In this paper, we consider the dynamics of a 2D target-searching agent performing Brownian motion under the influence of fluid shear flow and chemical attraction. The analysis is motivated by numerous situations in biology where these effects are present, such as broadcast spawning of marine animals and other reproduction processes or workings of the immune systems. We rigorously characterize the limit of the expected hit time in the large flow amplitude limit as corresponding to the effective one-dimensional problem. We also perform numerical computations to characterize the finer properties of the expected duration of the search. The numerical experiments show many interesting features of the process and in particular existence of the optimal value of the shear flow that minimizes the expected target hit time and outperforms the large flow limit.