EQ-NAVIER-STOKES · Fluid Dynamics
Navier–Stokes Momentum Equation (1D, incompressible)
P_x - g*rho - mu*u_xx + rho*u*u_x + rho*u_t = 0
Derivative form
P_x - g*rho - mu*Derivative(u(x, t), (x, 2)) + rho*u(x, t)*Derivative(u(x, t), x) + rho*Derivative(u(x, t), t) = 0
Variables
variable
P_x
pressure gradient along flow
- Object
- fluid_parcel
- Property
- Pressure
- Context
- incompressible
- Constraint
- spatial_gradient
variable
g
gravitational acceleration
- Object
- fluid_parcel
- Property
- Acceleration
- Context
- incompressible
- Constraint
- body_force_acceleration
variable
mu
dynamic viscosity
- Object
- fluid_parcel
- Property
- DynamicViscosity
- Context
- incompressible
variable
rho
fluid density (constant for incompressible flow)
- Object
- fluid_parcel
- Property
- MassDensity
- Context
- incompressible
variable
u
local fluid velocity
- Object
- fluid_parcel
- Property
- Velocity
- Context
- incompressible
variable
u_t
partial time derivative of velocity
- Object
- fluid_parcel
- Property
- Acceleration
- Context
- incompressible
- Constraint
- local_time_derivative_of_velocity
variable
u_x
partial spatial derivative of velocity (rate of strain along flow)
- Object
- fluid_parcel
- Property
- Frequency
- Context
- incompressible
- Constraint
- velocity_gradient
variable
u_xx
second spatial derivative of velocity
- Object
- fluid_parcel
- Property
- Frequency
- Context
- incompressible
- Constraint
- velocity_laplacian
Axioms
classical constant_coefficients deterministic differential galilean_invariant homogeneous incompressible isotropic non_relativistic nonlinear viscous
Assumptions
- Incompressible flow (ρ constant; ∇·v = 0)
- Newtonian fluid (linear stress-strain-rate relation)
- Constant dynamic viscosity μ
- No thermal coupling (isothermal)
- Gravity is the only body force
Derivation
- Navier (1823) and Stokes (1845) via continuum mechanics + Newton's second law applied to a fluid element
- Conservation of mass + momentum with the Newtonian stress tensor τ_ij = −P δ_ij + μ(∂_i u_j + ∂_j u_i)
- Reduces to the heat equation ρ u_t = μ u_xx when the convective term u·u_x is negligible (low Reynolds number or small perturbation)
References
- Batchelor, An Introduction to Fluid Dynamics, §3.1
- Landau & Lifshitz, Fluid Mechanics, 2nd ed., §15
- Kundu & Cohen, Fluid Mechanics, 6th ed., Ch. 4