Asymptotic justification of the Reynolds equation for a spherical bearing

Technical Summary: Asymptotic Justification of the Reynolds Equation for a Spherical Bearing

Problem Statement
The paper addresses the lack of rigorous mathematical justification for the Reynolds equation in the context of spherical bearings. While the classical Reynolds equation (1886) assumes flat confining surfaces, practical spherical bearings involve curved geometries where neither bounding surface is flat. Previous derivations for such geometries (e.g., in [3], [6], [9], [15], [18], [19], [20], [21]) have largely relied on formal asymptotic expansions rather than rigorous convergence proofs. Although rigorous justifications exist for flat geometries and specific cases like journal bearings ([6]) or curved pipes ([17]), a rigorous proof for the convergence of the Stokes problem to a Reynolds-type equation between two arbitrary, closely spaced spherical surfaces was, to the authors' knowledge, unavailable.

Methodology
The authors employ an asymptotic analysis approach based on the convergence of the Stokes problem as the gap between the spheres approaches zero. The methodology proceeds through the following steps:

1. Domain Definition: The physical domain $\Omega_\varepsilon$ is defined as the region between an inner sphere of radius $R$ and an outer sphere separated by a small gap $h_\varepsilon(\phi, \theta) = \varepsilon h(\phi, \theta)$, where $\varepsilon$ is a small dimensionless parameter. The domain is restricted to a sector of the sphere to accommodate standard bearing coverage (typically 120°–150°).
2. Coordinate Transformation: To handle the $\varepsilon$-dependent domain, the authors map $\Omega_\varepsilon$ to a fixed reference domain $\Omega = D \times (0, 1)$ using a change of variables. This transformation decomposes the mapping into three steps: scaling the gap coordinate, adjusting the radial coordinate, and transforming to Cartesian coordinates.
3. Function Spaces and Norms: The analysis utilizes specific Sobolev spaces and norms adapted to the different coordinate systems. Crucially, the authors establish equivalence relations between norms in the physical domain and the reference domain, demonstrating how these norms scale with $\varepsilon$ (e.g., $L^2$ norms scale with $\sqrt{\varepsilon}$).
4. Boundary Conditions: The study assumes the inner sphere rotates with angular velocity $\omega$ while the outer sphere is stationary. Inflow and outflow conditions are specified at the sector boundaries ($\phi = \phi_0, \phi_1$). The authors construct a divergence-free lifting of the boundary conditions with an explicit bound of order $O(\sqrt{\varepsilon})$ in the $H^1$ norm.
5. Convergence Analysis:
   • Velocity: By deriving energy estimates and utilizing the constructed lifting, the authors prove that the velocity field $\vec{u}_\varepsilon$ converges weakly in a specific Hilbert space $V^*$ to a limit $\vec{u}^*$. They show that derivatives with respect to the angular coordinates vanish in the limit, while the derivative with respect to the gap coordinate remains bounded.
   • Pressure: The pressure $p_\varepsilon$ is shown to converge weakly after scaling by $\varepsilon^2$. The authors derive estimates for the pressure gradients, showing that the scaled pressure gradient in the gap direction vanishes, while the tangential gradients converge to the gradient of a limit pressure $p^*$.
6. Limit Identification: By passing to the limit in the weak formulation of the Stokes equations, the authors derive the explicit form of the limit velocity components ($u^*_\phi, u^*_\theta, u^*_\rho$) in terms of the limit pressure $p^*$. Finally, integrating the continuity equation across the gap yields the limit equation for $p^*$.

Key Contributions and Results
The primary contribution is the rigorous proof that the solution to the Stokes problem in a spherical bearing converges to the solution of a generalized Reynolds equation as the gap parameter $\varepsilon \to 0$.

• Convergence Theorems: The paper establishes the existence of subsequences of the velocity and pressure that converge weakly to limit functions $\vec{u}^*$ and $p^*$.
• Limit Velocity Profile: The limit velocity field is shown to have a parabolic profile in the gap direction (consistent with lubrication theory), with the tangential components depending on the pressure gradient and the rotation of the inner sphere. The radial component vanishes ($u^*_\rho = 0$).
• The Limit Equation: The limit pressure $p^*$ is proven to be the unique solution to a Neumann problem (equations 168–171). This equation is a generalized Reynolds equation for spherical coordinates:
  $$\frac{\partial}{\partial \xi_1} \left( \frac{h^3 \sin \xi_1}{12\mu} \frac{\partial p^*}{\partial \xi_1} \right) + \frac{\partial}{\partial \xi_2} \left( \frac{h^3}{12\mu \sin \xi_1} \frac{\partial p^*}{\partial \xi_2} \right) = \frac{R^2 \omega \sin \xi_1}{2} \frac{\partial h}{\partial \xi_2}$$
  This equation includes boundary conditions related to the supply flow rates at the sector edges.
• Uniqueness: The uniqueness of the limit solution ensures that the convergence holds for the entire sequence, not just a subsequence.

Significance
The paper claims to provide the first rigorous mathematical justification of the Reynolds equation for a spherical bearing where neither confining surface is flat. By demonstrating that the solution to the full Stokes problem converges to the solution of the derived Reynolds equation, the authors validate the use of this simplified model for spherical bearings. This result bridges the gap between formal asymptotic derivations found in engineering literature and rigorous mathematical analysis, confirming the validity of the model under the assumption of a small gap parameter. The work extends previous justifications limited to flat or cylindrical geometries to the more complex spherical case.