section_path: In deformable bodies and fluids/Conservation in a continuum
section_name: Conservation in a continuum
paragraph_text: Conservation in a continuum
  
  In fields such as fluid dynamics and solid mechanics, it is not feasible to follow the motion of individual atoms or molecules. Instead, the materials must be approximated by a continuum in which, at each point, there is a particle or fluid parcel that is assigned the average of the properties of atoms in a small region nearby. In particular, it has a density ρ and velocity v that depend on time t and position r. The momentum per unit volume is ρ v.
  
  Consider a column of water in hydrostatic equilibrium. All the forces on the water are in balance and the water is motionless. On any given drop of water, two forces are balanced. The first is gravity, which acts directly on each atom and molecule inside. The gravitational force per unit volume is ρ g, where g is the gravitational acceleration. The second force is the sum of all the forces exerted on its surface by the surrounding water. The force from below is greater than the force from above by just the amount needed to balance gravity. The normal force per unit area is the pressure p. The average force per unit volume inside the droplet is the gradient of the pressure, so the force balance equation is
  
  − ∇ p + ρ g = 0. {\displaystyle -\nabla p+\rho \mathbf {g} =0\,.}
  
  If the forces are not balanced, the droplet accelerates. This acceleration is not simply the partial derivative ∂ v / ∂ t because the fluid in a given volume changes with time. Instead, the material derivative is needed:
  
  D D t ≡ ∂ ∂ t + v ⋅ ∇. {\displaystyle {\frac {D}{Dt}}\equiv {\frac {\partial }{\partial t}}+\mathbf {v} \cdot {\boldsymbol {\nabla }}\,.}
  
  Applied to any physical quantity, the material derivative includes the rate of change at a point and the changes due to advection as fluid is carried past the point. Per unit volume, the rate of change in momentum is equal to ρ D v / D t. This is equal to the net force on the droplet.
  
  Forces that can change the momentum of a droplet include the gradient of the pressure and gravity, as above. In addition, surface forces can deform the droplet. In the simplest case, a shear stress τ, exerted by a force parallel to the surface of the droplet, is proportional to the rate of deformation or strain rate. Such a shear stress occurs if the fluid has a velocity gradient because the fluid is moving faster on one side than another. If the speed in the x direction varies with z, the tangential force in direction x per unit area normal to the z direction is
  
  σ z x = − μ ∂ v x ∂ z, {\displaystyle \sigma _{zx}=-\mu {\frac {\partial v_{x}}{\partial z}}\,,}
  
  where μ is the viscosity. This is also a flux, or flow per unit area, of x -momentum through the surface.
  
  Including the effect of viscosity, the momentum balance equations for the incompressible flow of a Newtonian fluid are
  
  ρ D v D t = − ∇ p + μ ∇ 2 v + ρ g. {\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-{\boldsymbol {\nabla }}p+\mu \nabla ^{2}\mathbf {v} +\rho \mathbf {g}.\,}
  
  These are known as the Navier–Stokes equations.
  
  The momentum balance equations can be extended to more general materials, including solids. For each surface with normal in direction i and force in direction j, there is a stress component σ. The nine components make up the Cauchy stress tensor σ, which includes both pressure and shear. The local conservation of momentum is expressed by the Cauchy momentum equation:
  
  ρ D v D t = ∇ ⋅ σ + f, {\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}+\mathbf {f} \,,}
  
  where f is the body force.
  
  The Cauchy momentum equation is broadly applicable to deformations of solids and liquids. The relationship between the stresses and the strain rate depends on the properties of the material (see Types of viscosity).