Analysis of a three-dimensional fluid flow in rotating cylinders

Abstract: Subject of consideration is the modelling and analysis of a capillary-driven three-dimensional rimming-flow problem. We present the derivation of a fourth-order quasilinear degenerate-parabolic partial differential equation for the height $h > 0$ of a fluid film coating the inner wall of a cylinder that rotates around a horizontal axis. The equation arises from a rescaled Navier-Stokes system for thin fluid films by means of a lubrication approximation and accounts for the physical effects of rotation, surface tension and gravity. The effect of the latter is measured by a non-dimensional parameter $0 \leq δ\ll 1$. We characterise the structure of the steady states depending on the ratio $\ell$ of the cylinder length to its radius. In the absence of gravity ($δ=0$), in the case $\frac{\ell}π \notin \mathbb{Z}$, steady states are unique. For $0 < δ\ll 1$, steady states are shown to be locally unique for any $\ell$. These steady states are stable for $\ell < π$, while they are unstable for $\ell > π$. Furthermore, in the absence of gravity, for all $\ell > 0$, we show that there exists a manifold of time-periodic solutions. In the critical case $\ell = π$, we study the dynamics of the solutions close to the manifold of periodic orbits in the critical case $\ell = π$ on the large time scale $τ= δ² t$. It turns out that in the time scale $τ$ this dynamics can be approximated by a system of ordinary differential equations.

• The paper develops a rigorous derivation of a fourth-order degenerate-parabolic PDE from the Navier-Stokes equations to model 3D thin-film rimming flow, incorporating surface tension, rotation, and gravity.
• It establishes local well-posedness, positivity preservation, and energy decay under weak gravity, while identifying unique steady states and stability criteria based on cylinder dimensions.
• Numerical and asymptotic analyses reveal slow manifold dynamics and bifurcation-induced instability thresholds, offering practical insights for industrial coating and pattern formation processes.

Analysis of Three-Dimensional Capillary-Driven Rimming Flow in Rotating Cylinders

Derivation and Structure of the Thin-Film Rimming Flow Equation

The paper develops a rigorous analysis of capillary-driven three-dimensional rimming flows—thin films of Newtonian fluid partially filling a horizontally rotating cylinder under the action of surface tension, cylinder rotation, and gravity. Starting from the incompressible Navier-Stokes equations with free-boundary conditions, the authors perform a lubrication approximation under the assumption that the mean film thickness is small compared to the cylinder radius. This formal asymptotic analysis leads to a fourth-order, quasilinear, degenerate-parabolic partial differential equation (PDE) for the film height $h(t, \theta, z)$:

$$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$

Here, $\gamma$ represents a (dimensionless) surface tension parameter, and $\delta$ captures the strength of the gravity relative to surface tension. The equation encapsulates three critical physical effects:

• Cylinder rotation, yielding a first-order advection term $h_\theta$.
• Surface tension, producing the dominant fourth-order degenerate-parabolic term.
• Gravity, incorporated as a lower-order nonlinearity via $\delta (h^3 \cos \theta)$.

Degeneracy arises due to the vanishing of parabolicity when the film ruptures ($h\to 0$), which is nontrivial given the absence of a maximum principle for equations of this order.

Functional Setting and Well-Posedness

After transforming the equation using a reflection principle for the $z$-direction and working on the double torus $T^2$, the authors recast the PDE with (artificial but analytically standard) Neumann-type boundary conditions:

$h_z = h_{zzz} = 0 \quad \text{at } z=0, \ell$

This allows the application of Fourier analysis and a robust functional-analytic framework. Utilizing real or fractional Sobolev spaces interpolated by Fourier coefficients, the main results on well-posedness include:

• Local existence and uniqueness of positive strong solutions for arbitrary initial data in $$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$0 with $$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$1.
• Positivity is maintained, and the mass is conserved.
• For $$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$2, the associated energy functional is non-increasing.
• Maximal solutions exist globally in time unless $$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$3-norm blow-up or finite-time rupture (loss of strict positivity) occurs.

The analytic semigroup theory of quasilinear parabolic PDEs is central to these proofs, leveraging sectoriality and Lipschitz continuity properties of the operators involved.

Classification and Stability of Steady States

Gravity-Free Case ($$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$4)

A complete characterization is obtained for positive steady-states in the absence of gravity. The solution structure depends crucially on the dimensionless ratio $$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$5, defined as the cylinder length-to-radius ratio:

• For $$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$6, the only positive steady-states are constant ($$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$7).
• For $$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$8, positive steady-states form a one-parameter family $$h_t + h_\theta + \gamma \operatorname{div}(h^3 \nabla (\Delta h + h)) = \delta (h^3 \cos \theta)$$9, $\gamma$0—“deformed cylinders” with $\gamma$1-dependent radii.

Linear stability analysis in a rotating coordinate frame reveals:

• Exponential orbital stability of constant states for $\gamma$2 (short cylinders), where solutions close in $\gamma$3 norm to $\gamma$4 converge exponentially to a travelling wave.
• Nonlinear instability for $\gamma$5 (long cylinders), linked to unstable eigenmodes, with solutions developing “bubbles” along the $\gamma$6-axis. The instability threshold coincides with that found in related 1D models.

Small Gravity Regime ($\gamma$7)

The introduction of weak gravity removes degeneracy in the steady-state set:

• For all $\gamma$8, locally unique positive steady-states $\gamma$9 exist, proved via the implicit function theorem in the generic case and Lyapunov-Schmidt reduction for $\delta$0.
• The expansion in $\delta$1 is provided up to quadratic order, revealing the selection of a unique solution by gravity out of the former $\delta$2-dimensional manifold of steady-states.

Stability persists as in the gravity-free case:

• For $\delta$3, steady-states $\delta$4 are exponentially stable.
• For $\delta$5, steady-states remain unstable for small $\delta$6.

These results clarify subtle questions arising in prior literature [Puk05] regarding the persistence of non-uniqueness and the effect of gravity.

Long-Time Dynamics, Slow Manifolds, and Multiple Time Scales

In the critical case $\delta$7, the dynamics in the presence of small gravity ($\delta$8) exhibit a separation of time scales, explained via a multiple-scale Poincaré-Lindstedt expansion:

• Solutions rapidly reach an $\delta$9-neighborhood of a three-dimensional slow manifold $h_\theta$0 within which the long-time dynamics are governed by an ODE system on the slow time scale $h_\theta$1.
• Fast convergence onto $h_\theta$2 is guaranteed for initial data uniformly bounded away from rupture.
• Linearization of the ODE system quantifies decay rates and reveals the rotational symmetry inherent in the limiting dynamics.

Importantly, the use of formal asymptotics and normal form transformations enables the precise derivation of reduced ODEs for the dynamics near criticality, confirming and extending insights from related bifurcation analyses in 1D reductions [KLV25, JLV26].

Numerical and Analytical Results

Numerical integration of the reduced ODE system demonstrates:

• Spiraling inward motion of the “centre” of the fluid profile on the slow time scale, with the cross-sectional profile approaching a circle centered at the cylinder axis.
• Nontrivial 3D dynamics when both axial and azimuthal deformation modes are activated, reflecting the interplay between gravity and geometry.

Theoretical analysis ensures that these numerical observations are supported by energy methods and stability theory for sectorial operators.

Implications and Prospects

Theoretical Implications

The results provide a comprehensive, mathematically rigorous treatment of 3D capillary rimming flow inside rotating cylinders, yielding:

• A clear diagnosis of instability/stability thresholds as a function of system parameters.
• An explicit mechanism by which gravity removes degeneracy of steady states and selects unique, stable configurations.
• A blueprint for rigorous time-scale separation analysis, with potential extensions to other high-dimensional, quasilinear degenerate-parabolic PDEs governing thin films.

Practical Implications

These findings are directly relevant in several industrial fluid mechanics contexts:

• Coating processes in rotating drums for manufacturing films, metal foils, or pharmaceutical tablets, where control of uniform layer deposition and avoidance of rupture/bubble formation is critical.
• Papermaking (e.g., Fourdrinier machines) and rotational casting/molding applications.
• Pattern formation and instability control in microfluidics and lab-on-chip devices with cylindrical geometries.

From an applied mathematics perspective, the methodology rigorously justifies the use of lubrication theory and associated boundary conditions for high-dimensional, complex geometry thin-film problems.

Future Developments

Potential research directions include:

• Extending the analysis to include inertia, non-Newtonian effects, or more complex boundary conditions, which are prevalent in applications.
• Investigation of global-in-time existence versus finite-time rupture criteria.
• Coupling to thermal or multi-phase effects, utilizing similar multiple-scale and functional-analytic techniques.
• Exploring stochastic perturbations and their impact on dynamics near critical manifolds.

Conclusion

This paper provides a mathematically rigorous and physically detailed account of three-dimensional capillary-driven rimming flows in rotating cylinders. It derives and analyzes a fourth-order degenerate-parabolic PDE for thin-film dynamics, classifies steady state structure and stability, elucidates slow manifold reductions, and rigorously identifies the bifurcation and selection effects introduced by gravity. Both the theoretical tools and the insights into physical instability mechanisms will be valuable for further study of advanced coupled fluid-mechanical systems and for optimization of relevant industrial processes.