# Question:Cannot solve with dsolve the self similarity problem for external natural convection (boundary layer and Boussinesq approximation)

## Question:Cannot solve with dsolve the self similarity problem for external natural convection (boundary layer and Boussinesq approximation)

Maple 2021

Hello,

I am trying to solve the self similarity problem for external natural convection which is the following coupled problem

•     3/4*F(eta)*Theta(eta)'  = Theta(eta)''
•     1/2 (F(eta)')^2 - 3/4 F(eta)*F(eta)'' = Pr*(Theta(eta) - F(eta)''')

where Pr is a constant and  ()' denotes differentiation with respect to eta. The boundary conditions are:

•     at eta = 0, F = 0, F'=0, Theta = 1
•     at eta -> infinity, F' = 0, Theta = 0

I tried the following but I cannot obtain a solution

 > Pr:=1
 (1)
 > etamax := infinity
 (2)

 > eq:= 3/4*F(eta)*diff(theta(eta),eta) = diff(theta(eta),eta,eta),1/Pr*(1/2*diff(F(eta),eta)^2 - 3/4*F(eta)*diff(F(eta),eta,eta)) = -diff(F(eta),eta,eta,eta) + theta(eta)
 (3)

 >

 > bcs := F(0) = 0, theta(0)=1, D(F)(0)=0,theta(etamax)=0,D(F)(etamax)=0
 (4)

 > sol:=dsolve([eq,bcs],[F(eta),theta(eta)])