Stability of Cappuccino Foams

Transcription

1 Stability of Cappuccino Foams Kaveh Laksari, Stephen Shank, Dong Zhou January 24, 2012 Abstract In this study, we want to investigate the longevity of steam-frothed milk foams. For foams in general, drainage, coalescence, collapse and the concentration of surfactant all contribute to the change of the foam longevity. In this report, a model for drainage and collapse in standing foams is presented. More specifically, we track liquid volume fraction of the foam and the locations of the foam-gas and foamliquid interfaces. We close with numerical results and a discussion of possibilities for future work. 1 Introduction Foams appear in everyday life and arise in plenty of industrial applications, but a full understanding of their behavior is yet to be developed. A natural first question to ask is: what are foams, and what can be said about them? A loose definition is as follows: foams are metastable systems comprised of gas bubbles dispersed in liquids. Foam formation requires the presence of gas, liquid, surfactant (or surface-active agent) and energy. On the bubble scale, empirical evidence shows that foams conform to four laws articulated by Joseph Plateau in the 19th century, known as Plateau s laws [14]: 1. Soap films are made of smooth surfaces 2. Average curvature is essentially constant on films 3. Soap films meet in threes along an edge, called a Plateau border (henceforth abbreviated as PB). Moreover, films meet at angle of cos 1 ( 1/2) = Plateau borders meet in fours at a vertex On a macroscopic scale, foams can be characterized by their liquid volume fraction [7], denoted ɛ. Freshly generated milk foams are wet foams (high liquid volume fraction) with spherical bubbles of similar (with the ideal, equal sized case referred to as monodispersive). Whether a foam retains its initial shape or decays is generally dictated by three processes, which may often be related [8]: 1. Drainage of liquid in foam to liquid phase 2. Coalescence (merging of two bubbles), or collapse (popping of bubbles) at the gas/foam interface 1

2 (a) (b) Figure 1: Pasteurized skim milk (a) immediately after foaming by mechanical agitation and (b) at half-life [10] (a) (b) Figure 2: (a) Cryo-SEM Picture of a PB [ phds/naduerra/index], and (b) a rendering of a PB and PB juncture[13] 2

3 3. Disproportionation or Ostwald ripening, or the tendency of larger bubbles to engulf smaller ones A standing foam becomes drier with time; namely, its liquid content decreases due to the drainage of liquid through the Plateau border channels under the action of gravity. Drainage is the dominant process in a freshly formed foam, as their liquid content is sufficiently high that films are not thin enough to rupture and bubbles are of roughly the same size so that ripening does not occur. As liquid drains, the capillary pressure resulting from the curvature of the Plateau border channel walls causes the liquid in the films to be sucked out. As a result, these films become thinner and finally rupture. Film rupture has two consequences: when it occurs at the top of the foam, gas escapes out of the foam and the foam height decreases, known as collapse. When rupture occurs within the foam, it causes two neighboring bubbles to merge together as one, henceforth called coalescence. Sauter mean bubble radius (i.e., the R such that a given bubble B satisfies 4 3 πr3 = vol(b)) is sufficiently small ( µm [9]) that a continuum treatment is possible [4]. We will assume that ɛ is uniform in x and y directions, and thus only depends on height and time. We orient z = 0 as the initial height of the foam and assume z increases in the downward direction. Finally, we let z 1 (t) and z 2 (t) denote locations of the gas/foam and foam/liquid interfaces (see Figure 3). z z 1 foam z 2 liquid Figure 3: Decaying foam. z = 0 corresponds to the initial height of the foam and increases in the downward direction. Therefore, we seek: A geometry whose structure conforms to Plateau s laws A partial differential equation describing the time evolution of ɛ(z, t), and ordinary differential equations describing the motion of z 1 (t) and z 2 (t) Bhakta and Ruckenstein 1 have proposed a plausible schematic for a PB cross-section and assume polyhedral bubble structure. Liquid drains through Plateau borders, and this 1 The geometry and theoretical model in this study have been taken from multiple works of the aforementioned authors [1, 2, 3, 5, 4] 3

4 film r p Figure 4: Schematic for cross section of Plateau border is where the majority of the liquid is held. Referring to Figure 2, a plausible structure for a cross section of a Plateau border is Here r p is called the Plateau border radius, which is related to the pressure difference and surface tension via the Young-Laplace law [14]. Cross-sectional area of a Plateau border is denoted by a p. Standard geometric considerations of the above configuration yield a p = δr 2 p, where δ = 3 π/ When modeling a foam, regular polygons are often used in place of bubbles. Our model uses uniform pentagonal dodecahedra (assuming monodispersed foam), which consist of 12 faces, each possessing 5 edges. (a) (b) Figure 5: (a) A pentagonal dodecahedron, generated using the open source software Antiprism [ and (b) the same figure with faces removed. These (nearly) tile the space and share many properties in common with Plateau s laws, as seen in Figure 6. Moreover, many quantities relevant to the foam can be calculated easily due to the simple geometric structure of such figures. For example, n p, the number of Plateau borders per bubble, can be calculated by counting arguments, such as # of faces # of edges n p = # of faces shared per edge # of edges shared per bubble = = 10 Edge length, denoted l, corresponds to the length of a PB channel. Assuming a monodisperse foam with bubble volume V, it can be shown that V = 1( )l 3, and 4 4

5 (a) (b) Figure 6: Pentagonal dodecahedra in space. (a) Tiling of the space and (b) edges meeting similarly to that of Plateau border junctures. so l = 0.816R, where R is again the Sauter mean radius. 2 Theoretical Model Following Bhakta and Ruckenstein [2], we describe a continuum model for drainage in a foam. Let N denote the number of bubbles per unit volume in an infinitesimal element of the foam. Assuming a monodisperse foam with (fixed) bubble volume V, it follows that NV = 1 ɛ. Liquid in the films is assumed to be negligible, as the majority of liquid is stored in the Plateau borders. We can then calculate the liquid volume fraction in such an element via ɛ = Nn p a p l (number of bubbles per unit volume number of PBs per bubble volume of a single PB). The model is essentially nothing other than flow in a PB channel combined with conservation of liquid, both of which can be described in terms of these quantities, listed again below for emphasis (note that all are functions of ɛ): N(ɛ) = 1 ɛ V Number of bubbles per unit volume a p (ɛ) = V r p (ɛ) = ɛ n pl 1 ɛ V ɛ δn pl 1 ɛ Cross-sectional PB area PB radius 2.1 Flow in a Plateau Border Channel Different milks have different viscosities µ and surface tensions σ, which are usually influenced by the temperature of the milk and the kind of protein that makes up the surfactant. Another related is the constant c v, which accounts for the effects of finite surface viscosity and generally has the value 1 [4]. Average velocity of liquid in a vertical PB channel is given by Desai and Kumar [6, 1]: u = c v 20 3µ a p ( ρg + σ z ( )) 1 r p 5

6 Let q p denote the average downward flow rate per unit area. Performing a spatial averaging over all possible PB orientations, it can be shown that q p = R ɛu [1]. Conservation 5l of liquid is then expressed as t ɛ = z q p (1) Simplifying the equations and substituting the required variables in the conservation of mass equation above yields the following nonlinear PDE for the liquid volume fraction ɛ(z, t) in the foam[1]: ( ) t ɛ + α ɛ(2 ɛ) 1 (1 ɛ) 2 z ɛ ɛ = α 2 z (1 ɛ) }{{} 3/2 z ɛ (2) }{{} advective term diffusive term where α 1 = V c v ρg n p lµ, α 2 = σc v δv n p lµ The time-derivative term shows the evolution of liquid volume fraction; the second term on the left hand side shows the advection of liquid downward as the flow in the Plateau borders takes the liquid out of the foam. The term on the right hand side represents the diffusion of liquid upward as the bubbles drag the liquid around them while they are moving towards the upper boundary of the foam. 2.2 Pre-Collapse Boundary Conditions We assume two different stages of behavior: first, prior to collapse (or bubble rupture) at the foam-gas interface, and after. These yield pre- and post-collapse treatments for the boundary conditions of the PDE and the ODEs for z 1 and z 2. Assuming (for now) that no collapse occurs, no liquid flows in through the top boundary, and so u(z 1 (t), t) = 0, which yields a mixed boundary condition for ɛ. The foam is assumed sufficiently wet at the liquid/foam interface, with nearly spherical bubbles, and so ɛ(z 2 (t), t) = C, where C =.261 is a constant related to the classical problem of sphere packing. As shown in Figure 3, prior to collapse, the foam-gas interface, initially z 1 (t) = 0, is immobile, and therefore dz 1 = 0. The bottom boundary, initially z dt 2 (0) = L 0, where L 0 denotes the initial height of the foam, moves up, due to additional liquid having drained out of the foam and into the liquid phase. Gas volume in the foam is conserved, so we require that d dt z2 (t) z 1 (t) (1 ɛ)dz = 0 Recall that in the pre-collapse stage, z 1 (t) = 0, so the above yields an ODE for z 2 (t). 2.3 Film Rupture and Collapse of Foam As liquid drains, films gets thinner and eventually rupture, causing collapse and coalescence of bubbles. The film thinning process is described by a Reynolds type equation (3) 6

7 [12] dx F = 2 ( ) σ Π x 3 dt 3µRF 2 F (4) r p where x F is the thickness of the film, R F is the radius of the film and Π is the disjoining pressure in the film. The disjoining pressure is essentially composed of molecular forces; it is positive if two surfaces of film repel each other, and is negative otherwise. The disjoining pressure in the film is a result of two forces, the van der Waals attractive forces and the repulsive double-layer forces (denoted Π V W and Π DL, respectively) as explained in [3]. Explicit formulas for calculating Π V W and Π DL, and therefore also for Π, are available in [4], but are somewhat lengthy and therefore omitted. Figure 7 shows the change of disjoining pressure Π in film thickness. If Π increases in response to the local film thinning, no rupture occurs. If Π decreases, the local thinning is accelerated and will lead to rupture. Figure 7: Change of disjoining pressure in film thickness at temperature T = 298K(25 C). If the capillary pressure (σ/r p ) is smaller than the maximum disjoining pressure (Π max ), then the difference σ/r p Π attains 0 before Π starts to decrease. As a result, film thinning stops and no film rupture occurs. Therefore, film rupture occurs only if σ/r p > Π max. Let r pc be the critical Plateau border radius such that σ r pc = Π max. Since r p is a function of ɛ, then there exists a critical value ɛ c corresponding to Π max. Assume that the film thinning process happens in a much smaller time scale compared with drainage, i.e., the film thickness attains its equilibrium instantaneously. Then we are able to use ɛ c to approximate the liquid volume fraction at film rupture. The critical value ɛ c is determined by maximum disjoining pressure Π max and the following relations σ r pc = Π max, r pc = 7 V ɛ c δn p l 1 ɛ c

8 ɛ c = σ 2 δn pl Π 2 max V 1 + σ2 Π 2 max δn pl V = σ 2 δn p l Π 2 maxv + σ 2 δn p l Notice that r p (ɛ) is an increasing function in ɛ and ɛ is non-decreasing in z, therefore rupture occurs (σ/r p > Π max ) only if ɛ < ɛ c, i.e., film rupture happens only at the top boundary of foam. When ɛ(z 1 (t), t) ɛ c a different treatment of the upper boundary is required. 2.4 Post-Collapse Boundary Conditions After collapse, the foam behaves the same in the interior and the foam/liquid interface, and so the PDE and boundary condition for ɛ(z 2 (t), t) remain unchanged. The ODE for z 2 (t) is derived similarly, via d dt z2 (t) z 1 (t) (1 ɛ)dz = dz 1 dt This still reflects the fact that gas-volume in the foam should be conserved, but now some gas escapes from the foam at the gas-foam interface due to collapse. The term on the right represents a drop in the height the foam, i.e., the amount amount of gas that escapes in an instant. Liquid at the foam-gas interface should return to the foam, and can only enter the PB channels, leading to ɛ(z 1 (t), t) dz 1 dt = q p z1 Collapse is now always assumed to occur at the top boundary, requiring ɛ(z 1 (t), t) = ɛ c Combining everything together, the interaction between the foam and the two interfaces is governed by the following closed system of equations: (5) ( ) t ɛ + α ɛ(2 ɛ) 1 (1 ɛ) 2 z ɛ = α ɛ 2 z (1 ɛ) 3/2 z ɛ ( ) dz 2 dt = α C 2 C 1 (1 C) + α ɛ 2 2 (1 C) 5/2 z ( z2 ) dz 1 dt = α ɛ c 1 ɛ 1 α 2 1 ɛ c (1 ɛ c ) 3/2 ɛ c z z1 (6a) (6b) (6c) with the initial conditions: ɛ(z, 0) = ɛ 0 (z) (7a) z 1 (0) = 0, z 2 (0) = L 0 (7b) and boundary conditions: before collapse (ɛ(z 1 (t), t) > ɛ c ) : u z=z1 = 0, ɛ(z 2 (t), t) = C (8a) after collapse (ɛ(z 1 (t), t) ɛ c ) : ɛ(z 1 (t), t) = ɛ c, ɛ(z 2 (t), t) = C (8b) 8

9 3 Numerical Approach The above results in a nonlinear PDE with moving boundaries. Generation of the foam was not considered, and so initial conditions were assumed constant, i.e., ɛ 0 (z) = C, denoting freshly formed, monodisperse, and sufficiently wet foam. The PDE was discretized in time semi-implicitly by taking the advection term and the coefficient of the diffusion term explicitly, while the diffusive term itself was discretized implicitly. The full discretizations were different for pre- and post-collapse; we only include the details of post-collapse here for brevity s sake, as it is the trickier of two. Pre-collapse was treated with similar techniques. The following change of variables was applied to the spatial variable z in order to keep the size of the domain fixed at all times: by letting ζ = z z 1 z 2 z 1 Then ɛ z ɛ t φ(ζ, t) = ɛ(ζ(z 2 z 1 ) + z 1, t) = ɛ(z, t). = φ dζ ζ dz + φ t = φ dζ ζ dt + φ t Therefore, the original system 6 becomes dt dz = 1 φ z 2 z 1 ζ dt dt = z 1 + (z 2 z 1)ζ z 2 z 1 φ ζ + φ t φ t dz 2 dt dz 1 dt = ( z 1 + (z 2 z 1)ζ α 1 f(φ) z 2 z 1 = C2 (1 C) 2 α 1 + = ɛ c (1 ɛ c ) α 1 ) φ ζ + α 2 (z 2 z 1 ) 2 ζ φ C (1 C) 5/2 α 2 (z 2 (t) z 1 (t)) 1 (1 ɛ c ) 3/2 ɛ 1/2 c α 2 (z 2 (t) z 1 (t)) ζ φ ζ ( g(φ) φ ζ ζ=1 ζ=0 ) (9a) (9b) (9c) where f(φ) = φ(2 φ) (1 φ) 2, g(φ) = φ (1 φ) 3/2 (10) The PDE was semi-discretized in space and the resulting ODE was solved semiimplicitly in time together with the ODE s for the moving boundaries. Below is the semi-discrete form of the system of equations given above: Let f = z 1 + (z 2 z 1)ζ α 1 f(φ) and 0 = ζ 1 < ζ 2 < < ζ N < ζ N+1 = 1. Then the z 2 z 1 semidiscrete form is dφ j dt = f φ j (t) φ j 1 (t) j + ζ α [ ] 2 g(φ (z 2 (t) z 1 (t)) 2 ζ 2 j+ 1 )(φ j+1 φ j ) g(φ 2 j 1 )(φ j φ j 1 ) 2 (11a) 9

10 ( ) dz 2 dt = α α 6 C φn 5 + (z 2 (t) z 1 (t)) ζ ( ) dz 1 dt = α α 8 φ2 ɛ c 7 (z 2 (t) z 1 (t)) ζ (11b) (11c) where j = 2,, N Notice that here j starts from 2. Above we have set: f j = z 1 + (z 2 z 1)ζ j α 1 f(φ j ), α 5 = z 2 z 1 ɛ c α 7 = (1 ɛ c ) α 1 1, α 8 = (1 ɛ c ) 3/2 ɛ 1/2 c α 2 C 2 (1 C) 2 α 1, α 6 = C (1 C) 5/2 α 2 Neglecting the equations for the moving boundaries, this gives a linear tridiagonal system that changes at each time step. The fully-discrete system of equations in the matrix form is shown below: 1 t φ n+1 2 φ n+1 3. φ n+1 N φ n 2 φ n 3 φ n N A n 22. = A n 32 A n A n N,N 1 A n NN φ n 2 φ n 3 φ n N. D22 n D n 23 D32 + n DN 1,N n + f 2 nɛc ζ 0. 0 D n N,N 1 D n NN φ n+1 2 φ n+1 3. φ n+1 N g(φ n )ɛ 1+ 1 c 2 + α 2 (z2 n z1 n ) 2 ζ 2 0. g(φ n )C N+ 1 2 where A n jj = f n j ζ, f n j A n j,j 1 = ζ, α 2 Djj n = (z2 n z1 n ) 2 ζ 2 (g(φn ) + g(φ n )), j+ 1 j D n j,j 1 = D n j 1,j = α 2 (z2 n z1 n ) 2 ζ 2 g(φn ) j 1 2 The values z 1 and z 2 were solved for explicitly from the values of φ. N = 800 gridpoints were used, as results for N = 800 and N = 1600 agreed to several decimal places, indicating proximity to the true values. 10

11 4 Results In this section, results of simulations carried out to examine the effect of varying a given parameter, such as the bubble radius R, surface tension σ, viscosity µ and temperature T and their effects on the behavior of the model will be discussed. A comparison of foam behaviors between skim milk and whole milk is then performed using actual parameters from various milks. 4.1 Parameter Study The parameter study is carried out by varying one parameter while keeping others fixed. In general, this does not make sense physically since changing one parameter would almost certainly lead to a change in the others. For example, temperature change affects surface tension and viscosity. However, for this part of the report, our goal is to investigate how each individual parameter affects the result without considering the actual underlying physics. Below we plot the positions of the boundaries z 1 (t) and z 2 (t) with respect to time and the foam height z 1 (t) z 2 (t). Actual parameters for milks were taken from the literature [9, 11]. A sample value for density and maximum disjoining pressure is ρ = 1030 kg/m 3 and Π max = N/m 2, with all values typically close to these Viscosity Figure 8: Effect of viscosity µ on drainage and collapse with σ = N/m, R = 200µm, T = 298K(25 C). Both coefficients α 1 and α 2 in equation (6a), (6b) and (6c) are multiples of 1 µ, therefore we can remove µ from equations by letting τ = t µ (12) where τ is the rescaled time variable. Figure 8 verifies that different values of µ just rescale the problem: larger value of µ slows down drainage and collapse, smaller value of µ accelerates the process. 11

12 Figure 9: Effect of bubble radius R on drainage and collapse with σ = N/m, µ = Pa s, T = 313K(40 C). Figure 10: Effect of surface tension on drainage and collapse with µ = Pa s, R = 400µm, T = 298K(25 C) Surface Tension and Bubble Radius Figure 9 and 10 show the motion of boundaries and change of foam height for different values of R and σ. From the previous section, the critical liquid volume fraction σ 2 δn p l ɛ c = Π 2 maxv + σ 2 δn p l σ 2 δn p (0.816)R = Π 2 max 4π 3 R3 + σ 2 δn p (0.816)R 1 = ( ) 2 R cπ 2 max + 1 σ 4π where c = 3δn p (0.816) is a constant. Therefore, for fixed temperature (fixed Π max), R and 1/σ have similar effect on ɛ c. Larger ratio R gives smaller value of ɛ σ c, which delays 12

13 collapse at the top boundary Temperature Temperature only enters the model from evaluating the maximum disjoining pressure Π max. From the discussion above, it can be observed that Π max has similar effect on the solution as the ratio R (Figure 11). σ Figure 11: Effect of temperature on drainage and collapse with σ = N/m, µ = Pa s, R = 200µm. A summary of the observations made from parameter study is as follows 1. Foam with smaller bubbles tends to decay slower (for t < 2000s), while foam with larger bubbles decays faster in this time period. 2. Small surface tensions delay collapse. 3. Viscosity only affects the time scale in this model, the more viscous the liquid is, the slower the foam decays. 4. Higher temperature tends to increase the rate of the decay. 4.2 Comparison of Actual Milk Parameters The actual milk parameters (surface tension and viscosity for different types of milk at different temperature) were taken from the Journal of Food Science and Journal of Dairy Science and are listed in Table 1 [9]. The bubble radius R is taken from {0.17mm, 0.2mm, 0.4mm, 0.6mm}. The first two values reflect real bubble size in steam frothed milk foam [11], and others are for comparison purpose. Figures are included after references. Before comparing numerical results for different milk, we first introduce a new quantity called average liquid volume fraction, ɛ(t), given by ɛ(t) = 1 z 2 (t) z 1 (t) z2 (t) z 1 (t) ɛ(z, t)dz. (13) 13

14 Milk type σ (N/m) µ (Pa s) whole milk (25 C) (25 C) (40 C) (40 C) skim milk (25 C) (25 C) (40 C) (40 C) Table 1: Parameters for whole milk and skim milk. This quantity, a spatial average of the liquid volume fraction ɛ(z, t) at a given time t, indicates how wet or dry the foam is as a whole without considering foam height and liquid distribution through foam. Figures show the motion of foam boundaries, change of foam height and average liquid volume fraction for different choices of bubble radius R. The following observations have been made for various bubble sizes: 1. Foam with smaller bubble radii are wetter than ones with bigger bubble radii. 2. Collapse happens later in foams with smaller bubble radii. 3. Foam with smaller bubble size decay slower for t < 2000s, and faster afterwards. A comparison between whole milk and skim milk (Figure 16) shows that foam generated from whole milk decays slower than skim milk and has a higher average liquid volume fraction before t 2000s. Based on experience, a good foam should maintain its initial state longer, which means a slower decay in foam height and a higher average liquid volume fraction. We took this as our measure of what constitutes a good milk foam. From these aspects, whole milk gives a better foam quality than skim milk. 5 Future work The model presented captures many essential features of a freshly formed foam, but is not without its deficiencies. Coalescence and Ostwald ripening are neglected, but play a larger role as time elapses. Models which track film thickness x F and Plateau border radius r p exist [3], in which liquid in films is no longer neglected and film thickness provides a much more natural way to formulate conditions for coalescence in the interior and collapse on the boundary. Further complexity can be added by taking surfactant concentration into account, and allowing for ripening, thus leading to change in mean bubble radii and polydisperse foams [5]. Moreover, temperature is only taken into account in the parameters used for the model; the change of temperature that results from the injection of steam into cool milk certainly has a role on the dynamics of the foam [9]. Also, the generation of the foam was not considered either, and more appropriate initial conditions could certainly have an influence on our prediction for the quality of the foam during drinking time. All these questions and more provide plenty of interesting avenues for future work. 14

15 6 Conclusions A model for drainage and collapse of foam is presented and the effect of parameters such as bubble radius (R), surface tension (σ), viscosity (µ) and temperature (T ) is examined. For milk foam in cappuccino, comparisons between different bubble sizes and different types of milk are performed. It is also shown that milk-related parameters (surface tension and viscosity) and bubble radius have effect on foam height and average liquid volume fraction. Within the drinking time (half an hour), foam height and average liquid volume fraction can be used to characterized quality of foam: slower decay rate in foam height and higher average liquid volume fraction indicate better foam quality. References [1] Ashok Bhakta and Eli Ruckenstein, Drainage of a standing foam, Langmuir 11 (1995), no. 5, [2], Foams and concentrated emulsions: Dynamics and phase behavior, Langmuir 11 (1995), no. 12, [3], Modeling of the generation and collapse of aqueous foams, Langmuir 12 (1996), no. 12, [4] Ashok Bhakta and Eli Ruckenstein, Decay of standing foams: drainage, coalescence and collapse, Advances in Colloid and Interface Science 70 (1997), no. 0, [5], Drainage and coalescence in standing foams, Journal of Colloid and Interface Science 191 (1997), no. 1, [6] Dilip Desai and Rajinder Kumar, Flow through a plateau border of cellular foam, Chemical Engineering Science 37 (1982), no. 9, [7] R. Hohler and S. Cohen-Addad, Rheology of liquid foam, Journal of Physics: Condensed Matter 17 (2005), no. 41, R1041. [8] T. Huppertz, Foaming properties of milk: A review of the influence of composition and processing, International Journal of Dairy Technology 63 (2010), no. 4, [9] Carlos A. Jimenez-Junca, Jean C. Gumy, Alexander Sher, and Keshavan Niranjan, Rheology of milk foams produced by steam injection, Journal of Food Science 76 (2011), no. 9, E569 E575. [10] Sapna Kamath, Thom Huppertz, Avis V. Houlihan, and Hilton C. Deeth, The influence of temperature on the foaming of milk, International Dairy Journal 18 (2008), no , [11] D. Kristensen, P.Y. Jensen, F. Madsen, and K.S. Birdi, Rheology and surface tension of selected processed dairy fluids: Influence of temperature, Journal of Dairy Science 80 (1997), no. 10,

16 [12] O. Reynolds, On the Theory of Lubrication and Its Application to Mr. Beauchamp Tower s Experiments, Including an Experimental Determination of the Viscosity of Olive Oil, Royal Society of London Philosophical Transactions Series I 177 (1886), [13] C. Schick, A mathematical analysis of foam films, Ph.D. thesis, Kaiserslautern, [14] D. Weaire and R. Phelan, The physics of foam, Oxford University Press,

17 Figure 12: Solution for whole milk with different bubble radii at T = 25 C. Figure 13: Solution for whole milk with different bubble radii at T = 40 C 17

18 Figure 14: Solution for skim milk with different bubble radii at T = 25 C Figure 15: Solution for skim milk with different bubble radii at T = 40 C 18

19 Figure 16: Comparison between whole milk and skim milk with R = 200µm at T = 40 C 19