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.

---

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: $\frac{d}{dt}\int_V \rho\phi\,dV + \oint_S (\rho\phi\mathbf{v} - \Gamma\nabla\phi)\cdot\mathbf{n}\,dA = \int_V S_\varphi\,dV,$ where:

• $\rho$: density,
• $\Gamma$: diffusion coefficient ($\mu, k$, or $D$depending on context),
• $S_\varphi$: volumetric source term.

Applying the divergence theorem and assuming constant $\rho$: $\rho\frac{D\varphi}{Dt} = \nabla\cdot(\Gamma\nabla\varphi) + S_\varphi.$ This is the general convection–diffusion (transport) equation.

---

2. Special Cases of $\varphi$

Quantity	$\varphi$	Diffusion coefficient $\Gamma$	Source term $S_\varphi$
Momentum	$u_i$	$\mu$	Pressure gradient + body forces
Thermal energy	T	k	Viscous dissipation or heat generation
Species concentration	c_i	$\rho D_i$	Reaction rate $r_i$

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

---

3. One-Dimensional Steady Convection–Diffusion Equation

For steady 1D transport in the x-direction: $\rho u\frac{d\varphi}{dx} = \frac{d}{dx}\left(\Gamma\frac{d\varphi}{dx}\right) + S_\varphi.$

3.1 Constant Properties and No Source

$\rho u\frac{d\varphi}{dx} = \Gamma\frac{d^2\varphi}{dx^2}.$

Introduce Peclet number: $Pe = \frac{\rho uL}{\Gamma}.$

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

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

Limit	Behavior
$Pe \to 0$	Diffusion-dominated: linear profile
$Pe \to \infty$	Convection-dominated: sharp boundary layer

---

4. Diffusion Equation (Transient, No Convection)

$\frac{\partial\varphi}{\partial t} = D \nabla^2 \varphi.$

4.1 1D Semi-Infinite Medium

Boundary conditions: $\varphi(0,t)=\varphi_s, \; \varphi(\infty,t)=\varphi_\infty$.

Solution: $\varphi(x,t) = \varphi_\infty + (\varphi_s - \varphi_\infty)\,\mathrm{erfc}\left(\frac{x}{2\sqrt{}{Dt}}\right).$

Characteristic diffusion length: $\delta_d(t) = \sqrt{}{Dt}.$

---

5. Convective Heat and Mass Transfer in Boundary Layers

Energy equation in steady 2D flow: $\rho c_p(u\partial T/\partial x + v\partial T/\partial y) = k \partial^2T/\partial y^2.$

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.

---

6. Dimensionless Form of the Transport Equation

Define characteristic scales: $U, L, \Delta\varphi$. Dimensionless variables: $\varphi^* = \frac{\varphi - \varphi_0}{\Delta\varphi}, \quad x_i^* = x_i/L, \quad t^* = tU/L.$

Then: $\frac{\partial\varphi^*}{\partial t^*} + \tilde{v}_i \frac{\partial\varphi^*}{\partial x_i^*} = \frac{1}{Pe} \nabla^{*2}\varphi^* + \tilde{S}_\varphi.$

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

---

7. Analogy Between Transport Processes

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

Property	Flux	Constitutive Law	Diffusivity
Momentum	$\tau = \mu du/dy$	Newton’s law	$\nu = \mu/\rho$
Heat	$q = -k dT/dy$	Fourier’s law	$\alpha = k/(\rho c_p)$
Mass	$j = -\rho D dy/dy$	Fick’s law	$D$

Dimensionless ratios:

Symbol	Definition	Interpretation
Pr	$\nu/\alpha$	Momentum vs. thermal diffusion
Sc	$\nu/D$	Momentum vs. mass diffusion
Le	$\alpha/D$	Thermal vs. mass diffusion
Pe	$Re\cdot Sc$or $Re\cdot Pr$	Convective vs. diffusive transport

---

8. Analytical Solutions to Canonical Problems

8.1 Steady Diffusion in a Slab (No Convection)

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

8.2 Cylindrical Coordinates (Radial Diffusion)

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

8.3 Spherical Coordinates

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

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

---

9. Unsteady Convective Transport — Advection–Diffusion Equation

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

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

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

---

10. Coupled Transport and Turbulent Diffusion

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

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

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

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

---

11. Boundary Conditions in Transport Problems

Type	Mathematical form	Physical meaning
Dirichlet	$\varphi = \varphi_s$	Prescribed value (temperature/concentration)
Neumann	$\partial\varphi/\partial n = q_s/\Gamma$	Prescribed flux
Robin (mixed)	$-\Gamma\partial\varphi/\partial n = h(\varphi - \varphi_\infty)$	Convective boundary
Symmetry	$\partial\varphi/\partial n = 0$	No gradient across centerline

---

12. Numerical Considerations (Conceptual Overview)

Finite-volume discretization in 1D steady convection–diffusion: $a_P \varphi_P = a_E \varphi_E + a_W \varphi_W + S_P.$

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 $Pe_\Delta = \rho u\Delta x/\Gamma$determines discretization choice.

---

13. Asymptotic Behavior and Limiting Regimes

Regime	Condition	Dominant Mechanism
Diffusion-dominated	$Pe \ll 1$	Nearly linear $\varphi$profile
Convection-dominated	$Pe \gg 1$	Thin boundary layer, steep gradients
Mixed regime	$Pe \approx 1$	Coupled transport

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

---

14. Summary Equations

Concept	Equation
General transport law	$\rho D\varphi/Dt = \nabla\cdot(\Gamma\nabla\varphi) + S_\varphi$
Diffusion equation	$\partial\varphi/\partial t = D\nabla²\varphi$
1D steady convection–diffusion	$\rho u d\varphi/dx = \Gamma d²\varphi/dx²$
Peclet number	$Pe = UL/D$
Nusselt number	$Nu_x = 0.332 Re_x^{1/2} Pr^{1/3}$
Sherwood number	$Sh_x = 0.332 Re_x^{1/2} Sc^{1/3}$

---

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.