Theory for all the Single-Phase Flow Interfaces

The single-phase fluid-flow interfaces are based on the Navier-Stokes equations, which in their most general form read

Equation 4-5 is the continuity equation and represents the conservation of mass. Equation 4-6 is a vector equation and represents the conservation of momentum. Equation 4-7 describes the conservation of energy, formulated in terms of temperature. This is an intuitive formulation that facilities specification of boundary conditions.3.1.1The Continuity Equation

Figure 13 Finite control volume fixed in space (Jiyuan, Yeoh and Liu 2008)Mass Conservation

By the conservation of mass, matter cannot be created or destroyed. Figure shows an arbitrary control volume V fixed in space and time. A fluid is considered to be moving through the fixed control volume and flows across the control surface. For mass conservation, the rate of change of mass within the control volume is equivalent to the mass flux crossing the surface S of volume V. Equation 3.1 shows the mass conservation in integral form where is the unit normal vector.

[3.1]By Gausss divergence theorem, the volume integral of a divergence of a vector is equated to an area integral over the surface that defines the volume. This is shown in equation 3.2.

[3.2]The surface integral of equation 3.1 can be replaced by the volume integral of equation 3.2. This gives

[3.3]

where . Recall:

We define the operator (pronounced del) by

We define the divergence of a vector field F, written divF or , as the dot product of del with F. So if F =, then

divF = Notice that divF is a scalar. Equation 3.3 is valid for any size of volume V. This implies that

[3.4]

is the fluid velocity which can be described by the local velocity Cartesian components u, v and w. The local velocity components are functions of location (x, y, z) and time (t). Equation 3.4 is the mass conservation equation for fluid flow which for a Cartesian coordinate system can be expressed as

[3.5]Expanding equation 3.5 using the chain rule and grouping the density terms gives

[3.6]

Or [3.7]

is the substantial derivative in Cartesian coordinates. is the time rate of change of density following a moving fluid element (Jiyuan, Yeoh and Liu 2008, 65-66).3.1.2The Momentum EquationForce Balance

For a general variable property per unit mass denoted as , the substantial derivative of with respect to time is given by

[3.8]

Equation 3.8 defines the rate of change of the variable property per unit mass. To obtain the rate of change of the variable property per unit volume, the substantial derivative of is multiplied by the density.

[3.9]

Equation 3.9 represents the non-conservation form of the rate of change of the variable property per unit volume.

Figure 14 Surface forces acting on the infinitesimal control volume for the velocity component u. Deformed fluid element due to the action of the surface forces (Jiyuan, Yeoh and Liu 2008)Figure 14 shows an infinitesimal control volume fluid element. The sum of the forces acting on the fluid element, by Newtons second law of motion, is equal to the product of its mass and acceleration. Newtons second law can be applied along the x, y and z directions. The x component of Newtons second law is given by

[3.10]Fx and ax are the force and acceleration along the x direction. The acceleration ax is the time rate change of u, given by its substantial derivative. Thus,

[3.11]

The mass of the fluid element, m is. Therefore, the rate of increase of x-momentum is

[3.12]The force on a moving fluid element is due to two sources, body forces and surface forces. The various forms of body forces that may affect the rate of change of fluid momentum include gravity, centrifugal, Coriolis and electromagnetic forces.

Figure 14 shows that the surface forces for the velocity component u, that cause deformation of the fluid element are due to the normal stress and tangential stresses and acting on the surfaces of the fluid element. Substituting the sum of the surface forces on the fluid element and the time rate change of u from equation 3.12 into equation 3.10, the x-momentum equation becomes

[3.13]Similarly, the y-momentum and z-momentum equations are

[3.14]

[3.15]

The normal stresses , and are due to the pressure p and normal viscous stress components , andacting perpendicular to the control volume. The remaining terms contain the tangential viscous stress components. In many fluid flows, a suitable model for the viscous stresses is introduced. They are usually a function of the local deformation rate (or strain rate) that is expressed in terms of the velocity gradients (Jiyuan, Yeoh and Liu 2008, 75-78).

To close the equation system Equation 4-5 through Equation 4-7, some constitutive relations are needed. A common relation is derived by assuming that the fluid is Newtonian. Together with Stokes assumption, the viscous stress tensor becomes:

The dynamic viscosity (SI unit: Pas) is allowed to depend on the thermodynamic state but not on the velocity field. All gases and many liquids can be considered Newtonian. Examples of non-Newtonian fluids are honey, mud, blood, liquid metals, and most polymer solutions. For modeling of flows of non-Newtonian fluids, use the Non-Newtonian Flow interface (see Non-Newtonian Flow). All other fluid-flow interfaces use stress tensors based on Equation 4-9.3.1.3Navier-Stokes EquationA three dimensional case of constant property fluid flow will be considered. Constant property fluid flow implies that the density is constant and body forces, particularly due to gravity are not considered in the equations. By applying the continuity equation and including the stress-strain relationships, the momentum equations can be reduced to following.x direction[3.16]y direction

[3.17]z direction

[3.18]Equations 3.16 to 3.18 derived from Newtons second law, where v is the kinematic viscosity (v = / ) describe the conservation of momentum in the fluid flow and is also known as the Navier-Stokes equations (Jiyuan, Yeoh and Liu 2008,77-78).The single-phase fluid low user interfaces in COMSOL Multiphysics are based on the Navier-Stokes equations. The Navier-Stokes equations solved by default in all the single phase flow interfaces are the compressible formulation of continuity (equation 3.4) and the momentum equations (equations 3.13 to 3.15). (COMSOL 2013, 83-87)