Basics of Fluid Mechanics, 2014a

304 CHAPTER 9. DIMENSIONAL ANALYSIS
Equation (9.36) show one common possibility of a dimensionless presentation of governing
equation. The significance of the large and small value of the dimensionless
parameters will be discuss later in the book. Without actually solving the problem,
Nusselt’s method provides several more parameters that were not obtained by the block
method. The solution of the governing equation is a function of all the parameters
present in that equation and boundaries condition as well the initial condition. Thus,
the solution is
(
U x = f x, y, Eu, Re, F r, θ, τ 0 ,f u ,f τ , U )
0y
(9.37)
U 0x
The values of x, y depend on h and hence the value of h is an important parameter.
It can be noticed with Buckingham’s method, the number of parameters obtained
was only three (3) while Nusselt’s method yields 12 dimensionless parameters. This is
a very significant difference between the two methods. In fact, there are numerous
examples in the literature that showing people doing experiments based on Buckingham’s
methods. In these experiments, major parameters are ignored rendering these
experiments useless in many cases and deceiving.
Common Transformations
Fluid mechanics in particular and Thermo–Fluid field in general have several common
transformations that appear in boundary conditions, initial conditions and equations
11 . It recognized that not all the possibilities can presented in the example shown
above. Several common boundary conditions which were not discussed in the above
example are presented below. As an initial matter, the results of the non dimensional
transformation depends on the selection of what and how is nondimensionalization carried.
This section of these parameters depends on what is investigated. Thus, one of
the general nondimensionalization of the Navier–Stokes and energy equations will be
discussed at end of this chapter.
Boundary conditions are divided into several categories such as a given value to
the function 12 , given derivative (Neumann b.c.), mixed condition, and complex conditions.
The first and second categories were discussed to some degree earlier and will
be expanded later. The third and fourth categories were not discussed previously. The
non–dimensionalization of the boundary conditions of the first category requires finding
and diving the boundary conditions by a typical or a characteristic value. The second
category involves the nondimensionalization of the derivative. In general, this process
involve dividing the function by a typical value and the same for length variable (e.g.
x) as
∂U
∂x =
(
l ∂
U 0
)
U
U 0
∂ ( x
l
) = l ∂U
U 0 ∂x
(9.38)
11 Many of these tricks spread in many places and fields. This author is not aware of a collection of
this kind of transforms.
12 The mathematicians like to call Dirichlet conditions