Multiphase Flows and Bubble Dynamics — Interfacial Transport and Continuum Formulation

Scope: from first principles of continuum mechanics and thermodynamics to engineering models for multiphase systems. Includes two-fluid modeling, interfacial transfer laws, nucleation thermodynamics, and bubble dynamics.

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1. Nature and Classification of Multiphase Flow

A multiphase flow involves more than one thermodynamic phase (gas, liquid, or solid) coexisting with distinct interfaces. Classification:

Type	Examples	Description
Gas–Liquid	Bubbly, slug, annular	Cavitation, boiling, reactors
Liquid–Solid	Slurries, sedimentation	Suspensions, crystallization
Gas–Solid	Fluidized beds	Pneumatic conveying
Three-phase	Oil–gas–water	Reactors, extraction

Topologically, systems may be dispersed (one phase as particles, droplets, or bubbles in a continuous medium) or separated (distinct regions divided by interfaces).

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2. Continuum Formulation — Volume Averaging

Each phase $k$ occupies a local volume fraction $α_k$, with $\sum_k α_k = 1.$

For any extensive quantity $ψ_k$: $⟨ψ⟩ = \frac{1}{V} ∫_{V_k} ψ_k dV.$

2.1 Averaged Continuity

$\frac{∂(α_k ρ_k)}{∂t} + ∇·(α_k ρ_k \mathbf{v}_k) = Γ_k,$ where $Γ_k$ is the mass transfer rate between phases (e.g., evaporation/condensation).

2.2 Averaged Momentum

$∂(α_k ρ_k \mathbf{v}_k)/∂t + ∇·(α_k ρ_k \mathbf{v}_k\mathbf{v}_k) = -α_k ∇p_k + ∇·(α_k \boldsymbol{τ}_k) + α_k ρ_k \mathbf{g} + \mathbf{M}_k,$ where $\mathbf{M}_k$ is the interfacial momentum exchange (drag, lift, virtual mass, etc.).

2.3 Averaged Energy

$∂(α_k ρ_k h_k)/∂t + ∇·(α_k ρ_k h_k \mathbf{v}_k) = α_k \frac{Dp_k}{Dt} + ∇·(α_k k_k ∇T_k) + Q_k,$ with $Q_k$ representing interfacial and volumetric heat transfer.

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3. Interfacial Area and Exchange Mechanisms

Define interfacial area concentration $a_i = A_i/V$ (m²/m³).

Interphase exchange terms: $\mathbf{M}_k = a_i (p_i \mathbf{n}_i + \boldsymbol{τ}_i), \quad Q_k = a_i q_i, \quad Γ_k = a_i j_i.$

3.1 Momentum Transfer

Dominated by drag force: $\mathbf{F}_D = \frac{1}{2} C_D ρ_c A_p |\mathbf{u}_r|\mathbf{u}_r,$ where $\mathbf{u}_r = \mathbf{v}_d - \mathbf{v}_c$.

Empirical drag coefficient (for bubbles): $C_D = \frac{24}{Re_p}(1 + 0.15Re_p^{0.687}), \quad Re_p = \frac{ρ_c d_p |\mathbf{u}_r|}{μ_c}.$

Other forces: lift, virtual mass, wall lubrication.

3.2 Heat and Mass Transfer

Interfacial heat flux: $q_i = h_i (T_s - T_∞).$ Mass flux due to phase change: $j_i = β_i (ρ_s - ρ_∞).$ Correlations depend on regime (nucleate, film, condensation, evaporation).

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4. Surface Tension and Interfacial Pressure

Surface tension arises from molecular imbalance across the interface.

4.1 Laplace Pressure Relation

$Δp = p_{in} - p_{out} = 2σ/R,$ where $σ$ is surface tension and $R$ is bubble radius.

4.2 Capillary Number

$Ca = \frac{μU}{σ}.$ Low $Ca$: surface tension dominates (small-scale flows); high $Ca$: inertia/viscosity dominates.

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5. Thermodynamics of Nucleation

Formation of a new phase nucleus requires work against surface tension and volume free energy.

Gibbs free energy change for a spherical nucleus: $ΔG = 4πr^2σ - \frac{4}{3}πr^3Δp,$ with $Δp = p_v - p_l$.

Critical radius: $r^* = \frac{2σ}{Δp}.$ Activation barrier: $ΔG^* = \frac{16πσ^3}{3(Δp)^2}.$

Nucleation rate: $J = J_0 \exp(-ΔG^*/kT).$

5.1 Heterogeneous Nucleation

Occurs on walls or impurities, reducing effective barrier by a factor $f(θ)$ dependent on contact angle $θ$.

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6. Bubble Dynamics — Rayleigh–Plesset Equation

A single spherical bubble in an infinite liquid obeys: $R \ddot{R} + \frac{3}{2} \dot{R}^2 = \frac{1}{ρ_l} [p_B - p_∞ - 2σ/R - 4μ_l \dot{R}/R].$

Where:

• $R$: instantaneous bubble radius,
• $p_B$: internal pressure (vapor + gas),
• $p_∞$: far-field pressure.

6.1 Static Equilibrium

$p_B = p_∞ + 2σ/R_0.$

6.2 Oscillatory Dynamics

Linearizing about equilibrium gives natural frequency (Minnaert, 1933): $f_n = \frac{1}{2πR_0} \sqrt{\frac{3γp_∞}{ρ_l}}.$

6.3 Collapse Dynamics

If external pressure suddenly rises ($p_∞>p_B$): $R\ddot{R} + \frac{3}{2}\dot{R}^2 = \frac{1}{ρ_l}(p_B - p_∞).$ Solution predicts violent collapse, relevant to cavitation erosion.

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7. Cavitation and Condensation Phenomena

7.1 Cavitation Inception

Occurs when local static pressure drops below vapor pressure.

Cavitation number: $σ_c = \frac{p_∞ - p_v}{0.5ρU^2}.$ Smaller $σ_c$ → higher likelihood of cavitation.

7.2 Cavitation Damage

Bubble collapse near surfaces generates microjets and shock waves → material erosion, noise, vibration.

7.3 Condensation Shock

In high-speed vapor flows, phase change occurs abruptly — requiring coupled mass, momentum, and energy analysis.

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8. Flow Regimes in Gas–Liquid Systems

Regime	Void Fraction Range	Description
Bubbly	<0.25	Discrete spherical bubbles
Slug	0.25–0.6	Large bullet-shaped bubbles
Churn	0.6–0.8	Unstable coalescence/breakup
Annular	>0.8	Continuous gas core, liquid film

Transition determined by coalescence, breakup, and surface tension forces.

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9. Population Balance and Interfacial Area Transport

Population balance for number density $n(v,t)$ of dispersed elements (volume v): $\frac{∂n}{∂t} + ∇·(n\mathbf{v}) = B - D,$ where $B$ and $D$ are birth and death terms due to coalescence and breakup.

Mean interfacial area evolution: $\frac{Da_i}{Dt} = S_{break} - S_{coal} + S_{nuc}.$

Statistical closure achieved via empirical or mechanistic kernel functions (Smoluchowski framework).

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10. Heat and Mass Transfer in Bubbly Flows

Heat transfer coefficient around single bubble: $Nu = 2 + 0.6 Re^{1/2} Pr^{1/3}.$ Mass transfer (Sherwood number): $Sh = 2 + 0.6 Re^{1/2} Sc^{1/3}.$

Ensemble-averaged volumetric rate: $Q = a_i h_i (T_l - T_v).$

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11. Noncondensable Gases and Stability Effects

Presence of a noncondensable gas alters bubble oscillations and collapse severity (increased compressibility, reduced collapse pressure). Gas diffusion across interface may control long-term bubble stability.

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12. Entropy Production and Irreversibility

Local entropy generation in multiphase flow: $σ_s = \sum_k \left[ \frac{\boldsymbol{τ}_k : ∇\mathbf{v}_k}{T_k} + \frac{k_k (∇T_k)^2}{T_k^2} + \frac{J_{mass}·∇μ}{T_k} \right].$

Cavitation, boiling, and condensation all generate entropy through irreversible phase change and viscous dissipation.

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13. Typical Engineering Correlations

Phenomenon	Correlation	Validity
Drag coefficient	$C_D = 24/Re_p(1+0.15Re_p^{0.687})$	Re<1000
Void fraction (homogeneous model)	$α = Q_g/(Q_g+Q_l)$	Low slip ratio
Slip ratio (drift flux)	$V_g - V_l = C_0 J_m + V_{gj}$	Bubbly, slug flow
Cavitation inception	$σ_c = (p_∞ - p_v)/(0.5ρU^2)$	Hydraulic systems

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14. Cross-Links

• 05_Turbulent_Combustion_and_Reactive_Flows.md — multiphase combustion, sprays, vaporization.
• Thermodynamics/09_Phase_Transitions_and_Critical_Phenomena.md — nucleation and criticality.
• Heat_Transfer/Boiling_and_Condensation.md — interfacial heat transfer and phase-change kinetics.
• Thermodynamics/10_NonEquilibrium_Thermodynamics.md — entropy production across interfaces.