Two-dimensional global stability analysis of elongated bubbles moving in a horizontal tube

The linear stability of an elongated axisymmetric gas bubble transported by a liquid in a capillary tube is analyzed through the use of numerical simulations. The study focuses on the influence of inertia, characterized by the Reynolds number (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>Re</a:mi></a:math>) and the imposed flow rate, characterized by a capillary number (<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mi>Ca</b:mi><b:mi mathvariant="normal">l</b:mi></b:msub></b:math>), on the stability of the bubble tail, which exhibits ripples at high Re. The numerical approach utilizes a boundary-fitted method in a reference frame anchored to the bubble, combined with a mixed spatial discretization based on spectral collocation in the radial direction and finite differences in the axial direction. This framework enables the computation of steady nonlinear solutions through a Newton iteration scheme and facilitates the linear stability analysis of these solutions. Direct numerical simulations using the volume of fluid method in OpenFOAM are also performed to corroborate the results of the stability analysis. We perform systematic simulations for <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msub><d:mi>Ca</d:mi><d:mi mathvariant="normal">l</d:mi></d:msub><d:mo>=</d:mo><d:mn>0.005</d:mn><d:mo>−</d:mo><d:mn>0.04</d:mn></d:mrow></d:math> and observe that the system becomes unstable, with the emergence of oscillations at the rear of the bubble, when the Reynolds number grows above a critical value, designated as <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msup><f:mi>Re</f:mi><f:mo>*</f:mo></f:msup></f:math>; this critical value is dependent on the capillary number. The instability is due to the increased inertia of the recirculating flow in the liquid behind the bubble, which impinges its rear meniscus. A modified Weber number <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi>We</g:mi><g:mi mathvariant="normal">p</g:mi></g:msub></g:math>, based on the relative velocity between the external flow and the bubble, is introduced to describe the competition between the destabilizing pressure force acting on the bubble rear and surface tension. Our results show that the bubble dynamics become unstable for a critical value, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:msubsup><i:mi>We</i:mi><i:mrow><i:mi mathvariant="normal">p</i:mi></i:mrow><i:mo>*</i:mo></i:msubsup><i:mo>≈</i:mo><i:mn>3.65</i:mn></i:mrow></i:math>, which remains quite uniform across the range of capillary numbers tested, and divides the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:msub><k:mi>Ca</k:mi><k:mi mathvariant="normal">l</k:mi></k:msub><k:mo>−</k:mo><k:mi>Re</k:mi></k:mrow></k:math> diagram into stable and unstable regimes. Our findings offer insights into the behavior of bubbles in microfluidic applications, with implications for heat transfer, mass transfer, and cleaning processes in microchannels.