Questions: The Navier-Stokes Equations

Question 1

In the Navier-Stokes momentum equation ρ(DV/Dt) = −∇P + μ∇²V + ρg, which term represents the diffusion of momentum due to fluid viscosity?

• A. ρ(DV/Dt)
• B. −∇P
• C. μ∇²V
• D. ρg

Answer: (C)

The term μ∇²V is the viscous diffusion term: it represents how momentum spreads through the fluid due to internal friction between fluid layers. The Laplacian ∇²V is analogous to the diffusion term in the heat equation. −∇P is the pressure force, ρg is the gravitational body force, and ρ(DV/Dt) is mass times acceleration.

Question 2

Removing the viscous term μ∇²V from the Navier-Stokes equations directly yields Bernoulli's equation.

• T. True
• F. False

Answer: False

Removing the viscous term gives Euler's equations for inviscid flow: ρ(DV/Dt) = −∇P + ρg. Bernoulli's equation is obtained by a further step: integrating Euler's equations along a streamline under steady, incompressible conditions. Bernoulli is a scalar energy relationship derived from Euler's vector equation, not the equation itself.

Question 3

Why do exact analytical solutions to the Navier-Stokes equations exist only for a small class of simple flows?

Model answer: The Navier-Stokes equations are nonlinear partial differential equations due to the convective acceleration term V·∇V in the material derivative. This nonlinearity prevents the superposition of solutions and makes general closed-form solutions impossible. Exact solutions require geometric simplifications (e.g., fully developed channel or pipe flow) that eliminate the nonlinear term.

The convective term V·∇V couples the velocity components together nonlinearly, meaning the equations cannot be solved by standard linear methods. For Couette or Poiseuille flow, the geometry forces the flow to be unidirectional, eliminating the convective acceleration entirely and leaving a linear ODE. In general 3D flow, no such simplification is available, requiring numerical methods (CFD).