An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid

Area of Science:

• Fluid Dynamics
• Computational Mathematics
• Fractional Calculus

Background:

• The Rayleigh-Stokes problem describes fluid motion under specific conditions.
• Generalized second-grade fluids exhibit complex rheological behavior.
• Fractional derivatives offer a more nuanced description of time-dependent processes.

Purpose of the Study:

• To analyze the Rayleigh-Stokes problem for a generalized second-grade fluid with a time-fractional derivative.
• To develop and validate numerical methods for solving this complex fluid dynamics problem.
• To establish theoretical error estimates for the proposed numerical schemes.

Main Methods:

• Utilizing Riemann-Liouville fractional derivatives in the fluid model.
• Developing space semidiscrete Galerkin and fully discrete schemes (Backward Euler, Backward Difference).
• Applying convolution quadrature for time discretization.
• Establishing Sobolev regularity for initial data.

Main Results:

• Optimal error estimates derived for both space semidiscrete and fully discrete approximations.
• Demonstrated accuracy and efficiency for smooth and nonsmooth initial data.
• Verification of convergence theory through numerical experiments.

Conclusions:

• The developed numerical methods accurately and efficiently solve the fractional Rayleigh-Stokes problem.
• The study provides robust theoretical underpinnings for fractional fluid dynamics simulations.
• The findings are applicable to modeling complex fluid behaviors with memory effects.