Two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability

Area of Science:

• Fluid Dynamics
• Biomechanics
• Mathematical Modeling

Background:

• Biological fluid transport often involves low seepage Reynolds numbers.
• Vessel contraction and expansion significantly impact fluid dynamics.
• Understanding flow in porous-walled channels is crucial for biological systems.

Purpose of the Study:

• To investigate viscous flow in a simulated channel with moving porous walls.
• To analyze the effects of wall contractions and expansions on fluid behavior.
• To develop and validate analytical and numerical solutions for this flow regime.

Main Methods:

• Utilized similarity transformations for exact solutions.
• Reduced the problem to a nonlinear differential equation.
• Employed double perturbation methods (permeation Reynolds number and wall expansion ratio) for analytical solutions.
• Validated analytical results against numerical solutions.

Main Results:

• Flow turning is rapid near the wall during suction-coupled contraction.
• Flow is delayed near the wall during injection-coupled expansion.
• Viscous boundary layers thicken with reduced injection or expansion rates.
• Pressure drop increases with contraction rate but decreases with expansion or permeation rates.

Conclusions:

• Analytical solutions closely match numerical results for small parameters.
• The study provides insights into flow behavior across various channel geometries, including collapsed states.
• Findings are applicable to understanding fluid transport in biological vessels.