[Paper Review] Existence, Uniqueness and Asymptotic Behavior of Regular Time-Periodic Viscous Flow around a Moving Body: Rotational Case

Giovanni P. Galdi|arXiv (Cornell University)|Jun 4, 2020

Navier-Stokes equation solutions24 references4 citations

TL;DR

This paper establishes the existence, uniqueness, and asymptotic behavior of regular time-periodic solutions to the Navier-Stokes equations for viscous flow around a rigid body undergoing combined translational and rotational motion with the same period. Using a contraction mapping argument on a function space with weighted decay, it proves that for small data, a unique solution exists, and when the body has a non-zero average translational velocity, the flow exhibits a wake-like structure analogous to steady-state flow.

ABSTRACT

We show existence and uniqueness for small data of regular time-periodic solutions to the Navier-Stokes problem in the exterior of a rigid body, $\mathscr B$, that moves by time-periodic translational motion of the same period along a constant direction, $\bfe_1$, and spins with constant angular velocity $\bfomega$ parallel to $\bfe_1$. We also study the spatial asymptotic behavior of such solutions and show, in particular, that if $\mathscr B$ has a net motion characterized by a non-zero average translational velocity $\bar{\bfxi}$, then the solution exhibit a wake-like behavior in the direction $-\bar{\bfxi}$ entirely analogous to that of a steady-state flow around a body that moves with velocity $\bar{\bfxi}$ and angular velocity $\bfomega$.

Motivation & Objective

• To establish the existence and uniqueness of regular time-periodic solutions to the Navier-Stokes equations for viscous flow around a rigid body moving with time-periodic translational and rotational motion.
• To analyze the spatial asymptotic behavior of such solutions, particularly in the presence of a non-zero average translational velocity.
• To remove restrictive assumptions on the period and on the time-dependence of the motion, such as constant velocity or parallel alignment, while maintaining small data conditions.
• To provide a rigorous mathematical foundation for the steady streaming phenomenon observed in fluid dynamics.
• To extend prior results by avoiding reliance on maximal regularity or $L^p$-$L^q$ estimates, instead using high-order energy estimates and a contraction mapping framework.

Proposed method

• Formulates the problem as a time-periodic Navier-Stokes system with a moving boundary condition corresponding to the body's translational and rotational motion.
• Introduces a function space $\mathscr{S}$ equipped with weighted norms to capture decay at infinity, ensuring regularity and integrability of solutions.
• Applies the Galerkin method combined with high-order energy estimates to prove existence of a regular $T$-periodic solution to the linearized problem.
• Uses a contraction mapping principle in the space $\mathscr{S}$, treating the nonlinear term $\mathbf{u} \cdot \nabla \mathbf{u}$ as a perturbation.
• Establishes a priori estimates via weighted $L^2$-type norms involving the average translational velocity $\lambda$, ensuring decay of solutions at infinity.
• Relies on the well-posedness of the linear problem and the boundedness of the nonlinear map to prove convergence of the iterative scheme.

Experimental results

Research questions

• RQ1 Under what conditions does a unique regular time-periodic solution exist for the Navier-Stokes equations around a rigid body undergoing time-periodic translational and rotational motion?
• RQ2 How does the spatial asymptotic behavior of the solution depend on the average translational velocity of the body?
• RQ3 Can the existence and uniqueness of solutions be established without assuming constant velocity or periodicity constraints on the driving motion?
• RQ4 What is the nature of the wake-like structure in the flow when the body has a non-zero average translational velocity?
• RQ5 How does the inclusion of rotational motion affect the long-term behavior and decay properties of the viscous flow?

Key findings

• For sufficiently small data in the norm $\mathsf{D}$, which includes the body force, its divergence, the translational velocity, and angular velocity, a unique $T$-periodic solution $({\mathbf{u}}, p)$ exists in the space $\mathscr{S} \times L^2(D^{1,2})$.
• The solution satisfies the a priori estimate $\|{\mathbf{u}}\|_{\mathscr{S}} \leq c \, \mathsf{D}$, with $c = c(\Omega, T)$, ensuring boundedness relative to data size.
• When the average translational velocity $\overline{\boldsymbol{\xi}} = \lambda \geq 0$ is non-zero, the flow exhibits a wake-like behavior in the direction $-\overline{\boldsymbol{\xi}}$, analogous to steady-state flow around a body moving with velocity $\overline{\boldsymbol{\xi}}$ and angular velocity $\omega$.
• The asymptotic decay of the solution is characterized by a distinctive profile at large distances, consistent with steady-state behavior in the wake region.
• The contraction mapping argument is valid under the condition $\mathsf{D} < 1/(16c_2^2)$ and $\|\mathbf{u}\|_{\mathscr{S}} < \delta = 4c_2 \mathsf{D}$, ensuring convergence of the iterative scheme.
• The method avoids restrictive assumptions such as $\boldsymbol{\omega} \equiv 0$, $\boldsymbol{\xi}$ constant, or period matching $2\pi\kappa/|\boldsymbol{\omega}|$, thus generalizing prior results.