Question

In the variational problem of fluid flow, what is the correct matrix form of C in the bilinear form Bv(w,v)=∫Ωe(Dw)^TC(Dv)dx?

(a) \(\begin{pmatrix}2&0&0\\0&2&0\\0&0&1\end{pmatrix}\)

(b) μ\(\begin{pmatrix}2&0&0\\0&1&0\\0&0&2\end{pmatrix}\)

(c) μ\(\begin{pmatrix}2&0&0\\0&2&0\\0&0&1\end{pmatrix}\)

(d) μ\(\begin{pmatrix}1&0&0\\0&2&0\\0&0&2\end{pmatrix}\)

This question was posed to me in final exam.

Origin of the question is Penalty topic in division Flows of Viscous Incompressible Fluids of Finite Element Method

Answer 1

The correct choice is (c) μ\(\begin{pmatrix}2&0&0\\0&2&0\\0&0&1\end{pmatrix}\)

Explanation: The complete restated form of the weak forms of viscous fluids flow equations is given by Bt(w,v)+Bv(w,v)-B̅p(w,P)=l(w) and –Bp(w3,v)=0. The bilinear form, Bv(w,v)=∫Ωe(Dw)^TC(Dv)dx is symmetric because it contains the symmetric matrix μ \(\begin{pmatrix}2&0&0\\0&2&0\\0&0&1\end{pmatrix}\), denoted by the letter C. C is the viscosity matrix since it contains the viscosity term μ.