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A series of primary drainage experiments was carried out in order to investigate nonequilibrium capillarity effects in two‐phase flow through porous media. Experiments were performed with tetrachloroethylene (PCE) and water as immiscible fluids in a sand column 21 cm long. Four drainage experiments were performed by applying large pressures on the nonwetting phase at the inlet boundary: 20, 30, 35, 38 kPa. Our results showed that the nonequilibrium local fluids pressure difference‐saturation curves are above the capillary pressure saturation curve. Moreover, the nonequilibrium pressure difference showed a nonmonotonic behavior with an overshoot that was more pronounced at higher injection pressures. The dynamic capillarity coefficient was calculated from measured local pressures and saturations (the scale of sensor devices, 0.7 cm). Its value was found to vary between 1.3 × 105 to 2 × 105 Pa s. Within the saturation range of 0.50 > Sw > 0.85, no clear dependency of the dynamic coefficient on the wetting saturation was observed. Also, no dependency of the dynamic capillarity coefficient on the applied boundary pressure was found. Averaged values of at the length scales of 11 and 18 cm were also estimated from averaged pressures and saturations. The upscaled dynamic coefficient was found to vary between 0.5 × 106 and 1.2 × 106 Pa s at the average window size of 11 cm. This is one order of magnitude larger than the local‐scale coefficient. Larger values were found for the length scale of 18 cm: 1.5 × 106 and 2.5 × 106 Pa s. This suggests that the value of dynamic coefficient increases with the scale of observation. Key Points The non‐equilibrium pressure difference shows a non‐monotonic behavior Pressure difference overshoot is more pronounced at higher injection pressures The dynamic capillarity coefficient τ is a length scale dependent

Classical Darcy's equation for multiphase flow assumes that gravity and the gradient in fluid pressure are the only driving forces and resistance to the flow is parameterized by (relative) permeability as a function of saturation. It is conceivable that, in multiphase flow, other driving forces may also exist. This would mean that such nonequilibrium effects are lumped into a permeability coefficient. Indeed, many studies have shown that the relative permeability coefficient generally depends not only on saturation but also on dynamics of the system. Through the application of rational thermodynamics, a theory of two‐phase flow had been developed in which interfacial areas were introduced as separate thermodynamic entities and their macroscale effects were explicitly included. This theory includes new driving forces whose significance needs still to be established. To study new terms in the theory, we employ a dynamic pore network model called DYPOSIT. A long pore network (several representative elementary volumes connected in series) is generated, which represents as a one‐dimensional porous medium column. This model provides pore scale distribution of local phase pressures, capillary pressure, interfacial area, saturation, and flow rate, which are averaged to obtain the macroscale distributions of these variables. Our analysis shows that there are discrepancies between the simulation results and the classical equations to describe the transient behavior, especially for the nonwetting phase transient permeability. The coefficients in the extended equations are quantified and parameterized. Although under the applied Dirichlet boundary conditions, flow varies significantly, there is a clear trend illustrating dependency of coefficients on saturation, independent of dynamic conditions. Furthermore, using the new coefficients, it is possible to explain either transient or steady state flow regimes, which is a new achievement. Key Points The classical Darcy's law for two‐phase flow has flaws The gradient of pressure alone cannot explain the two‐phase flow correctly Interfacial area can be the missing state variable in two‐phase flow

Numerical models have been widely used to simulate multiphase flow in porous media for a variety of applications (e.g., NAPL migration in subsurface aquifers, carbon sequestration, agriculture, paper production, and petroleum reservoir development). The relationship between the difference in phase pressures and saturation is used as one of the important constitutive relationships in numerical models. Theoretical studies have suggested that this relationship should include a damping coefficient or capillarity coefficient () on the basis of thermodynamic considerations. A literature review suggests that the magnitude of this capillarity coefficient varies by over three orders of magnitude. While recent experimental studies have explored the effect of porous medium properties, effect of domain size, hysteresis, and the imposed boundary conditions on the magnitude of , there has been no experimental study investigating the impact of fluid viscosity on . This study reports on a series of primary drainage experiments conducted under both static and dynamic conditions in F70 silica sand. Fluid pairs used included water and silicone oil with two differing viscosities and slightly different densities (used as model nonaqueous phase liquids) in addition to air. Water saturation and both wetting and nonwetting phase pressures were measured in a custom‐built aluminum column using EC‐5 probes and tensiometers at three levels. Results show a strong dependence of the magnitude of the capillarity coefficient on effective fluid viscosity. This implies that consideration should be given for the inclusion of a capillarity coefficient in modeling tools used to simulate multiphase flow when fluids saturations are changing rapidly and when fluids have a large viscosity ratio. Key Points Rate dependence of the phase pressure difference Capillarity coefficient is a function of fluid viscosity Capillarity coefficient is relatively constant when normalized to effective viscosity

Hybrid particle–continuum computational frameworks permit the simulation of gas flows by locally adjusting the resolution to the degree of non-equilibrium displayed by the flow in different regions of space and time. In this work, we present a new scheme that couples the direct simulation Monte Carlo (DSMC) with the lattice Boltzmann (LB) method in the limit of isothermal flows. The former handles strong non-equilibrium effects, as they typically occur in the vicinity of solid boundaries, whereas the latter is in charge of the bulk flow, where non-equilibrium can be dealt with perturbatively, i.e. according to Navier–Stokes hydrodynamics. The proposed concurrent multiscale method is applied to the dilute gas Couette flow, showing major computational gains when compared with the full DSMC scenarios. In addition, it is shown that the coupling with LB in the bulk flow can speed up the DSMC treatment of the Knudsen layer with respect to the full DSMC case. In other words, LB acts as a DSMC accelerator. This article is part of the themed issue ‘Multiscale modelling at the physics–chemistry–biology interface’.

Rayleigh-Taylor (RT) instability widely exists in nature and engineering fields. How to better understand the physical mechanism of RT instability is of great theoretical significance and practical value. At present, abundant results of RT instability have been obtained by traditional macroscopic methods. However, research on the thermodynamic non-equilibrium (TNE) effects in the process of system evolution is relatively scarce. In this paper, the discrete Boltzmann method based on non-equilibrium statistical physics is utilized to study the effects of the specific heat ratio on compressible RT instability. The evolution process of the compressible RT system with different specific heat ratios can be analyzed by the temperature gradient and the proportion of the non-equilibrium region. Firstly, as a result of the competition between the macroscopic magnitude gradient and the non-equilibrium region, the average TNE intensity first increases and then reduces, and it increases with the specific heat ratio decreasing; the specific heat ratio has the same effect on the global strength of the viscous stress tensor. Secondly, the moment when the total temperature gradient in y direction deviates from the fixed value can be regarded as a physical criterion for judging the formation of the vortex structure. Thirdly, under the competition between the temperature gradients and the contact area of the two fluids, the average intensity of the non-equilibrium quantity related to the heat flux shows diversity, and the influence of the specific heat ratio is also quite remarkable.

Based on the framework of our previous work [H.L. Lai et al., Phys. Rev. E, 94, 023106 (2016)], we continue to study the effects of Knudsen number on two-dimensional Rayleigh–Taylor (RT) instability in compressible fluid via the discrete Boltzmann method. It is found that the Knudsen number effects strongly inhibit the RT instability but always enormously strengthen both the global hydrodynamic non-equilibrium (HNE) and thermodynamic non-equilibrium (TNE) effects. Moreover, when Knudsen number increases, the Kelvin–Helmholtz instability induced by the development of the RT instability is difficult to sufficiently develop in the later stage. Different from the traditional computational fluid dynamics, the discrete Boltzmann method further presents a wealth of non-equilibrium information. Specifically, the two-dimensional TNE quantities demonstrate that, far from the disturbance interface, the value of TNE strength is basically zero; the TNE effects are mainly concentrated on both sides of the interface, which is closely related to the gradient of macroscopic quantities. The global TNE first decreases then increases with evolution. The relevant physical mechanisms are analyzed and discussed.

Using a direct numerical simulation (DNS), we investigate the onset of non-equilibrium effects and the subsequent emergence of a self-preserving state as a turbulent boundary layer (TBL) encounters a smooth-to-rough (STR) step change. The rough surface comprises over 2500 staggered cuboid-shaped elements where the first row is placed at 50 θ0 from the inflow. A Reθ=4500  value is attained along with δk≈35 as the TBL develops. While different flow parameters adjust at dissimilar rates that further depend on the vertical distance from the surface and perhaps on δSTR/k, an equilibrium for wall stress, mean velocity, and Reynolds stresses exists across the entire TBL by 35 δSTR after the step change. First-order statistics inside the inner layer adapt much earlier, i.e., at 10–15 δSTR after the step change. Like rough-to-smooth (RTS) scenarios, an equilibrium layer develops from the surface. Unlike RTS transitions, a nascent logarithmic layer is identifiable much earlier, at 4 δSTR after the step change. The notion of equivalent sandgrain roughness does not apply upstream of this fetch because non-equilibrium advection effects permeate into the inner layer. The emergent equilibrium TBL is categorized by a fully rough state (ks+≈120–130; ks/k≈2.8). Decomposition of wall stress into constituent parts reveals no streamwise dependence. Mean velocity in the outer layer is well approximated by Coles’ wake law. The wake parameter and shape factor are enhanced above their smooth-wall counterparts. Quadrant analysis shows that shear-stress-producing motions adjust promptly to the roughness, and the balance between ejections and sweeps in the outer layer remains impervious to the underlying surface.

In this work, the impact of local thermal non-equilibrium on thermophoresis and electrophoresis in the bioconvection flow of Boger nanofluid over a sheet containing porous media and gyrotactic microorganisms are examined. It can be used to enhance the design of microfluidic devices used in biomedical engineering, particularly targeted drug delivery systems that need precise particle manipulation. Additionally, it may be applied in environmental engineering to eliminate contaminants by means of particle deposition during wastewater treatment. Additionally, the model contributes to the optimization of cooling systems in microelectronics by increasing the competence of heat transmission in nanofluid-based cooling methods. Bioenergy systems are also impacted, especially when it comes to enhancing the production of biofuel through bioconvection processes. Ordinary differential equations (ODEs) were created from the constitutive equations using similarity variables, and MATLAB's bvp4c tool was used to solve them. The analysis shows that the concentration profile for electrophoretic particle deposition improves as the values increase, while the tendency for thermophoretic particle deposition is the opposite. Graphical abstract

Rayleigh–Taylor (RT) instability is a basic fluid interface instability that widely exists in nature and in the engineering field. To investigate the impact of the initial inclined interface on compressible RT instability, the two-component discrete Boltzmann method is employed. Both the thermodynamic non-equilibrium (TNE) and hydrodynamic non-equilibrium (HNE) effects are studied. It can be found that the global average density gradient in the horizontal direction, the non-organized energy fluxes, the global average non-equilibrium intensity and the proportion of the non-equilibrium region first increase and then reduce with time. However, the global average density gradient in the vertical direction and the non-organized moment fluxes first descend, then rise, and finally descend. Furthermore, the global average density gradient, the typical TNE intensity and the proportion of non-equilibrium region increase with increasing angle of the initial inclined interface. Physically, there are three competitive mechanisms: (1) As the perturbed interface elongates, the contact area between the two fluids expands, which results in an increasing gradient of macroscopic physical quantities and leads to a strengthening of the TNE effects. (2) Under the influence of viscosity, the perturbation pressure waves on both sides of the material interface decrease with time, which makes the gradient of the macroscopic physical quantity decrease, resulting in a weakening of the TNE strength. (3) Due to dissipation and/or mutual penetration of the two fluids, the gradient of macroscopic physical quantities gradually diminishes, resulting in a decrease in the intensity of the TNE.

In the top submerged lance (TSL) smelting process, flow boiling may occur in the lance’s inner pipe due to the heat coming from the furnace when liquid fuel is adopted. In the current study, a numerical simulation was carried out by coupling the Eulerian two-fluid model with the improved RPI wall boiling model to investigate the subcooled n-heptane flow boiling in the inner pipe. The effects of inlet velocity and pipe wall emissivity on two-phase flow and heat transfer are elucidated. The results show that, for pipes with inlet velocity ranging from 0.3 m·s−1 to 1.0 m·s−1, an increase in inlet velocity leads to a lower void fraction near the outlet, as well as a lower average velocity and a lower average temperature of each phase. Meanwhile, the Onset of Nucleate Boiling (ONB) position approaches to the outlet, and the total pressure drop of the entire pipe reduces when the inlet velocity increases. However, the opposite trends appear when increasing the pipe wall emissivity. The maximum wall temperature corresponding to the critical heat flux (CHF) point is slightly affected by inlet velocity but significantly affected by pipe wall emissivity. The non-equilibrium effect and the specific components of pressure drop are also further investigated.