For steady, fully developed flow of two immiscible liquid layers in a channel, the governing equations reduce to one momentum equation for each phase, coupled by wall and liquid-liquid interface conditions. For incompressible Newtonian fluids, mass conservation requires a divergence-free velocity field, while momentum conservation is described by the Navier-Stokes equations. In the common parallel-plate approximation, the axial velocity in each phase is quadratic across the channel, and the integration constants are determined by no-slip at the walls plus velocity and shear-stress continuity at the interface.

The velocity profiles are governed by pressure-gradient-driven momentum balances in each phase. The viscosity contrast affects the profile through the phase-specific momentum equations, while the interface conditions enforce physically consistent motion and stress transfer between the liquids.

What Must Be Solved

Continuity in Each Phase

For incompressible phases with constant density, the continuity equation is

[ \nabla \cdot \mathbf{v}^{(i)} = 0, \qquad i=1,2 ]

where (\mathbf{v}^{(i)}) is the velocity field of phase (i).

For a steady, fully developed, unidirectional channel flow,

[ \mathbf{v}^{(i)} = \left(v_x^{(i)}(y),0,0\right) ]

and continuity is automatically satisfied because the axial velocity varies only in the transverse direction.

Momentum Conservation

The general incompressible Newtonian momentum equation for each phase is

[ \rho_i \frac{D\mathbf{v}^{(i)}}{Dt}

\rho_i\mathbf{g} -\nabla P^{(i)} +\mu_i\nabla^2\mathbf{v}^{(i)} ]

where (\rho_i), (\mu_i), and (P^{(i)}) are the density, dynamic viscosity, and pressure of phase (i).

This equation includes transient and convective acceleration, gravity, pressure forces, and viscous momentum diffusion.

Fully Developed Axial Flow

When the flow is steady, fully developed, and driven primarily by an axial pressure gradient, the axial momentum equation becomes

[ 0

-\frac{dP^{(i)}}{dx} + \mu_i\frac{d^2v_x^{(i)}}{dy^2} ]

or, equivalently,

[ \frac{d^2v_x^{(i)}}{dy^2}

\frac{1}{\mu_i}\frac{dP^{(i)}}{dx} ]

for each liquid layer.

The resulting velocity distribution is

[ v_x^{(i)}(y)

\frac{1}{\mu_i}\frac{dP^{(i)}}{dx}\frac{y^2}{2} + a_i y+b_i ]

where (a_i) and (b_i) are integration constants determined by the boundary and interface conditions.

Boundary Conditions at Solid Walls

No-Slip Condition

At a stationary solid wall, the tangential fluid velocity equals the wall velocity. For the two-layer geometry in the reference configuration,

[ v_x^{(1)}(H_1)=0 ]

and

[ v_x^{(2)}(-H_2)=0 ]

where phase 1 occupies the region between the interface and the upper wall, and phase 2 occupies the region between the lower wall and the interface.

These conditions supply two of the four equations needed to determine the integration constants.

No-Penetration Condition

At an impermeable stationary wall, the normal velocity is zero:

[ \mathbf{n}\cdot\mathbf{v}^{(i)}=0 ]

This condition prevents either liquid from crossing the channel wall. In the idealized unidirectional solution, it is already represented by the absence of a wall-normal velocity component.

Moving or Porous Walls

If a wall moves, the tangential condition becomes

[ \mathbf{v}t=\mathbf{V}{w,t} ]

rather than zero. A porous or reactive wall requires a separate normal-flux condition, because fluid or species may cross the boundary.

Conditions at the Liquid-Liquid Interface

Kinematic Interface Condition

At a stationary interface located at (y=0), the normal velocity must be compatible with the interface position. For an impermeable, nonreacting interface,

[ \mathbf{n}\cdot\mathbf{v}^{(1)}

\mathbf{n}\cdot\mathbf{v}^{(2)} ]

For the parallel, fully developed configuration, both normal velocities are zero.

If the interface is moving or deforming, its velocity must instead satisfy the corresponding kinematic condition. The interface location then becomes part of the unknown solution.

Tangential Velocity Continuity

For two ordinary, immiscible Newtonian liquids without interfacial slip,

[ v_x^{(1)}(0)=v_x^{(2)}(0) ]

This condition ensures that the liquid layers have the same tangential velocity at their mutual boundary.

It supplies the third relation required to determine the integration constants.

Tangential Stress Continuity

When the interface has no tangential stress generated by surface-tension gradients, the viscous shear stresses are continuous:

[ \mu_1\frac{dv_x^{(1)}}{dy}(0)

\mu_2\frac{dv_x^{(2)}}{dy}(0) ]

This is essential when the viscosities differ. The velocity gradient generally changes across the interface so that the shear stress remains balanced.

It supplies the fourth relation for the four integration constants in the two quadratic velocity profiles.

Normal Stress and Surface Tension

A complete two-phase model also requires a normal-stress condition at the interface. In general form,

[ \mathbf{n}\cdot \left(\boldsymbol{\tau}^{(1)}-\boldsymbol{\tau}^{(2)}\right) \cdot\mathbf{n}

\gamma\kappa ]

with appropriate sign convention, where (\boldsymbol{\tau}) is the total stress tensor, (\gamma) is interfacial tension, and (\kappa) is interface curvature.

For a flat, stationary interface with constant interfacial tension, curvature is zero and the normal-stress condition reduces to equality of the normal stresses. This condition becomes especially important when the interface deforms, the channel is not planar, or capillary forces control the phase distribution.

Marangoni Effects

If interfacial tension varies with temperature or composition, tangential stress is not simply continuous. Its jump is balanced by the surface-tension gradient:

[ \mathbf{t}\cdot \left(\boldsymbol{\tau}^{(1)}-\boldsymbol{\tau}^{(2)}\right) \cdot\mathbf{n}

\nabla_s\gamma\cdot\mathbf{t} ]

where (\nabla_s) denotes the surface gradient.

For isothermal liquids with uniform composition at the interface, this term is normally neglected.

Conditions at Channel Inlets and Outlets

Specified Inlet Flow

Pilot-plant models commonly specify inlet velocity profiles, phase flow rates, or total volumetric flow rates. A fully developed inlet may be prescribed as

[ v_x^{(i)}(y)=v_{\text{in}}^{(i)}(y) ]

or through the phase flow rate,

[ Q_i=\int_{A_i}v_x^{(i)},dA ]

The inlet specification must be consistent with the phase distribution and interface position.

Specified Pressure Gradient

For a long, fully developed channel, the pressure gradient may be the controlling input:

[ \frac{dP^{(i)}}{dx} ]

In many co-current channel configurations, the axial pressure gradient is common to both phases, although the resulting velocity profiles differ because the viscosities and layer thicknesses differ.

Outlet Condition

At an outlet, the model may specify pressure, a developed-flow condition, or an appropriate convective outflow condition. The choice depends on whether the pilot plant is being modeled as pressure-driven, flow-controlled, or connected to downstream equipment.

The outlet condition must avoid imposing an artificial constraint on a developing interface or velocity field.

Why Viscosity Contrast Matters

Different Curvature of the Profiles

The curvature of each velocity profile is proportional to

[ \frac{1}{\mu_i}\frac{dP^{(i)}}{dx} ]

At the same pressure gradient, the less viscous phase develops a larger velocity curvature than the more viscous phase.

Stress Balance Controls the Gradient Change

Although the tangential velocity is continuous at the interface, the velocity gradients need not be. The stress condition requires

[ \mu_1\left.\frac{dv_x^{(1)}}{dy}\right|_{0}

\mu_2\left.\frac{dv_x^{(2)}}{dy}\right|_{0} ]

Thus, a viscosity ratio changes how momentum is distributed between the layers.

Consequences for Unit Operations

The resulting velocity field affects residence-time distribution, interfacial renewal, pressure drop, phase holdup, mass-transfer area, and separation performance.

These effects matter in liquid-liquid extraction modules, stratified-flow reactors, membrane-contacting devices, and other channel-type pilot equipment.

Understanding the Trade-offs

The Fully Developed Assumption

The quadratic solution is valid only after entrance effects have become negligible. Near inlets, distributors, bends, contractions, and phase-splitting devices, axial development and secondary motion may require the full multidimensional equations.

The Flat-Interface Assumption

A fixed interface at (y=0) excludes interface deformation and capillary evolution. This can be inadequate when flow rates are high, channel dimensions are small, interfacial tension is low, or phase properties produce significant pressure differences.

The Laminar-Flow Assumption

The simplified ordinary differential equation describes viscous, fully developed flow and is most directly applicable in laminar conditions. If the channel enters a transitional or turbulent regime, additional closure relations and turbulence modeling may be required.

The Reynolds number should be based on an appropriate hydraulic dimension, such as the hydraulic mean diameter

[ d_m=\frac{4A}{p} ]

for the relevant channel geometry.

The No-Slip Interface Assumption

Velocity continuity assumes negligible interfacial slip. In some specialized systems, surfactants, contamination, interfacial rheology, or microscale effects can alter the interface condition and require an interfacial constitutive model.

The Constant-Property Assumption

The reference equations assume constant viscosity and density. Strong temperature or composition gradients can make (\mu_i) and (\rho_i) variable, coupling momentum with energy and species balances.

Making the Right Choice for Your Goal

Use the following model-selection principles when representing a two-phase channel pilot plant:

If your primary focus is fully developed pressure-driven flow: Solve the phase-specific reduced momentum equations with no-slip at the walls, tangential velocity continuity, and tangential stress continuity at the interface.
If your primary focus is interface position or phase holdup: Add the kinematic interface condition and the normal-stress balance, including surface tension and curvature.
If your primary focus is entrance, distributor, or turning-region behavior: Retain the full transient or steady multidimensional Navier-Stokes equations and impose physically consistent inlet, outlet, wall, and interface conditions.
If your primary focus is mass transfer or reaction: Couple the flow equations to species conservation and impose concentration or flux conditions at inlets, walls, and the liquid-liquid interface.
If your primary focus is scale-up and pressure-drop prediction: Include the appropriate hydraulic diameter, Reynolds-number regime, wall roughness, phase flow rates, and property variations.

A reliable two-phase pilot-plant model is defined not only by its differential equations, but by boundary and interface conditions that accurately represent how the real equipment operates.

Summary Table:

Model element	Governing requirement	Purpose
Continuity	`∇ · v⁽ⁱ⁾ = 0`	Ensures incompressible mass conservation in each phase
Momentum	`ρᵢ Dv⁽ⁱ⁾/Dt = ρᵢg − ∇P⁽ⁱ⁾ + μᵢ∇²v⁽ⁱ⁾`	Describes pressure, gravity, inertia, and viscous effects
Fully developed flow	`0 = −dP⁽ⁱ⁾/dx + μᵢ d²vₓ⁽ⁱ⁾/dy²`	Produces quadratic velocity profiles in each layer
Solid walls	No-slip and no-penetration conditions	Defines fluid motion at stationary, moving, or porous walls
Liquid-liquid interface	Normal-velocity compatibility and tangential velocity continuity	Maintains a physically consistent phase boundary
Shear stress	`μ₁ dvₓ⁽¹⁾/dy = μ₂ dvₓ⁽²⁾/dy`	Balances momentum transfer across the interface
Normal stress	Stress jump balanced by `γκ`	Accounts for curvature and surface tension when the interface deforms
Marangoni effects	Tangential stress jump balanced by `∇ₛγ`	Represents surface-tension gradients caused by temperature or composition
Inlet and outlet	Prescribed flow rate, velocity, pressure, or developed-flow condition	Matches the model to actual pilot-plant operation
Extended models	Energy, species, turbulence, and variable-property equations	Supports mass transfer, reaction, scale-up, and non-ideal operating conditions

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