Fluid Mechanics &
Transport in Porous Media

Continuum Hypothesis & Fluid Properties

At macroscopic scales (length scales ≫ molecular spacing ∼ 0.3 nm), a fluid can be treated as a continuous medium characterised by field quantities: density ρ(r,t), velocity u(r,t), pressure p(r,t), and temperature T(r,t). This is the continuum hypothesis, valid when the Knudsen number Kn = λ/L ≪ 1 (λ = mean free path, L = characteristic flow length).

Dynamic viscosity μ quantifies a fluid’s resistance to shearing. Newtonian fluids (water, air, most simple liquids) obey Newton’s viscosity law: shear stress is proportional to shear rate.

Non-Newtonian fluids (blood, polymer melts, ketchup) have viscosity depending on shear rate. Shear-thinning (pseudoplastic) fluids become less viscous under high shear. Shear-thickening (dilatant) fluids become more viscous. Bingham plastics require a yield stress before flowing at all.

Continuity Equation & Mass Conservation

Mass cannot be created or destroyed. In differential form, this gives the continuity equation:

The incompressibility condition ∇·u = 0 greatly simplifies the equations of motion and is an excellent approximation for liquids and for gas flows with Mach number Ma < 0.3.

Navier-Stokes Equations & Momentum Conservation

Applying Newton’s second law to a fluid parcel in the continuum limit — accounting for pressure gradients, viscous stress, and body forces — yields the Navier-Stokes equations. For incompressible flow:

Solving the Navier-Stokes equations in full generality is one of the Millennium Prize Problems in mathematics — the existence and smoothness of solutions in 3D is not proven. In practice, engineers use DNS (direct numerical simulation), RANS (Reynolds-averaged), or LES (large-eddy simulation) at different scales of resolution.

Reynolds Number & the Laminar-Turbulent Transition

The Reynolds number Re is the most important dimensionless group in fluid mechanics. It is the ratio of inertial forces to viscous forces:

Turbulence involves a cascade of kinetic energy from large eddies (injected at scale L by mean flow shear) down to small Kolmogorov eddies (scale η = (ν³/ε)^1/4) where viscosity dissipates energy as heat. The ratio L/η scales as Re^3/4, which is why DNS of high-Re flows is computationally crushing.

Bernoulli’s Equation

For steady, incompressible, inviscid flow along a streamline, the Navier-Stokes equations reduce to Bernoulli’s equation:

Bernoulli’s equation explains aircraft lift (pressure difference between upper and lower wing surfaces), the operation of carburettors, aerofoil stall, and the Coandă effect. It fails where viscosity matters (boundary layers, pipe friction) or where there is significant flow unsteadiness.

Darcy’s Law & Flow in Porous Media

At small Reynolds numbers (Re < 1 based on grain size), viscosity completely dominates inertial forces. Flow through porous media (sand, gravel, rock) at these conditions obeys Darcy’s empirical law (1856):

Hydraulic head h = z + p/(ρg) combines elevation head and pressure head. Flow always goes from high head to low head, regardless of whether this corresponds to high or low elevation. Hydraulic conductivity K spans 13 orders of magnitude: from 10^-13 m/s in intact granite to 10⁰ m/s in coarse gravel.

Groundwater Flow Equations

Combining Darcy’s law with the continuity equation gives the groundwater flow equation:

The Theis solution assumes a uniform, isotropic, infinite confined aquifer. Real aquifers show anisotropy (K_x ≠ K_y), heterogeneity, unconfined conditions (free surface moves with pumping), and complex boundary conditions. Numerical models (MODFLOW, FEFLOW) are used for practical wellfield design.

Advection-Diffusion & the Péclet Number

Once we know the velocity field, we can calculate how a dissolved contaminant or tracer spreads. Transport is governed by two competing processes:

• Advection: transport by the mean flow velocity (directional, conserves mass).
• Diffusion/Dispersion: spreading by molecular diffusion + mechanical dispersion from pore-scale velocity variability.

Mechanical dispersion D_m = α_L v (longitudinal dispersivity α_L ∼ 0.1–100 m depending on heterogeneity scale). In aquifer remediation, advection-diffusion modelling predicts contaminant plume extent and pump-and-treat efficiency.

Boundary Layer Theory & Internal Flows

Near a solid surface, viscosity matters even at high Re: the boundary layer is a thin region where the flow transitions from zero velocity at the wall (no-slip condition) to the free-stream velocity U_∞. The boundary layer thickness δ grows along the plate:

The Hagen-Poiseuille result shows that flow resistance scales as R⁴ — halving a pipe’s radius increases resistance 16-fold. This has profound implications for blood flow in the cardiovascular system: small arterioles contribute most of the total vascular resistance despite carrying little of the total volume.