Dynamic Behaviours of a Filament in a Viscoelastic Uniform Flow

The dynamic behaviours of a filament in a viscoelastic uniform flow were investigated by an immersed boundary-lattice Boltzmann method. The effects of the Reynolds numbers (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>e</mi></mrow></semantics></math></inline-formula>, ranging from 10 to 200) and the Weissenberg number (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>i</mi></mrow></semantics></math></inline-formula>, ranging from 0 to 1.2) on the filament flapping motion and the drag and lift coefficients on the filament were studied. It was found that a higher inertial effect (larger <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>e</mi></mrow></semantics></math></inline-formula>) promotes the flapping motion of the filament. In addition, the major effect of the viscoelasticity of the Giesekus fluid is to decrease the critical Reynolds number for the flapping motion of the filament and to promote the flapping motion. The drag coefficient on the filament in a Giesekus uniform flow decreases with the increase of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>i</mi></mrow></semantics></math></inline-formula> at low <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>e</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>e</mi><mo><</mo><mn>100</mn></mrow></semantics></math></inline-formula>), and experiences oscillations with similar amplitudes at all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>i</mi></mrow></semantics></math></inline-formula> at a sufficiently high <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>e</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>e</mi><mo>></mo><mn>100</mn></mrow></semantics></math></inline-formula>). In contrast, the viscoelasticity of the FENE-CR fluid increases the critical Reynolds number at lower <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>i</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>i</mi><mo><</mo><mn>0.8</mn></mrow></semantics></math></inline-formula>), and shows little influence on the critical Reynolds number at higher <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>i</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>W</mi><mi>i</mi><mo>≥</mo><mn>0.8</mn></mrow></semantics></math></inline-formula>). In addition, the viscoelasticity of the FENE-CR fluid hinders the flapping motion of the filament, and increases the drag coefficient on the filament at low <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>e</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>e</mi><mo><</mo><mn>100</mn></mrow></semantics></math></inline-formula>).