Core-Annular WRIBL Model

• The core-annular WRIBL model is a reduced-order, one-dimensional formulation that represents axisymmetric two-phase flow with dynamically evolving core and annular layers.
• It employs weighted residual techniques to average long-wave Navier–Stokes equations, incorporating inertial, viscous, and capillary forces up to O(ε²) for accurate interfacial dynamics.
• The formulation bridges full diffuse-interface simulations and simpler lubrication models, providing an efficient tool for analyzing capillary instability and annular collar dynamics.

Searching arXiv for recent and foundational papers on core-annular WRIBL and related reduced models. The core-annular weighted residual integral boundary layer (WRIBL) model is a reduced-order, one-dimensional representation of axisymmetric two-phase flow in a cylindrical tube, designed for two hydrodynamically active immiscible Newtonian fluids separated by a deformable interface. In the formulation rederived algorithmically in 2025, the inner fluid is the core phase and the outer fluid is the annular phase; the interface is located at $r=d(z,t)$, and the principal dynamical variables are the interface position $d(z,t)$ and the phase flow rates $Q_c(z,t)$ and $Q_a(z,t)$ (Hazra et al., 7 Jul 2025). The model belongs to the WRIBL class because it derives one-dimensional evolution equations by averaging the long-wave Navier–Stokes equations over the thin-gap direction with carefully chosen weight functions, thereby retaining inertial effects, viscous coupling between phases, capillarity, longitudinal viscous diffusion, and nonlinear curvature at $O(\epsilon^2)$ while remaining substantially cheaper than direct simulation (Hazra et al., 7 Jul 2025). In the literature surrounding core-annular flows, it occupies an intermediate position between full diffuse-interface or sharp-interface simulations and simpler lubrication or slender-jet models; a closely related long-wave two-fluid model for a corrugated tube provides a lubrication-type limiting case rather than a genuine WRIBL formulation (Wang, 2020), while phase-field simulations provide high-fidelity benchmarks for laminar, linear, and nonlinear regimes (Song et al., 2019).

1. Physical setting and reduced variables

The model concerns axisymmetric core-annular flow in a cylindrical tube of radius $R$, with two immiscible Newtonian fluids of densities $\rho_c,\rho_a$, viscosities $\mu_c,\mu_a$, and interfacial tension $\gamma$. In cylindrical coordinates $(r,z)$, the interface is at

$d(z,t)$0

so that $d(z,t)$1 is the nondimensional interface position and the annular film thickness is $d(z,t)$2 (Hazra et al., 7 Jul 2025).

The long-wave parameter is

$d(z,t)$3

where $d(z,t)$4 is the axial length scale (Hazra et al., 7 Jul 2025). The nondimensionalization uses

$d(z,t)$5

$d(z,t)$6

with $d(z,t)$7 a characteristic axial velocity (Hazra et al., 7 Jul 2025). The principal dimensionless groups are

$d(z,t)$8

and

$d(z,t)$9

with $Q_c(z,t)$0 the dimensional axial body force per unit volume (Hazra et al., 7 Jul 2025).

A defining feature of the core-annular WRIBL formulation is that both phases are retained dynamically. The reduced model does not treat the outer phase as a passive gas or an algebraically eliminated lubricating film. Instead, it evolves the interface position together with the cross-section averaged flow rates

$Q_c(z,t)$1

so that the one-dimensional dynamical system represents both core and annulus as momentum-carrying layers (Hazra et al., 7 Jul 2025).

This suggests a central conceptual distinction from lubrication-type limits. In a slender-jet or lubrication reduction, one phase may be quasi-eliminated through a local closure. In the WRIBL model, by contrast, the annulus remains an active participant through its own flow-rate dynamics, interfacial stress transmission, and weighted residual projection (Hazra et al., 7 Jul 2025).

2. Long-wave equations and interfacial conditions

Retaining terms up to $Q_c(z,t)$2, the axisymmetric continuity equation in each phase is

$Q_c(z,t)$3

The axial momentum equation is

$Q_c(z,t)$4

These forms arise after using the $Q_c(z,t)$5 radial momentum balance to eliminate $Q_c(z,t)$6 in favor of interfacial values (Hazra et al., 7 Jul 2025).

At the interface $Q_c(z,t)$7, the velocity continuity conditions are

$Q_c(z,t)$8

Tangential stress continuity, to $Q_c(z,t)$9, is

$Q_a(z,t)$0

The normal stress balance is

$Q_a(z,t)$1

with curvature

$Q_a(z,t)$2

The kinematic condition is

$Q_a(z,t)$3

At the centerline $Q_a(z,t)$4,

$Q_a(z,t)$5

and at the wall $Q_a(z,t)$6,

$Q_a(z,t)$7

These equations remain two-dimensional and free-boundary; WRIBL reduces them to one-dimensional form (Hazra et al., 7 Jul 2025).

The derivation assumes an $Q_a(z,t)$8 model and additionally states that terms of order $Q_a(z,t)$9 are neglected. It also assumes $O(\epsilon^2)$0, $O(\epsilon^2)$1 are at most $O(\epsilon^2)$2 (Hazra et al., 7 Jul 2025). A plausible implication is that the model is intended for moderate inertial regimes where long-wave inertial transport is important but higher-order inertial corrections do not dominate.

3. Weighted residual construction

The derivation introduces a decomposition of the axial and radial velocities into quasi-steady local profiles plus corrections: $O(\epsilon^2)$3 Here $O(\epsilon^2)$4 is the quasi-steady, quasi-developed profile parametrized by the local interface position $O(\epsilon^2)$5 and local phase flow rate $O(\epsilon^2)$6, and $O(\epsilon^2)$7 is a correction (Hazra et al., 7 Jul 2025). The leading profile is chosen so that it exactly reproduces the local phase flow rates: $O(\epsilon^2)$8

At leading order, the axial profiles satisfy

$O(\epsilon^2)$9

with $R$0 independent of $R$1, subject to centerline, wall, and interfacial matching conditions (Hazra et al., 7 Jul 2025). The general solutions are

$R$2

and regularity at the centerline removes the logarithm from the core phase (Hazra et al., 7 Jul 2025).

The annular profile can be written exactly, although the explicit form is lengthy. For symbolic work it is rewritten as

$R$3

$R$4

where the coefficient functions are exact shorthand for rational-logarithmic functions of $R$5 (Hazra et al., 7 Jul 2025).

From continuity, the radial velocities are

$R$6

Enforcing continuity of $R$7 at the interface yields

$R$8

which is the exact one-dimensional phase-flow constraint (Hazra et al., 7 Jul 2025).

The weighted residual step then averages the momentum residual with phase-dependent weight functions $R$9 using

$\rho_c,\rho_a$0

The weights are chosen so that integration by parts closes the dominant transverse viscous term exactly and eliminates the unknown corrections $\rho_c,\rho_a$1 (Hazra et al., 7 Jul 2025). They satisfy

$\rho_c,\rho_a$2

with

$\rho_c,\rho_a$3

For the flow-rate equation, the integral constraint is

$\rho_c,\rho_a$4

This choice allows the interfacial pressure jump to enter through the normal-stress balance and removes explicit pressure-gradient dependence from the averaged momentum equation (Hazra et al., 7 Jul 2025).

This construction is characteristic of WRIBL methodology. The reduced equations are not obtained by direct asymptotic elimination alone, but by a weighted projection designed to preserve dominant transverse diffusion and to encode cylindrical geometry and two-phase coupling in closed coefficient functions.

4. Governing equations of the core-annular WRIBL model

The exact interface evolution law follows from continuity integrated across the annulus: $\rho_c,\rho_a$5 or equivalently,

$\rho_c,\rho_a$6

using $\rho_c,\rho_a$7 (Hazra et al., 7 Jul 2025).

The principal second-order WRIBL averaged-momentum equation is

$\rho_c,\rho_a$8

$\rho_c,\rho_a$9

Repeated indices are summed, and

$\mu_c,\mu_a$0

The coefficient sets

$\mu_c,\mu_a$1

are functions of $\mu_c,\mu_a$2 only (Hazra et al., 7 Jul 2025).

The curvature used in the capillary term is

$\mu_c,\mu_a$3

so that

$\mu_c,\mu_a$4

Accordingly, the capillary forcing term $\mu_c,\mu_a$5 contains both the classical cylindrical Rayleigh–Plateau contribution and the $\mu_c,\mu_a$6 nonlinear-curvature and axial-curvature corrections (Hazra et al., 7 Jul 2025).

A second weight choice yields a diagnostic pressure equation: $\mu_c,\mu_a$7

$\mu_c,\mu_a$8

The paper notes that the full phase pressures are radially uniform at this order because $\mu_c,\mu_a$9 follows from the $\gamma$0 radial momentum equation (Hazra et al., 7 Jul 2025).

The physical interpretation of the principal terms is explicit. The left-hand side

$\gamma$1

represents weighted inertial response, convective nonlinearities, and nonparallel geometric corrections. On the right-hand side, $\gamma$2 is the leading transverse-viscous closure; $\gamma$3 is capillary forcing; $\gamma$4 is differential forcing; and the $\gamma$5 terms encode $\gamma$6 geometric, interfacial-stress, and longitudinal-diffusion corrections (Hazra et al., 7 Jul 2025).

5. Relation to lubrication and slender-jet models

The core-annular WRIBL model should be distinguished from lubrication-type two-phase reductions. A particularly relevant comparison is the long-wave model for an axisymmetric viscous core thread surrounded by a less viscous annular fluid layer inside a cylindrical tube with axial wall corrugation (Wang, 2020). That model derives one-dimensional equations for the core radius $\gamma$7 and a core axial velocity $\gamma$8: $\gamma$9

$(r,z)$0

with wall coupling through

$(r,z)$1

In that formulation, the annular phase is not evolved through its own flow-rate equation; rather, annular stresses are collapsed into a local geometric resistance $(r,z)$2, under the singular scaling $(r,z)$3 and negligible annular inertia (Wang, 2020).

This model is therefore not a classical WRIBL model and not an explicit integral-boundary-layer system. It is more accurately described as a slender-jet / lubrication-type active-interface model with a local interface evolution law, a core axial momentum balance, capillary pressure closure, and wall-mediated annular drag (Wang, 2020). Its relevance to WRIBL lies in structural adjacency: it provides explicit interfacial stress balances, capillary closure, and a wall-coupling mechanism that could serve as a creeping-flow or weak-annulus-viscosity limit for a more general core-annular WRIBL model (Wang, 2020).

The thin-annulus limit in the corrugated-tube study reinforces this distinction. Under

$(r,z)$4

with $(r,z)$5, the reduced model yields a modified Hammond-type thin-film equation (Wang, 2020). This is explicitly a lubrication equation, not a WRIBL system.

A plausible implication is that the WRIBL framework should be viewed as a broader class capable of recovering lubrication or slender-jet closures in special limits while retaining two-phase flow-rate dynamics, systematic $(r,z)$6 inertial and viscous-diffusion effects, and a more explicit representation of cylindrical two-layer momentum exchange.

6. Benchmarking against sharp-interface and diffuse-interface studies

Phase-field simulations of core-annular pipe flow provide a separate but closely related reference frame for assessing reduced models (Song et al., 2019). That work studies upward core-annular flow in a vertical circular pipe using a diffuse-interface CHNS formulation, but it also supplies an analytic sharp-interface laminar solution, linear stability benchmarks, and nonlinear axisymmetric wave dynamics that are directly relevant to WRIBL validation (Song et al., 2019).

For a steady coaxial core-annular base state with interface radius

$(r,z)$7

the centerline velocity of laminar CAF is

$(r,z)$8

and the analytic sharp-interface velocity profile is

$(r,z)$9

Here

$d(z,t)$00

For matched density,

$d(z,t)$01

and the total volume flux is

$d(z,t)$02

These expressions provide exact steady-state closure targets for a reduced model (Song et al., 2019).

The phase-field study identifies three instability classes summarized from Hu and Joseph: at low $d(z,t)$03, capillary instability dominates; for $d(z,t)$04, instability is predominantly interfacial-friction-driven; and for $d(z,t)$05, it becomes predominantly inertial (Song et al., 2019). This taxonomy is valuable for WRIBL interpretation because the weighted residual model explicitly retains the mechanisms most relevant to capillary and viscosity-stratified interfacial dynamics, and at least the long-wave branch of inertial instabilities.

In the nonlinear regime, the CHNS simulations reproduce axisymmetric bamboo waves and provide comparison targets such as hold-up ratio, wave speed, and centerline velocity reduction (Song et al., 2019). For the CAF3 benchmark, the leading eigenvalue is complex and the laminar state undergoes a Hopf bifurcation to a traveling wave of the form

$d(z,t)$06

with computed wave speed $d(z,t)$07 compared with a sharp-interface result $d(z,t)$08 (Song et al., 2019). The paper also reports a diffuse-interface linear growth rate of about $d(z,t)$09 versus a sharp-interface result $d(z,t)$10 for the same case, with nonlinear effects becoming important around $d(z,t)$11 and a bamboo-wave pattern formed around $d(z,t)$12 (Song et al., 2019).

These benchmarks are significant for WRIBL models because they test exactly the kinds of observables the reduced system is intended to capture: phase-flux partition, interfacial-wave growth, traveling-wave speed, and transport degradation in a periodic pipe. At the same time, the phase-field study warns that laminar high-viscosity-contrast CAF has diffuse-interface error

$d(z,t)$13

so sharp-interface analytic solutions remain the preferred reference where available (Song et al., 2019).

7. Derivation workflow, numerical use, and scope

The 2025 derivation makes a methodological contribution by showing that the core-annular WRIBL model of Dietze and Ruyer-Quil can be reconstructed with SymPy through a sequence of symbolic substeps (Hazra et al., 7 Jul 2025). The workflow defines $d(z,t)$14, constants such as $d(z,t)$15, unknown fields $d(z,t)$16, $d(z,t)$17, $d(z,t)$18, and symbolic radial functions $d(z,t)$19; solves the ODEs for $d(z,t)$20 and the weight functions with dsolve and solve; abstracts cumbersome exact coefficients into functions $d(z,t)$21; computes $d(z,t)$22 by symbolic integration; assembles the weighted residual equation term by term; eliminates $d(z,t)$23 using

$d(z,t)$24

extracts the coefficients by symbolic expansion; and exports them for numerical use, including replacement of log with np.log for NumPy/SciPy implementation (Hazra et al., 7 Jul 2025).

For time integration, the model is reformulated using

$d(z,t)$25

where $d(z,t)$26 is the total flow rate, depending only on time, and the pressure equation is integrated over the periodic domain to obtain an ODE

$d(z,t)$27

The resulting numerical system comprises a PDE for $d(z,t)$28, a PDE for $d(z,t)$29, and an ODE for $d(z,t)$30 (Hazra et al., 7 Jul 2025). The discretization reported is:

• second-order central finite differences in $d(z,t)$31,
• a uniform grid with 500 points,
• time integration using SciPy’s solve_ivp with LSODA (Hazra et al., 7 Jul 2025).

The initial condition for Rayleigh–Plateau calculations is

$d(z,t)$32

on a periodic domain of nondimensional length $d(z,t)$33 (Hazra et al., 7 Jul 2025). The paper studies, among other cases, an air-core–mucus-annulus configuration with strong capillary-driven hump growth and eventual liquid-bridge or plug formation, and an air-core–water-annulus gravity-driven case in which the hump saturates into a stable annular collar or unduloid that translates down the tube (Hazra et al., 7 Jul 2025).

The scope and limitations are stated clearly. The model assumes:

• long-wave geometry $d(z,t)$34,
• axisymmetry,
• Newtonian, immiscible fluids,
• retention of terms up to $d(z,t)$35,
• Reynolds numbers small enough that $d(z,t)$36 terms are neglected,
• radially uniform pressure at this order,
• an interface that remains a single-valued function $d(z,t)$37 (Hazra et al., 7 Jul 2025).

It captures well:

• inertia,
• interfacial capillarity including nonlinear curvature,
• longitudinal viscous diffusion,
• two-phase coupling,
• collar or plug formation onset and pre-coalescence dynamics (Hazra et al., 7 Jul 2025).

Its limitations are equally important. Once the interface touches the wall or centerline and topology changes, the single-valued thin-film description breaks down. The basic WRIBL model cannot continue beyond true coalescence or dryout events without additional regularization; for post-plug dynamics, it may be augmented with a disjoining pressure to create pseudo-plugs or precursor films (Hazra et al., 7 Jul 2025). It is also not intended for short-wave, strongly non-slender, or fully three-dimensional non-axisymmetric regimes (Hazra et al., 7 Jul 2025).

Within those bounds, the core-annular WRIBL model occupies a precise role in the hierarchy of two-phase pipe-flow models. It is more general and dynamically richer than lubrication closures that quasi-eliminate one phase (Wang, 2020), yet substantially cheaper than full CHNS or sharp-interface DNS while retaining the essential multiscale physics needed for capillary instability, annular collar dynamics, and coupled phase-momentum transport (Hazra et al., 7 Jul 2025, Song et al., 2019).