Transport Equations — Convection, Diffusion, and Scalar Transport

Scope: unified derivation and analysis of scalar transport in fluids — including mass, momentum, energy, and species conservation. Covers convection–diffusion equations, dimensionless analysis, analytical solutions, and physical interpretation across regimes.

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1. General Conservation Principle

For any conserved scalar quantity $\phi$ (mass fraction, temperature, momentum component, etc.), the local conservation law over a control volume $V$ is: $\frac{d}{dt}\int_V \rho\phi\,dV + \oint_S (\rho\phi\mathbf{v} - \Gamma\nabla\phi)·\mathbf{n}\,dA = \int_V S_φ\,dV,$ where:

• $ρ$: density,
• $Γ$: diffusion coefficient ($μ, k$, or $D$ depending on context),
• $S_φ$: volumetric source term.

Applying the divergence theorem and assuming constant $ρ$: $ρ\frac{Dφ}{Dt} = ∇·(Γ∇φ) + S_φ.$ This is the general convection–diffusion (transport) equation.

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2. Special Cases of $φ$

Quantity	$φ$	Diffusion coefficient $Γ$	Source term $S_φ$
Momentum	$u_i$	$μ$	Pressure gradient + body forces
Thermal energy	T	k	Viscous dissipation or heat generation
Species concentration	c_i	$ρD_i$	Reaction rate $r_i$

Thus, heat, mass, and momentum transport share a common mathematical structure.

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3. One-Dimensional Steady Convection–Diffusion Equation

For steady 1D transport in the x-direction: $ρu\frac{dφ}{dx} = \frac{d}{dx}\left(Γ\frac{dφ}{dx}\right) + S_φ.$

3.1 Constant Properties and No Source

$ρu\frac{dφ}{dx} = Γ\frac{d^2φ}{dx^2}.$

Introduce Peclet number: $Pe = \frac{ρuL}{Γ}.$

Dimensionless form: $\frac{d^2φ^*}{dx^{*2}} - Pe\frac{dφ^*}{dx^*} = 0.$

Solution: $φ^*(x^*) = \frac{e^{Pe x^*} - 1}{e^{Pe} - 1}.$

Limit	Behavior
$Pe → 0$	Diffusion-dominated: linear profile
$Pe → ∞$	Convection-dominated: sharp boundary layer

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4. Diffusion Equation (Transient, No Convection)

$\frac{∂φ}{∂t} = D ∇^2 φ.$

4.1 1D Semi-Infinite Medium

Boundary conditions: $φ(0,t)=φ_s, \; φ(∞,t)=φ_∞$.

Solution: $φ(x,t) = φ_∞ + (φ_s - φ_∞)\,\mathrm{erfc}\left(\frac{x}{2√{Dt}}\right).$

Characteristic diffusion length: $δ_d(t) = √{Dt}.$

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5. Convective Heat and Mass Transfer in Boundary Layers

Energy equation in steady 2D flow: $ρc_p(u∂T/∂x + v∂T/∂y) = k ∂^2T/∂y^2.$

Define local Nusselt number: $Nu_x = \frac{h_x x}{k} = 0.332 Re_x^{1/2} Pr^{1/3} \quad (\text{laminar flat plate}).$

For mass transfer, Sherwood number: $Sh_x = \frac{k_m x}{D} = 0.332 Re_x^{1/2} Sc^{1/3}.$

These arise directly from boundary-layer solutions of the convection–diffusion equation.

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6. Dimensionless Form of the Transport Equation

Define characteristic scales: $U, L, Δφ$. Dimensionless variables: $φ^* = \frac{φ - φ_0}{Δφ}, \quad x_i^* = x_i/L, \quad t^* = tU/L.$

Then: $\frac{∂φ^*}{∂t^*} + \tilde{v}_i \frac{∂φ^*}{∂x_i^*} = \frac{1}{Pe} ∇^{*2}φ^* + \tilde{S}_φ.$

The single parameter $Pe = UL/D$ determines the dominant transport mechanism.

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7. Analogy Between Transport Processes

The governing equations for heat, mass, and momentum are identical in form:

Property	Flux	Constitutive Law	Diffusivity
Momentum	$τ = μ du/dy$	Newton’s law	$ν = μ/ρ$
Heat	$q = -k dT/dy$	Fourier’s law	$α = k/(ρc_p)$
Mass	$j = -ρD dy/dy$	Fick’s law	$D$

Dimensionless ratios:

Symbol	Definition	Interpretation
Pr	$ν/α$	Momentum vs. thermal diffusion
Sc	$ν/D$	Momentum vs. mass diffusion
Le	$α/D$	Thermal vs. mass diffusion
Pe	$Re·Sc$ or $Re·Pr$	Convective vs. diffusive transport

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8. Analytical Solutions to Canonical Problems

8.1 Steady Diffusion in a Slab (No Convection)

$\frac{d^2φ}{dx^2} = 0 \Rightarrow φ = A + Bx.$

8.2 Cylindrical Coordinates (Radial Diffusion)

$\frac{1}{r} \frac{d}{dr}\left(r\frac{dφ}{dr}\right) = 0 \Rightarrow φ = A\ln r + B.$

8.3 Spherical Coordinates

$\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dφ}{dr}\right) = 0 \Rightarrow φ = A + \frac{B}{r}.$

Used in heat conduction from cylinders/spheres and mass transfer around droplets.

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9. Unsteady Convective Transport — Advection–Diffusion Equation

General form: $\frac{∂φ}{∂t} + u\frac{∂φ}{∂x} = D\frac{∂^2φ}{∂x^2}.$

Solution via method of characteristics or Fourier transform: $φ(x,t) = \frac{1}{2√{πDt}} \exp\left[-\frac{(x - ut)^2}{4Dt}\right].$

Represents a Gaussian packet convected with velocity $u$ and spreading by diffusion.

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10. Coupled Transport and Turbulent Diffusion

Reynolds decomposition for scalar quantity: $φ = \bar{φ} + φ'.$ Averaging: $\overline{v_i φ} = \bar{v}_i \bar{φ} - \overline{v_i'φ'}.$

Define eddy diffusivity $D_t$: $-\overline{v_i'φ'} = D_t ∂\bar{φ}/∂x_i.$

Effective diffusion coefficient: $D_{eff} = D + D_t.$

For turbulent heat and mass transfer: $\frac{D_t}{ν_t} ≈ 1 \Rightarrow Pr_t ≈ 1, \quad Sc_t ≈ 1.$

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11. Boundary Conditions in Transport Problems

Type	Mathematical form	Physical meaning
Dirichlet	$φ = φ_s$	Prescribed value (temperature/concentration)
Neumann	$∂φ/∂n = q_s/Γ$	Prescribed flux
Robin (mixed)	$-Γ∂φ/∂n = h(φ - φ_∞)$	Convective boundary
Symmetry	$∂φ/∂n = 0$	No gradient across centerline

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12. Numerical Considerations (Conceptual Overview)

Finite-volume discretization in 1D steady convection–diffusion: $a_P φ_P = a_E φ_E + a_W φ_W + S_P.$

Coefficient definitions depend on differencing scheme:

Scheme	Accuracy	Stability	Notes
Central difference	2nd order	Unstable for high Pe	Symmetric diffusion
Upwind	1st order	Always stable	Introduces numerical diffusion
Power-law	Empirical	Moderately stable	Bridges both regimes

Numerical Peclet number $Pe_Δ = ρuΔx/Γ$ determines discretization choice.

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13. Asymptotic Behavior and Limiting Regimes

Regime	Condition	Dominant Mechanism
Diffusion-dominated	$Pe ≪ 1$	Nearly linear $φ$ profile
Convection-dominated	$Pe ≫ 1$	Thin boundary layer, steep gradients
Mixed regime	$Pe ≈ 1$	Coupled transport

In high-Pe flows, analytical methods (boundary-layer or matched asymptotic expansions) describe steep concentration/temperature fronts.

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14. Summary Equations

Concept	Equation
General transport law	$ρ Dφ/Dt = ∇·(Γ∇φ) + S_φ$
Diffusion equation	$∂φ/∂t = D∇²φ$
1D steady convection–diffusion	$ρu dφ/dx = Γ d²φ/dx²$
Peclet number	$Pe = UL/D$
Nusselt number	$Nu_x = 0.332 Re_x^{1/2} Pr^{1/3}$
Sherwood number	$Sh_x = 0.332 Re_x^{1/2} Sc^{1/3}$

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15. Cross-Links

• 02_Boundary_Layers_and_Separation.md — boundary-layer convection–diffusion coupling.
• Thermodynamics/10_NonEquilibrium_Thermodynamics.md — entropy generation and flux–force analogy.
• 03_Numerical_Methods.md — finite-volume discretization and stability criteria for transport equations.