--- title: Core-Annular WRIBL Model url: https://www.emergentmind.com/topics/core-annular-wribl-model type: topic --- # Core-Annular WRIBL Model 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 \(Q_a(z,t)\) [2507.05103]. 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 [2507.05103]. 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 [2007.03145], while phase-field simulations provide high-fidelity benchmarks for laminar, linear, and nonlinear regimes [1902.07351]. ## 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 \[ r=d(z,t), \] so that \(d\) is the nondimensional interface position and the annular film thickness is \(1-d\) [2507.05103]. The long-wave parameter is \[ \epsilon=\frac{R}{\Lambda}\ll 1, \] where \(\Lambda\) is the axial length scale [2507.05103]. The nondimensionalization uses \[ r^* = R,\qquad z^* = \Lambda,\qquad t^* = \frac{R}{U},\qquad d^* = R, \] \[ u_i^* = U,\qquad v_i^* = \epsilon U,\qquad p_i^* = \frac{\mu_i U}{\epsilon R}, \] with \(U\) a characteristic axial velocity [2507.05103]. The principal dimensionless groups are \[ Re_i = \frac{R U \rho_i}{\mu_i},\qquad \Pi_\mu=\frac{\mu_c}{\mu_a},\qquad Ca=\frac{\gamma}{\mu_a U}, \] and \[ \mathcal{B}_i = \frac{b_i R^2}{\mu_i U}, \] with \(b_i\) the dimensional axial body force per unit volume [2507.05103]. 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** \[ Q_c(z,t),\qquad Q_a(z,t), \] so that the one-dimensional dynamical system represents both core and annulus as momentum-carrying layers [2507.05103]. 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 [2507.05103]. ## 2. Long-wave equations and interfacial conditions Retaining terms up to \(O(\epsilon^2)\), the axisymmetric continuity equation in each phase is \[ \frac{1}{r}\partial_r (r v_i) + \partial_z u_i = 0. \] The axial momentum equation is \[ \epsilon Re_i\left(\partial_t u_i + v_i\partial_r u_i+ u_i\partial_z u_i\right) = -\partial_z \left(p_i |_{d}\right) -\epsilon^2 \partial_z \left(\partial_z u_i |_{d}\right) +\frac{1}{r}\partial_r \left(r\partial_r u_i\right) +2\epsilon^2 \partial_{zz} u_i+\mathcal{B}_i . \] These forms arise after using the \(O(\epsilon^2)\) radial momentum balance to eliminate \(\partial_z p_i\) in favor of interfacial values [2507.05103]. At the interface \(r=d(z,t)\), the velocity continuity conditions are \[ u_a=u_c,\qquad v_a=v_c. \] Tangential stress continuity, to \(O(\epsilon^2)\), is \[ \partial_r u_a- \Pi_{\mu} \partial_r u_c = \left[2\epsilon^2 \partial_z d\left(\partial_z u_a- \partial_r v_a\right)-\epsilon^2 \partial_z v_a\right] -\Pi_{\mu}\left[2\epsilon^2 \partial_z d\left(\partial_z u_c-\partial_r v_c\right)-\epsilon^2 \partial_z v_c\right]. \] The normal stress balance is \[ p_a - \Pi_{\mu} p_c = - Ca\, \kappa + 2\epsilon^2\left(\partial_r v_a- \partial_r u_a \partial_z d \right) -2\epsilon^2 \Pi_{\mu}\left(\partial _r v_c- \partial_r u_c \partial_z d \right), \] with curvature \[ \kappa = \frac{1}{d}-\frac{\epsilon^2(\partial_z d)^2}{2d}-\epsilon^2 \partial_{zz} d. \] The kinematic condition is \[ \partial_t d = v_a - u_a\partial_z d. \] At the centerline \(r=0\), \[ v_c=0,\qquad \partial_r u_c=0, \] and at the wall \(r=1\), \[ v_a=0,\qquad u_a=0. \] These equations remain two-dimensional and free-boundary; WRIBL reduces them to one-dimensional form [2507.05103]. The derivation assumes an \(O(\epsilon^2)\) model and additionally states that terms of order \(O(Re_i\epsilon^2)\) are neglected. It also assumes \(u_i'\), \(v_i'\) are at most \(O(\epsilon)\) [2507.05103]. 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: \[ u_i(r,z,t)=\hat u_i(r;d,Q_i)+u_i'(r,z,t),\qquad v_i(r,z,t)=\hat v_i(r;d,Q_i)+v_i'(r,z,t). \] Here \(\hat u_i\) is the quasi-steady, quasi-developed profile parametrized by the local interface position \(d\) and local phase flow rate \(Q_i\), and \(u_i'\) is a correction [2507.05103]. The leading profile is chosen so that it exactly reproduces the local phase flow rates: \[ \int_0^d \hat u_c\, r\,dr = Q_c,\qquad \int_d^1 \hat u_a\, r\,dr = Q_a. \] At leading order, the axial profiles satisfy \[ \frac{1}{r}\partial_r\left(r\partial_r \hat u_c\right)=A_c,\qquad \frac{1}{r}\partial_r\left(r\partial_r \hat u_a\right)=A_a, \] with \(A_c,A_a\) independent of \(r\), subject to centerline, wall, and interfacial matching conditions [2507.05103]. The general solutions are \[ \hat u_c = \frac{A_c r^2}{4}+C_{11}+C_{12}\ln r, \qquad \hat u_a = \frac{A_a r^2}{4}+C_{21}+C_{22}\ln r, \] and regularity at the centerline removes the logarithm from the core phase [2507.05103]. The annular profile can be written exactly, although the explicit form is lengthy. For symbolic work it is rewritten as \[ \hat{u}_a = r^2\big(A_1(d)Q_c+A_2(d)Q_a\big) +\log(r)\big(A_3(d)Q_c+A_4(d)Q_a\big) +\big(A_5(d)Q_c+A_6(d)Q_a\big), \] \[ \hat{u}_c = r^2\big(B_1(d)Q_c+B_2(d)Q_a\big) +\big(B_5(d)Q_c+B_6(d)Q_a\big), \] where the coefficient functions are exact shorthand for rational-logarithmic functions of \(d\) [2507.05103]. From continuity, the radial velocities are \[ \hat v_c = -\frac{1}{r}\int_0^r \partial_z \hat u_c\, r\,dr, \qquad \hat v_a = \frac{1}{r}\int_r^1 \partial_z \hat u_a\, r\,dr. \] Enforcing continuity of \(\hat v_i\) at the interface yields \[ \partial_z Q_a + \partial_z Q_c = 0, \] which is the exact one-dimensional phase-flow constraint [2507.05103]. The weighted residual step then averages the momentum residual with phase-dependent weight functions \(w_i(r)\) using \[ \langle p|q\rangle = \Pi_\mu \int_0^d p_c q_c\, r\,dr + \int_d^1 p_a q_a\, r\,dr. \] The weights are chosen so that integration by parts closes the dominant transverse viscous term exactly and eliminates the unknown corrections \(u_i'\) [2507.05103]. They satisfy \[ \frac{1}{r}\partial_r(r\partial_r w_c)=C_c,\qquad \frac{1}{r}\partial_r(r\partial_r w_a)=-1, \] with \[ w_a=0\quad \text{at } r=1,\qquad \partial_r w_c=0\quad \text{at } r=0,\qquad \partial_r w_a=\Pi_\mu \partial_r w_c\quad \text{at } r=d. \] For the flow-rate equation, the integral constraint is \[ \int_0^d w_c\, r\,dr = -\int_d^1 w_a\, r\,dr. \] 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 [2507.05103]. 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: \[ d\,\partial_t d = \partial_z Q_a, \] or equivalently, \[ d\,\partial_t d = -\partial_z Q_c, \] using \(\partial_z Q_a+\partial_z Q_c=0\) [2507.05103]. The principal second-order WRIBL averaged-momentum equation is \[ \Pi_{\mu i}Re_i\biggl( S_{ij}\partial_t Q_j + F_{ijk} Q_j\,\partial_z Q_k + G_{ijk}Q_jQ_k\,\partial_z d \biggr) = \Pi_{\mu}C_cQ_c-Q_a -Ca\,(\partial_z\kappa)\,I +\bigl[\Pi_{\mu}\mathcal{B}_c-\mathcal{B}_a\bigr]I \] \[ \qquad +J_jQ_j(\partial_z d)^2 +K_j(\partial_z Q_j)(\partial_z d) +L_jQ_j\,\partial_z^2 d +M_j\partial_z^2Q_j. \] Repeated indices are summed, and \[ \Pi_{\mu a}=1,\qquad \Pi_{\mu c}=\Pi_\mu. \] The coefficient sets \[ S_{ij},\ F_{ijk},\ G_{ijk},\ I,\ J_j,\ K_j,\ L_j,\ M_j \] are functions of \(d\) only [2507.05103]. The curvature used in the capillary term is \[ \kappa = \frac{1}{d}-\frac{\epsilon^2(\partial_z d)^2}{2d}-\epsilon^2 \partial_{zz} d, \] so that \[ \partial_z\kappa = -\frac{\partial_z d}{d^2} -\epsilon^2\partial_z\!\left(\frac{(\partial_z d)^2}{2d}\right) -\epsilon^2 \partial_{zzz}d. \] Accordingly, the capillary forcing term \(-Ca(\partial_z\kappa)I\) contains both the classical cylindrical Rayleigh–Plateau contribution and the \(O(\epsilon^2)\) nonlinear-curvature and axial-curvature corrections [2507.05103]. A second weight choice yields a diagnostic pressure equation: \[ \Pi_{\mu i}Re_i\biggl( \tilde S_{ij}\partial_t Q_j +\tilde F_{ijk}Q_j\partial_z Q_k +\tilde G_{ijk}Q_jQ_k\partial_z d \biggr) = \Pi_{\mu}\tilde C_cQ_c-Q_a -2\Pi_{\mu}\,\partial_z p_c|_d\,\tilde I +Ca(\partial_z\kappa)\tilde I +\bigl[\Pi_{\mu}\mathcal{B}_c+\mathcal{B}_a\bigr]\tilde I \] \[ \qquad +\tilde J_jQ_j(\partial_z d)^2 +\tilde K_j(\partial_z Q_j)\partial_z d +\tilde L_jQ_j\partial_z^2 d +\tilde M_j\partial_z^2Q_j. \] The paper notes that the full phase pressures are radially uniform at this order because \(\partial_r p_i=0\) follows from the \(O(\epsilon^2)\) radial momentum equation [2507.05103]. The physical interpretation of the principal terms is explicit. The left-hand side \[ \Pi_{\mu i}Re_i\left(S_{ij}\partial_tQ_j + F_{ijk}Q_j\partial_zQ_k + G_{ijk}Q_jQ_k\partial_z d\right) \] represents weighted inertial response, convective nonlinearities, and nonparallel geometric corrections. On the right-hand side, \(\Pi_\mu C_cQ_c-Q_a\) is the leading transverse-viscous closure; \(-Ca(\partial_z\kappa)I\) is capillary forcing; \([\Pi_\mu \mathcal B_c-\mathcal B_a]I\) is differential forcing; and the \(J_j,K_j,L_j,M_j\) terms encode \(O(\epsilon^2)\) geometric, interfacial-stress, and longitudinal-diffusion corrections [2507.05103]. ## 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** [2007.03145]. That model derives one-dimensional equations for the core radius \(S(z,t)\) and a core axial velocity \(w(z,t)\): \[ S_t+\frac{1}{2}Sw_z+wS_z=0, \] \[ \bar R_e(w_t+ww_z)=3\frac{(S^2w_z)_z}{S^2}-\kappa_z-2\lambda_0\frac{G(S,d)}{S^2}w, \qquad \kappa=\frac1S-\epsilon^2S_{zz}, \] with wall coupling through \[ G(S,d)=\frac{S^2+d^2}{S^2-d^2-(S^2+d^2)\ln(S/d)}. \] 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 \(G(S,d)\), under the singular scaling \(\lambda=\epsilon^2\lambda_0\) and negligible annular inertia [2007.03145]. 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 [2007.03145]. 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 [2007.03145]. The thin-annulus limit in the corrugated-tube study reinforces this distinction. Under \[ d=1+\delta(1+\sigma'\phi),\qquad S=1+\delta-\delta h, \] with \(\delta\ll1\), the reduced model yields a modified Hammond-type thin-film equation [2007.03145]. 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 \(O(\epsilon^2)\) 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 [1902.07351]. 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 [1902.07351]. For a steady coaxial core-annular base state with interface radius \[ \eta=\frac{R_I}{R}, \] the centerline velocity of laminar CAF is \[ U = (f-\rho_o g)\frac{R_I^2}{4\mu_o} + (f-\rho_w g)\frac{R^2-R_I^2}{4\mu_w} - (\rho_o-\rho_w)g \frac{R_I^2}{2\mu_w}\log\frac{R}{R_I}, \] and the analytic sharp-interface velocity profile is \[ u_z(r)= \begin{cases} \left(\dfrac{1}{\eta^2}-\dfrac{r^2}{\eta^2}-2(K_f-1)\log r\right)\dfrac{1}{A}, & 0