Boundary layer in laminar fluids (Grupo 1B)
| Trabajo realizado por estudiantes | |
|---|---|
| Título | Boundary layer in laminar fluids. Grupo 1-B |
| Asignatura | Ecuaciones Diferenciales |
| Curso | Curso 2013-14 |
| Autores | Sandro Andrés Martínez
David Ayala Díez Claudia Cózar Coarasa Lorena de la Fuente Sanz Marino Rivera Muñoz José Manuel Torres Serrano |
| Este artículo ha sido escrito por estudiantes como parte de su evaluación en la asignatura | |
In this numerical project we have studied what happens when we introduce a flat plate in a laminar fluid whose speed and viscosity are constant.
Blasius equation
First, we must assume that the fluid velocity before reaching the plate is constant, as in remote areas after passing the plate.
In our case we take this constant as [math]2[/math], such that
[math]\overrightarrow{u} = u_0\cdot\overrightarrow{i}, u_0 = 2 [/math]
Then, we must define the fluid stream function
[math]\psi(x,y) = \sqrt[]{ \nu \cdot\ u_0 \cdot x} f(\eta)[/math]
Where we take the viscosity [math] \nu [/math] as a unit value and
[math]\eta = y \sqrt[]{ \frac{u_0}{\nu x}}[/math]
[math] f(\nu) [/math] Satisfies the Blasius equation, and therefore we will raise the initial value problem associated with this equation with the following initial conditions
[math] \begin{cases} f’’’(\eta)+\frac{1}{2}f(\eta)f’’(\eta)=0 ; \\ f(\eta)=f’(\eta)=0, \lim_{\eta \to \infty}f’(\eta)= 1 ; \end{cases} [/math]
However, on time of programming we cannot introduce a conditional limit, so we have to replace them by the condition [math] f’’(\eta)=k [/math], and vary the values of k to find the one that satisfies the limit.
We cannot solve the differential equation like such, to apply the numerical methods, we need to pass it to a system of equations:
[math]
f(\eta)=y_1,f’(\eta)=y_2,f’’(\eta)=y_3\\
\begin{cases}
y_1’=y_2;\\
y_2’=y_3;\\
y_3’=-\frac{1}{2}y_1y_3;\\
y_1(0)=y_2(0)=0; y_3(0)=k
\end{cases}
[/math]
Once the system has been formulated , we start to solve it.
