Defect turbulence in a dense suspension of polar, active swimmers

We study the effects of inertia in dense suspensions of polar swimmers. The hydrodynamic velocity field and the polar order parameter field describe the dynamics of the suspension. We show that a dimensionless parameter <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>R</a:mi></a:math> (ratio of the swimmer self-advection speed to the active stress invasion speed []) controls the stability of an ordered swimmer suspension. For <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>R</b:mi></b:math> smaller than a threshold <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>R</c:mi><c:mn>1</c:mn></c:msub></c:math>, perturbations grow at a rate proportional to their wave number <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>q</d:mi></d:math>. Beyond <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi>R</e:mi><e:mn>1</e:mn></e:msub></e:math> we show that the growth rate is <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi mathvariant="script">O</f:mi><f:mo>(</f:mo><f:msup><f:mi>q</f:mi><f:mn>2</f:mn></f:msup><f:mo>)</f:mo></f:mrow></f:math> until a second threshold <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>R</h:mi><h:mo>=</h:mo><h:msub><h:mi>R</h:mi><h:mn>2</h:mn></h:msub></h:mrow></h:math> is reached. The suspension is stable for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>R</i:mi><i:mo>></i:mo><i:msub><i:mi>R</i:mi><i:mn>2</i:mn></i:msub></i:mrow></i:math>. We perform direct numerical simulations to characterize the steady-state properties and observe defect turbulence for <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:mi>R</j:mi><j:mo><</j:mo><j:msub><j:mi>R</j:mi><j:mn>2</j:mn></j:msub></j:mrow></j:math>. An investigation of the spatial organization of defects unravels a hidden transition: for small <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mrow><k:mi>R</k:mi><k:mo>≈</k:mo><k:mn>0</k:mn></k:mrow></k:math> defects are uniformly distributed and cluster as <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:mi>R</l:mi><l:mo>→</l:mo><l:msub><l:mi>R</l:mi><l:mn>1</l:mn></l:msub></l:mrow></l:math>. Beyond <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>R</m:mi><m:mn>1</m:mn></m:msub></m:math>, clustering saturates and defects are arranged in nearly stringlike structures. Published by the American Physical Society 2024