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Biological growth is a common phenomenon in nature, and some organisms such as DNA molecules and bacterial filaments grow in viscous media. The growth induced instability of morphoelastic rod in a viscous medium is studied in this paper. Based on the Kirchhoff kinetic analogy method, the mechanical model for growing elastic thin rod in the viscous medium is established. A perturbation analysis is used to analyze the stability of the growing elastic rod in the viscous medium. We apply the results into planar growing ring and get its criterion of instability. Take the criterion into DNA ring to discuss the influence of viscous resistance on its instability.