Continuity and Momentum Equations

The equations describing fluid flow emerge from the conservation laws for mass, momentum and energy. In this section, the equations for the conservation of mass and momentum are discussed. The scalar equation for energy transport will be discussed in a section presented later.

Continuity Equation (Conservation of mass):

(37.1)

Momentum Equation (Conservation of momentum):

(37.2)

Here ρ is the mass density, is the velocity field, is the diffusion flux vector (or surface traction) and is the source vector from conservative forces (emerging from pressure gradients and gravity). In Cartesian co-ordinates ( x¹, x², x³ ) with orthonormal basis vectors i , j and k, the vectors , and can be expressed as,

Here

(with j as dummy index, j =1,2,3)

(37.3)

and

(37.4)

Physically, σ_i is the i-th component (i =1,2,3) of the diffusion flux vector (or surface traction) at a point on the surface of local unit normal of the infinitesimal element, and represents the stress tensor component acting on the j-th plane in the i-th direction. The source vector can be interpreted as a conservative force arising out of the absolute pressure gradients (in which gravity effects are included), and P represents absolute pressure.

For any continuum where net distributive moments are absent, the stress tensor is symmetric ( = ). For a Newtonian fluid, the stress tensor components are given by

(37.5)

where is the Kronecker delta, ( =1 for i= j, and 0 for i ≠j ), μ is the molecular viscosity and λ is the Bulk viscosity, conventionally given by . The term represents the component of the strain rate tensor, given by

(37.6)

For incompressible fluid flow, . Hence for incompressible flow of a Newtonian fluid, the stress tensor components can be expressed as

(37.7)

The continuity equation is a scalar equation, and can be expanded in the Cartesian system of co-ordinates ( x¹ = x, x² = y, x³ = z ) into a single equation

(37.8)

The momentum equation is a vector equation, and therefore is expandable into three independent equations,

(37.9a)

(37.9b)

(37.9c)