Approximate solution techniques for the sorption of a finite amount of swelling solvent in a glassy polymer

This study examines a one-dimensional Stefan problem describing the sorption of a finite amount of swelling solvent in a glassy polymer. The polymer is initially in a dry non-swollen state, where the polymer network is dense. The polymer is then injected with a critical concentration of a swelling solvent, causing polymer chain relaxation to occur. A moving boundary separating the swollen rubbery polymer from the dry glassy polymer is created, whose speed is defined by a kinetic law. The form of the kinetic law is typically assumed to be linear, but this is a nonphysical restriction, and thus we consider the case for a general exponent. We present formal asymptotic expansions, for both small and large times as well as for small and large values of the control parameter, as well as considering the heat balance integral method. These approximations are compared with a numerical scheme, which uses a boundary immobilisation technique and correctly identifies the appropriate starting solution

Sinking bubbles in stout beers

A surprising phenomenon witnessed by many is the sinking bubbles seen in a settling pint of stout beer. Bubbles are less dense than the surrounding fluid so how does this happen? Previous work has shown that the explanation lies in a circulation of fluid promoted by the tilted sides of the glass. However, this work has relied heavily on computational fluid dynamics (CFD) simulations. Here, we show that the phenomenon of sinking bubbles can be predicted using a simple analytic model. To make the model analytically tractable, we work in the limit of small bubbles and consider a simplified geometry. The model confirms both the existence of sinking bubbles and the previously proposed mechanism

Approximate solution techniques for the sorption of a finite amount of swelling solvent in a glassy polymer

This study examines a one-dimensional Stefan problem describing the sorption of a finite amount of swelling solvent in a glassy polymer. The polymer is initially in a dry non-swollen state, where the polymer network is dense. The polymer is then injected with a critical concentration of a swelling solvent, causing polymer chain relaxation to occur. A moving boundary separating the swollen rubbery polymer from the dry glassy polymer is created, whose speed is defined by a kinetic law. The form of the kinetic law is typically assumed to be linear, but this is a nonphysical restriction, and thus we consider the case for a general exponent. We present formal asymptotic expansions, for both small and large times as well as for small and large values of the control parameter, as well as considering the heat balance integral method. These approximations are compared with a numerical scheme, which uses a boundary immobilisation technique and correctly identifies the appropriate starting solution

Sinking bubbles in stout beers

A surprising phenomenon witnessed by many is the sinking bubbles seen in a settling pint of stout beer. Bubbles are less dense than the surrounding fluid so how does this happen? Previous work has shown that the explanation lies in a circulation of fluid promoted by the tilted sides of the glass. However, this work has relied heavily on computational fluid dynamics (CFD) simulations. Here, we show that the phenomenon of sinking bubbles can be predicted using a simple analytic model. To make the model analytically tractable, we work in the limit of small bubbles and consider a simplified geometry. The model confirms both the existence of sinking bubbles and the previously proposed mechanism

Analysis of a model for the formation of fold-type oscillation marks in the continuous casting of steel

This paper investigates the different possible behaviours of a recent asymptotic model for oscillation-mark formation in the continuous casting of steel, with particular focus on how the results obtained vary when the heat transfer coefficient (m), the thermal resistance (Rmf ) and the dependence of the viscosity of the flux powder as a function of temperature, μf (T) , are changed. It turns out that three different outcomes are possible: (I) the flux remains in molten state and no solid flux ever forms; (II) both molten and solid flux are present, and the profile of the oscillation mark is continuous with respect to the space variable in the casting direction; (III) both molten and solid flux are present, and the profile of the oscillation mark is discontinuous with respect to the space variable in the casting direction. Although (I) gave good agreement with experimental data, it suffered the drawback that solid flux is typically observed during actual continuous casting; this has been rectified in this work via alternative (II). On the other hand, alternative (III) can occur as a result of hysteresis-type phenomenon that is encountered in other flows that involve temperature-dependent viscosity; in the present case, this manifests itself via the possibility of multiple states for the oscillation-mark profile at the instants in time when solid flux begins to form and when it ceases to form

Analysis of a model for the formation of fold-type oscillation marks in the continuous casting of steel

This paper investigates the different possible behaviours of a recent asymptotic model for oscillation-mark formation in the continuous casting of steel, with particular focus on how the results obtained vary when the heat transfer coefficient (m), the thermal resistance (Rmf ) and the dependence of the viscosity of the flux powder as a function of temperature, μf (T) , are changed. It turns out that three different outcomes are possible: (I) the flux remains in molten state and no solid flux ever forms; (II) both molten and solid flux are present, and the profile of the oscillation mark is continuous with respect to the space variable in the casting direction; (III) both molten and solid flux are present, and the profile of the oscillation mark is discontinuous with respect to the space variable in the casting direction. Although (I) gave good agreement with experimental data, it suffered the drawback that solid flux is typically observed during actual continuous casting; this has been rectified in this work via alternative (II). On the other hand, alternative (III) can occur as a result of hysteresis-type phenomenon that is encountered in other flows that involve temperature-dependent viscosity; in the present case, this manifests itself via the possibility of multiple states for the oscillation-mark profile at the instants in time when solid flux begins to form and when it ceases to form

Asymptotic analysis of the dominant mechanisms in the coffee extraction process

Extraction of coffee solubles from roast and ground coffee is a highly complex process, depending on a large number of brewing parameters. We consider a recent, experimentally validated, model of coffee extraction, describing extraction from a coffee bed using a double porosity model, which includes dissolution and transport of coffee. It was shown that this model can accurately describe coffee extraction in two situations: extraction from a dilute suspension of coffee grains and extraction from a packed coffee bed. Despite being based on some simplifying assumptions, this model can only be solved numerically. In this paper we consider asymptotic solutions of the model describing extraction from a packed coffee bed. Such solutions can explicitly relate coffee concentration to the process parameters. For an individual coffee grain, extraction is controlled by a rapid dissolution of coffee from the surface of the grain, in conjunction with a slower diffusion of coffee through the intragranular pore network to the grain surface. Extraction of coffee from the bed also depends on the speed of advection of coffee from the bed. We utilize the small parameter resulting from the ratio of the advection timescale to the grain diffusion timescale to construct asymptotic solutions using the method of matched asymptotic expansions. The asymptotic solutions are compared to numerical solutions and data from coffee extraction experiments. The asymptotic solutions depend on a small number of dimensionless parameters and so are useful to quickly fit extraction curves and investigate the influence of various process parameters on the extraction

Asymptotic analysis of the dominant mechanisms in the coffee extraction process

Extraction of coffee solubles from roast and ground coffee is a highly complex process, depending on a large number of brewing parameters. We consider a recent, experimentally validated, model of coffee extraction, describing extraction from a coffee bed using a double porosity model, which includes dissolution and transport of coffee. It was shown that this model can accurately describe coffee extraction in two situations: extraction from a dilute suspension of coffee grains and extraction from a packed coffee bed. Despite being based on some simplifying assumptions, this model can only be solved numerically. In this paper we consider asymptotic solutions of the model describing extraction from a packed coffee bed. Such solutions can explicitly relate coffee concentration to the process parameters. For an individual coffee grain, extraction is controlled by a rapid dissolution of coffee from the surface of the grain, in conjunction with a slower diffusion of coffee through the intragranular pore network to the grain surface. Extraction of coffee from the bed also depends on the speed of advection of coffee from the bed. We utilize the small parameter resulting from the ratio of the advection timescale to the grain diffusion timescale to construct asymptotic solutions using the method of matched asymptotic expansions. The asymptotic solutions are compared to numerical solutions and data from coffee extraction experiments. The asymptotic solutions depend on a small number of dimensionless parameters and so are useful to quickly fit extraction curves and investigate the influence of various process parameters on the extraction

On the formation of fold-type oscillation marks in the continuous casting of steel

Asymptotic methods are employed to revisit an earlier model for oscillation-mark formation in the continuous casting of steel. A systematic non-dimensionalization of the governing equations, which was not carried out previously, leads to a model with 12 dimensionless parameters. Analysis is provided in the same parameter regime as for the earlier model, and surprisingly simple analytical solutions are found for the oscillation-mark profiles; these are found to agree reasonably well with the numerical solution in the earlier model and very well with fold-type oscillation marks that have been obtained in more recent experimental work. The benefits of this approach, when compared with time-consuming numerical simulations, are discussed in the context of auxiliary models for macrosegregation and thermomechanical stresses and strains