A boundary integral equation approach to computing eigenvalues of the Stokes operator

Abstract

The eigenvalues and eigenfunctions of the Stokes operator have been the subject of intense analytical investigation and have applications in the study and simulation of the Navier–Stokes equations. As the Stokes operator is second order and has the divergence-free constraint, computing these eigenvalues and the corresponding eigenfunctions is a challenging task, particularly in complex geometries and at high frequencies. The boundary integral equation (BIE) framework provides robust and scalable eigenvalue computations due to (a) the reduction in the dimension of the problem to be discretized and (b) the absence of high-frequency “pollution” when using Green’s function to represent propagating waves. In this paper, we detail the theoretical justification for a BIE approach to the Stokes eigenvalue problem on simply- and multiply-connected planar domains, which entails a treatment of the uniqueness theory for oscillatory Stokes equations on exterior domains. Then, using well-established techniques for discretizing BIEs, we present numerical results which confirm the analytical claims of the paper and demonstrate the efficiency of the overall approach.

Appendix A: Dirichlet eigenvalues and eigenfunctions on the annulus

In this section, we compute some of the Dirichlet eigenvalues corresponding to a subset of the radially symmetric eigenfunctions on the annulus. In polar coordinates (r,𝜃), consider the annulus defined by R₁ < r < R₂. Suppose that u is of the form:

$$ {\boldsymbol u} = \nabla^{\perp}\left( \alpha H_{0}(kr) + \upbeta J_{0}(kr) \right) , $$

(60)

and p = 0.

Clearly, this pair satisfies the oscillatory Stokes equation with parameter k, since J₀(kr) and H₀(kr) satisfy the Helmholtz equation on the annulus.

Let $\hat {r},\hat {\theta }$ denote the unit vectors in polar coordinates. A simple calculation shows that:

$$ \begin{array}{@{}rcl@{}} u_{r} &=& {\boldsymbol u} \cdot \hat{r} = 0 \\ u_{\theta} &=& {\boldsymbol u} \cdot \hat{\theta} = k(\alpha H_{0}^{\prime}(kr) + \upbeta J_{0}^{\prime}(kr)) . \end{array} $$

(61)

This in particular implies that on r = R₁, u_𝜃 takes on the constant value $k(\alpha H_{0}^{\prime }(kR_{1}) + \upbeta J_{0}^{\prime }(kR_{1}))$. Similarly, on r = R₂, u_𝜃 takes on the constant value $k(\alpha H_{0}^{\prime }(kR_{2}) + \upbeta J_{0}^{\prime }(kR_{2}))$.

Thus, if k satisfies:

$$ H_{0}^{\prime}(kR_{1}) J_{0}^{\prime}(kR_{2}) - H_{0}^{\prime}(kR_{2})J_{0}^{\prime}(kR_{1}) = 0 , $$

(62)

and for those values of k if α,β are non-zero solutions to the system of equations:

$$ \left[\begin{array}{ll} H_{0}^{\prime}(kR_{1}) & J_{0}^{\prime}(kR_{1}) \\ H_{0}^{\prime}(kR_{2}) & J_{0}^{\prime}(kR_{2}) \end{array}\right] \left[\begin{array}{ll} \alpha \\ \upbeta \end{array}\right] = \left[\begin{array}{ll} 0 \\ 0 \end{array}\right] , $$

(63)

then k is a Dirichlet eigenvalue and u defined by (60) is the corresponding eigenfunction.

Appendix B: Neumann eigenvalues and eigenfunctions on the unit disk

In this section, we derive an analytical expression which can be used to compute some of the radially symmetric Neumann eigenvalues on the unit disk for the Stokes operator.

Suppose that u is of the form

$$ {\boldsymbol u} = \nabla^{\perp} J_{0}(kr) , $$

(64)

and the pressure is given by p = 0, as u satisfies (Δ + k²)u = 0.

Then, the surface traction t on the circle of radius r is given by:

$$ {\boldsymbol t} = \left( -\frac{k}{r^{2}}J_{0}^{\prime}(kr) + \frac{k^{2}}{r} J_{0}^{\prime\prime}(kr)\right) \left[\begin{array}{ll} {\sin{(\theta)}} \\ -{\cos{(\theta)}} \end{array}\right] . $$

(65)

Thus, the values k which satisfy:

$$ -k J_{0}^{\prime}(kr) + k^{2} J_{0}^{\prime\prime}(kr) = 0 , $$

(66)

are Neumann eigenvalues on the unit disk. The first 4 roots of (66) are in Table 1.

Table 1 Roots of (7)