Brinkman equation

quick question. what's the correct way to express the Brinkman equation for steady state and incompressible flow?
is it :
(1) -miu_br*(u2x_x + u2y_y) + miu_br/kap_br*u2  + p2_x = Fx_br

or is it
(2) -miu_br*(u2x_x + u2y_y) + miu_br/kap_br*u2_t  + p2_x = Fx_br

the only difference is u2 versus u2_t.

is u2_t a time derivative? why does FEATool use this format u2_t?

Thanks
Sergio

Sergio wrote

What is the correct way to express the Brinkman equations for steady state and incompressible flow in FEATool Multiphysics's equation syntax?

Simply remove the terms with time derivative (single quoted) from the equations u'/v':

    u-equation:           - miu_br*(2*ux_x + uy_y + vx_y) + miu_br/kap_br*u_t  + p_x = Fx_br
    v-equation:           - miu_br*(vx_x + uy_x + 2*vy_y) + miu_br/kap_br*v_t  + p_y = Fy_br
    continuity-equation:    ux_t + vy_t = 0

Note, that if you use the stationary solver in FEATool Multiphysics this term will be ignored/omitted in the FEA discretization so there is usually no need to manually remove it (unless you want to use the time-dependent solver to solve a stationary problem).

Sergio wrote

Is u2_t a time derivative?

No, as stated above single quote denotes the time derivative of a dependent varibale.

In contrast, the postfix _t denotes multiplication with a (finite element) test function. For example:

    a*u*u_t => a*u*test_function(u) => implicit linear (LHS) FEM discretization => more computationally efficient

    a*u     => a*u                  => explicit non-linear (RHS) FEM discretization => less computationally efficient 

See also the equation syntax table in the custom equation section of the FEATool Multiphysics documentation.