The behaviour of foams at rest, but particularly during fluid mechanical transport is not sufficiently investigated yet. The present article deals with protein foams as they have a great importance in food production. In the first part, the foaming process of a highly viscous liquid due to gaseous materials dispersed under pressure in the liquid and mass transport of volatile components dissolved in the liquid is considered. The aim is to calculate the foam volume and the concentration of the dissolved, volatile components as a function of the material and process parameters. In the second part, material equations for bubble suspensions with gas volume fractions φ ≤ 0.6 and small bubble deformations (i.e. NCa 1) are presented. The basics form two constitutive laws which are used for describing a steady shear flow. If the rates of work of the two models are compared, material equations for the shear viscosity and the normal stress differences can be derived. Foaming processThe model presented in [4] and developed further in this article describes the foaming process of a highly viscous liquid due to gaseous materials dispersed under pressure in the liquid and mass transport of volatile components dissolved in the liquid. The model is divided into two model phases, which describe spherical foam and polyhedron foam, respectively. The increase of the foam volume is calculated by solving problem-specific conservation equations for mass, momentum and energy. Alike, concentration equations for the dissolved, volatile components, which are diffusing into the bubbles during the foaming process, are solved. The equations are subjected to a Lagrangian coordinate transformation and transformed into dimensionless equations by means of a set of dimensionless π-numbers. The equation of motion for the spherical foam is given bywhere R B is the bubble radius, R S is the radius of the liquid volume surrounding the bubble, P g is the internal pressure of the bubble, P ∞ is the ambient pressure, T Y Y and T θθ denote the normal stresses, T is the time and Y is the transformed spatial coordinate. The integral on the right side of equation (1) characterises the viscous elongation stresses around the bubble. The rheological behaviour of the liquid is described by the constitutive equation for Newtonian liquids or the generalised contravariant Maxwell-Oldroyd model for viscoelastic liquids. In the latter case, the extra stress tensor T results from the sum of the partial stresses T k based on a discrete spectrum that covers relaxation times λ R,k and viscosities η k , the sum of which is equal to the zero shear viscosity η 0 of the liquid:The function a takes account of viscosity changes due to changes in the temperature Θ and the concentration ξ i of the volatile components dissolved in the liquid. A similar differential equation is derived for the other relevant normal stress component T θθ of the extra stress tensor T. The equations for the partial stresses and the equation (1) are combined, resulting in a large system of non-linear ...

The steady laminar pipe flow of a suspension with a gas volume fraction ≤ 0.5 and small or intermediate bubble deformations in long, and straight sections of a circular pipe is calculated. The calculations are based on the constitutive equation that was originally derived for dilute emulsions and further developed for concentrated suspensions containing bubbles. In contrast to the literature, an analytical procedure is used to determine the solution of a pipe flow more accurately. The results are presented and discussed with respect to the Reynolds number Re and capillary number Ca. If Ca < 1 or Ca > 1, a bubble suspension has a parabolic velocity profile indicating a Newtonian rheology. If Ca ≈ 1, two regimes of flow are observed in agreement with the literature; that is, an inner plug flow where deformation rates are low and an outer flow where deformation rates are high. These results imply that, if Ca < 1, the Reynolds number Re decreases with increasing gas volume fraction and that, if Ca ≥ 1, the opposite effect occurs; that is, the Reynolds number Re increases with increasing gas volume fraction.

The rheological behaviour of protein solutions containing bubbles at rest, but particularly during fluid mechanical transport is not sufficiently investigated yet. Protein foams have a great importance in food production because of their special sensory properties. A suspension that contains bubbles in a Newtonian liquid exhibits inherently a complex rheological behaviour, such as elastic effects, a shear‐ and time‐dependent viscosity and normal stress differences. The deformation state of the bubbles in a suspension subjected to a steady shear is a function of the gas volume fraction ϕ and the capillary number NCa. In the present article, material equations for protein solutions with gas volume fractions ϕ ≤ 0.75 and small bubble deformations, i. e. NCa ≪ 1, are analysed and further developed. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The literature model studied in this article describes bubble formation and growth in a highly viscous polymer liquid with support of a gaseous matter dispersed under pressure before foaming. The foam growth is induced by the application of vacuum and mass transport of volatile components dissolved in the polymer liquid. Based on this literature model, aeration processes are calculated for intermediate viscosity and low viscosity biological systems, as they are of interest for biomatter foams, in particular for food foams in industrial processes. At the end of this article, the numerical results are presented and discussed.