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

1. Status of the One-Dimensional Template

Equation (4) of Ref. [1] is the general solution of Stokes’ first problem [2,3]: a plate translating in its own plane in a semi-infinite fluid. The experiment concerns a plate translating normal to its own plane, on whose front face the surface flow is a stagnation flow that runs outward from the stagnation point and is stripped off at the edge; the authors say as much (their Figure 1(e,f), “resembling Hiemenz flow” [4]).

The template might therefore be expected to be simply the wrong problem. It is not. As shown in Section 3.1, the velocity defect of the accelerating stagnation layer obeys the heat equation with a wall value equal to the local slip speed, so Equation (4) of Ref. [1] is the leading-order solution of the correct problem, and their coefficient $C=2/\sqrt{\pi }$ is the correct one. What the one-dimensional route misses is threefold: it assigns the wall value V rather than $VU\left(x\right)$, so it is pointwise wrong everywhere on the face and survives only through the identity ${\int }_{\mathrm{face}}U\mathrm{d}A=A$ derived in Section 4; it retains no convective term, so it provides no criterion for its own validity and no correction; and it interprets a shear tangential to the face, whose net is zero by symmetry, as a force along the direction of motion.

Supplying those three items requires the unsteady Hiemenz layer whose outer strain rate grows linearly in time. No classical solution exists for it: the unsteady two-dimensional boundary layer resisted Stewartson [5], Dennis [6] and others [7] for seven decades. That is precisely the gap filled by the similarity transformation of Sun [8], which we apply below.

2. Formulation and Reduced Equation

Figure 1 defines the geometry. Work in the plate frame; let x run along the front face from the stagnation point, y normal to it, and let h denote the half-height of the plate in the measurement plane. Potential flow past a normal lamina [9] gives the slip velocity

$$\begin{array}{cc}\hfill u_e(x,t)& =V\left(t\right)U\left(x\right),\hfill \\ \hfill U\left(x\right)& ={\displaystyle \frac{x}{\sqrt{h^2-x^2}}}\simeq {\displaystyle \frac{x}{h}}(x\ll h),\hfill \end{array}$$

(1)

so that near the stagnation point $u_e\simeq B\left(t\right)x$ with $B=V/h$. For $V=at$ write $B=bt$ with $b=a/h$. The boundary-layer equation on the face [10] is

$$u_t+u{u}_x+v{u}_y=\dot{B}x+B^{2}x+\nu u_{yy}.$$

(2)

Sun’s transformation for $U\propto x^m$ has $m=1$ here, which makes the layer thickness independent of x; his $\delta \left(x\right)$ becomes a $\delta \left(t\right)$, and his dimensionless diffusion time closes the problem,

$$\begin{array}{c}\psi =B\left(t\right)x\delta \left(t\right)f(\eta ,\tau ),\delta =\sqrt{\nu /B},\\ \eta =y/\delta ,\tau =\nu t/{\delta }^2=Bt.\end{array}$$

(3)

Then $u=Bx{f}_{\eta }$, $v=-B\delta f$, continuity is satisfied identically, and, using $\dot{B}/B^2=1/\tau$, $\dot{\delta }/\delta =-1/\left(2\tau \right)$ and ${\tau }_t/B=2$, every occurrence of x cancels from Equation (2):

$$f_{\eta \eta \eta }+f{f}_{\eta \eta }+1-f_{\eta }^2=2f_{\eta \tau }+{\displaystyle \frac{1}{\tau }}\left[f_{\eta }-1+\frac{\eta }{2}{f}_{\eta \eta }\right],$$

(4)

The boundary and initial conditions are

$$\begin{array}{cccccc}\hfill & \mathrm{no}\mathrm{slip}\text{:}\hfill & \hfill & u(x,0,t)=0\hfill & \hfill & \Rightarrow f_{\eta }(0,\tau )=0,\hfill \\ \hfill & \mathrm{no}\mathrm{penetration}\text{:}\hfill & \hfill & v(x,0,t)=0\hfill & \hfill & \Rightarrow f(0,\tau )=0,\hfill \\ \hfill & \mathrm{matching}\text{:}\hfill & \hfill & u\to u_e(x,t)\hfill & \hfill & \Rightarrow f_{\eta }(\infty ,\tau )=1,\hfill \\ \hfill & \hfill & & (y\to \infty )\hfill & \hfill & \\ \hfill & \mathrm{rest}\mathrm{at}t=0:\hfill & \hfill & u\equiv 0.\hfill & \hfill \end{array}$$

(5)

The plate frame is non-inertial, but its acceleration is directed along y, so the x-momentum equation (2) acquires no additional term and only $\partial p/\partial y$ is affected, which is already discarded in the boundary-layer approximation. Note that Equation (2) is second order in y at leading order, so the wall shear is fixed by the no-slip and matching conditions alone; no-penetration serves only to fix the constant of integration in $f={\int }_0^{\eta }{f}_{\eta }\mathrm{d}{\eta }^{\prime }$ and does not enter the force. For a general power-law program $V\propto t^n$ the same reduction gives

$$f_{\eta \eta \eta }+f{f}_{\eta \eta }+1-f_{\eta }^2=(n+1)f_{\eta \tau }+{\displaystyle \frac{n}{\tau }}\left[f_{\eta }-1+\frac{\eta }{2}{f}_{\eta \eta }\right],$$

(6)

which for $n=0$ is the classical impulsively started Hiemenz equation and for $\tau \to \infty$ degenerates to the steady Hiemenz equation $f^{‴}+f{f}^{″}+1-f^{\prime 2}=0$ [10], itself the $\beta =1$ member of the Falkner–Skan family [11]. Equation (4) is thus the unsteady Blasius-type equation [12] for the accelerating normal plate.

Interpretation of $t^{*}$.

During the ramp $\tau =Bt=Vt/h$, whereas the experimental dimensionless time [1] is $t^{*}=h^{-1}\int V\mathrm{d}t=Vt/2h$. Hence

$$\tau =2t^{*}(t<t_a),\tau =t^{*}+t_a^{*}(t>t_a),{\tau }_a={\displaystyle \frac{1}{a^{*}}},$$

(7)

with $a^{*}\equiv ah/V_a^2$. The empirical “number of plate heights travelled” is, up to a factor two, Sun’s diffusion time. This is not evident from Ref. [1] and explains why all their curves collapse in $t^{*}$.

3. Exact Leading-Order Solution

At early times the layer thickness is $\sqrt{\nu t}$, i.e. $\zeta \equiv \eta /\sqrt{\tau }=y/\sqrt{\nu t}$. Writing $f=\sqrt{\tau }F(\zeta ,\tau )$, Equation (4) becomes

$$F_{\zeta \zeta \zeta }+\frac{\zeta }{2}{F}_{\zeta \zeta }-F_{\zeta }+1=\tau \left[2F_{\zeta \tau }-F{F}_{\zeta \zeta }-1+F_{\zeta }^2\right].$$

(8)

At $O\left(1\right)$, with $G\equiv F_{\zeta }=u/u_e$,

$$G^{″}+\frac{\zeta }{2}{G}^{\prime }-G+1=0,G\left(0\right)=0,G(\infty )=1,$$

(9)

whose decaying solution is a Kummer function of the second kind [8,13]. Here ${\mathrm{i}}^n\mathrm{erfc}$ denotes the nth repeated integral of the complementary error function, defined by ${\mathrm{i}}^{-1}\mathrm{erfc}z=(2/\sqrt{\pi })e^{-z^2}$, ${\mathrm{i}}^0\mathrm{erfc}z=\mathrm{erfc}z$ and ${\mathrm{i}}^n\mathrm{erfc}z={\int }_z^{\infty }{\mathrm{i}}^{n-1}\mathrm{erfc}s\mathrm{d}s$, with ${\mathrm{i}}^n\mathrm{erfc}0=1/\left[2^n\mathrm{\Gamma }(1+n/2)\right]$; explicitly,

$$\begin{array}{cc}\hfill G\left(\zeta \right)& =1-{\displaystyle \frac{1}{\sqrt{\pi }}}{e}^{-{\zeta }^2/4}U\left(\frac{3}{2},\frac{1}{2},\frac{{\zeta }^2}{4}\right)=1-4{\mathrm{i}}^2\mathrm{erfc}\left(\frac{\zeta }{2}\right)\hfill \\ \hfill & =1-\left(1+\frac{{\zeta }^2}{2}\right)\mathrm{erfc}\frac{\zeta }{2}+\frac{\zeta }{\sqrt{\pi }}{e}^{-{\zeta }^2/4},\hfill \end{array}$$

(10)

with

$$\begin{array}{c}\hfill G^{\prime }\left(0\right)={\displaystyle \frac{2}{\sqrt{\pi }}},G^{″}\left(0\right)=-1,\\ \hfill {\delta }^{*}={\displaystyle \frac{4}{3\sqrt{\pi }}}\sqrt{\nu t}=0.7523\sqrt{\nu t}.\end{array}$$

(11)

For $V\propto t^n$ the profile obeys $G^{″}+\frac{\zeta }{2}{G}^{\prime }-nG+n=0$, so

$$G_n\left(\zeta \right)=1-{\displaystyle \frac{{\mathrm{i}}^{2n}\mathrm{erfc}(\zeta /2)}{{\mathrm{i}}^{2n}\mathrm{erfc}\left(0\right)}},G_n^{\prime }\left(0\right)={\displaystyle \frac{\mathrm{\Gamma }(n+1)}{\mathrm{\Gamma }(n+\frac{1}{2})}},$$

(12)

which returns $\mathrm{erf}(\zeta /2)$ and $1/\sqrt{\pi }$ for the impulsive start ($n=0$), and Equation (10) with $2/\sqrt{\pi }$ for constant acceleration ($n=1$). The whole hierarchy of corrections lives in repeated integrals of the complementary error function [13], i.e. in Kummer functions—the structure Sun found for the flat-plate problem [8]. Figure 2(a) contrasts G with the Rayleigh profile: constant acceleration produces a fuller profile and a wall shear larger by exactly a factor two.

3.1. Equivalence with the One-Dimensional Template

Define the velocity defect $w=u_e-u$. Because $\partial u_e/\partial t$ is independent of y, the forcing cancels identically and Equation (2) reduces at leading order to

$$\begin{array}{cc}\hfill w_t& =\nu w_{yy},\hfill \\ \hfill w(x,0,t)& =u_e(x,t),\hfill \\ \hfill w(x,\infty ,t)& =0,\hfill \end{array}$$

(13)

which is Stokes’ first problem for a wall started from rest with velocity $u_e(x,t)=V\left(t\right)U\left(x\right)$. For $V=at$ its solution is $w=u_e[1-G\left(\zeta \right)]$ with G given by Equation (10), i.e. precisely Equation (4) of Ref. [1] with the plate velocity replaced by the local slip velocity; we have verified the two expressions agree to machine precision. The equivalence resides entirely in the boundary condition: the defect at the wall is the local slip speed, not the speed of the plate. This establishes the status of the one-dimensional template asserted in Section 1. The equivalence is special to the leading order—at $O\left(\tau \right)$ the convective terms enter through $S\left(\zeta \right)$ in Equation (19) and no one-dimensional analogue exists.

4. The Diffusive Force and Its Memory Kernel

The diffusive momentum transfer from the face is ${\int }_{\mathrm{face}}{\tau }_w\mathrm{d}A$ with ${\tau }_w=\rho \nu u_y{|}_0=\rho VU\left(x\right)G^{\prime }\left(0\right)\sqrt{\nu /t}$. The geometric integral obeys an exact identity. For the two-dimensional lamina,

$${\int }_{\mathrm{face}}U\mathrm{d}A={\mathcal{l}}_a{\int }_{-h}^h{\displaystyle \frac{\left|x\right|\mathrm{d}x}{\sqrt{h^2-x^2}}}={\mathcal{l}}_a\left(2h\right)=A,$$

(14)

and for a disk of radius R, where potential flow gives $U=(2/\pi )r/\sqrt{R^2-r^2}$,

$${\int }_0^R{\displaystyle \frac{2}{\pi }}{\displaystyle \frac{r}{\sqrt{R^2-r^2}}}2\pi r\mathrm{d}r=\pi R^2=A.$$

(15)

The mean potential slip speed over the face equals V exactly, for both shapes.

Equations (14)–(15) assume attached potential flow, which is singular at the sharp edge; neither the present treatment nor Ref. [1] imposes a Kutta-type condition there. Capping the slip at $u_e^{max}=MV$, as separation requires, gives the closed form

$${\displaystyle \frac{1}{h}}{\int }_0^{h}U\mathrm{d}x=\mathrm{\Lambda }\left(M\right)\equiv 1+M-\sqrt{1+M^2},$$

(16)

equal to $0.586$, $0.783$ and $0.901$ for $M=1$, $2.2$ and 5, and tending to unity as $M\to \infty$. With the measured $M=2.2$ of Section 7 the leading coefficient falls from $2/\sqrt{\pi }=1.128$ to $0.884$. This is the only boundary condition in the chain that alters the result, and it does so by $22\%$. It must not be applied together with the shape factor $\mathrm{\Phi }\left(\tau \right)$ of Section 5, which already carries part of the same edge physics; the two are alternative treatments and are used below only to bracket the estimate, a bracket far narrower than the quantity it is compared with. Hence

$$F_a^{\mathrm{diff}}\left(t\right)=\rho AV\sqrt{\nu /t}{\displaystyle \frac{\mathrm{\Gamma }(n+1)}{\mathrm{\Gamma }(n+\frac{1}{2})}}\stackrel{n=1}{\to }{\displaystyle \frac{2}{\sqrt{\pi }}}\rho Aa\sqrt{\nu t}.$$

(17)

Because the leading-order problem $u_t=U\dot{V}+\nu u_{yy}$ is linear, an arbitrary velocity program gives a memory integral with the Basset kernel [14,15],

$$F_a^{\mathrm{diff}}\left(t\right)={\displaystyle \frac{\rho A\sqrt{\nu }}{\sqrt{\pi }}}{\int }_0^t{\displaystyle \frac{a\left(s\right)}{\sqrt{t-s}}}\mathrm{d}s.$$

(18)

For the experimental ramp this is $(2/\sqrt{\pi })\rho Aa\sqrt{\nu }[\sqrt{t}-H(t-t_a)\sqrt{t-t_a}]$: identically Equation (5) of Ref. [1], with the same constant and the same $t^{-1/2}$ relaxation tail, which is also the impulsively started asymptote [16]. The one-dimensional template gave the right number for the wrong reason; the two-dimensional calculation returns $2/\sqrt{\pi }$ only because of Equations (14)–(15). Equation (17) further predicts a shape-insensitive leading coefficient, consistent with the measured $C_a=291,308,311$ for rectangle, square and disk [1].

5. Corrections to the Leading Order

5.1. The $O\left(\tau \right)$ Term

With $F=F_0+\tau F_1+\cdots$ and $G_1=F_1^{\prime }$, Equation (8) gives

$$G_1^{″}+\frac{\zeta }{2}{G}_1^{\prime }-3G_1=S\left(\zeta \right),S\equiv G^2-1-F_0{G}^{\prime },$$

(19)

with $G_1\left(0\right)=G_1(\infty )=0$ and $F_0\left(\zeta \right)=\zeta -\frac{4}{3\sqrt{\pi }}+8{\mathrm{i}}^3\mathrm{erfc}(\zeta /2)$. The homogeneous operator $h^{″}+\frac{\zeta }{2}{h}^{\prime }-\lambda h=0$ has decaying solution ${\mathrm{i}}^{2\lambda }\mathrm{erfc}(\zeta /2)$, so $h_1={\mathrm{i}}^6\mathrm{erfc}(\zeta /2)$ with $h_1\left(0\right)=1/384$. The operator is self-adjoint with weight $w=e^{{\zeta }^2/4}$, so Green’s identity yields the wall value without solving the boundary-value problem,

$$\begin{array}{cc}\hfill G_1^{\prime }\left(0\right)& =-384{\int }_0^{\infty }{e}^{{\zeta }^2/4}{\mathrm{i}}^6\mathrm{erfc}\left(\frac{\zeta }{2}\right)S\left(\zeta \right)\mathrm{d}\zeta \hfill \\ \hfill & =0.4808307846\dots \hfill \end{array}$$

(20)

(the integrand decays as ${\zeta }^{-7}$ times a Gaussian). An integer-relation search against $\{1,{\pi }^{-1/2},{\pi }^{-1},\sqrt{2}/\pi ,ln(1+\sqrt{2})\}$ found no closed form. Therefore

$$f_{\eta \eta }(0,\tau )={\displaystyle \frac{2}{\sqrt{\pi }}}{\tau }^{-1/2}\left[1+c_1\tau +O\left({\tau }^2\right)\right],$$

(21)

with $c_1={\displaystyle \frac{\sqrt{\pi }}{2}}{G}_1^{\prime }\left(0\right)=0.42613$, i.e. $F_a=(2/\sqrt{\pi })\rho Aa\sqrt{\nu t}[1+0.852t^{*}]$. Direct numerical solution of Equation (4) returns a small-$\tau$ slope of $0.425$ [Figure 2(b)], confirming both.

5.2. Local Strain

The leading solution is local, so B may be replaced by the local strain $u_e^{\prime }\left(x\right)=V{h}^2{(h^2-x^2)}^{-3/2}$, giving ${\tau }_{\mathrm{loc}}=\tau {(1-x^2/h^2)}^{-3/2}$. Two consequences follow. First, the face is never in a single regime: ${\tau }_{\mathrm{loc}}=1$ at $1-x/h\simeq \frac{1}{2}{\tau }^{2/3}$, so an outer sliver of relative width ${\tau }^{2/3}/2$ is strain-saturated while the core is still Rayleigh; since $\int U\mathrm{d}x$ near the edge behaves as $\sqrt{h^2-x^2}$, that sliver carries a fraction ${\tau }^{1/3}$ of the shear integral. Second, separating the instantaneous quasi-steady wall shear (which is the sharp-edge friction singularity and must be cut off by separation) from the memory excess $R_{\mathrm{mem}}$, the history force acquires a shape factor,

$$\begin{array}{cc}\hfill F_a& ={\displaystyle \frac{2}{\sqrt{\pi }}}\rho Aa\sqrt{\nu t}\mathrm{\Phi }\left(\tau \right),\hfill \\ \hfill \mathrm{\Phi }& ={\displaystyle \frac{1}{A}}{\int }_{\mathrm{face}}U{R}_{\mathrm{mem}}\left({\tau }_{\mathrm{loc}}\right)\mathrm{d}A,\hfill \end{array}$$

(22)

with ${\mathrm{\Phi }}_{\mathrm{strip}}={\int }_0^{1}u{(1-u^2)}^{-1/2}{R}_{\mathrm{mem}}\left(\tau {(1-u^2)}^{-3/2}\right)\mathrm{d}u$ and ${\mathrm{\Phi }}_{\mathrm{disk}}=\frac{4}{\pi }{\int }_0^1{r}^2{(1-r^2)}^{-1/2}{R}_{\mathrm{mem}}(·)\mathrm{d}r$, both plotted in Figure 2(b). If the edge singularity is capped at $u_e^{max}=MV$, then ${\tau }_{\mathrm{loc}}\le \tau {(1+M^2)}^{3/2}$ and $\mathrm{\Phi }\approx 1$ requires $\tau {(1+M^2)}^{3/2}\lesssim 0.1$.

5.3. The Displacement–Transpiration Pressure

The leading profile is x-independent, so ${\delta }^{*}=k_{\delta }\sqrt{\nu t}$ with $k_{\delta }=4/\left(3\sqrt{\pi }\right)$ is uniform over the face and the equivalent blowing is $v_w={\partial }_x\left(u_e{\delta }^{*}\right)=k_{\delta }\sqrt{\nu t}V\left(t\right)U^{\prime }\left(x\right)$. For a rigid screen this drives a half-space Neumann potential ${\phi }^{\prime }{|}_{\mathrm{wall}}=-\frac{1}{2\pi }\int v_w\left({\mathit{x}}^{\prime }\right){|\mathit{x}-{\mathit{x}}^{\prime }|}^{-1}\mathrm{d}{A}^{\prime }$, and unsteady Bernoulli [17] gives an excess pressure $p^{\prime }=-\rho {\partial }_t{\phi }^{\prime }>0$ on the face. Using ${\int }_{\mathrm{disk}}\mathrm{d}A/|\mathit{x}-{\mathit{x}}^{\prime }|=4RE(r^{\prime }/R)$ the double integral collapses to a single one, and with ${\partial }_t\left(V\sqrt{t}\right)=\frac{3}{2}a\sqrt{t}$,

$$\begin{array}{cc}\hfill \Delta F& ={\displaystyle \frac{16I\left(M\right)}{{\pi }^{5/2}}}\rho A\sqrt{\nu aV},\hfill \\ \hfill I\left(M\right)& ={\int }_0^{\sqrt{1-M^{-2}}}{\displaystyle \frac{rE\left(r^2\right)}{{(1-r^2)}^{3/2}}}\mathrm{d}r,\hfill \end{array}$$

(23)

so that $\Delta F/F_a^{\mathrm{diff}}=8I\left(M\right)/{\pi }^2$, equal to $0.58$, $1.09$, $2.03$, $3.77$ for $M=1.5,2,3,5$. This is the same $\sqrt{\nu aV}$ group as Equation (17), but obtained as a pressure force normal to the plate, which removes the inconsistency of identifying a tangential shear with a drag and makes the shape-only dependence of $C_a$ natural, since M is a geometric quantity.

6. Numerical Solution of the Reduced Equation

Equation (4) was integrated as a parabolic problem for $W=f_{\eta }$,

$$2W_{\tau }=W_{\eta \eta }+f{W}_{\eta }+1-W^2-{\displaystyle \frac{1}{\tau }}\left[W-1+\frac{\eta }{2}{W}_{\eta }\right],$$

(24)

with $f={\int }_0^{\eta }W\mathrm{d}{\eta }^{\prime }$, on $\eta \in [0,20]$ with 5000 intervals, backward Euler with a tridiagonal solve, geometric time steps $\mathrm{d}\tau =0.002\tau$, started at $\tau ={10}^{-4}$ from the exact profile (10). The solution converges to $f_{\eta \eta }\left(0\right)\to 1.2340$ against the Hiemenz value $1.232588$ ($0.1\%$). Figure 2(b) shows the wall-shear amplification relative to the pure $\sqrt{t}$ law, $R\left(\tau \right)\equiv f_{\eta \eta }(0,\tau )\sqrt{\pi \tau }/2$: unity to within $5\%$ for $\tau \lesssim 0.15$, then a crossover near $\tau \simeq 1$ to the strain-saturated branch $R\to 1.0923\sqrt{\tau }$, where the force loses its acceleration dependence entirely, $F_a^{\mathrm{diff}}\propto \rho A{V}^{3/2}\sqrt{\nu /h}$.

7. Analysis of the Supplemental Video

The supplemental animation of Ref. [1] (1073 frames, $640\times 336$ pixels) was analyzed frame by frame: the plate, the two edge vortex cores and the three panel markers were tracked, and the force and circulation curves were digitized from the axes.

Calibration. 

The marker in the $V\left(t\right)$ panel gives $a=0.80\mathrm{m}{\mathrm{s}}^{-2}$, $V_a=0.40\mathrm{m}{\mathrm{s}}^{-1}$, $t_a=0.50\mathrm{s}$. The plate spans exactly two units in $y^{*}$ and its displacement in $x^{*}$ equals $t^{*}$ to within $1.5\%$ over the whole run, so the normalizing length is the half-height, $h=0.10$ m, the in-plane height is $0.20$ m, and $a^{*}=ah/V_a^2=0.50$, $t_a^{*}=1.0$, ${\tau }_a=2$. This is the run of Figure 1 of Ref. [1].

What cannot be measured. 

The upstream face is blank in every frame: no resolved vorticity and no resolved near-wall vectors. With $\sqrt{\nu t_a}\approx 0.7$ mm against a vector pitch of $\approx 2$ mm this is expected, so B and M are not directly accessible. We note that the profile in Figure 1(f) of Ref. [1] rises over 2–5 mm, three to seven times thicker than $\sqrt{\nu t}$; whatever that figure resolves, it is not the laminar Hiemenz layer.

Vortex kinematics. 

The pair remains attached and at constant spacing: $b_v/h=2.0$–$2.24$ for $0.25\le t^{*}\le 3$, with cores within $0.1$–$0.5h$ of the plate [Figure 3(b)]. Beyond $t^{*}\approx 3$ the tracker fails, which is itself the reported breakup into a three-dimensional wake [18].

Impulse test. 

Since ${\mathcal{l}}_a{b}_v\simeq A$, the impulse of the vortex pair [9,19,20,21] predicts $F\simeq \rho A\dot{\mathrm{\Gamma }}$. Comparing the digitized $F_D$ with $\rho {\mathcal{l}}_a\mathrm{d}\left(\mathrm{\Gamma }{b}_v\right)/\mathrm{d}t$, built from the independently digitized ${\mathrm{\Gamma }}^{*}$ and the independently tracked $b_v$ with no fitted parameters, gives, over $0.25<t^{*}<1.3$, a mean deviation of $-2.5\%$, an rms deviation of $10\%$ and a maximum deviation of $15\%$, with the peak reproduced [Figure 3(a)]. The residual scatter is dominated by differentiating a curve digitized from a 137-pixel-wide panel; the quasi-steady level carries a separate uncertainty, $F_{QS}=1.6$–$2.1$ N for $C_D=1.0$–$1.3$, which propagates to ${\widehat{F}}_a=3.3$–$2.8$ N. After $t^{*}\gtrsim 1.5$ the agreement degrades, as it must: the pair begins to convect and shed, and the tracked $\mathrm{\Gamma }$ no longer represents the whole wake. Since $F_D\simeq \rho A\dot{\mathrm{\Gamma }}$ while $F_{QS}=\frac{1}{2}{C}_D\rho A{V}_a^2$ corresponds to the steady shear-layer feeding rate $\frac{1}{2}{C}_D{V}_a^2$ [22], the empirical $\sqrt{a}$ law is equivalent to a statement about how far the edge shear layer over-feeds the vortex during the acceleration.

Edge velocity ratio. 

The shear layer feeds the vortex at $\dot{\mathrm{\Gamma }}=\frac{1}{2}{u}_{e,max}^2=\frac{1}{2}{M}^2{V}^2$. Integrating to $t_a$ and matching the measured $\mathrm{\Gamma }\left(t_a\right)={\mathrm{\Gamma }}^{*}{V}_{a}h=0.063{\mathrm{m}}^2{\mathrm{s}}^{-1}$ gives $M^2\simeq 4.7$, i.e. $M=u_e^{max}/V\simeq 2.2$, the quantity required by Equations (22) and (23). It gives $8I\left(M\right)/{\pi }^2=1.28$ and ${\tau }_{\mathrm{loc}}^{max}\simeq 15\tau$.

8. Absence of a Turbulent Contribution

Since the derived coefficient falls two orders of magnitude short of the measured $C_a$, it is natural to ask whether a gradient-transport closure, $\nu \to \nu +{\nu }_t$, could close the gap: matching $C_a=291$ against $2/\sqrt{\pi }$ requires only ${\nu }_{\mathrm{eff}}/\nu =6.7\times {10}^4$, i.e. ${\nu }_{\mathrm{eff}}=0.067{\mathrm{m}}^2{\mathrm{s}}^{-1}$, which is of the same order as $\mathrm{\Gamma }\left(t_a\right)\approx 0.065{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and as $Vh$. On magnitude alone the idea is attractive. It fails on three independent counts.

Exponents. Write the closure in the most general admissible form ${\nu }_t=\kappa V^{\alpha }{a}^{\beta }{\mathcal{l}}^{\gamma }$; dimensional consistency demands $\alpha +2\beta =1$ and $\alpha +\beta +\gamma =2$. Substituting in Equation (17) gives $F_a\propto \rho A{V}^{(\alpha +1)/2}{a}^{(\beta +1)/2}{\mathcal{l}}^{\gamma /2}$, so the measured exponents $a^{0.51\pm 0.05}{V}_a^{0.52\pm 0.04}$ force $\alpha =\beta =0$ and hence $\alpha +2\beta =0\ne 1$. No eddy viscosity built from the flow scales can reproduce the observed law. The two natural closures are special cases: the Prandtl mixing length [23,24] with outer scales, ${\nu }_t=cV\mathcal{l}$, gives $F_a\propto V^1{a}^{1/2}$, whose V exponent lies twelve standard deviations from the fit of Ref. [1]; and a circulation-based closure ${\nu }_t=c\mathrm{\Gamma }\simeq c{M}^2{V}^3/6a$ gives $F_a\propto \rho A{V}^2$, which collapses the history force into the form drag and removes the acceleration dependence altogether. Structurally, the measured law requires ${\nu }_{\mathrm{eff}}$ to be independent of V and a, so it must carry its own time scale; the problem offers only $\mathcal{l}/V$ and $V/a$, and the sole quantity supplying an independent one is a material property. This is why $\nu$ must appear in Equation (3) of Ref. [1], and simultaneously why it cannot be a turbulent $\nu$.

Stability. For the video run at $t=t_a$ the exact profile (10) gives $\theta =0.348\sqrt{\nu t_a}=0.246$ mm and ${\delta }_{99}=2.05$ mm, with shape factor $H={\delta }^{*}/\theta =2.16$. With $u_e\simeq M{V}_a=0.88\mathrm{m}{\mathrm{s}}^{-1}$ this is ${\mathrm{Re}}_{\theta }=217$ and ${\mathrm{Re}}_{{\delta }_{99}}=1.8\times {10}^3$, far below any transition threshold [10]. Decisively, the acceleration parameter $K=\nu u_e^{\prime }/u_e^2$ runs from $2.5\times {10}^{-5}$ at the plate edge to $6\times {10}^{-4}$ near the stagnation point, one to two orders above the relaminarization threshold $K\approx 3\times {10}^{-6}$ [25,26]. A stagnation-point layer under this much favorable strain relaminarizes flows that arrive already turbulent; and here the layer has existed for $0.5$ s from rest.

Nature of the transport. The enhancement over molecular diffusion is real but coherent: the vorticity generated on the face is advected into the attached edge vortex, as Section 7 shows quantitatively. Gradient-transport modeling is the wrong representation of a two-vortex velocity field, whose momentum transport is entirely non-local. An eddy viscosity is appropriate only in the relaxation branch, $t^{*}\gtrsim 6$, once the vortex breaks up into a three-dimensional turbulent wake [18] and the measured $F_a$ decays faster than Equation (18) predicts [1]; that affects the tail, not the peak, and leaves the $\sqrt{a}$ law untouched.

9. Test of the Empirical Law

9.1. The Reported Coefficient Is Not Constant

The law ${\widehat{F}}_a=C_a\rho A\sqrt{\nu a{V}_a}$ rests entirely on $C_a$ being independent of a and $V_a$; if it is not, the grouping and with it the appearance of $\nu$ lose their basis. Figure 5 of Ref. [1] reports $C_a$ for every run against $a^{*}$, with color encoding the target velocity. We rendered that figure at 400 dpi, classified marker pixels by color against the four reference values, and took cluster centroids, obtaining 47 resolved points. Restricting to the band $0.15<a^{*}<1.5$ shared by all four velocities, so that the correlation between $V_a$ and $a^{*}$ cannot masquerade as a trend, the medians are 268, 304, 329 and 342 for $V_a=0.30$, $0.45$, $0.60$ and $0.75\mathrm{m}{\mathrm{s}}^{-1}$ respectively. A joint fit $C_a=K{V}_a^p{a}^{\ast q}$ over all points gives

$$p=0.36[0.28,0.45],q=0.10\pm 0.02,\mathrm{rms}8\%,$$

(25)

the interval being a bootstrap $95\%$ confidence interval. The stratification by target velocity is evident in Figure 4(a). The value $p=0$ demanded by the reported law is excluded at about seven standard deviations; $p=\frac{1}{2}$, predicted by any inviscid force of the form $\rho AV\sqrt{ah}$ or $\rho A{\left(aVh\right)}^{2/3}$, is excluded at about three. The implied force exponents are ${\widehat{F}}_a\propto a^{0.60\pm 0.02}{V}_a^{0.66\pm 0.06}$, against the quoted $0.51\pm 0.05$ and $0.52\pm 0.04$.

Three caveats apply. Markers overlap in Figure 5 of Ref. [1], so merged centroids are biased toward the denser part of a cluster; the three plate shapes, whose reported means are 291, 308 and 311, are pooled here and contribute about $4\%$ scatter; and only four of the seven target velocities appear in that figure. None of these produces a monotone ordering of medians within a fixed $a^{*}$ band. The raw table would settle the exponent immediately, and we would encourage its publication.

9.2. Why the Boundary-Layer Force Is Small

Independently of the exponent, the diffusive force derived above is far below the measurement. For the video run, Section 4 and Section 5 give

$$F_a=\underset{0.0128\mathrm{N}}{\underbrace{{\displaystyle \frac{2}{\sqrt{\pi }}}\rho Aa\sqrt{\nu t_a}}}\times \underset{0.073\mathrm{or}0.783}{\underbrace{\mathrm{\Phi }\left(2\right)\mathrm{or}\mathrm{\Lambda }\left(2.2\right)}}\times \underset{1.85}{\underbrace{[1+2c_1]}}\times \underset{\mathrm{transp}.}{\underbrace{(1+1.28)}},$$

(26)

i.e. $0.004$–$0.042$ N depending on the edge treatment, against ${\widehat{F}}_a={\widehat{F}}_D-F_{QS}=4.9-1.9=3.0$ N. The shortfall in the coefficient is $C_a/(2/\sqrt{\pi })=258$ at leading order. That number is not a deficiency of the boundary-layer solution; it factorizes. Writing the two forces side by side, with $\delta \equiv \sqrt{\nu t_a}=\sqrt{\nu V_a/a}$ and $m_v=\rho Ah$,

$$F_a^{\mathrm{diff}}={\displaystyle \frac{2}{\sqrt{\pi }}}\rho Aa\delta ,m_{v}a=\rho Aah,$$

(27)

they are the same expression with a different length: the boundary layer accelerates fluid over $\sqrt{\nu t}$, the plate over h. Hence

$${\displaystyle \frac{C_a}{C}}={\displaystyle \frac{{\widehat{F}}_a}{F_a^{\mathrm{diff}}}}={\displaystyle \frac{\sqrt{\pi }}{2}}{\displaystyle \frac{h}{\delta }}{\displaystyle \frac{{\widehat{F}}_a}{m_{v}a}},{\displaystyle \frac{h}{\delta }}=\sqrt{a^{*}{\mathrm{Re}}_h},$$

(28)

with ${\mathrm{Re}}_h=V_{a}h/\nu$. For the video run $h/\delta =141.4$ and ${\widehat{F}}_a/\left(m_{v}a\right)=2.06$, and $0.886\times 141.4\times 2.06=257.9$, reproducing $C_a/C$ exactly. Only the second factor is unexplained physics; the first is geometry, $\delta =0.71$ mm against $h=100$ mm. This identity is not a prediction of $C_a$, and we do not present it as one: it states that a viscous-diffusive mechanism is the wrong order of magnitude for this experiment by a ratio of lengths, which no refinement of the boundary-layer solution and no gradient-transport closure (Section 8) can alter.

9.3. The Dominant Mechanism

What does account for ${\widehat{F}}_a$ is the vortex impulse. Section 7 shows that $\rho {\mathcal{l}}_a\mathrm{d}\left(\mathrm{\Gamma }{b}_v\right)/\mathrm{d}t$ reproduces the measured $F_D$ to an rms of $10\%$ through the acceleration with no fitted parameter, using the measured circulation and the tracked vortex spacing. This is a diagnostic rather than a closed theory—it consumes $\mathrm{\Gamma }\left(t\right)$ as input—but it identifies where a predictive theory must be sought: in the roll-up of the edge shear layer [16,21,27], not in the diffusion of vorticity from the face.

Figure 4(b) recasts the same data in the form implied by Equation (28). Since ${\widehat{F}}_a/\left(m_{v}a\right)=C_a\delta /h$, the reported law predicts a single line of slope $-1$ through $C_a=291$; the measured points instead follow lines displaced by target velocity, the $V_a=0.30$ and $V_a=0.75$ sets bracketing the reported law. The ratio ${\widehat{F}}_a/\left(m_{v}a\right)$ spans an order of magnitude across the grid and does not reach unity, which would require $a/V_a=8.5$ against a maximum of $5.5$; added mass therefore underestimates throughout the dataset while approaching sufficiency at the high-$a^{*}$ corner, as conjectured in Ref. [1] and anticipated by von Kármán [28].

The positive content of Section 2, Section 3 and Section 4 is unaffected by any of this: Equation (4) is the correct similarity reduction for the accelerating normal plate, Equation (10) its exact leading-order solution, and Equation (18) the correct history force for the vorticity generated on the face—a force that is real, now derived rather than asserted, and small. Its natural test is the opposite regime, low ${\mathrm{Re}}_h$ or high $a^{*}$, where $h/\delta$ is of order unity.

10. Conclusions

Applying Sun’s similarity transformation to the front face of a uniformly accelerated normal plate yields the following.

(i) The two-dimensional unsteady boundary-layer equation for this flow reduces to the single equation (4) in $(\eta ,\tau )$, with coefficients independent of the streamwise coordinate and with $\tau =Bt$ the diffusion time. The reduction extends to any power-law velocity program, Equation (6), and contains the impulsively started and steady Hiemenz problems as limits. Its leading-order solution is exact, Equation (10), and belongs to the Kummer family, as do all higher corrections.

(ii) The transformation identifies the experimental dimensionless time as the diffusion time, $\tau =2t^{*}$, which accounts for the collapse observed in Ref. [1], and the identity ${\int }_{\mathrm{face}}U\mathrm{d}A=A$ returns their history force with the coefficient $2/\sqrt{\pi }$ unchanged. That coefficient is correct, and Section 3.1 shows why: the one-dimensional template is the leading-order solution of the correct problem once its wall value is read as the local slip velocity. The two-dimensional formulation adds the streamwise distribution, the correction $c_1=0.42613$, the shape factor $\mathrm{\Phi }\left(\tau \right)$, the crossover to strain-saturated behavior at $\tau \simeq 1$, and the generalization to arbitrary $a\left(t\right)$ as the Basset integral (18).

(iii) That history force is two orders of magnitude below the measured ${\widehat{F}}_a$, and Equation (28) shows the shortfall to be the ratio $h/\delta =\sqrt{a^{*}{\mathrm{Re}}_h}$ of plate size to diffusion length, together with a residual factor of two. A viscous-diffusive mechanism is simply the wrong order of magnitude for this experiment, and no gradient-transport closure recovers it (Section 8).

(iv) 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 while also excluding a pure $V_a^{1/2}$ dependence at about three. The measured exponents lie between the viscous and inviscid alternatives, so the existing data determine that the law is not $\rho A\sqrt{\nu aV}$ without determining what it is.

(v) Frame-by-frame analysis of the supplemental video shows that $F_D\simeq \rho {\mathcal{l}}_a\mathrm{d}\left(\mathrm{\Gamma }{b}_v\right)/\mathrm{d}t$ to an rms of $10\%$ through the acceleration with no fitted parameter, with $b_v$ constant at the plate height, identifying ${\widehat{F}}_a$ as a vortex-impulse force.

One experiment would close the question. Repeating a subset of runs in a glycerol–water mixture at $\nu \approx 5$–$10{\nu }_{\mathrm{water}}$, at fixed A, a and $V_a$, changes ${\widehat{F}}_a$ by $\sqrt{5}$–$\sqrt{10}$ if the law is viscous and leaves it unchanged if it is not; because $\nu$ was never varied, no measurement in the existing dataset can separate these. Two further checks use data already in hand: publishing the raw ${\widehat{F}}_a$ table would replace the digitization of Section 9.1 with the exponent itself, and extending the circulation and vortex-spacing tracking of Section 7 to the full grid would test whether $\rho {\mathcal{l}}_a\mathrm{d}\left(\mathrm{\Gamma }{b}_v\right)/\mathrm{d}t$ reproduces the measured $F_D$ everywhere, which is a load-cell against particle-image-velocimetry consistency check and the natural starting point for a predictive theory of the edge shear layer.

Data availability. 

The numerical solution of Equation (24), the quadrature (20), and the frame-by-frame analysis of the supplemental video of Ref. [1] are available from the author on reasonable request. The video itself is distributed with Ref. [1].