Derivation of the Navier-Stokes equation

Lecture material – Environmental Hydraulic Simulation Page 50The first two terms are the substantial derivative of the density over time. It is composed of the localpart for unsteady flows and the convective part due to change in place.This yields the continuity equation:Dρ⎛ ∂u∂v∂w⎞+ρ0Dt⎜ + +x y z⎟ =⎝ ∂ ∂ ∂ ⎠Eq. 2-95We now assume an incompressible fluid 1 . The first term of Eq. 2-95, that is the substantial derivativeof the density over time, is zero under this precondition. The result of the continuity equation is thenthat incompressible fluids are solenoidal and the continuity equation can be written as:⎛ ∂u∂v∂w⎞div v = 0 or ⎜ + + ⎟ = 0Eq. 2-96⎝ ∂x∂y∂z⎠The constant density in the flow field is a commensurate, but not a necessary condition forincompressible flows.2.2.1.1.2.2 Momentum / motion equationsThe momentum equation is a basic law of mechanics. It states that mass times acceleration is equal tothe sum of forces that act on a volume unit. We differentiate between mass forces (weight forces) andsurface forces (pressure and friction forces).dI = dm t ⋅v( )1dVdIdt=1dVddt ∫ ρ ⋅v dV = F + PVEq. 2-97F : mass force per volume unitP : surface force per volume unitWith further simplification we get:1 A fluid, which does not change it’s density at the pressure from outside, is called incompressible. In nature, allfluids are always compressible. For most calculations the fluid may still be granted as an incompressible fluid, as theerror is negligibly small. Moreover, this assumption simplifies the calculation enormously. In the technical hydraulicsthe compressibility, for example, of a hydraulic fluid should be kept clearly in mind.

Lecture material – Environmental Hydraulic Simulation Page 511dVddt∫ρ ⋅v dV =1dV∫DDt( ρ ⋅v)dV1dVDDt⎛⎜1 ⎜ Dρ dVdV⎜ + Dt⎜= 0,⎝ incompressible( ρ ⋅v dV ) = v ( ρ dV )⎞ ⎟Dv ⎟Dt⎟⎟⎠Eq. 2-98Dv= ρDtThe last expression contains the substantial acceleration that is composed of the local and theconvective acceleration: Dv∂vdv ∂v = + = + ( v ⋅ grad)vEq. 2-99Dt ∂tdt ∂tThis equation makes the transition from the Lagrange point of view, that has no reference to space, butonly regards changes in time, to the Eularian point of view, which accounts for the translation part byconvective transport. The momentum equation with these changes is: Dv⎛ ∂v ⎞ ρ = ρ ⎜ + ( v ⋅ grad) v⎟=F+PEq. 2-100Dt ⎝ ∂t⎠The mass forces are to be seen as given, outside, forces, in opposition to the surface forces that dependon the deformation state of the fluid. The total of surface forces makes a stress state. Now a connectionbetween the stress state and the deformation state has to be made. This is done with help of thetransport equation.All further considerations will be limited to isotropic Newton fluids. All gases and many liquids,especially water, belong to this category. If the relation between all components of the stress tensorand the deformation velocity tensor is the same for all directions, the fluid is considered isotropic. Ifthis relation is linear it is considered a Newton fluid. The equation for the momentum transport iscalled the Newton or Stokes friction law in this case.