# Combined buoyancy and viscous effects in liquid–liquid flows in a vertical pipe

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## Abstract

This paper presents experimental and numerical results of interfacial dynamics of liquid–liquid flows when an immiscible core liquid is introduced into a continuous liquid flow. The fully developed flow model predicts multiple solutions of the jet diameter over a range of dimensionless numbers: flow rate ratio, viscosity ratio, Bond and Capillary numbers. Experiments have been carried out using Polyethylene Glycol (PEG) and Canola oil to investigate the realizability of the three possible solutions predicted by the fully developed flow model. The measured values of inner fluid radii agree very well with the lower branch of the three branched solution while deviating from the top branch beyond a critical flow ratio value. This deviation is attributed to the fact that the flow develops a non-axisymmetric solution at this critical point. Computational fluid dynamics simulations have also been performed to examine the developing core annular flow and to compare the analytical solution results of liquid jet radius. The results predicted by numerical simulations agree very well with both the lower and upper branches of solution predicted by the analytical theory.

## Keywords

Capillary Number Viscosity Ratio Solution Branch Multiple Steady State Vertical Pipe## List of symbols

*b*Fractional area occupied by the jet, \({\frac{R_1}{R_2}}\)

*f*Pressure gradient in axial direction, Pa/m

*g*The magnitude of gravitational acceleration, ms

^{−2}*P*Pressure of fluid

*r*Radial co-ordinate direction

*t*Subscript,

*t*represents the partial derivative of variable with respect to*t***V**Velocity vector

*z*Axial co-ordinate direction

*R*(*θ*,*z*,*t*)Radius of inner fluid with respect to

*θ*,*z*,*t**R*_{1}Fully developed radius of the inner fluid, m

*R*_{2}Tube radius, m

*R*_{i}Nozzle inner radius, m

*R*_{o}Nozzle outer radius, m

*L*_{1}Length of nozzle, m

*L*_{2}Length of tube, m

*W*_{i}Axial velocity of fluid phase

*i*, m/s;*i*= 1, 2 for inner and outer fluid phases, respectively*V*_{i}Fully developed axial velocity of fluid phase

*i*, m/s;*i*= 1, 2 for inner and outer fluid phases, respectively*P*_{i}Pressure of fluid phase

*i*, Pa;*i*= 1, 2 for inner and outer fluid phases, respectively*Q*_{i}Flow rate of fluid phase

*i*, m^{3}/s;*i*= 1, 2 for inner and outer fluid phases, respectively*Re*_{i}Reynolds number, \({\frac{\rho _i\overline{{W}}_iR_2}{\mu_i}; i=1,2}\) for inner and outer fluid phases, respectively

- \({\overline{{\bf U}}}\)
Base flow velocity vector

- \({\overline{{W}}_1}\)
Average velocity of injected phase, m/s; \({\frac{Q_1}{\pi R_i^2 }}\)

- \({\overline{{W}}_2}\)
Average velocity of continuous phase, m/s, \({\frac{Q_2}{\pi R_2^2}}\) (here, \({R_{\rm o}/R_{2}\ll 1)}\)

- Bo/Ca
Ratio of Bond and Capillary numbers, \({\frac{\pi (\rho _2-\rho_1)gR_2^4}{\mu _2Q_2}}\)

*η*Ratio of inner and outer fluid viscosities, \({\frac{\mu_1}{\mu _2}}\)

*α*Volume fraction function

*λ*Flow ratio of inner and outer fluids, \({\frac{Q_1}{Q_2}}\)

*κ*Curvature, m

^{−1}*μ*Weighted average viscosity in the interface region, Pa s

*ρ*Weighted average density in the interface region, kg/m

^{3}*σ*Surface tension, N/m

*θ*Angular co-ordinate direction

- λ
^{*} Critical value of λ

*η*^{*}Critical value of

*η*- Bo/Ca
^{*} Critical value of Bo/Ca

*μ*_{i}Viscosity of fluid phase

*i*, Pa s;*i*= 1, 2 for inner and outer fluid phases, respectively*ρ*_{i}Density of fluid phase

*i*, kg/m^{3};*i*= 1, 2 for inner and outer fluid phases, respectively

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