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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.

Foams become more important in industrial applications, but also in food production because of many advantages, including low weight, thermal insulation and special texture characteristics. Relevant constitutive equations were developed by Oldroyd [

Oldroyd [

( S + λ 1 D S D t ) = 2 η 0 ( E + λ 2 D E D t ) . (1)

D / D t denotes the Jaumann time derivative defined by

where A is an arbitrary tensor of order two, W is the rotation velocity tensor and D / D t is the material time derivative of A. The three constants involved in the constitutive Equation (1), the zero-shear viscosity η 0 , the relaxation time λ 1 and retardation time λ 2 of the bubble suspension, are given by

η 0 = η c ( 1 + ϕ ) , (3)

λ 1 = 6 5 ( η c R σ ) ( 1 + 16 15 ϕ ) , (4)

λ 2 = 6 5 ( η c R σ ) ( 1 − 8 5 ϕ ) , (5)

where η c is the shear viscosity of the liquid phase, R is the bubble radius, σ is the interface tension between liquid and gas phases and ϕ is the gas volume fraction. The relaxation time λ 1 and retardation time λ 2 for the system vary proportionally with the bubble radius R and inversely proportionally with the interface tension σ . In a steady shear flow, the velocity vector is assumed to be u = ( γ ˙ x 2 , 0 , 0 ) , where γ ˙ is the constant shear rate. The Oldroyd model (1) - (5) gives expressions for the shear viscosity η and the normal stress differences N 1 and N 2 of a bubble suspension in the form

η = η 0 [ 1 + λ 1 r λ 2 r ( 6 5 N Ca ) 2 1 + λ 1 r 2 ( 6 5 N Ca ) 2 ] , (6)

N 1 = 2 η 0 γ ˙ [ ( λ 1 r − λ 2 r ) ( 6 5 N Ca ) 1 + λ 1 r 2 ( 6 5 N Ca ) 2 ] , (7)

N 2 = − 1 2 N 1 , (8)

where η 0 r = η 0 / η c , λ 1 r = λ 1 / λ c , and λ 2 r = λ 2 / λ c are the relative zero-shear viscosity, the reduced relaxation time and reduced retardation time, respectively, and λ c = 6 η c R / 5 σ is the relaxation time of a single bubble. The capillary number is defined as N Ca = η c γ ˙ R / σ [

Pal [

The maximum packing volume fraction of undeformed bubbles depends on the size distribution of initial undeformed bubbles. One way to take account for the effect of the bubble size distribution on the rheological properties of bubble suspensions is to include the parameter ϕ m . Hence, the new equations for η 0 , λ 1 and λ 2 [

η 0 = η c ( 1 − ϕ ϕ m ) − ϕ m , (9)

λ 1 = λ c ( 1 − ϕ ϕ m ) − ( 16 / 15 ) ϕ m , (10)

λ 2 = λ c ( 1 − ϕ ϕ m ) ( 8 / 5 ) ϕ m . (11)

The pipe flow represents a fluid flow forced by a pressure difference through a pipe line. The most important flow is the steady laminar flow of a Newtonian liquid through a straight, circular pipe. The mathematical formulation is done in cylindrical coordinates ( r , φ , z ) . The fully developed flow is characterized by the fact that the velocity field consists only of the axial component w ( r ) as a function of the radial variable r. Hence, the shear rate γ ˙ = d w / d r and the shear stress τ depend only on the radial variable r. The fully developed flow in a circular pipe is a balance between pressure forces and viscous forces [

τ = − Δ p 2 L r , (12)

in which Δ p is the pressure difference along the pipe, L is the pipe length and R c is the pipe radius. The constitutive equation prescribes how the shear stress is related to the fluid velocity. A viscous fluid sticks at the pipe wall so that w = 0 at r = ± R c . Hence, a Newtonian fluid with shear viscosity μ has the velocity profile

w ( r ) = W max [ 1 − ( r R c ) 2 ] , (13)

with the maximal velocity W max = Δ p R c 2 / 4 μ L in the centerline of the pipe. The flow volume rate [

Q = π R c 4 Δ p 8 L μ . (14)

In the following, the steady laminar flow of a bubbly suspension is calculated using the Oldroyd model (1) with the new expressions (9)-(11) and ϕ m = 0.637 . Based on the Equation (6), a cubic equation for the material behavior is obtained, i.e.

η 0 x ( r ) ⋅ 1 + λ 1 λ 2 x 2 ( r ) 1 + λ 1 2 x 2 ( r ) = − Δ p 2 L r , (15)

where the shear rate x ( r ) is the function to be determined and C : = Δ p / 2 L . Transformation gives the normal form of a cubic equation

x 3 + a x 2 + b x + c = 0 , (16)

with the coefficients

a = ( C r ) λ 1 η 0 λ 2 , b = 1 λ 1 λ 2 , c = C r η 0 λ 1 λ 2 . (17)

By the substitution x = z − a / 3 , the square term in Equation (16) is eliminated, and the reduced form of the cubic equation

z 3 + p z + q = 0 (18)

is obtained with coefficients

p = b − a 2 3 , (19)

⇒ p ( r ) = 3 η 0 2 λ 2 − ( C r ) 2 λ 1 3 3 η 0 2 λ 1 λ 2 2 ,

q = 2 a 3 27 − a b 3 + c , (20)

⇒ q ( r ) = 2 ( C r ) 3 λ 1 4 − 9 ( C r ) η 0 2 λ 1 λ 2 + 27 ( C r ) η 0 2 λ 2 2 27 η 0 3 λ 1 λ 2 3 .

The solutions of Equation (18) are calculated by Cardano’s formula [

u = − q 2 + Δ 3 , (21)

v = − q 2 − Δ 3 , (22)

in which Δ is the discriminant:

Δ : = ( q 2 ) 2 + ( p 3 ) 3 . (23)

Δ determines the behavior of the solutions significantly, that is, depending on whether Δ is greater, smaller than or equal to zero, the solutions are calculated differently. The third roots u and v have to be chosen so that the condition − p = 3 u v is met. The analytical procedure can be summarized as follows:

z = { u + v , 2 u , equation ( 27 ) or ( 28 ) , if Δ > 0 if Δ = 0 if Δ < 0

The last case Δ < 0 depends on the sign of the solution. The physically correct solution is given by the fact that the velocity profile is continuously differentiable.

If Δ > 0 , there is one real solution and two conjugated complex solutions. A solution can simply be determined by solving the formulas for u and v, and subsequent adding. The other solutions are obtained by multiplying u and v with the primitive third unit roots. Therefore, the following formulas for the solutions of the reduced cubic Equation (18) are to be specified [

z 1 = u + v , (24)

z 2 = ( − 1 2 + i 1 2 3 ) u + ( − 1 2 − i 1 2 3 ) v , (25)

z 3 = ( − 1 2 − i 1 2 3 ) u + ( − 1 2 + i 1 2 3 ) v . (26)

If Δ = 0 , it has to be distinguished which values p and q have, since the multiplicity of the solution depends on these values. If p = q = 0 , then z 1 , 2 , 3 = 0 . If p ≠ q ≠ 0 , then u = v = − ( q / 2 ) 1 / 3 . According to the formulas given in the Equations (24)-(26), there is a simple real solution and a double real solution: z 1 = 2 u = ( − 4 q ) 1 / 3 = 3 q / p and z 2 , 3 = − u = ( q / 2 ) 1 / 3 = − 3 q / 2 p [

If Δ < 0 (i.e. also p < 0 ), the third roots u and v are conjugated complex to each other. Since u + v = u + u ¯ = 2 Re ( u ) , three different real solutions are obtained. The third roots are determined using trigonometric functions. They are obtained in the form:

z 1 = ( − 4 3 p ) cos ( 1 3 arccos ( − q 2 − 27 p 3 ) ) , (27)

z 2 , 3 = − ( − 4 3 p ) cos ( 1 3 arccos ( − q 2 − 27 p 3 ) ± π 3 ) . (28)

The solutions of the original Equation (16) are determined by the re-substitution x = z − a / 3 .

The real solution is always selected. Rejecting the complex solutions can already be realized when determining u and v. Unfortunately, there is no common procedure to determine the sign of a real solution in advance. However, note that the shear rate x ( r ) decreases or increases from the pipe centerline to the pipe wall, that is x ( r ) < 0 for r > 0 and x ( r ) > 0 for r < 0 . The discriminant Δ is positive in the example calculations with capillary number Ca < 1 while the discriminant also becomes negative in the example calculations with capillary number Ca ≥ 1. Based on the shear rate x ( r ) , the velocity profile is calculated by the composed trapezoidal rule, which is an integration method with accuracy order two [

As a reference example, the steady laminar pipe flow of a two-phase system consisting of an intermediate viscosity liquid and air bubbles is considered. Small, spherical, monodisperse air bubbles are dispersed in the liquid. The liquid is assumed to have a Newtonian rheology and the density and surface tension of water, that is ρ c = 1000 kg / m 3 and σ = 0.072 N / m at 20˚C. The systems considered here are representative for food systems in industrial processes. The pressure difference along the pipe is postulated to be constant and moderate. The necessary material data and process data are given in

The dimensionless numbers, the Reynolds number Re and capillary number Ca [

Material quantity | Value | Unity |
---|---|---|

Density ρ_{c} | 1000 | kg/m^{3 } |

Shear viscosity η_{c} | 1 | Pas^{ } |

Surface tension σ_{ } | 0.072 | N/m |

Process quantity | Value | Unity |
---|---|---|

Temperature ϑ | 20 | ˚C |

Pipe length L_{ } | 1 | m^{ } |

Pipe radius R_{c}_{ } | 0.025 | m |

Pressure difference Δp | 2500^{ } | Pa |

Bubble radius R | 0.0001; 0.0005; 0.002 | m |

Gas volume fraction 𝜙 | 0.15; 0.30; 0.45; 0.50 | m^{3}/m^{3} |

Parameter 𝜙_{m } | 0.637 | m^{3}/m^{3} |

Re = Q ν R c , (29)

Ca = η c γ ˙ ( R c ) R σ , (30)

where Q is the volume flow rate, ν = η ( γ ˙ ( R c ) ) / ρ is the kinematic viscosity of the bubble suspension and γ ˙ ( R c ) is the shear rate at the pipe wall. The density of the suspension is given by ρ = ( 1 − ϕ ) ⋅ ρ c + ϕ ⋅ ρ g with the air density ρ g = 1.2 kg / m 3 at 20˚C. Llewellin, Mader, Wilson and Manga [

The bubble radius R of initial undeformed monodisperse bubbles is varied, and the effect on the rheological behavior of suspensions is studied. In Figures 1-3, the velocity profiles of the two-phase systems with gas volume fraction 𝜙 = 0.15 (red solid line), 𝜙 = 0.30 (black solid line), 𝜙 = 0.45 (green solid line), and 𝜙 = 0.50 (blue solid line) are shown. Additionally, the velocity profiles of the corresponding Newtonian two-phase systems (dashed line) are shown for purposes of comparison. The Newtonian systems differ from the non-Newtonian systems in that the shear viscosity η is replaced by the constant zero-shear viscosity η 0 and the volume flow rate Q is calculated according to formula (14).

At small capillary numbers Ca < 1, suspensions of bubbles in a pipe line show a parabolic velocity profile indicating a Newtonian rheology [

with a low gas volume fraction flow faster along the pipe than suspensions with a high gas volume fraction. At intermediate capillary numbers Ca ≥ 1, the suspensions of bubbles deviate strongly from the corresponding Newtonian systems. As explained by Llewellin, Mader and Wilson [

regimes of flow, an inner plug flow where deformation rates are low and an outer flow where deformation rates are high. In this case, a relatively undeformed plug is surrounded by a rapidly deforming outer flow [

The material behavior is explained by _{r} is greater than unity at capillary numbers Ca < 1 and is less than unity at capillary numbers Ca > 1. With the increase in 𝜙, the relative viscosity η_{r} increases at Ca < 1 and decreases at Ca > 1. Llewellin, Mader and Wilson [

For intermediate or high capillary numbers Ca ≥ 1, bubbles are stretched so that they form free slip surfaces in the suspension, and the viscosity decreases.

If the capillary number Ca is less than unity, the Reynolds number Re decreases with increasing gas volume fraction. If the capillary number Ca is greater than unity, the opposite effect occurs, that is, the Reynolds number Re increases with the gas volume fraction. In Tables 3-5, the values of relevant material and process quantities relating to Figures 1-3 are specified: the capillary number Ca, the Reynolds number Re, the shear viscosity η ( γ ˙ ( R c ) ) and shear rate γ ˙ ( R c ) at the pipe wall, the volume flow rate Q as well as the velocity w ( 0 ) in the pipe centerline. Relevant results relating to the rheological behavior of suspensions containing bubbles are also summarized in the articles [

The steady laminar pipe flow of a suspension with a gas volume fraction ≤ 0.5 and small or intermediate bubble deformations is calculated in long, and straight sections of a circular pipe. The calculations are based on the constitutive equation that was originally derived by Oldroyd for dilute emulsions and further developed by Pal for concentrated suspensions of bubbles. The pipe flows are characterized by a Reynolds number and a capillary number defined in this article. If the capillary number is less or greater than unity, the suspension of bubbles shows a Newtonian rheology. If the capillary is approximately unity, the

Quantity | 𝜙 = 0.15 | 𝜙 = 0.30 | 𝜙 = 0.45 | 𝜙 = 0.50 |
---|---|---|---|---|

Ca | 0.0366 | 0.0290 | 0.0199 | 0.0164 |

Re | 9.2420 | 4.7698 | 1.7727 | 1.0848 |

η(γ’(R_{c})) [Pas] | 1.1853 | 1.4973 | 2.1773 | 2.6540 |

γ’(R_{c}) [s^{−1}] | 26.3640 | 20.8707 | 14.3524 | 11.7748 |

Q [m^{3}/s] | 0.000322132 | 0.000254937 | 0.000175271 | 0.000143782 |

w(0) [m/s] | 0.3287 | 0.2601 | 0.1788 | 0.1467 |

Quantity | 𝜙 = 0.15 | 𝜙 = 0.30 | 𝜙 = 0.45 | 𝜙 = 0.50 |
---|---|---|---|---|

Ca | 0.1876 | 0.1520 | 0.1069 | 0.0883 |

Re | 9.6227 | 5.1628 | 1.9898 | 1.2322 |

η(γ’(R_{c})) [Pas] | 1.1570 | 1.4275 | 2.0301 | 2.4565 |

γ’(R_{c}) [s^{−1}] | 27.0099 | 21.8912 | 15.3932 | 12.7212 |

Q [m^{3}/s] | 0.000327379 | 0.000263077 | 0.000183430 | 0.000151160 |

w(0) [m/s] | 0.3327 | 0.2663 | 0.1850 | 0.1523 |

Quantity | 𝜙 = 0.15 | 𝜙 = 0.30 | 𝜙 = 0.45 | 𝜙 = 0.50 |
---|---|---|---|---|

Ca | 0.9641 | 1.3410 | 2.8733 | 4.1791 |

Re | 15.0029 | 21.2071 | 79.0661 | 160.6685 |

η(γ’(R_{c})) [Pas] | 0.9004 | 0.6473 | 0.3021 | 0.2077 |

γ’(R_{c}) [s^{−1}] | 34.7069 | 48.2766 | 103.4396 | 150.4483 |

Q [m^{3}/s] | 0.000397227 | 0.000490020 | 0.001084688 | 0.001666643 |

w(0) [m/s] | 0.3878 | 0.4359 | 0.8777 | 1.3766 |

suspension shows two regimes of flow, an inner plug flow and a rapidly deforming outer flow. These results were already found out by Llewellin, Mader, Wilson and Manga, who used the Frankel and Acrivos model. They explained different velocity profiles by bubble deformations depending on the capillary number. In this article, an analytical method and a physical more sensible model were used to determine solutions for pipe flows of suspensions containing bubbles more accurately. Furthermore, it can be shown: If the capillary number is less than unity, the Reynolds number decreases with increasing gas volume fraction. If the capillary number is equal to or greater than unity, the Reynolds number increases with increasing gas volume fraction.

This research project was supported by the German Ministry of Economics and Technology and the FEI (Forschungskreis der Ernährungsindustrie e.V., Bonn). Project AiF 17125 N. This project is part of the cluster “Proteinschäume in der Lebensmittelproduktion: Mechanismenaufklärung, Modellierung und Simulation” funded by the FEI and the AiF.

The authors declare no conflicts of interest regarding the publication of this paper.

Gladbach, K., Delgado, A. and Rauh, C. (2018) Pipe Flow of Suspensions Containing Bubbles. World Journal of Mechanics, 8, 417-429 https://doi.org/10.4236/wjm.2018.810030