For the mass m_{it} moving with the velocity v_{i;} the following balance is valid

Application of the Reynolds transport theorem results in the well-known continuity equation for phase i,

where m_{i} is the rate of production of phase i, by mass transfer or chemical reaction. Conservation of mass requires that (Gidaspow 1994)

If the fluid is incompressible and there are no phase changes, V_{t}, the volume of phase i remains constant. Then, application of the Reynolds transport theorem results in the following incompressible continuity equation in multiphase flow

Conservation of Momentum

The rate of change in the momentum of a multiphase system moving with the velocity v_{i} equals the sum of the forces acting on the system including the forces of interaction between the phases. Other forces acting on the system are surface forces, external forces, and momentum exchange due to phase change. Therefore, the momentum balance for phase i can be written as (Bowen 1976; Gidaspow 1994)

Application of the Reynolds transport theorem on the right-hand side of the above balance followed by the application of the divergence theorem where,

I

on the left-hand side, gives the three momentum balances for each phase i, as follows

By differentiating and using the continuity equation for phase i, it is easy to show that the momentum balance for phase i can be expressed as

The term on the left-hand side accounts for acceleration of phase i. The terms on the right-hand side are momentum in-flow due to surface forces, body forces, and interaction forces, respectively.

Therefore, by expressing interaction forces K_{i} in terms of a friction coefficient в for solid-fluid interaction, the momentum equation for phase k becomes

The sum of the interaction forces K_{i} is clearly zero

The stress tensor T_{i} for phase i is given by

where the elements, say, T_{ixy}, are the ith force in the x direction per unit area of the yth face.

The simplest expression for the stress in an inviscid flow, analogous to the single-phase potential flow theory, is through the definition of a phase pressure p_{i}, via the identity I

For incompressible viscous flows, where there are frictional forces due to differences in phase velocities, the traction T is a function of the symmetric gradient of the velocity. The driving force for the transfer of shear is the symmetric gradient of velocity rather than the ordinary gradient because of the need to satisfy invariance under a change of frame of reference under rotation, called objectivity in continuum mechanics, or the Galileo relativity principle. To meet the requirement of objectivity, let

Linearization of the T_{k} gives

For incompressible fluids, A_{k} is chosen to be the negative of the pressure of fluid k, and the derivative of the traction with respect to the symmetric gradient is the viscosity of fluid k, as shown below,

and

Using the tensor identity, k-phase stress tensor can be expressed as

For a constant phase viscosity p_{k}, the incompressible k-phase (e_{k}p_{k} = constant and consequently div(v_{k}) = 0) Navier-Stokes equation can be rewritten as

For a more general case of compressible viscous flow with negligible phase change, A_{k} is

Therefore, the traction for phase k becomes

where the first term represents the k-phase pressure, the second term represents the k-phase viscous shear, and the third term represents the compression or expansion acting on k-phase by /'-phase by deforming the k-velocity field, with

Then, for compressible k-phase in a multiphase system, the Navier-Stokes equation in convective form is given as