Physics of Swimming: Sperm vs. Sperm Whale — Reynolds Number Explained

Jellyfish-inspired micro-robots mimic biological swimming efficiency, trapping drugs fluidically without mechanical holding and releasing them at targets. They can anchor to surfaces using silk needles that degrade over time, delivering drugs under tissue for bladder cancer treatment. Soft magnetic capsule robots (0.5 mm) can be assembled with 3D printing and jigs, containing nozzles for drug injection. They roll on tissue surfaces and release drugs on demand when high magnetic fields are applied. These designs demonstrate how nature-inspired engineering creates versatile medical robots for targeted drug delivery.

In situ chemistry enables targeted drug delivery by encapsulating two non-toxic precursors in microdroplets (using microfluidics) and combining them only at the tumor site using ultrasound, which vaporizes the perfluorocarbon shell to release the precursors where they react to form the therapeutic drug, thereby minimizing systemic toxicity compared to traditional approaches that deliver pre-formed toxic compounds.

Targeted drug delivery uses nanoparticles (50-150 nm) decorated with biomimetic peptides that mimic natural protein structures to specifically bind cancer cell receptors, enabling efficient drug delivery with reduced side effects; the design requires balancing steric repulsion polymers for prolonged circulation with targeting ligands that exploit tumor-specific markers, as demonstrated by successful colorectal cancer treatment in animal models using only 50 ng/mL DNA payload compared to 25 μg/mL conventional methods.

This lecture presents two complementary approaches to advancing drug delivery systems: (1) Nature-inspired microfluidics that utilize biological principles such as aqueous two-phase systems, vascular architecture, and plant-responsive mechanisms to create novel microfluidic devices with unique properties like self-assembling membranes, deformable structures, and environmentally responsive behaviors; and (2) Microphysiological systems (MPS) that replicate human organ functions in vitro, specifically demonstrating how human BBB MPS with stem cell engineering and physiological shear stress can enable reliable screening of BBB shuttles like aptamers for enhanced drug delivery across the blood-brain barrier.

Since the early 2000s, micro/nanorobotics has drawn extensive inspiration from nature to overcome scaling challenges. Bacterial flagellar propulsion, which uses corkscrew-like motion for movement, has been replicated using artificial helices controlled by external magnetic fields. This bioinspiration strategy enables autonomous locomotion without complex internal mechanisms. In therapeutic applications, microbots address the critical challenge of targeted drug delivery, where many treatments fail due to systemic toxicity from high required dosages. Drugs are packaged in nanoliposomes (similar to cell membranes, also used in mRNA vaccines) and attached to microbots. Biohybrid systems combine living bacteria with magnetic responsiveness, leveraging natural biological intelligence with external control. Targeting is achieved through rotational magnetic fields that guide microbots to extravasate from blood vessels and penetrate target tissues like cancerous areas, with clinical implementation involving pre-scan imaging to identify target locations.

This section develops boundary layer theory and Reynolds number dependencies. Kinematic viscosity (ν = μ/ρ) governs mass action flows, with speeds inversely proportional to kinematic viscosity. In Poiseuille flow through uniform tubes, velocity distribution remains parabolic regardless of density because acceleration is absent in uniform zones. However, at tube inlets, acceleration occurs and parabolic profiles develop gradually downstream, with development distance increasing directly with Reynolds number. Similarly, flow past immersed bodies shows deformation zones expanding at lower speeds and contracting at higher speeds. At high Reynolds numbers, this region becomes called a boundary layer. Pressure distribution uses irrotational flow outside the boundary layer, while shear follows fluid deformation within. The Kingsbury thrust bearing demonstrates viscous lubrication: inclined boundaries create pressure buildup forcing surfaces apart, with metal-to-metal contact occurring only when motion nearly ceases.

Boundary layer theory recognizes that viscosity affects only a thin layer adjacent to walls, allowing outer flow to be treated as inviscid. For flat plates with zero pressure gradient, boundary layers start laminar at the leading edge and grow thicker downstream due to viscous diffusion. The Reynolds number (Re_x = ρU∞X/μ) characterizes boundary layer behavior, increasing along the plate. The critical Reynolds number (~5×10^5 for flat plates) marks the transition from laminar to turbulent flow, with turbulent layers growing faster than laminar ones.

The critical Reynolds number Re_crit ≈ 5 × 10⁵ is the threshold at which laminar boundary layer transition to turbulence occurs for flow over a flat plate. Below this Reynolds number, the boundary layer remains laminar; above it, the boundary layer becomes turbulent. The critical Reynolds number depends on the reference length (x for flat plate), reference velocity (U∞), and fluid properties. Different geometries have different critical Reynolds numbers (e.g., Re_crit ≈ 2000 for pipe flow based on diameter and average velocity). For turbulent boundary layer flow, the velocity profile is approximated by U/U∞ = (y/Δ)^(1/7), which is a curve fit of experimental data. The turbulent boundary layer has a much fuller velocity profile compared to laminar flow, with higher velocities near the wall.

The transition from laminar to turbulent boundary layer depends on the Reynolds number. At low Reynolds numbers, the boundary layer remains laminar; at high Reynolds numbers, it becomes turbulent, increasing skin friction drag. Laminar flow airfoils are designed to maintain laminar boundary layer over larger chord portions (60-70% vs 30-35% for conventional airfoils) by positioning maximum thickness further aft. This extends the region of positive acceleration where laminar flow can be maintained, reducing drag but reducing the range of angles of attack where laminar flow is maintained. Boundary layer suction involves removing fluid from the boundary layer through slots in the airfoil surface to prevent separation. This maintains boundary layer thickness below critical values, delaying or preventing flow separation. In flight tests, this technique reduced drag by 5-6 times while maintaining laminar flow over most of the wing. Boundary layer tripping artificially promotes transition from laminar to turbulent flow using devices like rings on spheres or trip strips. Tripping moves the separation point downstream, reducing drag for bluff bodies.

When fluid flows over a flat plate, a boundary layer forms where velocity transitions from zero at the surface to free stream velocity. Initially laminar (smooth, orderly flow), the boundary layer transitions to turbulent (disorganized) at Reynolds number ~5×10⁵. The Reynolds number varies along the plate length (using X as distance from leading edge) and increases progressively. For laminar conditions, the Prandtl-Blasius solution provides analytical expressions for boundary layer thickness and local shear stress as functions of position. The coefficient of drag for laminar flow depends on Reynolds number at the trailing edge, giving a single value for the entire plate.

Modern naval architecture can be revolutionized by mimicking nature's hydrodynamic solutions, as demonstrated by the Leviathan super yacht design which uses biomimetic principles—such as curved hulls with overlapping titanium scales that break surface tension, tapered sterns that reduce wake, and structural dorsal fins that integrate function with form—to create vessels that move through water with the efficiency and silence of apex predators, moving beyond the traditional 'floating wedding cake' approach of stacked decks and straight lines.

Humans have historically created innovations by observing nature, as exemplified by Leonardo da Vinci's robot prototype inspired by human anatomy and flying machine inspired by birds and bats. This principle of biomimicry extends to submarine design. Early submarines before World War II had pointed bows similar to boats because they spent most time on the surface, using diesel engines requiring air for combustion. This limited underwater operation to about 12 hours at 4-5 knots. The transition to nuclear propulsion fundamentally changed submarine capabilities, enabling speeds over 30 knots and submerged endurance exceeding 90 days.

Effective submarine design follows hydrodynamic principles that minimize water resistance. The cigar-shaped hull design reduces drag by allowing water to flow smoothly around the vessel. Bright yellow concrete serves as the primary hull material because its saturated color remains visible even in dark underwater conditions, improving safety and identification. Windows use light blue stained glass to create an oceanic atmosphere while maintaining visibility. The combination of streamlined shape and high-visibility materials demonstrates how form and function must work together in underwater vehicle design.

Hydrodynamic testing uses large wave tanks (14x14 meters, 4 meters deep) with independently controlled wave generators to simulate ocean conditions. Scale models (1:100) are tested to predict full-scale vessel behavior. Computational Fluid Dynamics (CFD) software complements physical testing by simulating fluid behavior around vessels, though complex simulations can take up to 10 days. Submarines require different hydrodynamic calculations than surface ships due to vertical movement through water. Propulsion must overcome drag forces, with power requirements calculated based on vessel weight and desired speed. Ballast tanks provide stability and depth control.

Submarine design involves fundamental principles of hydrodynamics and buoyancy. Submarines use ballast tanks to control buoyancy based on Archimedes' law - water is pumped into tanks to increase density for diving, and compressed air expels water to surface. Single-hull submarines have ballast tanks integrated into the hull, providing less reserve buoyancy but more internal space. Double-hull submarines have a separate inner pressure hull surrounded by an outer hydrodynamic hull, providing greater reserve buoyancy and safety. Hydrodynamic efficiency affects speed and noise, with wider submarines experiencing more drag. The trade-off between speed, volume, and noise is a fundamental design challenge.