Mathematical model of gas-liquid or liquid-liquid Taylor flow with non-Newtonian continuous liquid in microchannels

Abstract

A mathematical model of two-phase Gas-Liquid or Liquid-Liquid flow hydrodynamics for non-Newtonian (shear thickening or shear thinning) continuous liquid in microchannels was build up. Governing equations are: Navier-Stokes and continuity equations in the axisymmetric problem statement, the empirical correlation for liquid film thickness. An analytical solution was obtained for velocity profiles within the liquid slug and in the liquid film, and in the plug (bubble/droplet) of dispersed phase. Following to Abiev (CES, 2017) it was assumed that pressure gradients in the liquid slug and in the liquid film, and in the plug (bubble/droplet) are not the same in general. The proposed model allows calculating shear stresses and velocity profiles, as well as mean velocities and pressure gradients in the liquid slug, in the liquid film, and in the plug. The model could be applied for non-Newtonian continuous liquids, e.g. for some solutions of high-molecular compounds and for precipitation of micro- and nanosized particles, which accumulate in the continuous phase.

Appendix 1. Assumptions and conditions for simplification of the Eq. (41)

The Eq. (41) is nonlinear and contains two terms. For certain conditions one of them could be disregarded, then the analytical solution of the integral of Eq. (41a) could be found.

The conditions when the second term in parentheses in Eq. (41) is negligible are:

$$ {C}_{1f}<<{B}_{1f}\frac{r^2}{2}. $$

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For the small film thickness (this is usual for typical flow regimes in microchannels (Ca < 0.01...0.1)) δ/R_d ≈ δ/R < < 1, i.e. within the film r ≈ R_d. Hence, the condition (a) could be rewritten as

$$ {C}_{1f}<<{B}_{1f}\frac{R_d^2}{2}. $$

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Taking into account Eq. (38), one gets from (b):

$$ {B}_2<<2{B}_{1f}, $$

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or

$$ -{\rho}_2{g}_x+{\left(\frac{\partial p}{\partial x}\right)}_2<<2\left[-{\rho}_1{g}_x+{\left(\frac{\partial p}{\partial x}\right)}_{1f}\right]. $$

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For gas-liquid systems ρ₂ < < ρ₁, then the term ρ₂g_x could be disregarded. For liquid-liquid systems ρ₂ ≈ ρ₁, so that the gravitation term ρ₂g_x is again smaller than doubled term 2ρ₁g_x. Thus, the pressure gradient plays more pronounced role.

For gas-liquid systems the viscosity of gases is obviously much smaller than that for liquids, therefore the pressure gradient in the liquid film is much larger than that in the gas bubble. Hence, for G-L systems the condition (d) is definitely fulfilled. This conclusion is corroborated by the data presented in Table 1: the values of (∂p/∂x)_1f are one or two orders of magnitude larger than (∂p/∂x)₂.

For liquid-liquid systems, even if the viscosities (effective viscosities for non-Newtonian liquids) are comparable, the pressure drop in the liquid film is usually larger due to the small thickness of the liquid film (compared to the liquid plug). This effect is extensively discussed in our further paper, which is to be published soon.

Thus, both for gas-liquid and liquid-liquid systems the condition (a) allowing to use an approximation (41a) in place of Eq. (41) is confirmed.

The limitations of the used approach are:

1) Low capillary numbers; for Ca > 0.1 the thickness of the liquid film is large:

δ/R ≈ 1, so that in the film r ≠ R_d.

2) Low viscosity of the plug (dispersed phase); if the viscosity of the dispersed phase is much larger than that for continuous phase, the assumption (d) is not applicable.

If this limitations are not fulfilled, the Eq. (41) should be either solved in a full form (e.g. numerically) or, in case of very high viscosity of dispersed phase, the first term could be considered as negligible in Eq. (41), so that instead of Eq. (41a) following equation could be used:

$$ d{u}_{1f}\approx {\left(\frac{C_{1f}}{K_1}\frac{1}{r}\right)}^{\frac{1}{n}} dr={\left[\frac{R_d^2}{2{K}_1}\left({B}_2-{B}_{1f}\right)\frac{1}{r}\right]}^{\frac{1}{n}} dr $$

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for further integration. Such analytical solution for velocity profile could be easily found, but this solution is out of scope of this paper.