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In order to solve the near field description of a round jet, I want to work out the variables [tex] F(\eta) [/tex], [tex] \rho(\eta) [/tex] and [tex] Y(\eta) [/tex] which represents the self similar stream function, density, and mass fraction respectively. The system obtained is:
[tex] \Big(\frac{F'}{\rho}\Big)''+\frac{F}{2}\Big(\frac{F'}{\rho}\Big)'=0 [/tex] (1)
[tex] (\rho Y')'+\frac{F}{2}Y'=0 [/tex] (2)
besides a function [tex]\rho=\rho(Y)[/tex] which is known previously.
Boundary conditions are:
1) [tex] \eta\rightarrow+\infty[/tex]; [tex] \frac{F'}{\rho}\rightarrow 0[/tex]; [tex] Y \rightarrow 0[/tex];
2) [tex] \eta\rightarrow-\infty[/tex]; [tex] \frac{F'}{\rho}\rightarrow 1[/tex]; [tex] Y \rightarrow 1[/tex]; [tex] F\rightarrow \eta[/tex];
The first question I have is how can I transform (1) into a system of three first order ordinary differential equations. I have done it yet before with Blasius type equations, but here the density makes it a bit difficult. The aim of my question is to compute both coupled equations with a Non Linear Shooting Method.
Thanks in advance.
[tex] \Big(\frac{F'}{\rho}\Big)''+\frac{F}{2}\Big(\frac{F'}{\rho}\Big)'=0 [/tex] (1)
[tex] (\rho Y')'+\frac{F}{2}Y'=0 [/tex] (2)
besides a function [tex]\rho=\rho(Y)[/tex] which is known previously.
Boundary conditions are:
1) [tex] \eta\rightarrow+\infty[/tex]; [tex] \frac{F'}{\rho}\rightarrow 0[/tex]; [tex] Y \rightarrow 0[/tex];
2) [tex] \eta\rightarrow-\infty[/tex]; [tex] \frac{F'}{\rho}\rightarrow 1[/tex]; [tex] Y \rightarrow 1[/tex]; [tex] F\rightarrow \eta[/tex];
The first question I have is how can I transform (1) into a system of three first order ordinary differential equations. I have done it yet before with Blasius type equations, but here the density makes it a bit difficult. The aim of my question is to compute both coupled equations with a Non Linear Shooting Method.
Thanks in advance.