In the paper Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method by Shaher Momani and Zaid Odibat ($2006$), the Navier-Stokes equation was written in time-fractional form as $$\dfrac{\partial^{\alpha}u}{\partial t^{\alpha}}=\dfrac{\partial^2 u}{\partial t^2}+\dfrac{1}{r}\dfrac{\partial u}{\partial r}$$ where $0<\alpha\leq1$.
If $\alpha=1$ then we know that $\dfrac{\partial u}{\partial t}$ represents the acceleration and the above equation represents the Navier-Stokes equation. But if $\alpha$ is a fraction (e.g., $\alpha=\dfrac{1}{2}$) then does the equation still represent the Navier-Stokes equation? In that case, $\dfrac{\partial^{1/2}u}{\partial t^{1/2}}$ will not represent acceleration anymore. In fact, is the resulting model meaningful?
This question just came out of curiosity when I went through this paper. Apologies if the question is too elementary.