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A proper vertex colouring is called a 2-dynamic colouring, if for every vertex v with degree at least 2, the neighbours of v receive at least two colours. The smallest integer k such that G has a dynamic colouring with k colours denoted by $\chi _2(G) $. We denote the cartesian product of G and H by $G\Box H $. In this paper, we find the 2-dynamic chromatic number of cartesian product of complete graph with complete graph $K_{r} \Box K_{s} $, complete graph with complete bipartite graph $K_n \Box K_{1,s} $ and wheel graph with complete graph $W_l \Box K_n $.