The recently introduced paradigm of Physics-Informed Neural Networks (PINNs) has offered a possible method to target CFD: the idea behind it is to use the expressive power of NNs in order to represent the solution to the target PDE, upon a suitable training phase against a loss function informing how closely a candidate solution satisfies the known equations. We introduce Differentiable Quantum Circuits (DQC), a protocol inspired by PINNs which relies upon the expressive power of quantum circuits to effectively replace classical NNs. The variational nature of DQC and the application of analytical differentiation rules makes it amenable to near-term quantum devices. In this work we showcase investigations of the DQC potential specifically for well-established CFD use-cases. We start by solving (i) a quasi-1D approximation for the convergent-divergent nozzle, then address the paradigmatic (ii) lid-driven cavity modelling and the (iii) analysis of the flow around wing-shaped objects, both within a full 2D approach, using either solely the analytical contribution from the loss, or also the additional information provided by coarse solutions from numerical solvers. In (i) we investigate the subsonic-supersonic transition, as well as the possibility to generalize a solution computed for only a subdomain to the whole domain, with an additional training stage. In (ii-iii), we show solutions for the fluid velocity in a laminar case and investigate transient regimes. For all cases, we adopt small quantum registers involving less than 12 qubits, suggesting the possibility of testing simplified models with DQC on realistic, near-term quantum hardware. In conclusion, we report findings on several computational fluid dynamics instances, where physics-informed solutions represented by DQC might go beyond qualitatively/quantitatively competing solutions with standard approaches. In particular, we stress the additional benefits of fast inference of additional solutions in regimes where no training was performed, fast interpolation coarse-grid solutions to achieve (both pointing towards the generalizability of the approach and the possibility to use it for design optimization), as well as leveraging upon the fast developments in quantum computing to alleviate energy-demand when obtaining such benefits from GPU implementations.