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Flow velocity is an important characteristic of the fluidic mediums. In this paper, we introduce a molecular based flow velocity meter consisting of a molecule releasing node and a receiver that counts these molecules. We consider both flow velocity detection and estimation problems, which are employed in different applications. For the flow velocity detection, we obtain the maximum a posteriori (MAP) decision rule. To analyze the performance of the proposed flow velocity detector, we obtain the error probability, its Gaussian approximation and Chernoff information (CI) upper bound, and investigate the optimum and sub-optimum sampling times accordingly. We show that, for binary hypothesis, the sub-optimum sampling times using CI upper bound are the same. Further, the sub-optimum sampling times are close to the optimum sampling times. For the flow velocity estimation, we obtain the MAP and minimum mean square error (MMSE) estimators. We consider the mean square error (MSE) to investigate the error performance of the flow velocity estimators and obtain the Bayesian Cramer-Rao (BCR) and expected Cramer-Rao (ECR) lower bounds. Further, we obtain the optimum sampling times for each estimator. It is seen that the optimum sampling times for each estimator are nearly the same. The proposed flow velocity meter can be used to design a new modulation technique in molecular communication (MC), where information is encoded in the flow velocity of the medium instead of the concentration, type, or release time of the molecules. The setup and performance analysis of the proposed flow velocity detector and estimator for molecular communication system need further investigation.