Square-Root Acceleration Drag Law of a Uniformly Accelerated Normal Plate

Reijtenbagh, Tummers and Westerweel [Phys. Rev. Lett. 130, 174001 (2023)] measured the instantaneous drag on plates accelerated normal to their own plane, reported \(\hat F_a=C_a\rho A\sqrt{\nu aV_a}\) with \(C_a\simeq291\) independent of acceleration and velocity, and rationalized it with a history force built on the one-dimensional Stokes first problem. We solve the correct model problem---an unsteady Hiemenz layer whose strain rate grows linearly in time---using the similarity transformation of Sun [Phys. Fluids 36, 083616 (2024)]. The two-dimensional boundary-layer equation collapses to a single equation in $(\eta,\tau)$ whose coefficients are independent of the streamwise coordinate; its leading-order solution is exact and is a Kummer function of the second kind. The transformation identifies the experimental dimensionless time with the diffusion time, \(\tau=2t^{*}\), and an exact geometric identity returns the empirical law with the same coefficient \(2/\sqrt\pi\), generalized to arbitrary velocity programs as a Basset memory integral. The resulting force is smaller than the measurement by a factor that we show factorizes exactly into the ratio \(h/\delta=\sqrt{a^{*}\mathrm{Re}_h}\) of plate size to diffusion length and a residual factor of two: the boundary layer accelerates fluid over \(\sqrt{\nu t}\) where the plate accelerates it over \(h\). We then test the empirical law against the published data. Digitizing the reported coefficient for all runs shows that \(C_a\) is not constant but rises as \(V_a^{0.36\,[0.28,0.45]}\), excluding the assumed \(\nu^{1/2}\) scaling at about seven standard deviations. Frame-by-frame analysis of the supplemental video shows that \( F_D \approx \rho \, l_a \, \frac{d(\Gamma b_v)}{dt} \) to within $10\%$ throughout the acceleration, with no fitted parameter. The unsteady force on a starting plate is therefore vortical, and a single experiment---repeating a few runs at elevated viscosity---would settle the exponent.