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Integrating $Y^{\prime \prime }+2a\eta Y^{\prime }=0$ once gives $Y^{\prime }=K{e}^{-a{\eta }^2}$. A second integration expresses the solution using the error function when the value at zero is prescribed, and the complementary error function when decay at infinity is prescribed. Write ${\theta }_{\lambda }={\theta }^s(\lambda )$ and take $\lambda >0$, $Le>0$. The wall condition and the interface value giveFor the exterior liquid, continuity at the interface fixes ${\theta }^l(\lambda )={\theta }_{\lambda }-1+{\mathcal{R}}_{\theta }$ and ${\mathcal{C}}^l(\lambda )={\theta }_{\lambda }-1$. Thus the two decaying similarity solutions areThe factor $\sqrt{Le}$ is required because solute and heat diffuse on different length scales. Both denominators are positive for a finite positive $\lambda$. These formulas use the differential equations, wall/far-field data and continuity; the remaining three flux conditions determine their unknown constants.

For the approximate similarity profile, $F^{\prime }=A\mathrm{erf}(\xi /\sqrt{2})$, and thereforeFor the prescribed undisturbed horizontal streams, a suitable choice is $D=0$, which gives zero transverse velocity at both far-field limits. Then $V=-\Delta U\sqrt{CS}B{e}^{-{\xi }^2/2}$ inside the layer. The nonzero internal crossflow is consistent with the streamwise widening of the velocity transition.

Another possible convention is to make the line $y=0$ a streamline, imposing $V(0)=0$. This requires $D=-B$, equivalently $F(0)=0$, and gives $V(\pm \infty )=\Delta U\sqrt{CS}B$. It therefore requires a uniform outer crossflow relative to the coordinates used for the centred error function profile. A single constant cannot impose both zero far-field crossflow and $V(0)=0$ unless $\Delta U=0$. The physical lateral boundary conditions, or the chosen material centreline, determine which convention is appropriate. Since $B,D=O(r)$, these transverse velocities are of order $\Delta U^2/U_c$ for fixed $C,S$.

The two external stream velocities imposeLet $r=\Delta U/U_c\ll 1$ and seek $F=O(r)$ over $\xi =O(1)$. The nonlinear term $F{F}^{\prime \prime }$ is then $O(r^2)$, so the leading ordinary differential equation isFor $G=F^{\prime \prime }$, this is the first-order linear differential equation $G^{\prime }+\xi G=0$. Its integrating factor is $e^{{\xi }^2/2}$, giving $F^{\prime \prime }=M{e}^{-{\xi }^2/2}$. An integration using the error function givesThe two far-field boundary conditions require $E=0$ and $M=r/\sqrt{2\pi }$. A second integration usesConsequentlyThe relation between $A$ and $B$ cancels the extra $\xi e^{-{\xi }^2/2}$ term when differentiating. It gives the familiar error function mean velocity profileto first order in $r$. The additive constant $D=O(r)$ does not affect this velocity profile at the retained order; its interpretation comes from the transverse velocity.

At the interface, the tangential velocity is $u(x,0)=x{f}^{\prime }(0)$ and the normal velocity remains zero. The preceding error function expression givesHence, within the leading boundary layer model,Allowing the formally discarded finite-Reynolds number terms gives an additional order-${\delta }^2$ error, smaller than the order-$\gamma$ correction under the assumed ordering. The coefficient is $\sqrt{\pi /8}\approx 0.626657$.

Thus positive surface shear increases the outward surface speed linearly: for $x>0$ it increases $u$, and for $x<0$ it makes $u$ more negative. The stagnation point remains at $x=0$. A negative coefficient would result from using the printed $+\infty$ condition literally, and would describe the different half-line problem identified in part (ii).

Set $g=f_1^{\prime }$. The boundary value problem becomesFor $y_1=1+{\zeta }^2$, direct substitution gives $y_1^{\prime \prime }+\zeta y_1^{\prime }-2y_1=2+2{\zeta }^2-2(1+{\zeta }^2)=0$. This growing polynomial solution is useful for reduction of order.

Write $g=y_{1}v$. Substitution into the linear ordinary differential equation cancels the terms proportional to $v$ and givesAn integrating factor for this first-order equation for $v^{\prime }$ yieldsThe far-field boundary condition selects the convergent integral from $-\infty$:Any additional multiple of $1+{\zeta }^2$ would grow at $-\infty$. The chosen expression tends to zero: for $r=-\zeta \ge 1$,Here $(1+t^2{)}^{-2}\le (1+r^2{)}^{-2}$ and ${\int }_r^{\infty }{e}^{-t^2/2}dt\le r^{-1}{\int }_r^{\infty }t{e}^{-t^2/2}dt$. Thus the far-field boundary condition is verified without a cancellation of large terms.

At $\zeta =0$, differentiating gives $g^{\prime }(0)=C$, so $C=1$. Using the antiderivative supplied in the paper gives the explicit answerHere $\mathrm{erf}$ is the error function. Finally,recovers the unique correction satisfying $f_1(0)=0$. The decay bound also shows that $f_1$ approaches a finite constant at $-\infty$, as allowed by the normal velocity matching. This is the weak surface-shear perturbation of stagnation flow.

If the printed $+\infty$ condition is instead imposed literally on a positive half-line, reduction of order givesIt satisfies $g_{+}^{\prime }(0)=1$ but grows as $\zeta \to -\infty$, so it cannot describe the fluid domain specified in this paper. This makes the consequence of the printed sign error explicit.

A photon-recoil lag correlation asks how often a photon record occurs at a specified time displacement from an electron record. Let $r_e,r_{\gamma }$ be observed separate-channel rates, $S$ the rate of detected genuine pairs before a timing cut, and $f(\ell )$ their normalized timing probability density function. For an original dilute, stationary model with observation time $T$, neglecting end effects and detector dead time, the expected pair density isThis counts all pairs of records, rather than at most one detector trigger per electron. It uses independent unrelated detections for the flat term and an added same-event contribution. A normal distribution for timing spread is a chosen approximation, not a claim about the historical instrument. Intrinsic emission delay, counter response and readout uncertainty can all affect the observed probability density function.

Define the accepted lag interval by $|\ell |<w/2$: $w$ is its full width. Integrating the flat baseline gives the existing accidental coincidence expectation $T{r}_e{r}_{\gamma }w$. The additional area is $TS{\int }_{-w/2}^{w/2}f(\ell )d\ell$. Moving a window of the same full width away from the peak tests the baseline under the model's stationarity assumption. It does not remove correlated instrumental backgrounds or changing rates automatically.

For the lag model, choose $T=1000\mathrm{s}$, $r_e=2{\mathrm{s}}^{-1}$, $r_{\gamma }=0.5{\mathrm{s}}^{-1}$, $S=0.1{\mathrm{s}}^{-1}$ and timing standard deviation $\sigma =1\mathrm{ms}$. A full width $w=2\mathrm{ms}$ gives $2$ expected accidental coincidences and $100\mathrm{erf}(1/\sqrt{2})\simeq 68.27$ accepted genuine pairs, using the error function. The total is approximately $70.27$, whereas the displaced interval from $4$ to $6\mathrm{ms}$ has approximately $2.003$ expected pairs. A finite sample fluctuates around these expectations; these values are not measured counts or a significance calculation.

The right panel isolates the information lost by retaining singles totals alone. In a separate idealized model, stipulate $M=10^6$ independent $1\mathrm{ms}$ bins, with Bernoulli distribution indicators $X,Y$ for electron and photon detection in a bin. Set $p_e=\mathrm{Pr}(X=1)=0.002$ and $p_{\gamma }=\mathrm{Pr}(Y=1)=0.0005$. Independence predicts $\mathrm{Pr}(X=1,Y=1)=p_e{p}_{\gamma }=10^{-6}$, hence one expected double detection. Alternatively chooseThese nonnegative probabilities sum to one and preserve both marginal probabilities, yet predict $101$ double detections in expectation. Their covariance is $0.000101-p_e{p}_{\gamma }=0.000100$. This is an exact joint probability distribution illustrating pairing; it does not model the timing spread in the left panel or reconstruct either historical hypothesis quantitatively. Its lesson is why two singles totals cannot decide whether the observed events share a cause.

A planar turbulent mixing layer develops between two parallel streams of different mean velocities. Their shear flow generates turbulence that transports streamwise linear momentum across the interface. A thin-layer eddy viscosity model with ${\nu }_T=C\Delta U\delta (x)$ has thickness proportional to downstream distance. For $\Delta U/U_c\ll 1$, its leading mean velocity transition is an error function.

In $\epsilon y^{\prime \prime }+2x{y}^{\prime }-2xy=0$, the drift vanishes at the endpoint $x=0$. Balancing diffusion against drift gives the inner expansion coordinate $z=x/\sqrt{\epsilon }$. The leading layer equation is $Y^{\prime \prime }+2z{Y}^{\prime }=0$, with a bounded error function profile. A regular outer first-order calculation alone cannot impose both endpoint values.

The standard normal distribution function is the cumulative distribution function of a standard normal random variable $Z$:Here $\varphi$ is the standard normal density and $\mathrm{erf}$ is the error function. The symmetric probability distribution gives $\Phi (-x)=1-\Phi (x)$ and $\Phi (0)=1/2$, while differentiation gives ${\Phi }^{\prime }(x)=\varphi (x)$. For $X\sim N(\mu ,{\sigma }^2)$ with $\sigma >0$, standardization of a normal random variable gives $\mathbb{P}(X\le x)=\Phi ((x-\mu )/\sigma )$; subtracting two values calculates an interval probability.

Set $\delta =\sqrt{2\kappa /\alpha }$, $R=\alpha h/V$ and $S=L/[c_p(T_0-T_E)]$. In the regular limit $R\gg S\gg 1$, at fixed finite thermal strain number $\alpha h^2/\kappa$ and bounded phase fraction, the heat equation reduces to $\kappa T^{\prime \prime }+\alpha z{T}^{\prime }=0$. Integration gives the displayed error function profile. Matching the roof temperature gradient $G$ yieldsThe product composition is $C_s=C_0-\alpha (T_0-T_E)\delta [1-e^{-(h/\delta {)}^2}]/[mV\sqrt{\pi }\mathrm{erf}(h/\delta )]$. The asymptotic reduction requires admissible phase fractions. If the thermal strain number grows singularly with $R$, the two ordering inequalities alone do not control boundary-layer errors; the full strain-driven steady mush equations retain the latent-heat term.

For equal phase thermal properties, a roof undercooling $\Delta T$ and liquid superheat $\Theta$, the planar Stefan problem has interface $a=2\lambda \sqrt{\kappa t}$. The solid error function profile and liquid complementary-error-function profile give $S\lambda =e^{-{\lambda }^2}[1/\mathrm{erf}\lambda -(\Theta /\Delta T)/\mathrm{erfc}\lambda ]/\sqrt{\pi }$. Here $S=L/(c_p\Delta T)$ is the latent-to-sensible heat ratio. Liquid heat supplied to the interface reduces crust growth.

For a planar turbulent mixing layer, let $U_c=(U_a+U_b)/2$, $\Delta U=U_a-U_b$, ${\nu }_T=C\Delta U\delta$, $\delta =S\Delta Ux/U_c$, and $\xi =\sqrt{S/C}y/\delta$. The stream function $\psi =U_c[y+\delta \sqrt{C/S}F(\xi )]$ enforces mean incompressibility. Substitution into $U{U}_x+V{U}_y={\nu }_T{U}_{yy}$ gives the displayed ordinary differential equation, with $F^{\prime }(\pm \infty )=\pm \Delta U/(2U_c)$. When $\Delta U/U_c$ is small, its linear approximation is $F^{\prime \prime \prime }+\xi F^{\prime \prime }=0$, so $F^{\prime \prime }$ is a Gaussian function and $F^{\prime }$ an error function.

For the stress-driven stagnation-point boundary layer on $\zeta <0$, a regular perturbation $f=\zeta +\gamma f_1+O({\gamma }^2)$ gives $g=f_1^{\prime }$ satisfying $g^{\prime \prime }+\zeta g^{\prime }-2g=0$, $g^{\prime }(0)=1$ and $g(-\infty )=0$. Reduction of order with the known solution $1+{\zeta }^2$ givesIndeed, writing $g=(1+{\zeta }^2)v$ gives $v^{\prime }=C{e}^{-{\zeta }^2/2}/(1+{\zeta }^2{)}^2$; decay at $-\infty$ fixes the lower endpoint and $g^{\prime }(0)=1$ fixes $C=1$. The integral at zero is $\sqrt{2\pi }/4=\sqrt{\pi /8}$, as follows by differentiating the antiderivativeHere $\mathrm{erf}$ is the error function. Thus positive imposed outward shear stress increases the outward surface velocity by the displayed first-order factor. Replacing decay at $-\infty$ by decay at $+\infty$ while leaving the equation and wall derivative unchanged reverses this coefficient and describes a different problem.