We prove discrete-to-continuum convergence for dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with cost functional having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of $1$-Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.

Homogenisation is a widely used technique in manufacturing powdered milk with a direct impact on product solubility, and the homogenisation pressure is a central attribute of this process. We aimed to understand the effect of increasing homogenisation pressures (0/0, 15/5, and 75/5 MPa, 1st/2nd stages) on particle-size distribution during homogenised whole milk powder manufacture and rehydration of the final product. The fluid milk was thermally treated, homogenised, concentrated by rotary evaporation, and then dried using a spray dryer. Particle size (Dv90) was monitored at all stages of the manufacturing process. The final product (milk powder) was analysed using particle-size distribution, electronic scanning microscopy, water activity, and isotherms. The results demonstrated that increasing the homogenisation pressure leads to milk powder with smaller particle size when rehydrated (Dv90 values: 6.08, 1.48 and 0.64 μm for 0, 20 and 80 MPa, respectively). Furthermore, the volume (%) of the particles in the ‘sub-micro’ region (smaller than 1.0 μm) presented an inversely proportional profile to the homogenisation pressure (homogenised fluid milk: 86.1, 29.3 and 2.4%; concentrated milk: 86.1, 26.5 and 5.7%, and reconstituted milk powder: 84.2, 31.8 and 10.9%). Surprisingly, this pattern was not observed in the SPAN value (which corresponds to the width or range of the size distribution based on the volume). Additionally, the increase in the homogenisation pressure did not affect the sorption isotherm pattern. These results demonstrate that increasing the homogenisation pressure decreases the particle size of the reconstituted powdered milk, indicating the potential for future studies on how this phenomenon affects its physicochemical and final product properties.

This research paper aimed to locate protein modifications caused by treatment of milk and determine if the modification locations were consistent. The majority of milk for consumption is homogenised using pressure and heat, and this causes changes in the location of proteins in the milk as well as protein modifications. To investigate these proteomic changes, raw milk was pasteurised (72°C, 15 s), then, to separate the treatment for homogenisation, heated at these different pressures and temperatures: 45°C without no pressure applied, 45°C with 35 MPa, 80°C without pressure applied and 80°C, with 35 MPa. Proteomic analysis was done after separating the milk into three fractions: whey, casein and cream. Protein modifications in each fraction were examined and we found Maillard products as well as oxidation to be of interest. The proteins were also further identified and characterised to compare protein modification sites and differences in proteins present in the cream resulting from homogenisation and/or pasteurisation. This experiment showed that both heat and pressure during homogenisation can cause increases in protein modifications as a result of oxidation or the Maillard reaction. Total cysteine oxidation and total proline oxidation differed between treatments although this was only significantly different for cysteine. It was observed that protein modifications occurred in the same location in the protein sequence rather than in random locations which we highlighted by examining α-S1-casein, lactadherin and β-lactoglobulin.

Evaporation within porous media is both a multiscale and interface-driven process, since the phase change at the evaporating interfaces within the pores generates a vapour flow and depends on the transport of vapour through the porous medium. While homogenised models of flow and chemical transport in porous media allow multiscale processes to be modelled efficiently, it is not clear how the multiscale effects impact the interface conditions required for these homogenised models. In this paper, we derive a homogenised model, including effective interface conditions, for the motion of an evaporation front through a porous medium, using a combined homogenisation and boundary layer analysis. This analysis extends previous work for a purely diffusive problem to include both gas flow and the advective–diffusive transport of material. We investigate the effect that different microscale models describing the chemistry of the evaporation have on the homogenised interface conditions. In particular, we identify a new effective parameter, $\mathcal{L}$, the average microscale interface length, which modifies the effective evaporation rate in the homogenised model. Like the effective diffusivity and permeability of a porous medium, $\mathcal{L}$ may be found by solving a periodic cell problem on the microscale. We also show that the different microscale models of the interface chemistry result in fundamentally different fine-scale behaviour at, and near, the interface.

We investigate a reaction–diffusion problem in a two-component porous medium with a nonlinear interface condition between the different components. One component is connected and the other one is disconnected. The ratio between the microscopic pore scale and the size of the whole domain is described by the small parameter $\epsilon$. On the interface between the components, we consider a dynamic Wentzell-boundary condition, where the normal fluxes from the bulk domains are given by a reaction–diffusion equation for the traces of the bulk solutions, including nonlinear reaction kinetics depending on the solutions on both sides of the interface. Using two-scale techniques, we pass to the limit $\epsilon \to 0$ and derive macroscopic models, where we need homogenisation results for surface diffusion. To cope with the nonlinear terms, we derive strong two-scale convergence results.

This paper presents the current state of mathematical modelling of the electrochemical behaviour of lithium-ion batteries (LIBs) as they are charged and discharged. It reviews the models developed by Newman and co-workers, both in the cases of dilute and moderately concentrated electrolytes and indicates the modelling assumptions required for their development. Particular attention is paid to the interface conditions imposed between the electrolyte and the active electrode material; necessary conditions are derived for one of these, the Butler–Volmer relation, in order to ensure physically realistic solutions. Insight into the origin of the differences between various models found in the literature is revealed by considering formulations obtained by using different measures of the electric potential. Materials commonly used for electrodes in LIBs are considered and the various mathematical models used to describe lithium transport in them discussed. The problem of upscaling from models of behaviour at the single electrode particle scale to the cell scale is addressed using homogenisation techniques resulting in the pseudo-2D model commonly used to describe charge transport and discharge behaviour in lithium-ion cells. Numerical solution to this model is discussed and illustrative results for a common device are computed.

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

The retention of human milk (HM) fat in nasogastric probes of infusion pumps can be observed during the feed of infants unable to suck at the mother’s breast. The lack of homogenisation of HM could contribute to the fat holding. Therefore, the present study evaluated (i) the influence of homogenisation on milk fat retaining in infant feeding probes and (ii) the in vivo effect of the homogenisation on lipid absorption by Wistar rats. The animals were fed with HM treated following two processing conditions, that is, pasteurised and homogenised–pasteurised. The animals were randomly subdivided into four experimental groups: water-fed (control), pasteurised milk, homogenised–pasteurised milk and pasteurised–skimmed milk. The results of food consumption, mass body gain, corporate metrics and plasma blood levels of total cholesterol did not show any difference (P < 0·05) among the three types of HM used in the experiments. The liver, intestine and intra-abdominal adipose tissue of the four groups of animals presented normal and healthy histology. The composition of fatty acids in the brain tissue of animals fed with homogenised HM increased when compared with the groups fed with non-homogenised HM. These values were 11·08 % higher for arachidonic acids, 6·59 % for DAH and 47·92 % for nervous acids. The ingestion of homogenised HM promoted higher absorption of milk nutrients. Therefore, the addition of the homogenisation stage in HM processing could be an alternative to reduce fat retention in probes and to improve the lipids’ absorption in the body.

Poroelastic effects have been of great interest in the seismic literature as they have been identified as a major cause of wave attenuation in heterogeneous porous media. The observed attenuation in the seismic wave can be explained in part by energy loss to fluid motion in the pores. On the other hand, it is known that the attenuation is particularly pronounced in stratified structures where the scale of spatial heterogeneity is much smaller than the seismic wavelength. Understanding of poroelastic effects on seismic wave attenuation in heterogeneous porous media has largely relied on numerical experiments. In this work, we present a homogenisation technique to obtain an upscaled viscoelastic model that captures seismic wave attenuation when the sub-seismic scale heterogeneity is periodic. The upscaled viscoelastic model directly relates seismic wave attenuation to the material properties of the heterogeneous structure. We verify our upscaled viscoelastic model against a full poroelastic model in numerical experiments. Our homogenisation technique suggests a new approach for solving coupled equations of motion.

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

The paper is devoted to the existence and rigorous homogenisation of the generalised Poisson–Nernst–Planck problem describing the transport of charged species in a two-phase domain. By this, inhomogeneous conditions are supposed at the interface between the pore and solid phases. The solution of the doubly non-linear cross-diffusion model is discontinuous and allows a jump across the phase interface. To prove an averaged problem, the two-scale convergence method over periodic cells is applied and formulated simultaneously in the two phases and at the interface. In the limit, we obtain a non-linear system of equations with averaged matrices of the coefficients, which are based on cell problems due to diffusivity, permittivity and interface electric flux. The first-order corrector due to the inhomogeneous interface condition is derived as the solution to a non-local problem.

Several prospective studies have shown an association between cows’ milk consumption and the risk of islet autoimmunity and/or type 1 diabetes. We wanted to study whether processing of milk plays a role. A population-based birth cohort of 6081 children with HLA-DQB1-conferred risk to type 1 diabetes was followed until the age of 15 years. We included 5545 children in the analyses. Food records were completed at the ages of 3 and 6 months and 1, 2, 3, 4 and 6 years, and diabetes-associated autoantibodies were measured at 3–12-month intervals. For milk products in the food composition database, we used conventional and processing-based classifications. We analysed the data using a joint model for longitudinal and time-to-event data. By the age of 6 years, islet autoimmunity developed in 246 children. Consumption of all cows’ milk products together (energy-adjusted hazard ratio 1·06; 95 % CI 1·02, 1·11; P = 0·003), non-fermented milk products (1·06; 95 % CI 1·01, 1·10; P = 0·011) and fermented milk products (1·35; 95 % CI 1·10, 1·67; P = 0·005) was associated with an increased risk of islet autoimmunity. The early milk consumption was not associated with the risk beyond 6 years. We observed no clear differences based on milk homogenisation and heat treatment. Our results are consistent with the previous studies, which indicate that high milk consumption may cause islet autoimmunity in children at increased genetic risk. The study did not identify any specific type of milk processing that would clearly stand out as a sole risk factor apart from other milk products.

The decontamination of hazardous chemical agents from porous media is an important and critical part of the clean-up operation following a chemical weapon attack. Decontamination is often achieved through the application of a cleanser, which reacts on contact with an agent to neutralise it. While it is relatively straightforward to write down a model that describes the interplay of the agent and cleanser on the scale of the pores in the porous medium, it is computationally expensive to solve such a model over realistic spill sizes.

In this paper, we consider the homogenisation of a pore-scale model for the interplay between agent and cleanser, with the aim of generating simplified models that can be solved more easily on the spill scale but accurately capture the microscale structure and chemical activity. We consider two situations: one in which the agent completely fills local porespaces and one in which it does not. In the case when the agent does not completely fill the porespace, we use established homogenisation techniques to systematically derive a reaction–diffusion model for the macroscale concentration of cleanser. However, in the case where the agent completely fills the porespace, the homogenisation procedure is more in-depth and involves a two-timescale approach coupled with a spatial boundary layer. The resulting homogenised model closely resembles the microscale model with the effect of the porous material being incorporated into the parameters. The two models cater for two different spill scenarios and provide the foundation for further study of reactive decontamination.

Tissue engineering aims to grow artificial tissues in vitro to replace those in the body that have been damaged through age, trauma or disease. A recent approach to engineer artificial cartilage involves seeding cells within a scaffold consisting of an interconnected 3D-printed lattice of polymer fibres combined with a cast or printed hydrogel, and subjecting the construct (cell-seeded scaffold) to an applied load in a bioreactor. A key question is to understand how the applied load is distributed throughout the construct. To address this, we employ homogenisation theory to derive equations governing the effective macroscale material properties of a periodic, elastic–poroelastic composite. We treat the fibres as a linear elastic material and the hydrogel as a poroelastic material, and exploit the disparate length scales (small inter-fibre spacing compared with construct dimensions) to derive macroscale equations governing the response of the composite to an applied load. This homogenised description reflects the orthotropic nature of the composite. To validate the model, solutions from finite element simulations of the macroscale, homogenised equations are compared to experimental data describing the unconfined compression of the fibre-reinforced hydrogels. The model is used to derive the bulk mechanical properties of a cylindrical construct of the composite material for a range of fibre spacings and to determine the local mechanical environment experienced by cells embedded within the construct.

The objective of the research presented in this paper was to investigate how different characteristics of whey protein microparticles (MWP) added to milk as fat replacers influence intermolecular interactions occurring with other milk proteins during homogenisation and heating. These interactions are responsible for the formation of heat-induced aggregates that influence the texture and sensory characteristics of the final product. The formation of heat-induced complexes was studied in non- and low-fat milk model systems, where microparticulated whey protein (MWP) was used as fat replacer. Five MWP types with different particle characteristics were utilised and three heat treatments used: 85 °C for 15 min, 90 °C for 5 min and 95 °C for 2 min. Surface characteristics of the protein aggregates were expressed as the number of available thiol groups and the surface net charge. Intermolecular interactions involved in the formation of protein aggregates were studied by polyacrylamide gel electrophoresis and the final complexes visualised by darkfield microscopy. Homogenisation of non-fat milk systems led to partial adsorption of caseins onto microparticles, independently of the type of microparticle. On the contrary, homogenisation of low-fat milk resulted in preferential adsorption of caseins onto fat globules, rather than onto microparticles. Further heating of the milk, led to the formation of heat induced complexes with different sizes and characteristics depending on the type of MWP and the presence or not of fat. The results highlight the importance of controlling homogenisation and heat processing in yoghurt manufacture in order to induce desired changes in the surface reactivity of the microparticles and thereby promote effective protein interactions.

A drift-diffusion model for charge transport in an organic bulk heterojunction solar cell, formed by conjoined acceptor and donor materials sandwiched between two electrodes, is formulated. The model accounts for (i) bulk photogeneration of excitons, (ii) exciton drift and recombination, (iii) exciton dissociation (into polarons) on the acceptor–donor interface, (iv) polaron recombination, (v) polaron dissociation into a free electron (in the acceptor) and a hole (in the donor), (vi) electron/hole transport and (vii) electron–hole recombination on the acceptor–donor interface. A finite element method is employed to solve the model in a cell with a highly convoluted acceptor/donor interface. The solutions show that, with physically realistic parameters, and in the power generating regime, the solution varies little on the scale of the micro-structure. This motivates us to homogenise over the micro-structure; a process that yields a far simpler one-dimensional effective medium model on the cell scale. The comparison between the solution of the full model and the effective medium (homogenised) model is very favourable for applied voltages less than the built-in voltage (the power generating regime) but breaks down as the applied voltages increases above it. Furthermore, it is noted that the homogenisation technique provides a systematic way to relate effective medium modelling of bulk heterojunctions [19, 25, 36, 37, 42, 59] to a more fundamental approach that explicitly models the full micro-structure [8, 38, 39, 58] and that it allows the parameters in the effective medium model to be derived in terms of the geometry of the micro-structure. Finally, the effective medium model is used to investigate the effects of modifying the micro-structure geometry, of a device with an interdigitated acceptor/donor interface, on its current–voltage curve.

In this work we study the homogenisation problem for nonlinear elliptic equations involving $p$-Laplacian-type operators with sign-changing weights. We study the asymptotic behaviour of variational eigenvalues which consist of a double sequence of eigenvalues. We show that the $k$th positive eigenvalue goes to infinity when the average of the weights is nonpositive, and converges to the $k$th variational eigenvalue of the limit problem when the average is positive for any $k\geq 1$.

Asymptotic homogenisation via the method of multiple scales is considered for problems in which the microstructure comprises inclusions of one material embedded in a matrix formed from another. In particular, problems are considered in which the interface conditions include a global balance law in the form of an integral constraint; this may be zero net charge on the inclusion, for example. It is shown that for such problems care must be taken in determining the precise location of the interface; a naive approach leads to an incorrect homogenised model. The method is applied to the problems of perfectly dielectric inclusions in an insulator, and acoustic wave propagation through a bubbly fluid in which the gas density is taken to be negligible.

Water flow in plant tissues takes place in two different physical domains separated bysemipermeable membranes: cell insides and cell walls. The assembly of all cell insides andcell walls are termed symplast and apoplast,respectively. Water transport is pressure driven in both, where osmosis plays an essentialrole in membrane crossing. In this paper, a microscopic model of water flow and transportof an osmotically active solute in a plant tissue is considered. The model is posed on thescale of a single cell and the tissue is assumed to be composed of periodicallydistributed cells. The flow in the symplast can be regarded as a viscous Stokes flow,while Darcy’s law applies in the porous apoplast. Transmission conditions at the interface(semipermeable membrane) are obtained by balancing the mass fluxes through the interfaceand by describing the protein mediated transport as a surface reaction. Applyinghomogenization techniques, macroscopic equations for water and solute transport in a planttissue are derived. The macroscopic problem is given by a Darcy law with a force termproportional to the difference in concentrations of the osmotically active solute in thesymplast and apoplast; i.e. the flow is also driven by the local concentration differenceand its direction can be different than the one prescribed by the pressure gradient.

We study the homogenisation of the Stokes system with a non-homogeneous Fourier boundary condition on the boundary of the holes, depending on a parameter γ. Such systems arise in the modelling of the flow of an incompressible viscous fluid through a porous medium under the influence of body forces. At the limit, by using the periodic unfolding method in perforated domains, we obtain, following the values of γ, different Darcy's laws of type Mu = −N∇p + F with suitable matrices M and N with F depending on the right-hand side in the bulk term and in the boundary condition.