Particle image velocimetry of the flow field inside in-axis and off-axis stretched liquid bridges

Abstract We present a novel experimental technique for investigating the internal flow field of liquid bridges under controlled stretching conditions. The key innovation of our setup is embedding the liquid bridge, which is held by two cylindrical plates, in an immiscible, refractive-index-matched continuous phase, providing unobstructed optical access to the entire liquid bridge volume throughout the extension process until breakup. This approach eliminates image distortions caused by the curved interface, allowing for accurate flow field measurements even as the bridge undergoes significant deformation. Using particle image velocimetry, we investigated Newtonian liquid bridges under both in-axis (purely extensional) and off-axis (combined extensional and shear) deformation across plate speeds ranging from 0.05 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>mm</mml:mi> <mml:mstyle scriptlevel="0"/> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> to 5 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>mm</mml:mi> <mml:mstyle scriptlevel="0"/> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , corresponding to capillary numbers Ca between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1.5</mml:mn> <mml:mo>⋅</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1.5</mml:mn> <mml:mo>⋅</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . All internal flows present with axisymmetric counter-rotating vortices that form a characteristic ‘butterfly’ pattern in the vorticity field. In-axis stretched liquid bridges exhibit axisymmetric shear rate fields with a zero-shear region at the minimum diameter. In contrast, off-axis stretching disrupts this symmetry. As the plate velocity increases, we observe a significant enhancement of in-plane velocities, shear gradients, and convective acceleration from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi>Ca</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . This corresponds to the transition from a quasi-static capillary regime to a dynamic regime, which was already documented in existing literature for research concerning liquid transfer by liquid bridge rupture. Our measurement technique addresses the limitations of previous studies by enabling high-resolution velocity field measurements throughout the entire liquid bridge volume, establishing a foundation for future investigations of more complex liquids.