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The purpose of this study is to perform a detailed review on the numerical modeling of multiphase and multicomponent flows in microfluidic system using phase-field method. The phase-field method is of emerging importance in numerical computation of transport phenomena involving multiple phases and/or components. This method is not only used to model interfacial phenomena typical to multiphase flows encountered in engineering and nature but also turns out to be a promising tool in modeling the dynamics of complex fluid-fluid interfaces encountered in physiological systems such as dynamics of vesicles and red blood cells). Intrinsically, a priori unknown topological evolution of interfaces offers to be the most concerning challenge toward accurate modeling of moving boundary problems. However, the numerical difficulties can be tackled simultaneously with numerical convenience and thermodynamic rigor in the paradigm of the phase field method.

The phase-field method replaces the macroscopically sharp interfaces separating the fluids by a diffuse transition layer where the interfacial forces are smoothly distributed. As against the moving mesh methods (Lagrangian) for the explicit tracking of interfaces, the phase-field method implicitly captures the same through the evolution of a phase-field function (Eulerian). In contrast to the deployment of an artificially smoothing function for the interface as used in the volume of a fluid or level set method, however, the phase-field method uses mixing free energy for describing the interface. This needs the consideration of an additional equation for an order parameter. The dynamic evolution of the system (equation for order parameter) can be described by Allen–Cahn or Cahn–Hilliard formulation, which couples with the Navier–Stokes equation with the aid of a forcing function that depends on the chemical potential and the gradient of the order parameter.

In this review, first, the authors discuss the broad motivation and the fundamental theoretical foundation associated with phase-field modeling from the perspective of computational microfluidics. They subsequently pinpoint the outstanding numerical challenges, including estimations of the model-free parameters. They outline some numerical examples, including electrohydrodynamic flows, to demonstrate the efficacy of the method. Finally, they pinpoint various emerging issues and futuristic perspectives connecting the phase-field method and computational microfluidics.

This paper gives unique perspectives to future directions of research on this topic.

The paper aims to develop an efficient data-driven modeling approach for the hydroelastic analysis of a semi-circular pipe conveying fluid with elastic end supports. Besides the structural displacement-dependent unsteady fluid force, the steady one related to structural initial configuration and the variable structural parameters (i.e. the variable support stiffness) are considered in the modeling.

The steady fluid force is treated as a pipe preload, and the elastically supported pipe-fluid model is dealt with as a prestressed hydroelastic system with variable parameters. To avoid repeated numerical simulations caused by parameter variation, structural and hydrodynamic reduced-order models (ROMs) instead of conventional computational structural dynamics (CSD) and computational fluid dynamics (CFD) solvers are utilized to produce data for the update of the structural, hydrodynamic and hydroelastic state-space equations. Radial basis function neural network (RBFNN), autoregressive with exogenous input (ARX) model as well as proper orthogonal decomposition (POD) algorithm are applied to modeling these two ROMs, and a hybrid framework is proposed to incorporate them.

The proposed approach is validated by comparing its predictions with theoretical solutions. When the steady fluid force is absent, the predictions agree well with the “inextensible theory”. The pipe always loses its stability via out-of-plane divergence first, regardless of the support stiffness. However, when steady fluid force is considered, the pipe remains stable throughout as flow speed increases, consistent with the “extensible theory”. These results not only verify the accuracy of the present modeling method but also indicate that the steady fluid force, rather than the extensibility of the pipe, is the leading factor for the differences between the in- and extensible theories.

The steady fluid force and the variable structural parameters are considered in the data-driven modeling of a hydroelastic system. Since there are no special restrictions on structural configuration, steady flow pattern and variable structural parameters, the proposed approach has strong portability and great potential application for other hydroelastic problems.

When the literature is reviewed carefully, the analytical solutions of these types of models are missing. First using appropriate similarity transformation, the equations are reduced to dimensionless form (NODE). To solve the reduced models, ansatz-based methods are considered. Finally, the explicit form solutions are obtained and the effects of material parameters and Prandtl number on the velocity and temperature profiles are shown in figures by the exact solutions. This study aims to discuss the aforementioned solution.

One of the non-Newtonian fluids is Eyring-Powell (EP) fluid which is derived from the kinetic theory of fluids. Two variations of EP model are considered to obtain the exact solutions that are missing in the literature. In order to obtain exact solutions, one of the ansatz-based methods is considered. The effects of material parameters and Prandtl number on the velocity and temperature profiles are shown in figures by the exact solutions. The results will guide to develop the model to predict the velocity profile and temperature profile when experimental data for dimensionless material parameters of EP fluid are available.

Finally, the explicit form solutions are obtained and the effects of material parameters and Prandtl number are shown in the figures. The results will guide to develop of the model to predict the velocity profile and temperature profile when experimental data for dimensionless material parameters of EP fluid are available. For the modified EP models, only special cases are considered. The generalized form, i.e. the modified EP models, which include deformation parameters, will be considered in the authors’ future work.

When the literature is reviewed carefully, the analytical solutions of these types of models are missing so by this work, the gap in the literature is filled. The explicit form solutions are obtained and the effects of material parameters and Prandtl number on the velocity and temperature profiles are shown in figures.

This paper sets out to present a fully explicit smoothed particle hydrodynamics (SPH) method to solve non‐Newtonian fluid flow problems.

The governing equations are momentum equations along with the continuity equation which are described in a Lagrangian framework. A new treatment similar to that used in Eulerian formulations is applied to viscous terms, which facilitates the implementation of various inelastic non‐Newtonian models. This approach utilizes the exact forms of the shear strain rate tensor and its second principal invariant to calculate the shear stress tensor. Three constitutive laws including power‐law, Bingham‐plastic and Herschel‐Bulkley models are studied in this work. The imposition of the incompressibility is fulfilled using a penalty‐like formulation which creates a trade‐off between the pressure and density variations. Solid walls are simulated by the boundary particles whose positions are fixed but contribute to the field variables in the same way as the fluid particles in flow field.

The performance of the proposed algorithm is assessed by solving three test cases including a non‐Newtonian dam‐break problem, flow in an annular viscometer using the aforementioned models and a mud fluid flow on a sloping bed under an overlying water. The results obtained by the proposed SPH algorithm are in close agreement with the available experimental and/or numerical data.

In this work, only inelastic non‐Newtonian models are studied. This paper deals with 2D problems, although extension of the proposed scheme to 3D is straightforward.

This study shows that various types of flow problems involving fluid‐solid and fluid‐fluid interfaces can be solved using the proposed SPH method.

Using the proposed numerical treatment of viscous terms, a unified and consistent approach was devised to study various non‐Newtonian flow models.

China has experienced a fast economic growth and shown a significant rise in income inequality in the past decades. During the same period, fluid milk consumption in urban areas has rapidly expanded. The objective of this paper is to analyze and simulate the influence of income distribution changes on fluid milk consumption of households in urban China.

This study applies an inverse hyperbolic sine (IHS) double-hurdle model to modeling at-home fluid milk consumption of households across different income strata based on a sample of 11,861 urban households in five provinces in China, and simulating the impact of changing income distribution, including five income growth patterns, on fluid milk consumption of total households as well as specific household groups.

The fluid milk consumption in urban China will continue to increase, with the unconditional income elasticity being 0.334 for the full sample and 0.347, 0.335 and 0.162 for the low-, middle-, and high-income groups, respectively. The simulation results show an evidence that, compared with distribution-neutral and disparity-enlarging income growth patterns, a rising income equality would lead to a more significant increase in fluid milk consumption. And the inequality-reducing income growth pattern has a larger impact on fluid milk consumption of households with seniors and no children, as well as households having no local urban household registration (hukou).

The government should strengthen the supply measures of fluid milk in urban areas, enlarge domestic dairy production, and diversify the sources of milk imports. It is also necessary to subsidize low-income families, especially households with seniors or households migrated from other areas without getting local urban hukou, which could simultaneously improve nutritional benefits and alleviate financial pressures.

A simulation considering the evolution of income distribution as well as different household groups is conducted. Widely distributed data with a large sample size and detailed demographic information are used. The problems of zero consumption and non-normal distribution are addressed by the IHS double-hurdle model.

This paper aims to investigate the flow of two-layered non-Newtonian fluids with different viscosities in an axisymmetric elastic tube.

A mathematical model was considered for this study to describe the flow characteristics of two-layered non- Newtonian Jeffrey fluids in an elastic tube. Because Jeffrey fluid model is a better model for the description of physiological fluid motion. Further, this model is a significant generalization of Newtonian fluid model. Analytical expressions for flux, stream functions, velocities and interface velocity have been derived in terms of elastic parameters, inlet, outlet and external pressures. The effects of various pertinent parameters on the flow behavior have been studied.

The volumetric flow rate was calculated by different models of Mazumdar (1992) and Rubinow and Keller (1972); from this it was found that the flux of Jeffrey fluid is more in the case of Rubinow and Keller model than Mazumdar. A comparative study is made between single-fluid and two-fluid models of Jeffrey fluid flows and it was observed that more flux and higher velocities were observed in the case of two-fluid model rather than single-fluid model. Furthermore, when both the Jeffrey parameter tends to zero and ratios of viscosities and radii are unity, the results in this study agree with those of Rubinow and Keller (1972).

To describe the fluid flow in an elastic tube with two-layered systems, the models and solutions developed here are very important. These results will be highly suitable in analyzing the rheological characteristics of blood flow in a small blood vessel because of their elastic nature.

The purpose of this paper is to systematically investigate the fluid lag phenomena and its influence in the hydraulic fracturing process, including all stages of fluid-lag evolution, the transition between different stages and their coupling with dynamic fracture propagation under common conditions.

A plane 2D model is developed to simulate the complex evolution of fluid lag during the propagation of a hydraulic fracture driven by an impressible Newtonian fluid. Based on the finite element method, a fully implicit solution scheme is proposed to solve the strongly coupled rock deformation, fluid flow and fracture propagation. Using the proposed model, comprehensive parametric studies are performed to examine the evolution of fluid lag in various geological and operational conditions.

The numerical simulations predict that the lag ratio is around 5% or even lower at the beginning stage of hydraulic fracture under practical geological conditions. With the fracture propagation, the lag ratio keeps decreasing and can be ignored in the late stage of hydraulic fracturing for typical parameter combinations. On the numerical aspect, whether the fluid lag can be ignored depends not only on the lag ratio but also on the minimum mesh size used for fluid flow. In addition, an overall mixed-mode fracture propagation factor is proposed to describe the relationship between diverse parameters and fracture curvature.

In this study, relatively simple physical models such as linear elasticity for solid, Newtonian model for fluid and linear elasticity fracture mechanics for fracture are used. The current model does not account for such effects like leak off, poroelasticity and softening of rock formations, which may also visibly affect the fluid lag depending on specific reservoir conditions.

This study helps to understand the effect of fluid lag during hydraulic fracturing processes and provides numerical experience in dealing with the fluid lag with finite element simulation.

Most studies of power‐law fluids are carried out using stress‐based system of Navier‐Stokes equations; and least‐squares finite element models for vorticity‐based equations of power‐law fluids have not been explored yet. Also, there has been no study of the weak‐form Galerkin formulation using the reduced integration penalty method (RIP) for power‐law fluids. Based on these observations, the purpose of this paper is to fulfill the two‐fold objective of formulating the least‐squares finite element model for power‐law fluids, and the weak‐form RIP Galerkin model of power‐law fluids, and compare it with the least‐squares finite element model.

For least‐squares finite element model, the original governing partial differential equations are transformed into an equivalent first‐order system by introducing additional independent variables, and then formulating the least‐squares model based on the lower‐order system. For RIP Galerkin model, the penalty function method is used to reformulate the original problem as a variational problem subjected to a constraint that is satisfied in a least‐squares (i.e. approximate) sense. The advantage of the constrained problem is that the pressure variable does not appear in the formulation.

The non‐Newtonian fluids require higher‐order polynomial approximation functions and higher‐order Gaussian quadrature compared to Newtonian fluids. There is some tangible effect of linearization before and after minimization on the accuracy of the solution, which is more pronounced for lower power‐law indices compared to higher power‐law indices. The case of linearization before minimization converges at a faster rate compared to the case of linearization after minimization. There is slight locking that causes the matrices to be ill‐conditioned especially for lower values of power‐law indices. Also, the results obtained with RIP penalty model are equally good at higher values of penalty parameters.

Vorticity‐based least‐squares finite element models are developed for power‐law fluids and effects of linearizations are explored. Also, the weak‐form RIP Galerkin model is developed.

Sand production is a challenging issue during hydrocarbon production in the oil and gas industry. This paper aims to investigate one sand production process, i.e. transient sand production, using a novel bonded particle lattice Boltzmann method. This mesoscopic technique provides a unique insight into complicated sand erosion process during oil exploitation.

The mesoscopic fluid-particle coupling is directly approached by the immersed moving boundary method in the framework of lattice Boltzmann method. Bonded particle method is used for resolving the deformation of solid. The onset of grain erosion of rocks, which are modelled by a bonded particle model, is realised by breaking the bonds simulating cementation when the tension or tangential force exceeds critical values.

It is proved that the complex fluid–solid interaction occurring at the pore/grain level can be well captured by the immersed moving boundary scheme in the framework of the lattice Boltzmann method. It is found that when the drawdown happens at the wellbore cavity, the tensile failure area appears at the edge of the cavity. Then, the tensile failure area gradually propagates inward, and the solid particles at the tensile failure area become fluidised because of large drag forces. Subsequently, some eroded particles are washed out. This numerical investigation is demonstrated through comparison with the experimental results. In addition, through breaking the cementation, which is simulated by bond models, between bonded particles, the transient particle erosion process is successfully captured.

A novel bonded particle lattice Boltzmann method is used to investigate the sand production problem at the grain level. It is proved that the complex fluid–solid interaction occurring at the pore/grain level can be well captured by the immersed moving boundary scheme in the framework of the lattice Boltzmann method. Through breaking the cementation, which is simulated by bond models, between bonded particles, the transient particle erosion process is successfully captured.

To form and develop a new mode of mixed elastohydrodynamic lubrication (mixed EHL) which is more realistic and of more application values to a practical elastohydrodynamic contact on gears, cams and roller bearings than the previous and current existing mixed EHL models.

The representative theoretical and experimental studies on elastohydrodynamic lubrication (EHL) and mixed EHL carried out in the previous and recent time, including those of the author, are reviewed. The obtained results on EHL and mixed EHL in those studies and the viewpoints on the mode of mixed EHL based on those results developed in those studies are compared and evaluated. Strong proves are formed on the new mode of mixed EHL proposed in the present paper based on these comparisons and evaluations.

Strong viewpoints are formed on the mode of the occurrence of dry contact in EHL in a practical concentrated contact. A new mode of mixed EHL is proposed by incorporating this mode of the occurrence of dry contact in EHL. Also, comments and evaluations on the previous researches on mixed EHL are made.

A very useful material for the engineers who are engaged in the design of EHL on gears, cams and roller bearings, and for the tribology scientists who thrust efforts in studying EHL and mixed EHL both by theoretical modeling and by experiments.

A new mode of mixed EHL is originally proposed by incorporating the finding of a more realistic mode of the occurrence of dry contact in EHL. This new mode of mixed EHL should become the direction of the theoretical research of mixed EHL in the future time. It provides a clearer way to this research.