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Navier-Stokes and Bernoulli equations with terms

Short answer

Navier-Stokes terms: local and convective acceleration, body force, pressure gradient, viscous force per unit volume. Bernoulli’s terms: pressure head p/γ, velocity head V²/2g, elevation z sum to constant total head along a streamline for steady inviscid incompressible flow.

Write the Navier-Stokes and Bernoulli's equation (derivation not required). Explain each terms in the equations with physical meaning.

Navier-Stokes equation

For an incompressible Newtonian fluid with constant viscosity, in vector form:

ρ(∂V⃗∂t+(V⃗⋅∇)V⃗)=ρg⃗−∇p+μ∇2V⃗\rho\left(\frac{\partial\vec V}{\partial t} + (\vec V\cdot\nabla)\vec V\right) = \rho\vec g - \nabla p + \mu\nabla^2\vec V

x-component in Cartesian form:

ρ(∂u∂t+u∂u∂x+v∂u∂y+w∂u∂z)=ρgx−∂p∂x+μ(∂2u∂x2+∂2u∂y2+∂2u∂z2)\rho\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}\right) = \rho g_x - \frac{\partial p}{\partial x} + \mu\left(\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} + \frac{\partial^2u}{\partial z^2}\right)

(The y and z equations are similar.) Together with continuity, ∇⋅V⃗=0\nabla\cdot\vec V = 0, it describes the flow.

TermPhysical meaning
ρ ∂V⃗/∂t\rho\,\partial\vec V/\partial tLocal (unsteady) inertia force per unit volume
ρ(V⃗⋅∇)V⃗\rho(\vec V\cdot\nabla)\vec VConvective inertia force per unit volume (change of velocity with position)
ρg⃗\rho\vec gBody (gravity) force per unit volume
−∇p-\nabla pPressure force per unit volume
μ∇2V⃗\mu\nabla^2\vec VViscous force per unit volume

Bernoulli's equation

For steady, incompressible, frictionless flow along a streamline:

pγ+V22g+z=constant\frac{p}{\gamma} + \frac{V^2}{2g} + z = \text{constant}
TermMeaning
p/γp/\gammaPressure head (flow work per unit weight)
V2/2gV^2/2gVelocity (kinetic) head
zzElevation (potential) head
SumTotal head, i.e. total energy per unit weight, constant along the streamline

Each term has the unit of length (metres of fluid). Multiplying by γ\gamma gives the pressure form, static + dynamic + hydrostatic pressure: p+12ρV2+γz=p + \tfrac12\rho V^2 + \gamma z = constant.

✓ Answer checked by Claude Opus 5.5 (high effort)

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