Can you someone please help to write last term of the first and third equation in the variational form

Here , is the variational form:

I have tried to write last term of the first equation and third equation in 2D but i am getting product of two unknows present there which is not solvable by FreeFem++.

My trial: (cv, nabla xi)=(c(v,vv), (xi_{x}, xi_{y}))
=((cv,cvv),(xi_{x}, xi_{y}))
=cvxi_{x}+cvv xi_{y}), here product of two unkown is present.
Simillarly, i have tried for (c*theta, nabla w)

Please help how write this term in linear form so that FF can solve easily using the command verf.

Dear Monirul,
The problem is non-linear (which is what you mean by ’ i am getting product of two unknows’).
You must solve it iteratively. You may look at this example: Newton Method for the Steady Navier-Stokes equations

I have seen this example. This is not helpful for my work as i have seen. There is no term in the equation you refer brother.
I need a equation where nable.(cv) term present where v is a vector in R^2. Moreover, if it is a non-linear term of one unknown then i can handel using fixed -point method. Here it is product of two unknowns.

The vector form of v is not an issue, just work with its components.
You need to linearize the equation first then apply Newton’s method. You can look for instance in these lecture notes:
In the example I referred to, u is a vector and there are 3 unknowns (line 56, [du1, du2, dp]). In your case, you have 5 unknowns [c, mu, v1, v2, p].