooke’s law, the two-way coupling between the bubble and the vessel is achieved by solving the equations of motions for the liquid medium separating them. Blood shear-thinning is predicted to increase the vessel’s peak shear stress by roughly 30%. Shear-thinning also doubles the frequency corresponding to the peak shear stress. For shear-thinning liquids, the velocity profiles are predicted to contain inflection points suggesting that the flow induced in the liquid is vulnerable to hydrodynamic instability. A parameter study reveals that the time-constant in the Carreau model strongly controls the flow kinematics and dynamics during bubble oscillations. The conclusion is that blood shear-thinning behavior should be considered in cases where bubbles are used as contrast agents.

封装的球形气泡越来越多地被用作超声成像中的造影剂。主要的限制是在气泡振荡期间血管壁所经历的剪切应力，因为它可能损害其内皮细胞。以前估计壁面剪切应力的尝试将血液视为牛顿流体。在本工作中，我们使用careau流变模型，数值研究了血液剪切变薄行为对壁面剪切应力的影响。为了确保气泡在振荡过程中保持接近球形，模拟仅限于小振幅的声脉冲。为了计算气泡的瞬时半径，我们依靠de Jong的模型来计算一个由Kelvin-Voigt材料制成的线性粘弹性壳。假设微容器服从胡克定律，通过求解分离气泡与容器的液体介质运动方程，实现气泡与容器的双向耦合。预计血液剪切变薄会使血管的峰值剪切应力增加约30%。剪切减薄也使峰值剪应力对应的频率加倍。对于剪切变薄的液体，预测速度剖面包含拐点，表明液体中诱导的流动容易受到水动力不稳定的影响。参数研究表明，careau模型中的时间常数对气泡振荡过程中的流动运动学和动力学具有很强的控制作用。结论是，在使用气泡作为对比剂的情况下，应考虑血液剪切稀释行为。

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引用次数: 0

A fluctuating lattice Boltzmann method for viscoelastic fluid flows 粘弹性流体流动的波动晶格玻尔兹曼方法

IF 2.8 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics

Pub Date : 2026-01-01 Epub Date: 2025-11-19 DOI: 10.1016/j.jnnfm.2025.105532

Juanyong Wang , Xinyue Liu , Lei Wang , Yuan Yu , Yiran Ji

This study introduces a novel fluctuating lattice Boltzmann (LB) method for simulating viscoelastic fluid flows governed by the Oldroyd-B model. In contrast to conventional LB approaches that explicitly compute the divergence of the polymer stress tensor using finite-difference schemes, the proposed method incorporates the polymer stress implicitly by introducing a polymer stress fluctuation term directly into the evolution equation. This treatment avoids the need for stress-gradient computations, and preserves the physical characteristics of viscoelastic fluid flows. The proposed method is validated against five classical benchmark problems: the simplified four-roll mill, planar Poiseuille flow, unsteady Womersley flow, flow past a cylinder, and the three-dimensional Taylor–Green vortex. The numerical results show excellent agreement with analytical solutions and previous numerical results, confirming the method’s reliability in viscoelastic fluid dynamics. Moreover, performance evaluations demonstrate that the present model reduces the memory occupancy and enhances computational efficiency, highlighting its potential for large-scale simulations of complex viscoelastic flows systems.

本文提出了一种新的波动晶格玻尔兹曼（LB）方法来模拟Oldroyd-B模型下的粘弹性流体流动。传统的LB方法使用有限差分格式显式计算聚合物应力张量的散度，而该方法通过将聚合物应力波动项直接引入演化方程来隐式地包含聚合物应力。这种处理方法避免了应力梯度计算的需要，并保留了粘弹性流体流动的物理特性。针对简化的四辊轧机、平面泊泽维尔流、非定常沃默斯利流、过筒流和三维泰勒-格林涡这5个经典基准问题对该方法进行了验证。数值结果与解析解和前人的数值结果吻合良好，证实了该方法在粘弹性流体力学中的可靠性。此外，性能评估表明，该模型减少了内存占用，提高了计算效率，突出了其在复杂粘弹性流动系统的大规模模拟中的潜力。

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引用次数: 0

Turbulence characteristic evolution in jets interacting with viscoplastic fluids 射流与粘塑性流体相互作用的湍流特性演化

IF 2.8 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics

Pub Date : 2025-12-01 Epub Date: 2025-09-13 DOI: 10.1016/j.jnnfm.2025.105494

H. Hassanzadeh , M.H. Moosavi , I.A. Frigaard , S.M. Taghavi

We experimentally study the dynamics of a horizontal jet of a Newtonian fluid injected into a viscoplastic ambient fluid (Carbopol gel) to simulate jet cleaning in plug and abandonment operations of oil and gas wells. The jet flow is analyzed using high-speed imaging, planar laser induced fluorescence, and time-resolved tomographic particle image velocimetry techniques to capture concentration and velocity fields with high spatial and temporal resolution. By varying the Reynolds and Bingham numbers, we analyze three recently identified flow regimes, i.e., mixing, mushroom, and fingering, focusing on their mixing index, velocity fields, fluctuation intensity, half-radius, vorticity, Reynolds stresses, probability density functions, and statistical moments (skewness and kurtosis). In the mixing regime, velocity and vorticity symmetry, axisymmetric mixing, and dominant axial Reynolds stresses align with Newtonian empirical correlations. The mushroom regime shows slight asymmetry, reduced mixing from turbulence suppression by yield stress, and moderate turbulence, while the fingering regime features pronounced asymmetry, erratic fluctuations, and suppressed velocity due to viscoplastic resistance. Self-similarity analysis of velocity, concentration, and Reynolds stress profiles confirms strong scaling in the mixing regime, partial scaling in the mushroom regime, and deviations in the fingering regime, where viscoplastic effects disrupt jet structure and turbulence.

实验研究了将牛顿流体注入粘塑性环境流体（Carbopol凝胶）的水平射流动力学，以模拟油气井封堵和弃井作业中的射流清洗。利用高速成像、平面激光诱导荧光和时间分辨层析粒子图像测速技术对射流进行分析，以获得高时空分辨率的浓度场和速度场。通过改变雷诺数和宾厄姆数，我们分析了最近发现的三种流动形式，即混合流、蘑菇流和指指流，重点研究了它们的混合指数、速度场、波动强度、半半径、涡度、雷诺应力、概率密度函数和统计矩（偏度和峰度）。在混合状态下，速度和涡度对称、轴对称混合和主导轴向雷诺应力与牛顿经验相关。蘑菇模式表现出轻微的不对称，屈服应力抑制湍流导致的混合减少，湍流适度，而指法模式则表现出明显的不对称、不稳定波动和粘塑性阻力导致的速度抑制。速度、浓度和雷诺应力分布的自相似分析证实了混合状态下的强结垢，蘑菇状态下的部分结垢，指法状态下的偏差，粘塑性效应破坏了射流结构和湍流。

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引用次数: 0

Modelling viscoplastic interfacial flows inclusive of curvature effects 包含曲率效应的粘塑性界面流动建模

IF 2.8 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics

Pub Date : 2025-12-01 Epub Date: 2025-09-27 DOI: 10.1016/j.jnnfm.2025.105498

P.T. Griffiths , D. Xu , L. Davoust

This study presents a theoretical and numerical investigation of interfacial flows of oxidised liquid metals in a shallow annular channel in the absence of inertial effects. For the first time, viscoplastic surface behaviour induced by oxidation is modelled using a bi-viscosity law within a framework that also accounts for interfacial curvature governed by the Young–Laplace equation. By solving a coupled bulk-surface flow system, the effects of surface rheology, contact angle, and dimensionless capillary length on surface velocity and surface shear rate profiles are quantified. Results highlight the competing influences of hydrophobicity and viscoplasticity on surface and bulk flow characteristics and demonstrate that accurate modelling of such systems necessitates inclusion of both curvature and non-Newtonian surface effects. In appropriate limits, our numerical results are validated against semi-analytical solutions. Our findings offer insights relevant to metal casting applications.

本文对氧化液态金属在没有惯性效应的浅环形通道中的界面流动进行了理论和数值研究。第一次，由氧化引起的粘塑性表面行为在一个框架内使用双粘度定律进行建模，该框架也考虑了由Young-Laplace方程控制的界面曲率。通过求解一个体-面耦合流动系统，量化了表面流变性、接触角和无因次毛管长度对表面速度和表面剪切速率的影响。结果强调了疏水性和粘塑性对表面和体积流动特性的竞争影响，并表明此类系统的准确建模需要包括曲率和非牛顿表面效应。在适当的范围内，我们的数值结果对半解析解进行了验证。我们的发现提供了与金属铸造应用相关的见解。

引用次数: 0

Wrinkling of a viscoplastic plate on a viscous substrate 粘塑性板在粘性基材上起皱

IF 2.8 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics

Pub Date : 2025-12-01 Epub Date: 2025-09-27 DOI: 10.1016/j.jnnfm.2025.105496

Thomasina V. Ball , A.F. Bonfils , Jerome A. Neufeld

We study the formation of wrinkles in a clamped viscoplastic plate overlying a thin viscous substrate, which is compressed horizontally. When the plate is purely viscous, the compressive force in the plate is constant and is set by the velocity boundary conditions. The wavelength of the emergent wrinkles depends on the thicknesses of the two layers and the ratio of the viscosities. As the domain length is reduced relative to the characteristic wavelength, the spatial profile and growth rate start to depend heavily on the domain length and imposed clamped boundary conditions. Introducing a yield stress to the plate initially increases the compressive force, proportional to the Bingham number, due to the requirement for the plate to be yielded throughout the domain. As the bending moments increase in the plate, the compressive force is relieved and a transition occurs where the plate begins to yield through bending rather than compression. During this transition, the plate is dominated by plugged, unyielded regions leading to localisation and the formation of straight-sided wrinkles.

我们研究了夹紧粘塑性板上的薄粘性基材，这是水平压缩皱褶的形成。当板为纯粘性时，板内的压缩力是恒定的，由速度边界条件确定。出现皱纹的波长取决于两层的厚度和粘度的比例。随着畴长相对于特征波长的减小，空间轮廓和增长率开始严重依赖于畴长和施加的箝位边界条件。由于要求板在整个区域内屈服，向板引入屈服应力最初会增加与宾汉姆数成正比的压缩力。当板中的弯矩增加时，压缩力被解除，并且在板开始通过弯曲而不是压缩屈服的地方发生转变。在这一转变过程中，板块主要由堵塞的、未屈服的区域主导，导致局部化和直边褶皱的形成。

引用次数: 0

General evaluation of the pressure gradient for lubrication flows in varying channels with applications to linear and hyperbolic contractions 不同通道润滑流压力梯度的一般评价，并应用于线性和双曲收缩

IF 2.8 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics

Pub Date : 2025-12-01 Epub Date: 2025-09-01 DOI: 10.1016/j.jnnfm.2025.105487

Panagiotis Sialmas, Kostas D. Housiadas

Under the classic lubrication approximation, we develop a unified framework for evaluating the pressure gradient of an incompressible, isothermal viscoelastic fluid in a symmetric channel with slowly varying geometry, including inertia. Exploiting the independence of the pressure gradient from the wall-normal coordinate—a property absent in general 2D planar or 3D axisymmetric flows—we derive multiple integral expressions for the pressure gradient and the corresponding average pressure drop required to maintain a constant flow rate. The derivations use the momentum balance formulated via the extra-stress tensor, providing a flexible, formal, and rigorous procedure, and the physical significance of each expression is discussed.

To bypass choosing among these expressions, we introduce a new set of lubrication equations based on a streamfunction, mapped coordinates, and transformed polymer extra-stress components. This formulation automatically satisfies the continuity equation, the constraints due to fluid incompressibility, the boundary conditions, and the flow symmetries, allowing the pressure gradient to be determined a posteriori and providing a tool for consistency and accuracy checks.

The equivalence of the integral expressions is illustrated in two representative cases: (i) Newtonian inertial flow in a linearly contracting channel, and (ii) viscoelastic inertialess flow in a hyperbolic contraction. In both cases, the predicted average pressure drop agrees very well with high-order asymptotic solutions post-processed via Padé approximants, high-accuracy spectral simulations, and DNS results from the literature. The framework provides a rigorous, general, and computationally robust tool for analyzing lubrication flows of viscoelastic fluids and can be easily extended to other complex fluids and broader flow conditions.

在经典润滑近似下，我们开发了一个统一的框架来评估不可压缩的等温粘弹性流体在具有缓慢变化几何形状（包括惯性）的对称通道中的压力梯度。利用压力梯度与壁面法向坐标的独立性（这是一般二维平面或三维轴对称流所没有的特性），我们推导了压力梯度和保持恒定流速所需的相应平均压降的多个积分表达式。推导使用了通过额外应力张量形成的动量平衡，提供了一个灵活、正式和严格的程序，并讨论了每个表达式的物理意义。为了避免在这些表达式中进行选择，我们引入了一组新的基于流函数、映射坐标和转换聚合物额外应力分量的润滑方程。该公式自动满足连续性方程、流体不可压缩性约束、边界条件和流动对称性，允许后验确定压力梯度，并为一致性和准确性检查提供工具。积分表达式的等价性在两种典型情况下得到说明：(i)线性收缩通道中的牛顿惯性流动，以及（ii）双曲收缩通道中的粘弹性无惯性流动。在这两种情况下，预测的平均压降与通过pad近似、高精度谱模拟和文献中的DNS结果后处理的高阶渐近解非常吻合。该框架为分析粘弹性流体的润滑流动提供了一个严格、通用、计算稳健的工具，并且可以很容易地扩展到其他复杂流体和更广泛的流动条件。

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引用次数: 0

Experimental investigation of yield stress fluid flow in an abrupt 4:1 planar contraction using laser Doppler velocimetry: Effects of inertia and geometry 用激光多普勒测速法研究屈服应力流体在4:1突然平面收缩中的流动：惯性和几何的影响

IF 2.8 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics

Pub Date : 2025-12-01 Epub Date: 2025-10-14 DOI: 10.1016/j.jnnfm.2025.105510

Belaid Hadj Arab , Zineeddine Louna , Mohamed Mahfoud , Paulo R de Souza Mendes , Yamid J. Garcia-Blanco , Admilson T Franco

This paper presents an experimental study of yield stress fluid flow in a rectangular channel containing an abrupt contraction with an aspect ratio of 4:1. The flow dynamics analysis is based on Laser Doppler Velocimetry (LDV), which enables the local characterization of velocities. The longitudinal and transversal velocity components are measured upstream, at, and downstream of the contraction to assess the effect of this geometric change and inertia on the flow. Profiles of the longitudinal velocity component, u, highlight the existence of a dead zone at the corners of the contraction, where a slip line is identified and located at the same position as the contraction. The establishment length of the downstream flow is also determined, and a linear correlation is proposed. A plug zone, characterized by a constant velocity at the center of the channel, is observed for different flow rates and position conditions. The transversal velocity component, v, remains nearly zero throughout the domain, except in the contraction region, where elongation effects become significant. Analysis of velocity distribution along the central axis reveals velocity peaks immediately downstream of the contraction, particularly at low flow rates. The elongation rate reaches a maximum at the contraction, and a linear correlation is established between this maximum value and the flow rate. A correlation describing the slip velocity at the wall is also proposed. Finally, the study is completed by establishing correlations between the friction factor, f, the Euler number, and the Reynolds number, which are evaluated at the upstream and downstream regions of the contraction.

本文进行了屈服应力流体在宽高比为4:1的含突然收缩矩形通道内流动的实验研究。流动动力学分析是基于激光多普勒测速（LDV），它可以实现速度的局部表征。在收缩的上游、处和下游测量纵向和横向速度分量，以评估这种几何变化和惯性对流动的影响。纵向速度分量u的剖面突出表明，在收缩的角落存在一个死区，在那里可以识别出一条滑移线，并与收缩位于同一位置。确定了下游水流的建立长度，并提出了线性相关关系。在不同的流量和位置条件下，在通道中心观察到一个以恒定速度为特征的堵塞区域。横向速度分量，v，在整个区域保持接近零，除了在收缩区域，伸长效应变得明显。沿中轴线的速度分布分析显示，收缩的下游立即出现速度峰值，特别是在低流速时。伸长率在收缩时达到最大值，且该最大值与流量呈线性相关。还提出了一种描述壁面滑移速度的关系式。最后，通过建立摩擦系数f、欧拉数和雷诺数之间的相关性来完成研究，并在收缩的上游和下游区域进行评估。

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引用次数: 0

Note on the theoretical analysis of the energy equation for viscoelastic flows under non-isothermal material properties 非等温材料特性下粘弹性流动能量方程的理论分析注记

IF 2.8 2区工程技术 Q2 MECHANICS

Journal of Non-Newtonian Fluid Mechanics

Pub Date : 2025-12-01 Epub Date: 2025-09-24 DOI: 10.1016/j.jnnfm.2025.105497

Carlos Veiga Rodrigues , Francisco Vide , Alexandre M. Afonso , Fernando T. Pinho

A theoretical analysis of the energy equation for viscoelastic flows is performed, following the work of Peters and Baaijens [Peters GWM, Baaijens FPT, 1997 J. Non-Newt. Fluid Mech., 68:205–224], and aimed at extending it to deal with fluids having temperature-dependent and time-dependent fluid properties, which give rise to new terms in the transport equation that are not present in the original Peters and Baaijens work. Lastly, an application of the extended energy equation is carried out using the PTT network model.

根据Peters和Baaijens的工作，对粘弹性流动的能量方程进行了理论分析[Peters GWM， Baaijens FPT, 1997 J. Non-Newt.]流体机械。[j]，旨在将其扩展到处理具有温度依赖和时间依赖流体性质的流体，这在彼得斯和Baaijens的原始工作中没有出现的输运方程中产生了新的项。最后，利用PTT网络模型对扩展后的能量方程进行了应用。

引用次数: 0