7.1 Navier-Stokes Equations

Derivation and Components of the Navier-Stokes Equations

The Navier-Stokes equations govern the motion of viscous fluids by combining conservation of mass and momentum into a coupled system of partial differential equations. They form the foundation of differential analysis in fluid mechanics, and nearly every analytical or computational fluid flow solution starts from some form of these equations.

That said, they come with built-in assumptions: the fluid is treated as a continuum and (in the standard incompressible form) as Newtonian. When those assumptions break down, you'll need modified approaches. This section covers the derivation, the physical meaning of each term, common simplifications, and the key limitations you should know.

Derivation of the Navier-Stokes Equations

The derivation rests on two conservation principles applied to an infinitesimal fluid element.

Conservation of mass gives the continuity equation:

$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0$

This says the rate of change of density $\rho$ at a point plus the net mass flux out of that point must equal zero. Mass is neither created nor destroyed.

Conservation of momentum starts from Newton's second law, $\vec{F} = m\vec{a}$, applied to a fluid element. The forces acting on that element fall into two categories:

• Surface forces: pressure acting normal to the element's faces, and viscous stresses caused by velocity gradients between adjacent fluid layers.
• Body forces: forces that act on the entire volume of the element, most commonly gravity (but also electromagnetic forces in certain problems).

Combining these forces with the acceleration of the fluid element (which includes both local and convective parts) yields the Navier-Stokes equations in their standard incompressible form:

$\rho \left(\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u}\right) = -\nabla p + \mu \nabla^2 \vec{u} + \rho \vec{g}$

where $p$ is pressure, $\mu$ is dynamic viscosity, and $\vec{g}$ is the gravitational acceleration vector (or more generally, body force per unit mass).

This is a vector equation, so in 3D it represents three scalar equations (one for each coordinate direction). Together with the continuity equation, you have four equations for four unknowns: three velocity components and pressure.

Components of the Navier-Stokes Equations

Each term in the equation has a distinct physical meaning. Understanding these is critical for knowing which terms you can drop when simplifying a problem.

• Local acceleration $\rho \frac{\partial \vec{u}}{\partial t}$: The rate of change of velocity at a fixed point in space. This term is zero for steady flows.
• Convective acceleration $\rho (\vec{u} \cdot \nabla \vec{u})$: The change in velocity that a fluid particle experiences as it moves through a spatially varying velocity field. Even in steady flow, a particle can accelerate if it moves into a region of higher or lower velocity (think of flow speeding up as it enters a constriction).
• Pressure gradient $-\nabla p$: Fluid accelerates from regions of high pressure toward regions of low pressure. The negative sign reflects this: a positive pressure gradient in the $x$-direction pushes fluid in the negative $x$-direction.
• Viscous diffusion $\mu \nabla^2 \vec{u}$: The Laplacian of velocity captures how momentum diffuses through the fluid due to viscous shear stresses. Where velocity gradients are sharp (near walls, for instance), this term is large. In inviscid approximations (Euler equations), this entire term is dropped.
• Body force $\rho \vec{g}$: External forces acting on the fluid volume. For most problems this is just gravity, but it can include electromagnetic body forces in magnetohydrodynamics.

Application and Limitations of the Navier-Stokes Equations

Application to Fluid Flow Problems

Solving the full Navier-Stokes equations analytically is only possible for a handful of simple geometries and flow conditions. The general approach is:

1. Identify valid simplifications based on the physics of the problem:

   • Steady-state flow: Set $\frac{\partial \vec{u}}{\partial t} = 0$, eliminating the local acceleration term.
   • Incompressible flow: Treat density $\rho$ as constant, which simplifies the continuity equation to $\nabla \cdot \vec{u} = 0$.
   • Newtonian fluid: Assume shear stress is linearly proportional to strain rate, $\tau = \mu \frac{\partial u}{\partial y}$, which is already built into the standard form above.
   • Fully developed flow: Velocity profiles no longer change in the streamwise direction, eliminating several convective terms.

2. Write out the component equations in the appropriate coordinate system (Cartesian, cylindrical, or spherical) and cancel terms that are zero based on your simplifications and geometry.

3. Apply boundary conditions:

   • No-slip condition: Fluid velocity equals the wall velocity at a solid boundary (usually zero for a stationary wall).
   • Free surface condition: Normal and tangential stresses must be continuous across the interface.
   • Symmetry conditions: Velocity gradients normal to a symmetry plane or axis are zero.

4. Integrate the reduced equations to obtain the velocity profile, then use it to find quantities like flow rate, wall shear stress, or pressure drop.

Two classic analytical solutions worth knowing:

• Poiseuille flow (fully developed, steady, laminar flow in a pipe): Produces a parabolic velocity profile with $u(r) = \frac{1}{4\mu}\left(-\frac{dp}{dx}\right)(R^2 - r^2)$, where $R$ is the pipe radius.
• Couette flow (flow between two parallel plates, one moving): Produces a linear velocity profile when there's no pressure gradient, or a combination of linear and parabolic profiles with a pressure gradient.

Limitations of the Navier-Stokes Equations

• Continuum assumption: The equations treat the fluid as a continuous medium with well-defined properties at every point. This breaks down for rarefied gases where the mean free path of molecules becomes comparable to the characteristic length of the flow. The Knudsen number $Kn = \frac{\lambda}{L}$ (ratio of mean free path $\lambda$ to characteristic length $L$) quantifies this. When $Kn > 0.1$, the continuum assumption becomes questionable, and kinetic theory or molecular dynamics approaches are needed.
• Newtonian fluid assumption: The standard form assumes a linear relationship between stress and strain rate. Non-Newtonian fluids like blood, polymer solutions, and many slurries exhibit shear-thinning, shear-thickening, or viscoelastic behavior that requires modified constitutive equations.
• Turbulence: Direct Numerical Simulation (DNS) resolves all scales of turbulent motion using the unmodified Navier-Stokes equations, but the computational cost scales roughly as $Re^{9/4}$ in 3D, making DNS impractical for most engineering flows at high Reynolds numbers. Instead, approximate turbulence models are used:
  • RANS (Reynolds-Averaged Navier-Stokes): Time-averages the equations and models the effect of turbulent fluctuations through additional closure models (e.g., $k$-$\epsilon$, $k$-$\omega$).
  • LES (Large Eddy Simulation): Resolves large-scale eddies directly and models only the smallest scales. More accurate than RANS but more expensive.
• Compressibility: The incompressible form shown above doesn't account for density variations driven by pressure or temperature changes. For compressible flows (typically $Ma > 0.3$), you need the full compressible Navier-Stokes equations, which couple in the energy equation and an equation of state relating pressure, density, and temperature.