Plant nutrients uptake from soil through the root system is essential for the completion of plant life cycle.Mathematical modeling study for plant root nutrients uptake is important task,for intensive farming,efficient food production of food agriculturally plants.Not only that,it is important also in the practice,that application of growing non-food,hyper-accumulator plants,which can take up large amounts of heavy metals,or remove radioactivity contaminations from polluted soil,or be used for mining of gold.In this paper,we shall develop a mathematical model for the dynamical process of minor elements up taken by a steadily growing plant root,and derive the uniformly valid asymptotic expansion solution for the problem,in terms of the slender body approximation theory.