second derivative temperature boundary condition in outlet
 

Hello my friends,
As we know in constant heat flux boundary condition, the fully developed BC in the outlet for temperature is grad(T)=grad(Tbulk)=grad(Twall)=constant. But I found in many papers that we can use the second derivative of temperature equal to zero in the outlet. At first, we know that this BC can be obtained by derivating from first relation. But is the second derivative well posed based on second order energy equation? or Is this BC means that grad(T)=constant not zero?
Thank you

yes, it means that n.grad T = constant along n

Thanks Prof. Denaro. Is second derivative of temperature equal to zero valid in the outlet?

yes, if the flow is fully developed you can assume also the second derivative of the velocity to vanish along the normal direction

I mean the second derivative for temperature too. If I use this BC, is my solution well-posed?

Yes, it is a Neumann condition applied to the heat flux

But I think that when we have a second order differential equation we should use Bc up to one order less than the equation.so for energy equation(second order) is this rational to use zero second derivative temperature in the outlet?

You should still consider the BC in terms of the non-homogeneous Neumann type: you prescribe the first derivative equal to some function. At an outlet you don't know exactly this function but, for a developed flow condition, you are just assuming that this function is such to take constant the normal derivative.