Fluid Mechanics: Boundary Conditions and Dimensional Analysis

Viscous friction forces cause the fluid layers in direct contact with a solid surface to adhere to it, resulting in no relative motion. Additionally, there cannot be any relative motion between the fluid and the wall perpendicular to the surface. Therefore, the boundary condition for a solid wall is:

(Wall-fluid relative velocity). For an ideal fluid without frictional forces, the condition is:

The force acting on a solid wall can be expressed as:

The force per unit area is then:

This represents the normal force to the surface (outgoing and incoming fluid at the solid). The first term represents static pressure, and the second term represents the viscous friction force acting on the surface. For a surface separating two immiscible fluids, the velocity must be continuous across the interface, and the net force acting on the interface must be zero. The boundary condition for the interface between two fluids is:

For a liquid-gas interface, assuming an incompressible fluid where the velocity normal to the surface is zero, the boundary condition is:

Finally, for the free surface of a fluid, the boundary condition is:

We can write the force acting on a solid wall as limiting fluid

Then the force per unit area will

Which is the normal force to the surface (outgoing and incoming fluid in the solid). The first term represents the static pressure and the second is the force of friction due to viscosity acting on the surface. In the case of a surface separating two immiscible fluids, the speed must be continuous across the interface and a fluid force that makes the other must vanish. The boundary condition for the interface between two fluids is

For a liquid-gas interface, taking into account the fluid is incompressible, which

the surface is zero and that the boundary condition is. Finally, for the free surface of a fluid, is the boundary condition is.

To define the current function, we assume plane, incompressible flow:

To satisfy the Laplace equation:

we assume an ideal fluid and irrotational flow:

The fundamental theorem of dimensional analysis states that if a physical relationship is expressible by an equation involving n physical quantities, and these variables are expressed in terms of k dimensionally independent quantities, then the original equation can be rewritten as an equation with n - k dimensionless numbers constructed from the original variables.