Diffusion Equation and variable of Similitude

[mullzer]

I am studying fluid dynamics by using adimensional analysis to obtain an expression for a dynamic (thermal) boundary layer in a poiseuille flow. The fluid nearest the boundary (y = 0) is said to have adimenional temperature T = 1 and as y tends to infinity, the temperature is T = 0.

The final adimensionalised problem usually take the form of the diffusion equation, e.g.

2y(dT/dx) = (d²T/dy²)


The method preferred by my superior is use "invariance groups"- i.e. to brake each variable in the problem into two parts: e.g. y = y*.y^ which leads to:


2(y*.y^) (T*/x*)(dT^/dx^) = (T*/y*²)(d²T/dy²)

The priniciple here is that in order for the original expression to hold true, all the * variables must cancel out to 1, leaving the original equation.

Equating the * terms brings the relation: y*= x*^1/3.

The variable of similitude is then introduced, n= y/x^(1/3), and the temperature can be written T(x,y) = f(n).

What i can't grasp is how to formulate this f(n). In this example, it takes the form:

f'' + (2/3)n²f' = 0.

I am not only interested in how this specific example was formulated but also the method in general as I am quite unfamiliar with it.

Thanks

 

[gtoguy97]

!The method you are referring to is sometimes called the similarity transformation. This method is used to reduce a complicated partial differential equation to an ordinary one. In this case, the equation is a diffusion equation. The idea is to transform the equation in such a way that the new equation has the same solution as the original equation. To do this, you need to define a new variable, n, which is a function of both x and y. This is the "variable of similitude." Then, substitute this new variable into the original equation. This will result in a new equation with only one variable (n) instead of two variables (x and y). Then, you can solve the ordinary differential equation for f(n). In your example, the variable of similitude is defined as n = y/x^(1/3). Then, the equation is transformed to f''(n) + (2/3)n^2f'(n) = 0. This ordinary differential equation can then be solved to find f(n).