Saffman-Taylor Instability and the Inner Splitting Mechanism

Area of Science:

• Fluid dynamics
• Rheology
• Interfacial phenomena

Background:

• Hele-Shaw displacements are crucial in various scientific and industrial applications.
• Viscous fingering and interfacial instabilities significantly impact displacement efficiency.
• Streamwise vorticity has been identified as a key mechanism in fluid mixing and pattern formation.

Purpose of the Study:

• To investigate the origin of streamwise vorticity in viscously unstable, miscible Hele-Shaw displacements.
• To differentiate the roles of viscous instability and gravitational effects in generating streamwise vorticity.
• To elucidate the contribution of variable viscosity to the observed vorticity.

Main Methods:

• Three-dimensional Navier-Stokes simulations were employed.
• Comparisons were made between neutrally buoyant, viscously unstable displacements and gravitationally unstable, constant viscosity displacements.
• Analysis focused on the generation and characteristics of streamwise vorticity.

Main Results:

• Streamwise vorticity was exclusively observed in viscously unstable displacements.
• Gravitationally unstable, constant viscosity displacements did not generate streamwise vorticity.
• The generation of streamwise vorticity was attributed to the lateral displacement of more viscous fluid by less viscous fluid.

Conclusions:

• Viscous instability is the primary driver for streamwise vorticity in these Hele-Shaw flows.
• Variable viscosity terms play a dominant role in the generation of streamwise vorticity.
• Understanding this mechanism is key for controlling fluid mixing and pattern formation in such systems.