0.5 Equations of motion and energy in cartesian coordinates  (Page 10/12)

where the bracket denotes the jump condition across the interface, $H$ is the mean curvature of the interface, and $\sigma$ is the interfacial or surface tension. The jump condition on the normal component of the stress is the jump in pressure across a curved interface given by the Laplace-Young equation. The tangential components of stress are continuous if there are no surface tension gradients and surface viscosity. Thus the tangential stress at the clean interface with an inviscid fluid is zero.

For boundary conditions at a fluid interface with adsorbed materials and thus having interfacial tension gradients and surface viscosity, see Chapter 10 of Aris and the thesis of Singh (1996).

Boundary conditions for the potentials and vorticity. Some fluid flow problems are more conveniently calculated through the scalar and vector potentials and the vorticity.

The boundary condition on the scalar potential is that the normal derivative is equal to the normal component of velocity.

The boundary condition on the vector potential is that the tangential components vanish and the normal derivative of the normal component vanish (Hirasaki and Hellums, 1970). Wong and Reizes (1984) introduced a method where the need for calculation of the scalar potential is replaced by the use of an irrotational component of velocity.

In two dimensional or axisymmetric incompressible flow, it is not necessary to have a scalar potential and the single nonzero component of the vector potential is the stream function. The boundary condition on the stream function for flow in the $x_1$ , $x_2$ plane of Cartesian coordinates is

The boundary condition on the normal component of the vorticity of a fluid with a finite viscosity on a solid surface is determined from the tangential components of the velocity of the solid. It is zero if the solid is not rotating. If the boundary is an interface between two viscous fluids then the normal component of vorticity is continuous across the interface (C. Truesdell, 1960). If the interface is with an inviscid fluid, the tangential components of vorticity vanishes and the normal derivative of the normal component vanishes for a plane interface (Hirasaki, 1967).

If the boundary is at along a region of space for which the velocity field is known, the vorticity can be calculated from the derivatives of the velocity field.

If the boundary is a surface of symmetry, the tangential components of vorticity must vanish because the normal component of velocity and the normal derivative of velocity vanish. The normal derivative of the normal component of vorticity vanishes from the solenoidal property of vorticity.

Scaling, dimensional analysis, and similarity

We have already seen some examples of scaling and dimensional analysis when we determined when the continuity equations and equations of motion could be simplified. The concept of similarity states that the solution of transport problems do not need to be determined separately for each value of the parameters. Rather the variables and parameters can be grouped into dimensionless variables and dimensionless numbers and the solution will have fewer degrees of freedom. Also, in some cases the partial differential equations can have the independent variables combined to fewer independent variables and be expressed as ordinary differential equations. The concept of similarity does not apply only to mathematical solutions but is also used to design physical analogs of systems on a smaller scale or with different transport mechanism. For example, before numerical simulation the streamlines and pressure gradients for flow in petroleum reservoirs were studied by electrical conduction on a laboratory scale model that is geometrically similar.