As far as I am aware equation (1.18) in 4th edition of Nield and Bejan is what you need. That equation also has the Prandtl-Darcy term (the time derivative of velocity), the advective inertia term (i.e. u.grad(u), though written in conservation form), and the Forchheimer  form-drag term. In many applications some or all of these terms are absent. When all are absent then Nield and Bejan's equation (1.17) is found, which expresses a balance between pressure gradient, Darcy drag and the Brinkman term.

Refer the attached articles

Yes, equation 2 of Lauriat's article is consistent with Nield and Bejan's.

The question now is the selection of the method to solve those equations if one requires solutions in 3D!

My dears  Andrew and Abimanyu,

Thank you very much for your answers. These equations are acceptable, but I want these equations in dimensionless parameter for porosity. In other words, I want the equations without porosity.

Best Regards

I am not sure that you will be able to remove the porosity. A standard nondimensionalisation will yield the Darcy term, u, with a unit coefficent, likewise for the pressure gradient term. The u|u| term will then be multiplied by a coefficient which is often called the Forchheimer number, and the Brinkman term will be multiplied by the Darcy number.

If you are considering isothermal flows, then u_t is the only time derivative, and so a suitable scaling for time may be used to remove porosity from that term. But then the advective inertia term turns out to have to contain the porosity, because there are no more scalings that may be applied.  But if you assume the porosity to be small, then you could neglect the advective inertia term.

On the other hand, if you are considering convection, then there are two time derivatives, one in the momentum equation and one in the heat transport equation. It is usual then to scale time using the thermal diffusivity and a lengthscale, and this leaves the d(theta)/dt term with a unit coefficient. But the consequence the u_t term is multiplied by the Prandtl-Darcy number (more recently called the Vadasz number), which involves the porosity.

These matters are tricky, though. The permeability is formally a function of porosity and of the microscopic geometry of the medium. So an analysis of the effect of varying the porosity, for example, ought to mean that the permeability should change too. If it is insisted that the permeability remains constant as the porosity changes, then one is in the bizarre situation of insisting therefore that the microscopic geometry changes as the porosity changes in order to maintain the same permeability! As far as I am aware, many authors are happy to let one parameter vary without considering what effect that that might have on the values of some of the other parameters.

Of course, the porosity is already dimensionless!

I doubt if what I have said will be what you wanted to hear, but I think that it is a true reflection of how things are!

Dear Mr. Andrew,

Thank you so much for your advice.

Regards

Tanks you can also see the yadav's articles