We consider an approximating control design for optimal mixing of a non-dissipative scalar eld in unsteady Stokes ows. The objective of our approach is to achieve optimal mixing at a given nal time, via an active control of the ow velocity through boundary inputs. Due to the zero diusivity of the scalar eld, establishing the well-posedness of its G^aeaux derivative requires high regularity on velocity eld. This condition results in the need to penalize the time derivative of the boundary control in the cost functional. As a result, the optimality system becomes difficult to solve. Our current approximating approach will provide a more transparent optimality system, with the set of admissible controls being L^2 in both time and space. This is achieved by first introducing a small diusivity to the scalar equation and then establishing a rigorous analysis of convergence of the approximating control problem to the original one as the diffusivity approaches to zero. Uniqueness of the optimal solution is obtained for the two dimensional case.