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Direct-ink writing (DIW) has rapidly become a versatile 3D fabrication method due to its ability to deposit a wide range of complex fluids into customizable 3D geometries. This review highlights key fundamental fluid mechanics and soft matter challenges across the different stages of the DIW printing process. The rheology of fluids and suspensions governs the flow behavior through narrow nozzles, posing questions about extrudability, confined flow dynamics, and clogging mechanisms. Downstream, the formation and deposition of extruded filaments involve extensional flows and potential instabilities, while postdeposition dynamics introduces complexities related to yield stress and structural stability. These stages are inherently interdependent, as optimizing material composition without considering filament stability risks compromising the final structure. As DIW applications expand through advanced ink formulations, developing fundamental fluid mechanics frameworks is essential to replace trial-and-error approaches with predictive design methodologies to enable more precise control over and reliability of the printing process.
Porous media flows are generally viewed as inefficient mixers, where solutes may be dispersed yet poorly mixed, making mixing a critical limiting factor for a wide range of processes. The complexity and opacity of porous structures have long made these dynamics difficult to observe. With emerging experimental techniques, concepts and models of mixing in porous media are rapidly evolving. Recent advances link mixing dynamics to fluid deformation arising in flow through porous materials. Unlike diffusion and dispersion, which only dissipate chemical gradients, fluid shear and stretching amplify and sustain them. This review explores the role of fluid deformation in governing mixing, chemical reactions, and biological processes in porous media. We begin by highlighting key experimental observations that have improved our understanding of mixing in these systems. We then examine the fundamental concepts, models, and open questions surrounding fluid deformation and mixing in porous media, emphasizing their dependence on material structure, heterogeneity, dimensionality, and transient flow phenomena, as well as their interaction with chemical and biological processes.
The squeezing of blood cells and vesicles through narrow constrictions, such as splenic slits, pulmonary capillaries, vascular endothelial gaps, and microfluidic channels, is crucial in physiology and biotechnology, with fluid mechanics playing a central role. The diverse geometries of these constrictions, the associated flow conditions, and the unique mechanical properties of cells and vesicles create a rich subject in fluid mechanics emerging from nonlinear dynamics of fluid–structure interactions involving both lubrication and Marangoni flows. Advances in microfluidics, video microscopy, and computational modeling have enabled investigations into these complex processes. This review surveys the key features and approaches, recent prominent studies, and unresolved challenges related to these processes, offering insights for researchers across biomechanics, biomedical engineering, biological physics, hematology, physiology, and applied mathematics.
Internal waves, generated by wind and tides, are ubiquitous in the ocean. Their dissipation and the resulting vertical mixing play an important role in setting the ocean circulation, stratification, and energetics. Ocean models usually parameterize many or all of these effects. The current generation of parameterizations often relies on assumptions of uniform or slowly varying stratification profiles. Here, we review the growing theoretical, modeling, and observational evidence that vertical nonuniformity in the stratification profile can significantly modify the assumed wave dynamics. Linear scattering, wave–wave interactions, and solitary-like internal wave generation in idealized nonuniform stratification profiles are discussed. The nonuniform features in oceanic vertical stratification profiles are characterized, followed by a discussion of the validity of the slowly varying stratification assumption for such profiles. A concerted effort is made to synthesize research in both fluid dynamics and oceanography.
Geophysical and astrophysical fluid dynamics (GAFD) is an interdisciplinary field. It encompasses a wide range of fluid systems, from planetary atmospheres and the oceans of Earth and icy moons to the interiors of telluric planets, giant planets, and stars. It also spans vast timescales and space scales. Despite this diversity, GAFD is built on common challenges in fundamental fluid mechanics, requiring a multi-approach strategy that integrates theory, simulations, and experiments to explain observations. This review highlights the role of laboratory experiments in GAFD. We first emphasize recent advances in experimental design, methods, and metrology, including large-scale facilities as well as innovative and analog setups. We then focus on two areas where experiments have driven recent breakthroughs: rotating turbulence and flows involving multiphase and phase-change processes. Finally, we discuss emerging challenges and the potential of outreach experiments to stimulate interest in fluid mechanics among students and the public.
Microplastic pollution is now ubiquitous in marine environments, posing risks to ecosystem and human health. In order to assess and mitigate this threat, we require accurate prediction of microplastic fate and transport in the ocean. While progress has been made studying global-scale transport pathways, our models often fall short at smaller scales; processes such as vertical transport, horizontal dispersion, particle transformation, and boundary fluxes (e.g., at beaches and the air–sea interface) remain poorly understood. The difficulty lies in the physical features of plastic particles: namely, near-neutral buoyancy in seawater, finite size, and irregular shape. These complexities are compounded by the multiscale forcing from waves and turbulence near the ocean surface where microplastics tend to reside. This review synthesizes recent advances in the fluid dynamics of marine plastic transport, emphasizing the role of fluid–particle interactions in ocean flows and highlighting outstanding challenges.
Particulate suspensions, consisting of solid particles dispersed in a fluid, exhibit complex flow behaviors influenced by multiple factors, including particle interactions, concentration gradients, and external forces. Suspensions play an important role in diverse processes, from sediment transport to food processing, and display instabilities triggered by shear-driven effects, frictional interactions, and viscous forces. These instabilities can often be understood by identifying the key mechanical quantities that govern the dynamics. Following hydrodynamic tradition, such mechanics can be characterized by dimensionless numbers, which encapsulate the interplay between geometric, kinematic, and mechanical factors. Many of these numbers represent competitions between opposing pairs of mechanical quantities, which we discuss in detail while also considering a few phenomena that require more complex combinations. By emphasizing the underlying mechanical principles, this review provides a perspective for understanding pattern formation and flow instabilities in confined particulate suspensions across different flow geometries.
The objective of this contribution is to review more than 80 years of experimental measurements of the settling of snow particles and surrogates in natural and laboratory settings and suggest viable directions for future research. Under the broad category of frozen hydrometeors, snow particles are characterized by a variety of shapes and inertial properties that we broadly refer to as snow morphology attributes and depend on the micrometeorology of the air column, including temperature, relative humidity, wind speed, and turbulence. The uncertainty in the prediction of snow settling velocity is partly due to the significant variability in snow crystal shape, density, and drag properties, as well as the modulating effect of ambient turbulence, which has been observed to affect particle orientation and falling style and enhance or reduce the terminal velocity, as compared to quiescent flow conditions. Because of the complexity of finite-size, nonspherical particles’ interaction with turbulent flows at high Reynolds numbers, we stress the need for simultaneous flow and snow morphology measurements in the field and we review past and current experimental techniques and methodologies.
In liquid filtration, a particulate-laden feed solution is passed through a porous material (the filter), often a membrane, designed to capture the particulate matter. Usually, the filter has a complex interior structure of interconnected pores, through which the feed passes, and in many cases of interest, it may be reasonable to approximate this interior structure as a network of interconnected tubes. This idea, which dates back about 70 years, greatly simplifies the modeling and simulation of the filtration process. In this article, we review the use of networks as a framework for modeling and investigating filtration, describing the key ideas and milestones. We also discuss some promising areas for future development of this field, particularly concerning the design of next-generation filters.
This review first examines how urban wind flow impacts the sustainability and resilience of cities and identifies the three main challenges in predictive modeling of urban flows: the complexity of the flow physics, the variability and uncertainty in the flow conditions, and the diversity and multiscale nature of urban geometries. To review the complexity of the flow physics, the typical flow patterns observed in canonical urban flows are summarized, and related modeling challenges and opportunities in both wind tunnel experiments and simulations are highlighted. Next, opportunities to predict realistic urban flows by addressing the other challenges are explored through the lens of a modeling framework with uncertainty quantification. The important role of field measurements, supporting the more accurate characterization of uncertainties in the flow conditions, as well as enabling validation with real-world data, is emphasized. The review concludes with two specific examples that demonstrate how integrated use of field measurements and computational models can improve the understanding and modeling of real urban flows to ultimately support sustainable development goals for urban areas.
Publication date: Available online 13 November 2025
Source: Computers & Fluids
Author(s): Tyler Buchanan, Monica Lăcătuş, Alastair West, Richard P. Dwight
Publication date: Available online 13 November 2025
Source: Computers & Fluids
Author(s): Haiming Zhu, Yuan Yang, Zunfeng Du, Jianxing Yu
Publication date: Available online 12 November 2025
Source: Computers & Fluids
Author(s): Marica Pelanti
Publication date: Available online 15 November 2025
Source: Computers & Fluids
Author(s): Francesco Mario D’Afiero
Publication date: Available online 15 November 2025
Source: Computers & Fluids
Author(s): Jian Shen, Xianliang Chen, Lin Fu
Publication date: Available online 16 November 2025
Source: Computers & Fluids
Author(s): Lorenzo Botti, Daniele A. Di Pietro, Francesco Carlo Massa
Publication date: Available online 15 November 2025
Source: Computers & Fluids
Author(s): T. Van Gastelen, W. Edeling, B. Sanderse
Publication date: Available online 11 November 2025
Source: Computers & Fluids
Author(s): Anand Srinivasan, Perry Johnson, Jose Castillo
Publication date: Available online 11 November 2025
Source: Computers & Fluids
Author(s): Yimeng Du, Zhendong Jin, Chengjun Zhang, Yan Cui, Yulong Wang, Rongxuan Hu, Peng Gao, Martin Sommerfeld
Publication date: Available online 8 November 2025
Source: Computers & Fluids
Author(s): Rajat Kumar Sarkar, Vishal Jadhav, Venkataramana Runkana
Comparison of pressure at 10 seconds. Left–reference data, right–model predictions.
This paper examines the application of PINN models to solving a two-dimensional cylinder flow problem with limited data. Using data obtained by direct numerical simulation, a surrogate PINN model was developed and trained. The model utilizes the governing equations of fluid dynamics and heat transfer, enabling it to accurately predict flow parameters such as velocity components, pressure, and temperature. The direct computational flow model was numerically solved using the SIMPLE algorithm, which couples pressures and velocities. The results showed that the PINN model, which does not contain initial and boundary conditions from direct numerical simulation, is capable of reproducing complex dynamic processes such as the formation of a Kármán vortex street behind a cylinder. However, limitations were identified due to the lack of initial and boundary conditions, which led to increased errors at the boundaries of the computational domain. For example, from the data obtained using the PINN model, a very small absolute difference in error for the velocity and temperature components between the reference data and the predicted values can be noted. Thus, for the horizontal velocity component, the maximum relative error was no more than 2.5%. For the temperature component, the relative error was no more than 0.02%. However, the relative error for pressure was 60%–75%. The main reason for this large error is the lack of a reference pressure value or initial pressure conditions in the loss function. The results show that the PINN surrogate model with eight hidden layers of 200 neurons successfully copes with the task of modeling complex unsteady flow. The integration of physical laws made it possible to achieve relatively satisfactory accuracy using only 10,000 data points.
Bi-cubic Hemite-Bézier and reduced cubic Hsieh-Clough-Tocher finite elements, of class C1, are compared for the solution of a highly anisotropic diffusion equation. They are tested numerically for various ratios of the diffusion coefficients on different meshes, even aligned with the anisotropy. Results confirm a high reduction of the numerical diffusion when working on meshes that are aligned with the anisotropy. An example of a reduced model is provided to comment on some obtained results.
Heat transfer in magnetically confined plasmas is characterized by extremely high anisotropic diffusion phenomena. At the core of a magnetized plasma, the heat conductivity coefficients in the parallel and perpendicular directions of the induction field can be very different. Their ratio can exceed 108$$ 1{0}^8 $$, and the pollution by purely numerical errors can make the simulation of the heat transport in the perpendicular direction very difficult. Standard numerical methods, generally used in the discretization of classical diffusion problems, are rather inefficient. The present paper analyzes a finite element approach for the solution of a highly anisotropic diffusion equation. Two families of finite elements of class 𝒞1, namely bi-cubic Hermite-Bézier and reduced cubic Hsieh-Clough-Tocher finite elements, are compared. Their performances are tested numerically for various ratios of the diffusion coefficients, on different mesh configurations, even aligned with the induction field. The time stepping is realized by an implicit high-order Gear finite difference scheme. An example of a reduced model is also provided to comment on some obtained results.
Based on Hamilton's principle, the study focuses on a novel strategy for the modeling of non-Newtonian fluids with the help of internal variables. Here, the viscosity evolves locally in space and time. Three configurations are numerically implemented, namely channel flow, a benchmark, and a lid-driven cavity.
The modeling of fluids is an important field for mechanics of materials. In this work, we demonstrate that Hamilton's principle, which is well-known for the modeling of solids, can also be formulated to derive the Navier–Stokes equations, which paves the way for easy inclusion of complex material constraints. Furthermore, we expand Hamilton's principle to enable the introduction of “internal variables”, which describe the space- and time-dependent evolution of the material properties. Hereby, a novel strategy for the modeling of non-Newtonian fluids is given. Eventually, Hamilton's principle inherently enables a space-time formulation with the automatic derivation of the correct formal functional setting, which covers different scales of viscosity through the internal variable. The resulting system is a space-time multiscale model for fluid flow, which is based on an additional partial differential equation. The model constitutes thus a much more adaptive description of the complex processes in non-Newtonian fluid flow as possible for classical models based on algebraic constitutive laws. This also includes a spatially and temporally local evolution of the effective viscosity, depending on the local flow conditions rather than material parameters and resulting in both shear-thinning and shear-thickening behavior. Numerical examples substantiate our proposed setting by some studies from Newtonian flow to non-Newtonian regimes with fading or increasing viscosity.
The figure shows the pressure and temperature at t=0.75$$ t=0.75 $$. The curvilinear grid deforms in a stable manner with the flow because of the high order finite element, and the physical states remain steady.
In this article, a cell-centered discontinuous Galerkin (DG) method is presented for solving Lagrangian radiation hydrodynamic equations (RHE). The equations are separated into a hydrodynamic part and a radiation diffusion part. These two parts are written in Lagrangian forms. The hydrodynamic part is discretized by a cell-centered DG scheme in reference space using Taylor basis functions. An approximate Riemann solver is used for the velocity of vertices, and the radiation diffusion is solved using an interior penalty method. Due to the deformation of the basis functions in physical space, curvilinear mesh is formed. Numerical tests are presented to show its accuracy and robustness.
In this paper, the Navier–Stokes equations are solved using purely meshless method. No boundary nor domain discretization is required. The method of fundamental solutions together with the Monte Carlo integration technique are employed via a suitable penalty formulation. The results are verified using previously published ones.
This paper presents a novel mesh-free approach for solving the Navier–Stokes equations. The method makes use of the meshless method of fundamental solutions (MFS) and the Monte Carlo integration technique for computing the domain integral of the convective terms. No domain or boundary discretization is required. This approach facilitates numerical computation while ensuring accuracy and stability. By imposing a penalty parameter, the Navier–Stokes equations are transformed to resemble the Navier equations of elasticity. Hence, elasticity based fundamental solutions are employed. The proposed formulation is validated through numerical examples, demonstrating its efficacy in capturing steady-state flow phenomena through several examples. This highly parallelized system is then accelerated via GPU computing. Overall, the proposed method provides a promising paradigm for advancing computational fluid mechanics, offering a versatile framework with broad applicability in engineering and scientific domains.
In this paper, we consider the concept of discretely divergence-free finite elements (DDFFE) based on the Rannacher–Turek finite element pair to efficiently solve the three-dimensional incompressible Navier–Stokes equations. For this purpose, we first define a spanning set of DDFFE functions and then characterize a set of basis functions for arbitrary geometries. The discretized problem is finally solved without the need for Schur complement techniques using a geometric multigrid solution algorithm, which can employ a wide variety of preconditioners and a newly defined prolongation operator.
A geometric multigrid solution technique for the incompressible Navier–Stokes equations in three dimensions is presented, utilizing the concept of discretely divergence-free finite elements without requiring the explicit construction of a basis on each mesh level. For this purpose, functions are constructed in an a priori manner spanning the subspace of discretely divergence-free functions for the Rannacher–Turek finite element pair under consideration. Compared to mixed formulations, this approach yields smaller system matrices with no saddle point structure. This prevents the use of complex Schur complement solution techniques, and more general preconditioners can be employed. While constructing a basis for discretely divergence-free finite elements may pose significant challenges and prevent the use of a structured assembly routine, a basis is utilized only on the coarsest mesh level of the multigrid algorithm. On finer grids, this information is extrapolated to prescribe boundary conditions efficiently. Here, special attention is required for geometries introducing bifurcations in the flow. In such cases, so-called “global” functions with an extended support are defined, which can be used to prescribe the net flux through different branches. Various numerical examples for meshes with different shapes and boundary conditions illustrate the strengths, limitations, and future challenges of this solution concept.
It is the first to adopt the Moment-based boundary condition to combine with the interpolation-based scheme for curved boundary treatment. Through numerical experiment, the Moment-based scheme has better accuracy than both BB and NEBB schemes.
This paper proposes the unified interpolated-based scheme for curved boundary treatment of the discrete unified gas kinetic scheme (DUGKS). The construction of the proposed boundary scheme is the combination of interpolation and the straight boundary condition (i.e., bounce-back (BB), non-equilibrium bounce-back (NEBB), and moment-based scheme). To note that, this paper is the first to adopt the moment-based boundary condition to combine with the interpolation-based scheme for curved boundary treatment. The asymptotic analysis confirms that the proposed schemes are of first-order accuracy. Their feasibility and accuracy are examined for different spatial grid resolutions through several numerical tests. They are robust and easy to implement. The results agree well with the analytical solution and validate the first-order accuracy. It is found that the moment-based scheme has better accuracy than both BB and NEBB schemes.
This paper presents an optimization algorithm designed to effectively handles new general class of the nonlinear variable-order fractional partial differential equations (GCNV-OFPDEs) with nonlocal boundary conditions. Our approach involves utilizing a novel variant of the polynomials namely generalized Abel polynomials (GAPs) and also new operational matrices to approximate the solution of the GCNV-OFPDEs. A key aspect of our algorithm is the transformation of GCNV-OFPDEs, along with their respective nonlocal boundary conditions, into systems of nonlinear algebraic equations. By solving these systems, we can determine the unknown coefficients and parameters. To address the nonlinear system, we employ the Lagrange multipliers to achieve optimal approximations. The convergence analysis of the approach are discussed. To validate the effectiveness of our algorithm, we conducted numerous experiments using various examples. The results obtained demonstrate the exceptional accuracy of our approach and its potential for extension to more complex problems in the future.
This paper presents an optimization algorithm designed to effectively handle a new general class of the nonlinear variable-order fractional partial differential equations (GCNV-OFPDEs) with nonlocal boundary conditions. Our approach involves utilizing a novel variant of the polynomials, namely generalized Abel polynomials (GAPs), and also new operational matrices to approximate the solution of the GCNV-OFPDEs. A key aspect of our algorithm is the transformation of GCNV-OFPDEs, along with their respective nonlocal boundary conditions, into systems of nonlinear algebraic equations. By solving these systems, we can determine the unknown coefficients and parameters. To address the nonlinear system, we employ the Lagrange multipliers to achieve optimal approximations. The convergence analysis of the approach is discussed. To validate the effectiveness of our algorithm, we conducted numerous experiments using various examples. The results obtained demonstrate the exceptional accuracy of our approach and its potential for extension to more complex problems in the future.
This study investigates the significance of a 3D transient magnetohydrodynamic boundary layer flow analysis for an incompressible Casson fluid laden with nanoparticles. The analysis considers thermal conductivity, magnetic effects, and a first-order chemical reaction occurring over a permeable inclined stretching surface positioned horizontally. The fluid motion is induced by a multi-directional stretching surface, which exhibits its own directional velocity, as characterized by the given relation u*=ax*/1−ct*+γ∂u*/∂z*,v*=by*/1−ct*+γ∂v*/∂z*,andw*=0$$ {u}^{\ast }={ax}^{\ast }/1-{ct}^{\ast }+\gamma \partial {u}^{\ast }/\partial {z}^{\ast },{v}^{\ast }={by}^{\ast }/1-{ct}^{\ast }+\gamma \partial {v}^{\ast }/\partial {z}^{\ast },\mathrm{and}\ {w}^{\ast }=0 $$, where a,b$$ a,b $$ are positive constants (stretching rates). The governing equations of the system include the effects of internal heating, Brownian motion, thermophoretic diffusion, and heat source/sink. The physical attributes of the nanofluid are assumed to remain constant. The velocity components in the coordinate axes are represented as u*,v*,andw*$$ {u}^{\ast },{v}^{\ast },\mathrm{and}\ {w}^{\ast } $$ for their respective directions. In this context, the convective surface temperature and surface concentration of the nanofluid are denoted as Tw*andCw*$$ {T}_w^{\ast }\ \mathrm{and}\ {C}_w^{\ast } $$. The fluid temperature profile is symbolized by T*$$ {T}^{\ast } $$, while the concentration profile is represented by C*$$ {C}^{\ast } $$. The ambient temperature is signified by T∞*$$ {T}_{\infty}^{\ast } $$, and the ambient concentration is denoted by C∞*$$ {C}_{\infty}^{\ast } $$.
The induced convective flow of three-dimensional Casson nanofluid governed by a bi-directional stretching surface has potential practical implications in numerous engineering fields, such as heat exchangers, cooling systems for heat-generating devices, and more. This investigation aims to analytically examine the natural convection mechanism and heat transfer analysis of a Casson nanofluid inside a porous surface exposed to a uniform magnetic field. Moreover, this research explores the physical insights of thermal characteristics by incorporating the effects of chemical reactions, velocity slip, Brownian diffusion, and heat sources/sinks on the transient magnetohydrodynamic flow of the nanofluid. The proposed flow framework is described by a system of partial differential equations, which are transformed into dimensionless ordinary differential equations using appropriate variables. The closed-form solutions of a set of leading characteristic dimensionless equations are obtained analytically through the efficient homotopic analysis method. Furthermore, stability and convergence analyses of the series solutions are performed to validate the computational results explicitly. The computational findings reveal a significant decrease in flow velocity, temperature, and particle concentration profiles as the Casson fluid parameter increases. Additionally, the effects on skin friction, Nusselt number, and Sherwood number are discussed in detail. This study aims to enhance the understanding of flow dynamics and heat and mass transfer mechanisms across various applications, offering valuable insights for engineering and scientific advancements. The authors accept that all the computational outcomes in this research, both analytical and numerical, are authentic and not published elsewhere.
Publication date: 15 January 2026
Source: Journal of Computational Physics, Volume 545
Author(s): Yao Gahounzo, Michal Kopera, Robert L. Higdon, Eric P. Chassignet, Alan Wallcraft, Alexandra Bozec
Publication date: 15 January 2026
Source: Journal of Computational Physics, Volume 545
Author(s): Luis Pérez, Carlos Reales, Alberth Silgado, Iván Velásquez
Publication date: 15 January 2026
Source: Journal of Computational Physics, Volume 545
Author(s): Giuseppe Sirianni, Barbara Re
Publication date: 15 January 2026
Source: Journal of Computational Physics, Volume 545
Author(s): Di Qi, Jian-Guo Liu
Publication date: 15 January 2026
Source: Journal of Computational Physics, Volume 545
Author(s): Florian Setzwein, Peter Ess, Peter Gerlinger
Publication date: 15 January 2026
Source: Journal of Computational Physics, Volume 545
Author(s): J.A. Hopman, J. Rigola, F.X. Trias
Publication date: 15 January 2026
Source: Journal of Computational Physics, Volume 545
Author(s): Guanjie Wang, Qifeng Liao
Publication date: 15 January 2026
Source: Journal of Computational Physics, Volume 545
Author(s): Cheng Tang, Roy Y. He, Hao Liu
Publication date: 15 January 2026
Source: Journal of Computational Physics, Volume 545
Author(s): Ali Mohaghegh, Cheng Huang
Publication date: 15 January 2026
Source: Journal of Computational Physics, Volume 545
Author(s): Kyongmin Yeo, Hyomin Shin, Heechang Kim, Minseok Choi
