In this paper, we investigate the feasibility of automatically recognizing the fine-grained stenosis rate of arteriovenous fistula (AVF) by analyzing blood flow sounds. Due to the challenges of recruiting patients with varying stenosis rates of AVF for data collection, we initially developed an AVF animal model using Beagle dogs. We manipulated the stenosis rates in their AVFs and recorded the corresponding blood flow sound data, resulting in the creation of the BeagleAVF dataset. This dataset consists of 3,351 blood flow sound segments from five Beagle dogs, each labeled with six distinct stenosis rates using synchronized digital subtraction angiography. We also introduce a novel deep learning method, the Frequency-Aware Decoupling Network (FAD-Net), designed to address the challenge of recognizing fine-grained stenosis rates of AVF through blood flow sounds. The core concept of FAD-Net is to seek stenosis-related frequency bands within the spectrums of blood flow sounds, thereby facilitating the learning of stenosis rate-discriminative features. To evaluate the FAD-Net, we conducted extensive experiments using the BeagleAVF dataset. The results demonstrate the effectiveness and superiority of FAD-Net compared to several baseline deep neural networks. Furthermore, our work provides experimental evidence supporting the feasibility of recognizing fine-grained stenosis rates of AVF through the analysis of blood flow sounds.