ABSTRACT
Governing equations for flow from wavy surfaces are not truly different from those that are used when dealing with flat and smooth surfaces. The only difference lies in the use of an additional equation, which is needed for describing the profile of the wavy surface. Thus, the starting point is from the well-known Navier-Stokes equations. For example, for the simplest case of two-dimensional flow from a wavy wall, the relevant equations of continuity, momentum, and energy are written as follows:
∂u
∂x + ∂v
∂y = 0 (1.1)
u ∂u
∂x + v
∂u
∂y = gx − 1
ρ
∂p
∂x − 1 ρ
( ∂τxx ∂x
+ ∂τxy ∂y
) (1.2)
u ∂v
∂x + v
∂v
∂y = gy − 1
ρ
∂p
∂y − 1 ρ
( ∂τyx ∂x
+ ∂τyy ∂y
) (1.3)
u ∂T
∂x + v
∂T
∂y =
l
( ∂2T
∂x2 + ∂2T
) (1.4)
where τij, stress tensor, is related to the deformation tensor
∆ij = [(∂ui/∂xj) + (∂uj/∂xi)]
through the rheological equation of state, thereby specifying the type under consideration. The stress components for two-dimensional flow of Newtonian fluids as well as non-Newtonian power-law fluids are given in Table 1.1.
