Navier-Stokes Equations for Incompressible Flow - Chemical Engineering Knowledge Platform

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📊 Key Properties & Parameters

Ratio of dynamic viscosity to fluid density; quantifies resistance to shear-driven flow under gravity-independent conditions.

Controls laminar/turbulent transition (via Reynolds number) and dictates pump sizing and heat exchanger fouling rates.

Dimensionless ratio of inertial to viscous forces; determines flow regime (laminar, transitional, turbulent).

Directly governs friction factor selection in Darcy–Weisbach calculations and turbulence modeling requirements in CFD.

Spatial rate of change of static pressure; drives net fluid motion against viscous resistance.

Determines required pump head, influences seal integrity, and affects vapor–liquid equilibrium in flashing systems.

Magnitude of the local fluid velocity vector field, obtained by solving the momentum equations.

Dictates erosion-corrosion rates in carbon steel piping and sets minimum velocities to prevent solids settling in slurry lines.

Enforces mass conservation for constant-density flow.

Symbol	Name	Unit	Description
u	Velocity vector	m/s	Fluid velocity field in three-dimensional space
∇·	Divergence operator		Mathematical operator measuring the magnitude of a vector field's source or sink at a given point

ρ(∂u/∂t + u·∇u) = −∂p/∂x + μ∇²u + ρfₓ

Conservation of linear momentum in x-direction for Newtonian fluid.

Variables:

Symbol	Name	Unit	Description
ρ	fluid density	kg/m³	Mass per unit volume of the fluid
u	x-component of velocity	m/s	Velocity component in the x-direction
t	time	s	Time variable
p	pressure	Pa	Thermodynamic pressure
μ	dynamic viscosity	Pa·s	Measure of fluid's resistance to shear deformation
fₓ	x-component of body force per unit mass	m/s²	External body force (e.g., gravity) per unit mass in the x-direction

Typical Ranges:

Process piping CFD

Residual < 1×10⁻⁶ per iteration

⚠️ Relative error in force balance < 0.5% across control volumes

Re = ρUL/μ = UL/ν

Dimensionless indicator of flow regime dominance.

Variables:

Symbol	Name	Unit	Description
Re	Reynolds Number	dimensionless	Dimensionless indicator of flow regime dominance
ρ	Fluid Density	kg/m³	Mass per unit volume of the fluid
U	Characteristic Velocity	m/s	Typical flow velocity
L	Characteristic Length	m	Representative physical length scale
μ	Dynamic Viscosity	Pa·s	Measure of fluid's resistance to shear deformation
ν	Kinematic Viscosity	m²/s	Ratio of dynamic viscosity to fluid density