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Transport Phenomena Formula Derivation Compendium (Navier-Stokes to Fick’s Laws)

The Transport Phenomena Formula Derivation Compendium is a pedagogical and reference resource that systematically derives and interconnects the foundational conservation-based differential equations governing momentum, energy, and mass transport—specifically the Navier-Stokes equations, Fourier’s law of heat conduction, and Fick’s laws of diffusion—from first principles (e.g., continuum mechanics, conservation of mass/momentum/energy, and constitutive relations). It emphasizes mathematical rigor, physical interpretation, and dimensional consistency while bridging microscopic kinetic theory to macroscopic field equations.

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📖 Overview

This compendium serves as a unifying framework for graduate-level transport phenomena education and engineering analysis. It begins with the general balance equation—a cornerstone formalism expressing accumulation = inflow − outflow + generation—and applies it successively to mass, momentum, and energy to yield the continuity, Cauchy momentum, and energy equations. The derivation then introduces constitutive assumptions: Newtonian stress-strain rate linearization yields the Navier-Stokes equations; Fourier’s linear approximation for heat flux gives the conductive energy flux term; and Fick’s first law emerges from linearizing the gradient-driven diffusive mass flux under dilute or isothermal conditions. Crucially, the compendium highlights analogies across transport domains—e.g., the shared mathematical structure of diffusion-type equations (parabolic PDEs) and the role of transport coefficients (viscosity μ, thermal conductivity k, diffusivity D) as proportionality constants linking fluxes to driving gradients. It also addresses limitations and extensions: non-Newtonian rheology, turbulent closure models, multicomponent diffusion (Maxwell–Stefan formulation), and coupling effects such as thermodiffusion (Soret effect) or diffusioosmosis.

📑 Key Components

1 Conservation Balance Equations

2 Constitutive Relations (Newton, Fourier, Fick)

3 Dimensional Analysis and Scaling

🎯 Applications

✓ Computational Fluid Dynamics (CFD) model formulation
✓ Design of chemical reactors and separation processes
✓ Biomedical transport modeling (e.g., drug delivery, oxygen diffusion in tissues)

📐 Key Formulas

Continuity Equation (Mass Conservation)

∇·v + ∂ρ/∂t = 0

Expresses local mass conservation for a fluid continuum; v is velocity vector, ρ is density

Navier-Stokes Equation (Momentum Conservation)

ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v + ρg

Describes motion of Newtonian fluid under pressure, viscous, and body forces

Fourier’s Law (Heat Conduction)

q = −k∇T

Relates conductive heat flux q to temperature gradient ∇T via thermal conductivity k

Fick’s First Law (Diffusive Mass Flux)

J_A = −D_AB ∇c_A

Gives diffusive molar flux of species A proportional to its concentration gradient

Generalized Transport Equation (Balance Form)

∂(ρφ)/∂t + ∇·(ρvφ) = ∇·(Γ_φ ∇φ) + S_φ

Unified form for scalar transport (φ = velocity component, temperature, concentration); Γ_φ is transport coefficient, S_φ is source term

🔗 Related Concepts

Onsager Reciprocal Relations Buckingham Pi Theorem Stokes-Einstein Relation

📚 References

Analysis of Transport Phenomena (2nd Edition)

MIT OpenCourseWare: Transport Processes

#fluid-dynamics #mass-transfer #heat-transfer