Remarks on Regularized Stokeslets in Slender Body Theory

• Main Author: Laurel Ohm
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Fluids
• Subjects: slender body theory; regularized Stokeslets; error analysis
• Online Access: https://www.mdpi.com/2311-5521/6/8/283

We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. Denoting the regularization parameter by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>, we consider regularized SBT based on the most common regularized Stokeslet plus a regularized doublet correction. Given sufficiently smooth force data along the filament, we derive <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>∞</mo></msup></semantics></math></inline-formula> bounds for the difference between regularized SBT and its classical counterpart in terms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>, and the force data. We show that the regularized and classical expressions for the velocity of the filament itself differ by a term proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><mo>(</mo><mi>δ</mi><mo>/</mo><mi>ϵ</mi><mo>)</mo></mrow></semantics></math></inline-formula>; in particular, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>=</mo><mi>ϵ</mi></mrow></semantics></math></inline-formula> is necessary to avoid an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> discrepancy between the theories. However, the flow at the surface of the fiber differs by an expression proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><mo>(</mo><mn>1</mn><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup><mo>/</mo><msup><mi>ϵ</mi><mn>2</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula>, and any choice of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>∝</mo><mi>ϵ</mi></mrow></semantics></math></inline-formula> will result in an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> discrepancy as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϵ</mi><mo>→</mo><mn>0</mn></mrow></semantics></math></inline-formula>. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed <i>slender body PDE</i> which classical SBT is known to approximate. Numerics verify this <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> discrepancy but also indicate that the difference may have little impact in practice.