Scallop Theorem: Fundamentals of Motion and Efficiency in Microfluidic Environments

Chapters Brief Overview:

1: Scallop theorem: Introduces the scallop theorem, outlining its relevance in microswimmer dynamics.

2: Divergence theorem: Discusses how the divergence theorem is crucial for understanding fluid behavior.

3: Oseen equations: Covers Oseen's equations for describing low Reynolds number flows around microswimmers.

4: Stokes flow: Examines Stokes flow and its application in understanding motion at small scales.

5: Langevin equation: Explains the Langevin equation in the context of random motion in fluids.

6: Prandtl–Batchelor theorem: Discusses the Prandtl–Batchelor theorem in relation to fluid mechanics.

7: Total variation diminishing: Focuses on the method of total variation diminishing in fluid flow calculations.

8: Derivation of the Navier–Stokes equations: Offers a detailed derivation of the Navier–Stokes equations, essential for fluid dynamics.

9: Potential vorticity: Explores the concept of potential vorticity and its importance in fluid mechanics.

10: Reynolds number: Analyzes the Reynolds number and its significance in the behavior of fluid flow.

11: Cauchy momentum equation: Introduces the Cauchy momentum equation and its role in describing fluid motion.

12: Newtonian fluid: Details the properties of Newtonian fluids and their relationship with microswimmers.

13: Stokes' law: Explains Stokes' law and its applicability in the study of microswimmers and their motion.

14: Leray projection: Investigates the Leray projection in the context of incompressible flow equations.

15: Stream function: Focuses on the stream function and its usefulness in analyzing flow patterns.

16: Stokes' theorem: Describes Stokes' theorem and its relevance to the study of fluid motion at small scales.

17: Helmholtz decomposition: Discusses Helmholtz decomposition and its connection to fluid dynamics.

18: Fluid mechanics: Provides an overview of the field of fluid mechanics, setting the stage for the book’s deeper insights.

19: Stokes approximation and artificial time: Explores the Stokes approximation and its impact on the modeling of microswimmers.

20: Viscous vortex domains method: Introduces the viscous vortex domains method for studying complex fluid flows.

21: Navier–Stokes equations: Concludes with an indepth look at the Navier–Stokes equations, the cornerstone of fluid dynamics.

This book goes beyond theory, bridging abstract concepts with practical applications for microswimmers. Aimed at both seasoned professionals and students, it provides a unique perspective, helping readers grasp the complex interplay between mathematical models and physical phenomena. Whether you're working in fluid dynamics, robotics, or biophysics, Scallop Theorem will expand your expertise and deepen your understanding.