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The Navier-Stokes equations are the backbone of fluid mechanics. They describe how fluids move and interact, using principles of mass and momentum conservation. These equations are powerful tools for understanding everything from blood flow to weather patterns.

However, the Navier-Stokes equations have limitations. They assume fluids are continuous and Newtonian, which isn't always true. For complex flows like turbulence, we often need simplified models to make calculations manageable.

Derivation and Components of the Navier-Stokes Equations

Derivation of Navier-Stokes equations

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  • Conservation of mass principle expressed by the continuity equation ρt+(ρu)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0 relates the change in fluid density ρ\rho over time tt to the divergence of the mass flux ρu\rho \vec{u}
  • Conservation of momentum principle based on Newton's second law F=ma\vec{F} = m\vec{a} considers surface forces (pressure and viscous stresses) and body forces (gravity, electromagnetic forces) acting on a fluid element
  • Navier-Stokes equations derived by applying conservation of mass and momentum to a fluid element result in the general form ρ(ut+uu)=p+μ2u+ρg\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 pp is pressure, μ\mu is dynamic viscosity, and g\vec{g} represents body forces per unit mass (gravity)

Components of Navier-Stokes equations

  • Local acceleration term ρut\rho \frac{\partial \vec{u}}{\partial t} represents the change in velocity over time at a fixed point in the fluid
  • Convective acceleration term ρ(uu)\rho (\vec{u} \cdot \nabla \vec{u}) accounts for the change in velocity due to the motion of the fluid itself
  • Pressure gradient term p-\nabla p represents the force acting on the fluid due to spatial variations in pressure
  • Viscous diffusion term μ2u\mu \nabla^2 \vec{u} represents the force acting on the fluid due to viscous stresses arising from velocity gradients
  • Body force term ρg\rho \vec{g} represents the force acting on the fluid due to external body forces such as gravity or electromagnetic fields

Application and Limitations of the Navier-Stokes Equations

Application to fluid flow problems

  • Simplify the Navier-Stokes equations based on problem assumptions and constraints
    1. Steady-state flow ut=0\frac{\partial \vec{u}}{\partial t} = 0 eliminates the local acceleration term
    2. Incompressible flow assumes constant density ρ\rho simplifying the continuity equation
    3. Newtonian fluid assumption relates shear stress τ\tau linearly to strain rate uy\frac{\partial u}{\partial y} via dynamic viscosity μ\mu
  • Apply appropriate boundary conditions such as no-slip (zero fluid velocity at solid boundaries) and free surface (continuous normal and shear stresses across the interface) conditions
  • Obtain analytical solutions for simple geometries and flow conditions (Poiseuille flow through pipes, Couette flow between parallel plates)

Limitations of Navier-Stokes equations

  • Continuum assumption treats fluid as a continuous medium which may not be valid for rarefied gases or flows with high Knudsen numbers (ratio of mean free path to characteristic length)
  • Newtonian fluid assumption may not hold for non-Newtonian fluids with complex rheology (blood, polymers)
  • Turbulence modeling is required as direct numerical simulation (DNS) of turbulent flows using Navier-Stokes equations is computationally expensive, necessitating approximate models (RANS, LES)
  • Compressibility effects require additional terms and equations (energy equation, equation of state) to extend Navier-Stokes equations to compressible flows
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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