An Exact Solution of the Unsteady Laminar Thermal Boundary Layer on a Flat Plate

1. Introduction

The canonical description of laminar forced convection is the pair consisting of the Blasius momentum solution $f_B$ [2] and the Pohlhausen energy solution [7]. With ${\eta }_B=y\sqrt{U_0/\nu x}$ and $\psi =\sqrt{\nu U_{0}x}{f}_B\left({\eta }_B\right)$,

$$f_B^{\prime \prime \prime }+\frac{1}{2}{f}_B{f}_B^{\prime \prime }=0,{\theta }_B^{\prime \prime }+\frac{1}{2}\mathit{Pr}{f}_B{\theta }_B^{\prime }+\mathit{Pr}\mathit{Ec}{\left(f_B^{\prime \prime }\right)}^2=0,$$

(1)

from which ${\mathit{Nu}}_x=0.332{\mathit{Pr}}^{1/3}{\mathit{Re}}_x^{1/2}$ for $\mathit{Pr}\gtrsim 0.6$. Essentially every engineering correlation for laminar forced convection descends from (1).

A difficulty arises whenever the thermal problem is genuinely time dependent: a plate switched on, a wall temperature stepped, a start-up transient in a heat exchanger. The standard procedure retains the time derivative in the energy equation while inserting the steady Blasius velocity field,

$${\displaystyle \frac{\partial T}{\partial t}}+u_B{\displaystyle \frac{\partial T}{\partial x}}+v_B{\displaystyle \frac{\partial T}{\partial y}}=\chi {\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{\nu }{c_p}}{\left({\displaystyle \frac{\partial u_B}{\partial y}}\right)}^2.$$

(2)

Equation (2) is internally inconsistent. A fluid particle cannot convect a time-dependent temperature along a time-independent trajectory; if temporal effects matter for T they matter for u and v as well. The inconsistency has been tolerated for a simple reason: no solution of the two-dimensional unsteady laminar boundary-layer equations was available to replace $u_B$ with.

The history is well documented. Stokes [14] solved the one-dimensional impulsive problem, and Rayleigh [8] recast it. For the two-dimensional semi-infinite plate, Stewartson [11], Stewartson [12], Stewartson [13], Hassan [5], Stuart [15], Takuda [17], Hall [4], Dennis [3], Riley [9] and Ma Hui [6] produced asymptotic and numerical results but no similarity solution. Dennis [3] identified the absence of a complete analytical solution as the central difficulty, and Stewartson’s conjecture of a singularity at the junction between the Rayleigh and Blasius regimes remained unresolved. The surveys of Wang [18], Wang [19], Wang [20] record how few exact solutions of the Navier–Stokes system are known, and how small a fraction of those concern unsteady flow.

The thermal problem inherited that gap of necessity. An exact solution of the unsteady energy equation presupposes an exact velocity field to convect the temperature, so for as long as the momentum problem remained open the thermal problem was closed to exact treatment. The one-dimensional Stokes–Rayleigh problem with a heated wall is elementary in both fields, but it carries no streamwise dependence and hence none of the difficulty.

Sun [16] broke the deadlock. The essential step was dimensional rather than algebraic. With $\delta \left(x\right)$ the (as yet undetermined) boundary-layer thickness, the group $\{y,t,\delta ,\nu \}$ admits exactly two dimensionless combinations, $\eta =y/\delta$ and the diffusion time

$$\tau ={\displaystyle \frac{\nu t}{{\delta }^2\left(x\right)}},$$

(3)

${\delta }^2/\nu$ being the time for vorticity generated at the wall to diffuse across the layer [15]. With $\psi =U\left(x\right)\delta \left(x\right)f(\eta ,\tau )$ the three independent variables collapse to two, and for the flat plate the resulting equation admits a solution in Kummer functions.

The present paper takes that velocity field as its starting point and completes the thermal half of the problem. Section 2 sets out the equations; Section 3 transforms the energy equation; Section 4 restates the velocity solution compactly. Section 5 contains the structural result: the velocity solution is invariant under a one-parameter Lie group, and on group-invariant functions the convective operator of the energy equation loses all reference to the velocity field, so that the unsteady thermal problem reduces exactly to a linear second-order ordinary differential equation. Section 6 solves it, Section 7 extracts the engineering quantities, Section 8 quantifies the error committed by (2), and Section 9 sets out, quantitatively, the range in which the results apply.

2. Formulation

A thin plate is immersed at zero incidence in a stream of speed $U\left(x\right)$ (Figure 1). The fluid has kinematic viscosity $\nu =\mu /\rho$, thermal diffusivity $\chi =k/\rho c_p$ and Prandtl number $\mathit{Pr}=\nu /\chi$. Within the boundary-layer approximation the governing equations are

$$\begin{array}{c}{\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{\partial v}{\partial y}}=0,\end{array}$$

(4)

$$\begin{array}{c}{\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}=U{\displaystyle \frac{\mathrm{d}U}{\mathrm{d}x}}+\nu {\displaystyle \frac{{\partial }^{2}u}{\partial y^2}},\end{array}$$

(5)

$$\begin{array}{c}{\displaystyle \frac{\partial T}{\partial t}}+u{\displaystyle \frac{\partial T}{\partial x}}+v{\displaystyle \frac{\partial T}{\partial y}}=\chi {\displaystyle \frac{{\partial }^{2}T}{\partial y^2}}+{\displaystyle \frac{\Phi }{\rho c_p}}.\end{array}$$

(6)

The dissipation function is $\Phi =2\mu e_{ij}{e}_{ij}$ with $e_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})$. In the boundary layer the only non-negligible strain component is $e_{12}=e_{21}\simeq \frac{1}{2}{u}_{,y}$, so $e_{ij}{e}_{ij}\simeq 2e_{12}^2=\frac{1}{2}{u}_{,y}^2$ and

$${\displaystyle \frac{\Phi }{\rho c_p}}={\displaystyle \frac{\nu }{c_p}}{\left({\displaystyle \frac{\partial u}{\partial y}}\right)}^2.$$

(7)

The convention matters, because $\Phi$ is sometimes written using $S_{ij}=u_{i,j}+u_{j,i}$, twice the rate of strain, in which case the prefactor in (7) is $\nu /2c_p$ and the Eckert numbers of the two conventions differ by a factor of two. All results below use (7).

Introducing a stream function $u={\psi }_y$, $v=-{\psi }_x$ satisfies (4) identically and converts (5) into

$${\psi }_{ty}+{\psi }_y{\psi }_{xy}-{\psi }_x{\psi }_{yy}=U{\displaystyle \frac{\mathrm{d}U}{\mathrm{d}x}}+\nu {\psi }_{yyy}.$$

(8)

We consider the two classical wall conditions, isothermal $T(x,0,t)=T_w$ and adiabatic ${\partial T/\partial y|}_{y=0}=0$, together with $u(x,0,t)=U_0$, $v(x,0,t)=0$ and $u\to 0$, $T\to T_{\infty }$ as $y\to \infty$. The dimensionless temperature and the Eckert number are

$$\theta ={\displaystyle \frac{T-T_{\infty }}{T_w-T_{\infty }}},\mathit{Ec}={\displaystyle \frac{U^2}{c_p(T_w-T_{\infty })}}.$$

(9)

3. Similarity Transformation of the Energy Equation

3.1. The Transformation

Following Sun [16] we set

$$\psi =U\left(x\right)\delta \left(x\right)f(\eta ,\tau ),\eta ={\displaystyle \frac{y}{\delta \left(x\right)}},\tau ={\displaystyle \frac{\nu t}{{\delta }^2\left(x\right)}},$$

(10)

so that

$$\begin{array}{ccc}{\eta }_x=-\eta {\delta }_x/\delta ,\hfill & {\eta }_y=1/\delta ,\hfill & {\eta }_t=0,\hfill \\ {\tau }_x=-2\tau {\delta }_x/\delta ,\hfill & {\tau }_y=0,\hfill & {\tau }_t=\nu /{\delta }^2.\hfill \end{array}$$

(11)

The structure of (11) governs everything that follows: y activates $\eta$ alone, t activates $\tau$ alone, and x is the only direction that activates both. Equation (8) becomes [16]

$$f_{\eta \eta \eta }+\alpha f{f}_{\eta \eta }+\beta \left[1-f_{\eta }^2\right]=f_{\tau \eta }+\gamma \tau \left(f_{\tau }{f}_{\eta \eta }-f_{\eta }{f}_{\tau \eta }\right),$$

(12)

with

$$\alpha ={\displaystyle \frac{\delta }{\nu }}{\displaystyle \frac{\mathrm{d}\left(U\delta \right)}{\mathrm{d}x}},\beta ={\displaystyle \frac{{\delta }^2}{\nu }}{\displaystyle \frac{\mathrm{d}U}{\mathrm{d}x}},\gamma ={\displaystyle \frac{U}{\nu }}{\displaystyle \frac{\mathrm{d}{\delta }^2}{\mathrm{d}x}}=2(\alpha -\beta ).$$

(13)

Constant $\alpha ,\beta ,\gamma$ require the power-law family $U\left(x\right)=C{x}^m$, $m=\beta /(2\alpha -\beta )$, with $\delta \left(x\right)={[\nu (2\alpha -\beta )x/U\left(x\right)]}^{1/2}$.

3.2. Transformation of the Energy Equation

From (11), for any $T=T(\eta ,\tau )$,

$$T_t={\displaystyle \frac{\nu }{{\delta }^2}}{T}_{\tau },T_y={\displaystyle \frac{T_{\eta }}{\delta }},T_{yy}={\displaystyle \frac{T_{\eta \eta }}{{\delta }^2}},T_x=-{\displaystyle \frac{{\delta }_x}{\delta }}\left(\eta T_{\eta }+2\tau T_{\tau }\right),$$

(14)

while the velocity components follow from (10) as

$$u=U{f}_{\eta },v=-{\left(U\delta \right)}_{x}f+U{\delta }_x\left(\eta f_{\eta }+2\tau f_{\tau }\right).$$

(15)

Hence the two convective terms are

$$\begin{array}{c}u{T}_x=\underset{(★)}{\underbrace{-{\displaystyle \frac{U{\delta }_x}{\delta }}\eta f_{\eta }{T}_{\eta }}}-{\displaystyle \frac{2U\tau {\delta }_x}{\delta }}{f}_{\eta }{T}_{\tau },\end{array}$$

(16)

$$\begin{array}{c}v{T}_y=-{\displaystyle \frac{{\left(U\delta \right)}_x}{\delta }}f{T}_{\eta }\underset{-(★)}{\underbrace{+{\displaystyle \frac{U{\delta }_x}{\delta }}\eta f_{\eta }{T}_{\eta }}}+{\displaystyle \frac{2U\tau {\delta }_x}{\delta }}{f}_{\tau }{T}_{\eta }.\end{array}$$

(17)

The terms marked $(★)$ and $-(★)$ are equal and opposite and cancel identically. Their origins are distinct: in (16) the term comes from ${\eta }_x$ inside $T_x$, whereas in (17) it comes from ${\eta }_x$ inside ${\psi }_x$, that is, from the requirement that v be evaluated by differentiating $f(\eta ,\tau )$ with respect to x through $\eta$ rather than treating f as a function of x alone. Omitting the second contribution leaves a spurious $\frac{1}{2}\gamma \eta f_{\eta }{\theta }_{\eta }$ in the transformed equation and destroys every reduction obtained below. What survives is

$$u{T}_x+v{T}_y=-{\displaystyle \frac{{\left(U\delta \right)}_x}{\delta }}f{T}_{\eta }+{\displaystyle \frac{2U\tau {\delta }_x}{\delta }}\left(f_{\tau }{T}_{\eta }-f_{\eta }{T}_{\tau }\right).$$

(18)

Substituting into (6), multiplying by ${\delta }^2/\nu$ and using (13), (7) and (9) gives the transformed energy equation

$${\theta }_{\tau }-\alpha f{\theta }_{\eta }+\gamma \tau \left(f_{\tau }{\theta }_{\eta }-f_{\eta }{\theta }_{\tau }\right)={\displaystyle \frac{1}{\mathit{Pr}}}{\theta }_{\eta \eta }+\mathit{Ec}{f}_{\eta \eta }^2,$$

(19)

with the same coefficients $\alpha ,\beta ,\gamma$ as (12). The surviving convective contribution is $-\alpha f{\theta }_{\eta }$ together with a Jacobian $\gamma \tau \partial (f,\theta )/\partial (\tau ,\eta )$.

Two checks confirm (19). First, at unit Prandtl number the streamwise velocity is itself a passive scalar, so setting $\theta =f_{\eta }$, $\mathit{Pr}=1$, $\mathit{Ec}=0$ must return (12) with $\beta =0$; it does. With the spurious term retained one obtains instead an extra $\eta f_{\eta }{f}_{\eta \eta }$, which has no counterpart in the momentum equation. Second, an independent verification carried out entirely in physical $(x,y,t)$ variables, which uses none of the algebra above, is reported in Section 9.

Remark 1.

$\mathit{Ec}$ in (19) is a genuine constant only if U is constant: for $U=C{x}^m$ with $m\ne 0$ one has $\mathit{Ec}\propto x^{2m}$ and the energy equation does not close in $(\eta ,\tau )$. The flat plate is therefore the unique member of the family for which thecoupledmomentum–energy system reduces to two variables. For $m\ne 0$ the momentum equation still reduces but the thermal problem does not.

3.3. Specialisation to the Flat Plate

For $\beta =0$ we have $m=0$ and $U\left(x\right)=U_0$; taking $\alpha =1$ without loss of generality gives $\gamma =2$ and

$$\delta \left(x\right)={\left({\displaystyle \frac{2\nu x}{U_0}}\right)}^{1/2},\eta =y\sqrt{{\displaystyle \frac{U_0}{2\nu x}}},\tau ={\displaystyle \frac{U_{0}t}{2x}}.$$

(20)

The coupled system is then

$$\begin{array}{c}{f}_{\eta \eta \eta }+f{f}_{\eta \eta }=f_{\tau \eta }+2\tau \left(f_{\tau }{f}_{\eta \eta }-f_{\eta }{f}_{\tau \eta }\right),\end{array}$$

(21)

$$\begin{array}{c}\underset{A}{\underbrace{\left(1-2\tau f_{\eta }\right)}}{\theta }_{\tau }+\underset{B}{\underbrace{\left(2\tau f_{\tau }-f\right)}}{\theta }_{\eta }={\displaystyle \frac{1}{\mathit{Pr}}}{\theta }_{\eta \eta }+\mathit{Ec}{f}_{\eta \eta }^2.\end{array}$$

(22)

Note that $\tau =U_{0}t/2x$ is large near the leading edge or at late times and small far downstream; the line $\tau =1/2$, that is $x=U_{0}t$, separates the quasi-steady from the purely unsteady region (Figure 1).

4. The Velocity Field in Compact Form

Sun [16] obtained an exact solution of (21) written through six auxiliary functions built from Kummer functions of the first and second kind. Using the contiguous relations of Appendix A these collapse to single terms.

Proposition 1.

Define

$$\zeta =3\eta \tau +2,p={\displaystyle \frac{{\zeta }^2}{12{\tau }^3}},\omega =\sqrt{p}={\displaystyle \frac{3\eta \tau +2}{2\sqrt{3}{\tau }^{3/2}}},$$

(23)

and the constants

$$K={\displaystyle \frac{2}{3\Gamma (2/3)}}\sqrt{{\displaystyle \frac{\pi }{3}}}=0.5038097,c_1={\displaystyle \frac{4\pi }{15\Gamma (5/6)\Gamma (2/3)}}=0.5480878.$$

(24)

Then Sun’s solution and its derivatives are

$$\begin{array}{c}f={\displaystyle \frac{1}{5{\tau }^2}}+{\displaystyle \frac{\eta }{2\tau }}+c_1\sqrt{\tau }-{\displaystyle \frac{K\zeta }{6\tau }}{\mathrm{e}}^{-p}U\left(\frac{11}{6},\frac{3}{2},p\right),\end{array}$$

(25)

$$\begin{array}{c}{f}_{\eta }={\displaystyle \frac{1}{2\tau }}+K{\mathrm{e}}^{-p}U\left(\frac{5}{6},\frac{1}{2},p\right),\end{array}$$

(26)

$$\begin{array}{c}{f}_{\eta \eta }=-{\displaystyle \frac{K\zeta }{2{\tau }^2}}{\mathrm{e}}^{-p}U\left(\frac{5}{6},\frac{3}{2},p\right),\end{array}$$

(27)

$$\begin{array}{c}{f}_{\eta \eta \eta }=-K{\mathrm{e}}^{-p}\left[{\displaystyle \frac{3}{2\tau }}U\left(\frac{5}{6},\frac{3}{2},p\right)-{\displaystyle \frac{{\zeta }^2}{4{\tau }^4}}U\left(\frac{5}{6},\frac{5}{2},p\right)\right],\end{array}$$

(28)

where $U(a,b,z)$ is the Kummer function of the second kind.

Proof. 

Sun’s $h_{12}$ contains the group ${\eta }^2{\tau }^2+\frac{4}{3}\eta \tau +\frac{4}{9}={\zeta }^2/9=\frac{4}{3}{\tau }^{3}p$, so that $h_{12}={\mathrm{e}}^{-p}[(6p-1)U(\frac{5}{6},\frac{3}{2},p)-6U(-\frac{1}{6},\frac{3}{2},p)]$. Relation (A2) with $a=5/6$, $b=3/2$ gives $U(-\frac{1}{6},\frac{3}{2},p)=(\frac{1}{6}+p)U(\frac{5}{6},\frac{3}{2},p)-\frac{5}{18}U(\frac{11}{6},\frac{3}{2},p)$, whence $h_{12}={\mathrm{e}}^{-p}[-2U(\frac{5}{6},\frac{3}{2},p)+\frac{5}{3}U(\frac{11}{6},\frac{3}{2},p)]$, and (A3) then yields $h_{12}=-2{\mathrm{e}}^{-p}U(\frac{5}{6},\frac{1}{2},p)$; with $c_3=-K/2$ this is (26). The same two steps applied to $h_{11}$ give $h_{11}=(\zeta /3\tau ){\mathrm{e}}^{-p}U(\frac{11}{6},\frac{3}{2},p)$, that is (25). Equations (27) and (28) follow from (A5) together with $p_{\eta }=\zeta /2{\tau }^2$ and ${\zeta }_{\eta }=3\tau$. □

Substituting (25)–() into (21) and evaluating in 30-digit arithmetic gives residuals below ${10}^{-25}$ at every point tested (Table 2). Beyond convenience, the compaction is what exposes the group structure of Section 5: in the original form the invariance is invisible.

Three properties matter below. First, the wall and outer conditions are attained only as $\tau \to \infty$: $f_{\eta }(\infty ,\tau )=1/2\tau$ and $f_{\eta }(0,\tau )=1+O\left({\tau }^{-1}\right)$, a point developed quantitatively in Section 9. Second, the wall shear stress is

$${\tau }_w={\displaystyle \frac{\mu U_0}{\delta }}{f}_{\eta \eta }(0,\tau )⟶-C_{\tau }{\displaystyle \frac{\mu U_0}{\sqrt{\nu t}}},C_{\tau }={\displaystyle \frac{2\pi }{3\Gamma (2/3)\Gamma (5/6)}}=1.370219,$$

(29)

independent of x, so that $c_f=1.93778{\left(\tau {\mathit{Re}}_x\right)}^{-1/2}$. Third, the layer thickness follows from $p=O\left(1\right)$, that is $\eta \sim (2/\sqrt{3})\sqrt{\tau }$ or $y\sim 1.155\sqrt{\nu t}$ in physical variables: the layer is of Rayleigh type, growing in time and uniform in x. These features are shown in Figure 2.

5. Group Invariance and Exact Reduction

According to the Lie group theory [21,22], we have

Lemma 1.

Let $X=(\eta \tau +2){\partial }_{\eta }+2{\tau }^2{\partial }_{\tau }$. Then $Xp=0$, hence $X\omega =0$, and Sun’s solution satisfies

$$Xf=(\eta \tau +2)f_{\eta }+2{\tau }^2{f}_{\tau }=\tau f-\eta .$$

(30)

Proof. 

From ${\zeta }_{\eta }=3\tau$ and ${\zeta }_{\tau }=3\eta$ we obtain $X\zeta =3\tau (3\eta \tau +2)=3\tau \zeta$, so that

$$Xp={\displaystyle \frac{2\zeta \left(X\zeta \right)}{12{\tau }^3}}-{\displaystyle \frac{3{\zeta }^2}{12{\tau }^4}}2{\tau }^2={\displaystyle \frac{{\zeta }^2}{2{\tau }^2}}-{\displaystyle \frac{{\zeta }^2}{2{\tau }^2}}=0,$$

(31)

and X annihilates every function of p alone. Writing $\mathcal{L}=X-\tau$ and using $X{\tau }^n=2n{\tau }^{n+1}$, $X\eta =\eta \tau +2$,

$$\mathcal{L}\left[{\displaystyle \frac{1}{5{\tau }^2}}\right]=-{\displaystyle \frac{1}{\tau }},\mathcal{L}\left[{\displaystyle \frac{\eta }{2\tau }}\right]={\displaystyle \frac{1}{\tau }}-\eta ,\mathcal{L}\left[c_1\sqrt{\tau }\right]=0,\mathcal{L}\left[{\displaystyle \frac{\zeta }{3\tau }}G\left(p\right)\right]=0,$$

(32)

the last two because $X\sqrt{\tau }={\tau }^{3/2}$ and $X[\zeta /3\tau ]=\zeta /3$. Summing gives $\mathcal{L}f=-\eta$, which is (30). □

Differentiating (30) with respect to $\eta$ and using $[{\partial }_{\eta },X]=\tau {\partial }_{\eta }$ gives $X{f}_{\eta }=-1=X[1/2\tau ]$, so $f_{\eta }-1/2\tau$ is a group invariant and must be a function of $\omega$ alone. Comparison with () identifies it:

$${\displaystyle \frac{u}{U_0}}=f_{\eta }={\displaystyle \frac{1}{2\tau }}+F\left(\omega \right),F\left(\omega \right)=K{\mathrm{e}}^{-{\omega }^2}U\left(\frac{5}{6},\frac{1}{2},{\omega }^2\right),$$

(33)

with $F\left(0\right)=1$, $F(\infty )=0$ and

$$\begin{array}{cc}\hfill F^{\prime }\left(\omega \right)=& -2K\omega {\mathrm{e}}^{-{\omega }^2}U\left(\frac{5}{6},\frac{3}{2},{\omega }^2\right),\hfill \\ \hfill F^{\prime }\left(0\right)=& -{\displaystyle \frac{4\pi }{3\sqrt{3}\Gamma (2/3)\Gamma (5/6)}}=-1.582193.\hfill \end{array}$$

(34)

Since ${\omega }_{\eta }=\sqrt{3}/2\sqrt{\tau }$, Equation () is simply $f_{\eta \eta }={\omega }_{\eta }{F}^{\prime }\left(\omega \right)$.

Theorem 1.

Let $\theta =\Theta \left(\omega \right)$ with ω given by (23). Then the unsteady energy Equation () is satisfied for arbitrary $\mathit{Pr}$ and $\mathit{Ec}$ if and only if

$${\Theta }^{\prime \prime }\left(\omega \right)+2\mathit{Pr}\omega {\Theta }^{\prime }\left(\omega \right)+\mathit{Pr}\mathit{Ec}{\left[F^{\prime }\left(\omega \right)\right]}^2=0.$$

(35)

Proof. 

From (23),

$${\omega }_{\eta }={\displaystyle \frac{\sqrt{3}}{2\sqrt{\tau }}},{\omega }_{\eta \eta }=0,{\omega }_{\tau }=-{\displaystyle \frac{\sqrt{3}(\eta \tau +2)}{4{\tau }^{5/2}}}.$$

(36)

Dividing (30) by $\tau$ gives $B=2\tau f_{\tau }-f=-[\eta +(\eta \tau +2)f_{\eta }]/\tau$, so that with $A=1-2\tau f_{\eta }$,

$$\begin{array}{cc}\hfill A{\omega }_{\tau }+B{\omega }_{\eta }& =-\left(1-2\tau f_{\eta }\right){\displaystyle \frac{\sqrt{3}(\eta \tau +2)}{4{\tau }^{5/2}}}-{\displaystyle \frac{\eta +(\eta \tau +2)f_{\eta }}{\tau }}{\displaystyle \frac{\sqrt{3}}{2\sqrt{\tau }}}\hfill \\ \hfill & =-{\displaystyle \frac{\sqrt{3}}{4{\tau }^{5/2}}}\left[(\eta \tau +2)-2\tau f_{\eta }(\eta \tau +2)+2\tau \eta +2\tau (\eta \tau +2)f_{\eta }\right]\hfill \\ \hfill & =-{\displaystyle \frac{\sqrt{3}(3\eta \tau +2)}{4{\tau }^{5/2}}}.\hfill \end{array}$$

(37)

The velocity field has cancelled out identically. Since ${\omega }_{\eta }^2=3/4\tau$ and ${\omega }_{\eta \eta }=0$,

$${\displaystyle \frac{A{\omega }_{\tau }+B{\omega }_{\eta }}{{\omega }_{\eta }^2}}=-{\displaystyle \frac{3\eta \tau +2}{\sqrt{3}{\tau }^{3/2}}}=-2\omega .$$

(38)

With ${\theta }_{\tau }={\Theta }^{\prime }{\omega }_{\tau }$, ${\theta }_{\eta }={\Theta }^{\prime }{\omega }_{\eta }$, ${\theta }_{\eta \eta }={\Theta }^{\prime \prime }{\omega }_{\eta }^2$ and $f_{\eta \eta }^2={\omega }_{\eta }^2{\left[F^{\prime }\left(\omega \right)\right]}^2$, Equation () becomes

$${\omega }_{\eta }^2\left[-2\omega {\Theta }^{\prime }-{\displaystyle \frac{1}{\mathit{Pr}}}{\Theta }^{\prime \prime }\right]=\mathit{Ec}{\omega }_{\eta }^2{\left[F^{\prime }\left(\omega \right)\right]}^2,$$

(39)

and the common factor ${\omega }_{\eta }^2$ cancels, giving (35). Conversely any $\Theta$ satisfying (35) generates a solution of () by the same identities, so the condition is necessary and sufficient within the invariant class. □

Remark 2.

Equation (37) is stronger than required here. It states that on functions invariant under X the operator $A{\partial }_{\tau }+B{\partial }_{\eta }$ reduces to $-2\omega {\omega }_{\eta }^2\mathrm{d}/\mathrm{d}\omega$ whateverthe velocity field, provided only that it satisfies $Xf=\tau f-\eta$. Any passive scalar therefore obeys the same reduced operator, the flow entering only through the source term.

Remark 3.

Equation (35) is the unsteady counterpart of the Pohlhausen equation (1), under the correspondence $\frac{1}{2}\mathit{Pr}{f}_B\left({\eta }_B\right)\leftrightarrow 2\mathit{Pr}\omega$ and $\mathit{Pr}\mathit{Ec}{\left[f_B^{\prime \prime }\right]}^2\leftrightarrow \mathit{Pr}\mathit{Ec}{\left[F^{\prime }\right]}^2$. In the steady problem the convection coefficient is the Blasius function itself, known only numerically; in the unsteady problem it degenerates to the linear function $2\omega$, which is why the homogeneous solution here is elementary while its steady analogue is not. The degeneracy is a direct consequence of (37).

6. Exact Solutions

Equation (35) is linear, with integrating factor ${\mathrm{e}}^{\mathit{Pr}{\omega }^2}$ and general solution

$$\begin{array}{cc}\hfill \Theta \left(\omega \right)& =\Theta \left(0\right)\hfill \\ \hfill & +{\Theta }^{\prime }\left(0\right){\int }_0^{\omega }{\mathrm{e}}^{-\mathit{Pr}{s}^2}\mathrm{d}s-\mathit{Pr}\mathit{Ec}{\int }_0^{\omega }{\mathrm{e}}^{-\mathit{Pr}{s}^2}\left[{\int }_0^s{\mathrm{e}}^{\mathit{Pr}{u}^2}{\left[F^{\prime }\left(u\right)\right]}^2\mathrm{d}u\right]\mathrm{d}s.\hfill \end{array}$$

(40)

6.1. Isothermal Wall

Imposing $\Theta \left(0\right)=1$ and $\Theta (\infty )=0$ and interchanging the order of integration gives the closed form

$$\theta (\eta ,\tau )=erfc\left(\sqrt{\mathit{Pr}}\omega \right)+\mathit{Pr}\mathit{Ec}\Phi (\omega ;\mathit{Pr}),$$

(41)

$$\begin{array}{cc}\hfill \Phi =& {\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\pi }{\mathit{Pr}}}}[erfc\left(\sqrt{\mathit{Pr}}\omega \right){\int }_0^{\omega }g\left(u\right)erf\left(\sqrt{\mathit{Pr}}u\right)\mathrm{d}u\hfill \\ \hfill & +erf\left(\sqrt{\mathit{Pr}}\omega \right){\int }_{\omega }^{\infty }g\left(u\right)erfc\left(\sqrt{\mathit{Pr}}u\right)\mathrm{d}u],\hfill \end{array}$$

(42)

with $g\left(u\right)={\left[F^{\prime }\left(u\right)\right]}^2{\mathrm{e}}^{\mathit{Pr}{u}^2}$. Equation (42) is a Green’s function representation, the two branches being the decaying and growing homogeneous solutions of (35) matched at $u=\omega$; it follows that $\Phi \left(0\right)=\Phi (\infty )=0$, so dissipation reshapes the interior of the layer without disturbing either boundary value. The wall gradient is

$$\begin{array}{cc}\hfill {\Theta }^{\prime }\left(0\right)& =-2\sqrt{{\displaystyle \frac{\mathit{Pr}}{\pi }}}\left(1-\mathit{Pr}\mathit{Ec}{P}_{\infty }\right),\hfill \\ \hfill P_{\infty }\left(\mathit{Pr}\right)& ={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{\pi }{\mathit{Pr}}}}{\int }_0^{\infty }{\left[F^{\prime }\left(u\right)\right]}^2{\mathrm{e}}^{\mathit{Pr}{u}^2}erfc\left(\sqrt{\mathit{Pr}}u\right)\mathrm{d}u.\hfill \end{array}$$

(43)

For $\mathit{Ec}=0$ the temperature is a complementary error function of the group invariant,

$$\theta =erfc\left(\sqrt{\mathit{Pr}}{\displaystyle \frac{3\eta \tau +2}{2\sqrt{3}{\tau }^{3/2}}}\right)\underset{\tau \to \infty }{\to }erfc\left({\displaystyle \frac{\sqrt{3\mathit{Pr}}}{2}}{\displaystyle \frac{y}{\sqrt{\nu t}}}\right),$$

(44)

shown in Figure 3, with the effect of dissipation in Figure 4.

6.2. Adiabatic Wall and Recovery Factor

Setting ${\Theta }^{\prime }\left(0\right)=0$ in (40) and requiring $\Theta (\infty )=0$ gives

$$T_{aw}-T_{\infty }=\mathit{Pr}{P}_{\infty }{\displaystyle \frac{U_0^2}{c_p}},r\equiv {\displaystyle \frac{T_{aw}-T_{\infty }}{U_0^2/2c_p}}=2\mathit{Pr}{P}_{\infty }\left(\mathit{Pr}\right).$$

(45)

Two limits are analytic. As $\mathit{Pr}\to 0$ the kernel ${\mathrm{e}}^{\mathit{Pr}{u}^2}erfc\left(\sqrt{\mathit{Pr}}u\right)\to 1$ and

$$r\to \sqrt{\pi \mathit{Pr}}{\int }_0^{\infty }{\left[F^{\prime }\left(u\right)\right]}^2\mathrm{d}u=1.70457\sqrt{\mathit{Pr}},$$

(46)

recovering the classical $\sqrt{\mathit{Pr}}$ scaling with prefactor $1.705$ in place of unity. As $\mathit{Pr}\to \infty$ the kernel behaves as ${\left(\sqrt{\pi \mathit{Pr}}u\right)}^{-1}$, the integral is cut off at $u\sim {\mathit{Pr}}^{-1/2}$ and

$$r\simeq \frac{1}{2}{\left[F^{\prime }\left(0\right)\right]}^{2}ln\mathit{Pr}+const=1.2517ln\mathit{Pr}-0.97,$$

(47)

which reproduces the computed values to within $0.03$ for $\mathit{Pr}⩾{10}^4$. The logarithmic growth is a genuinely unsteady effect: dissipation is confined to the Stokes sublayer of thickness $\sqrt{\nu t}$ while the thermal layer shrinks as ${\mathit{Pr}}^{-1/2}$, so their overlap saturates rather than following the steady $\sqrt{\mathit{Pr}}$ law (Figure 5).

6.3. Two Further Exact Solutions at $\mathit{Pr}=1$

Theorem 1 applies to group-invariant temperature fields. At $\mathit{Pr}=1$ there exist exact solutions of () that are not invariant. Because $u/U_0=f_{\eta }$ obeys (21), which is exactly () with $\mathit{Pr}=1$, $\mathit{Ec}=0$, we have the unsteady Reynolds analogy

$$\theta =f_{\eta }={\displaystyle \frac{u}{U_0}}(\mathit{Pr}=1,\mathit{Ec}=0),$$

(48)

and, by direct substitution, the unsteady Crocco integral

$$\theta =a+b{f}_{\eta }-\frac{1}{2}\mathit{Ec}{f}_{\eta }^2(\mathit{Pr}=1),$$

(49)

for arbitrary constants a and b, the coefficient $-\mathit{Ec}/2$ being forced by the dissipation term. Both have been verified to ${10}^{-22}$. Since $X{f}_{\eta }=-1\ne 0$, neither belongs to the invariant class, so (48) and (44) are distinct solutions satisfying the same asymptotic conditions. The consequences are discussed in Section 9.

7. Heat Transfer Characteristics

With ${\theta }_{\eta }(0,\tau )={\Theta }^{\prime }\left({\omega }_0\right){\omega }_{\eta }$ and ${\omega }_0=\omega (0,\tau )=1/\sqrt{3}{\tau }^{3/2}$, and using $\delta \sqrt{\tau }=\sqrt{\nu t}$ and $x/\delta =\sqrt{{\mathit{Re}}_x/2}$,

$$\begin{array}{cc}\hfill {\mathit{Nu}}_x& ={\displaystyle \frac{q_{w}x}{k(T_w-T_{\infty })}}=\sqrt{{\displaystyle \frac{3\mathit{Pr}}{2\pi }}}\left(1-\mathit{Pr}\mathit{Ec}{P}_{\infty }\right)\sqrt{{\displaystyle \frac{{\mathit{Re}}_x}{\tau }}}\hfill \\ \hfill & =0.690988\left(1-\mathit{Pr}\mathit{Ec}{P}_{\infty }\right)\sqrt{{\displaystyle \frac{\mathit{Pr}{\mathit{Re}}_x}{\tau }}},\hfill \end{array}$$

(50)

valid for $\tau \gg 1$ so that ${\omega }_0\to 0$. Equivalently, in purely temporal form,

$$h={\displaystyle \frac{q_w}{T_w-T_{\infty }}}=\sqrt{{\displaystyle \frac{3\mathit{Pr}}{\pi }}}\left(1-\mathit{Pr}\mathit{Ec}{P}_{\infty }\right){\displaystyle \frac{k}{\sqrt{\nu t}}},$$

(51)

which is independent of x: a signature of the Rayleigh-type structure of the unsteady layer, and a qualitative departure from the steady $h\propto x^{-1/2}$ law.

Equation (50) vanishes at the critical Eckert number

$${\mathit{Ec}}^{*}\left(\mathit{Pr}\right)={\displaystyle \frac{1}{\mathit{Pr}{P}_{\infty }\left(\mathit{Pr}\right)}}={\displaystyle \frac{2}{r}},$$

(52)

above which frictional heating exceeds the imposed wall-to-fluid temperature difference and the heat flux reverses; ${\mathit{Ec}}^{*}=1.764$ for air ($\mathit{Pr}=0.71$) and $0.777$ for water at $\mathit{Pr}=7$ (Table 1, Figure 6). The identity ${\mathit{Ec}}^{*}=2/r$ states that reversal occurs when $T_w=T_{aw}$, as it must.

Combining (50) with (29) gives the Stanton number $\mathit{St}={\mathit{Nu}}_x/{\mathit{Re}}_x\mathit{Pr}$ and the Reynolds analogy

$${\displaystyle \frac{\mathit{St}}{c_f/2}}=0.713174{\mathit{Pr}}^{-1/2}\left(1-\mathit{Pr}\mathit{Ec}{P}_{\infty }\right).$$

(53)

Defining $99\%$ thicknesses through $F=0.01$ and $\theta =0.01$,

$${\omega }_{u,99}=1.669435,{\omega }_{T,99}={\displaystyle \frac{1.821386}{\sqrt{\mathit{Pr}}}},{\displaystyle \frac{{\delta }_T}{{\delta }_u}}=1.091018{\mathit{Pr}}^{-1/2},$$

(54)

an exact power law at all $\mathit{Pr}$. This is worth contrasting with the steady Prandtl–Blasius layer, in which the corresponding ratio crosses over from a ${\mathit{Pr}}^{-1/2}$ behaviour at small $\mathit{Pr}$ to ${\mathit{Pr}}^{-1/3}$ at large $\mathit{Pr}$. No such crossover occurs here: both layers in the unsteady problem are diffusive rather than convective in structure, and the single exponent $-1/2$ registers that difference (Figure 10b). The ${\mathit{Pr}}^{-1/2}$ scaling in (53), in place of the ${\mathit{Pr}}^{-2/3}$ of the Colburn analogy, has the same origin.

Because $\omega$ depends on x, y and t through $\omega =(\sqrt{3}/2)y/\sqrt{\nu t}+1/\sqrt{3}{\tau }^{3/2}$, the isotherms in a fixed-t snapshot are not parallel to the plate. Figure 7(b) shows that the thermal footprint is confined to $x\lesssim U_{0}t$: information from the impulsively started heated plate has not yet reached stations further downstream, which is the correct physical behaviour and is encoded automatically in the similarity variable.

8. Comparison with the Quasi-Steady Practice

We can now quantify what is lost by using (2). Comparing (50) at $\mathit{Ec}=0$ with ${\mathit{Nu}}_x^{st}=0.332{\mathit{Pr}}^{1/3}{\mathit{Re}}_x^{1/2}$,

$${\displaystyle \frac{{\mathit{Nu}}_x^{\mathrm{exact}}}{{\mathit{Nu}}_x^{\mathrm{steady}}}}={\displaystyle \frac{\sqrt{3/2\pi }}{0.332}}{\displaystyle \frac{{\mathit{Pr}}^{1/6}}{\sqrt{\tau }}}=2.0813{\mathit{Pr}}^{1/6}{\tau }^{-1/2},$$

(55)

the two agreeing only at $\tau =4.332{\mathit{Pr}}^{1/3}$, that is $\tau \simeq 3.9$ for air and $\tau \simeq 8.3$ for water. Since $\tau =U_{0}t/2x$, at a station x the quasi-steady formula underestimates the wall heat flux for all times $t<8.7{\mathit{Pr}}^{1/3}x/U_0$, and by a factor of two or more for $t<2.2{\mathit{Pr}}^{1/3}x/U_0$. For the start-up of an air flow at 1 m s⁻¹ over a $0.1$ m plate this is the whole of the first $\sim 0.9$ s, precisely the transient the unsteady energy equation was introduced to capture.

The profiles tell the same story. Figure 8(a) compares $\theta \left(y\right)$ from (44) with the Pohlhausen profile at three stations at a fixed instant. The unsteady thermal layer has an x-independent thickness $\propto \sqrt{\nu t}$, so it is thicker than the quasi-steady prediction near the leading edge and thinner far downstream; the crossover is again at $\tau =O\left(1\right)$. A calculation that freezes the velocity field at Blasius therefore misrepresents both the magnitude and the streamwise trend of the transient.

9. Verification and Range of Validity

9.1. Numerical Verification

Table 2 lists the residuals of the momentum Equation (21) evaluated with (25)–(), and of the full unsteady energy Equation () evaluated with (41), in 25-digit arithmetic with $\tau$ derivatives obtained by Richardson-extrapolated central differences.

Because the cancellation of Section 3 is the step in the derivation most easily got wrong, it was also checked in a way that uses none of the transformation algebra. The fields $u(x,y,t)$, $v(x,y,t)$ and $T(x,y,t)$ were assembled in physical coordinates from (15) and (41), and the original Equations (4)–() evaluated by high-order finite differences in x, y and t. Continuity and momentum are satisfied by construction, so their residuals fix the noise floor of the differentiation. Table 3 reports the outcome, together with the residual obtained when the term $\eta f_{\eta }{\theta }_{\eta }$ is reinstated: the correct energy equation sits on the same noise floor as continuity and momentum, while the alternative is larger by twelve orders of magnitude.

Table 2. Residuals of (21) and () at selected points, computed with 25 significant digits.

$\mathit{\eta }$	$\mathit{\tau }$	$\mathit{Pr}$	$\mathit{Ec}$	momentum residual	energy residual
0.5	1.0	0.71	0.0	$2.6\times {10}^{-26}$	$<{10}^{-30}$
1.0	2.0	0.71	1.0	$<{10}^{-30}$	$1.8\times {10}^{-26}$
0.4	1.5	7.0	0.0	$<{10}^{-30}$	$1.9\times {10}^{-17}$
2.0	3.0	7.0	2.0	$2.8\times {10}^{-27}$	$2.4\times {10}^{-27}$
0.2	0.9	0.1	1.0	$1.3\times {10}^{-26}$	$6.7\times {10}^{-19}$
3.0	5.0	1.0	0.5	$8.1\times {10}^{-28}$	$3.5\times {10}^{-28}$

Table 2. Residuals of (21) and () at selected points, computed with 25 significant digits.

$\mathit{\eta }$	$\mathit{\tau }$	$\mathit{Pr}$	$\mathit{Ec}$	momentum residual	energy residual
0.5	1.0	0.71	0.0	$2.6\times {10}^{-26}$	$<{10}^{-30}$
1.0	2.0	0.71	1.0	$<{10}^{-30}$	$1.8\times {10}^{-26}$
0.4	1.5	7.0	0.0	$<{10}^{-30}$	$1.9\times {10}^{-17}$
2.0	3.0	7.0	2.0	$2.8\times {10}^{-27}$	$2.4\times {10}^{-27}$
0.2	0.9	0.1	1.0	$1.3\times {10}^{-26}$	$6.7\times {10}^{-19}$
3.0	5.0	1.0	0.5	$8.1\times {10}^{-28}$	$3.5\times {10}^{-28}$

Table 3. Residuals of the original Equations (4)–() evaluated in physical $(x,y,t)$ variables with $\nu =0.7$, $U_0=1.3$, $\mathit{Pr}=0.71$, step $h={10}^{-6}$, 30-digit arithmetic. The last column is the energy residual obtained when the spurious $\eta f_{\eta }{\theta }_{\eta }$ term is retained.

$(\mathit{x},\mathit{y},\mathit{t})$	continuity	momentum	energy	energy, extra term
$(1.0,0.8,2.0)$	$1.3\times {10}^{-13}$	$3.1\times {10}^{-14}$	$8.7\times {10}^{-14}$	$9.2\times {10}^{-2}$
$(0.6,1.4,3.0)$	$6.3\times {10}^{-14}$	$4.2\times {10}^{-14}$	$3.5\times {10}^{-14}$	$1.3\times {10}^{-1}$
$(2.0,0.5,5.0)$	$1.9\times {10}^{-14}$	$2.7\times {10}^{-14}$	$1.0\times {10}^{-14}$	$4.1\times {10}^{-2}$

Table 3. Residuals of the original Equations (4)–() evaluated in physical $(x,y,t)$ variables with $\nu =0.7$, $U_0=1.3$, $\mathit{Pr}=0.71$, step $h={10}^{-6}$, 30-digit arithmetic. The last column is the energy residual obtained when the spurious $\eta f_{\eta }{\theta }_{\eta }$ term is retained.

$(\mathit{x},\mathit{y},\mathit{t})$	continuity	momentum	energy	energy, extra term
$(1.0,0.8,2.0)$	$1.3\times {10}^{-13}$	$3.1\times {10}^{-14}$	$8.7\times {10}^{-14}$	$9.2\times {10}^{-2}$
$(0.6,1.4,3.0)$	$6.3\times {10}^{-14}$	$4.2\times {10}^{-14}$	$3.5\times {10}^{-14}$	$1.3\times {10}^{-1}$
$(2.0,0.5,5.0)$	$1.9\times {10}^{-14}$	$2.7\times {10}^{-14}$	$1.0\times {10}^{-14}$	$4.1\times {10}^{-2}$

9.2. Sense in Which the Solution Is Exact

Throughout, exact is used in the sense standard for the Navier–Stokes literature [18,19,20]: the velocity and temperature fields satisfy the governing partial differential equations identically, for arbitrary viscosity and diffusivity, with no truncation, series expansion or numerical approximation. The boundary conditions, as in the parent velocity solution, are attained only asymptotically in $\tau$. These statements are independent and both hold; the second is made quantitative next.

9.3. Attainment of the Boundary Conditions

Sun [16] imposes $f_{\eta }(0,\tau )=1$ and $f_{\eta }(\infty ,\tau )=0$ at $\tau =\infty$. At finite $\tau$ the compact form (33) makes the defect explicit and shows that a single mechanism is responsible for both. Since $F(\infty )=0$,

$$f_{\eta }(\infty ,\tau )={\displaystyle \frac{1}{2\tau }},$$

(56)

a uniform drift superposed on the profile $F\left(\omega \right)$; and since the same drift is present at the wall, where ${\omega }_0=1/\sqrt{3}{\tau }^{3/2}$ and $F\left({\omega }_0\right)\simeq 1+F^{\prime }\left(0\right){\omega }_0$,

$$f_{\eta }(0,\tau )\simeq 1+{\displaystyle \frac{1}{2\tau }}-{\displaystyle \frac{|F^{\prime }\left(0\right)|}{\sqrt{3}}}{\tau }^{-3/2}=1+{\displaystyle \frac{1}{2\tau }}-0.9134{\tau }^{-3/2}.$$

(57)

Equation (57) agrees with the exact wall value to four decimal places for $\tau ⩾10$ (Figure 9a). Two consequences follow, and it is important to state them plainly.

First, the approach to no slip is not monotone. The ${\tau }^{-3/2}$ term dominates at small $\tau$ and the $1/2\tau$ term at large $\tau$, so $f_{\eta }(0,\tau )$ crosses unity at $\tau =3.338$, reaches a maximum of $1.02276$ at $\tau =7.13$, and decays thereafter. Between $\tau \approx 4$ and $\tau \approx 30$ the fluid at the wall moves up to $2.3\%$ faster than the plate. The excess is small but it is a genuine violation of the no-slip condition, and any use of the solution in that range should acknowledge it.

Second, the two defects are of the same order. Both wall and outer conditions are satisfied to $O(1/2\tau )$, so a single criterion governs both: $f_{\eta }(0,\tau )$ lies within $1\%$ of unity for $\tau \gtrsim 34$ and within $0.5\%$ for $\tau \gtrsim 80$ (Figure 9b). The temperature inherits the same behaviour through ${\omega }_0$,

$$\theta (0,\tau )=erfc\left(\sqrt{\mathit{Pr}}{\omega }_0\right)\simeq 1-2\sqrt{{\displaystyle \frac{\mathit{Pr}}{3\pi }}}{\tau }^{-3/2},$$

(58)

which approaches unity monotonically and from below, reaching $1\%$ accuracy at $\tau \gtrsim 20$ for air and $\tau \gtrsim 70$ for water (Table 4). Taken together, the quantitative statements of Section 7Section 8 should be read as carrying an $O(1/2\tau )$ uncertainty: a few per cent for $\tau \sim 10$, below $1\%$ for $\tau \gtrsim 50$. In terms of the physical variables this is $x\lesssim U_{0}t/100$, so the solution describes the region near the leading edge, or equivalently the late-time behaviour at fixed x.

9.4. Further Limitations

The solution grows as $\sqrt{\nu t}$ and does not saturate at the Blasius thickness $\sqrt{\nu x/U_0}$; it is an exact member of a class of unsteady boundary layers rather than a description of the complete Rayleigh-to-Blasius transition. This is why h is independent of x, and why the ratio (55) decays without bound as $\tau$ grows.

The line $\tau \to \infty$ is a degenerate characteristic of (): the coefficient $A=1-2\tau f_{\eta }$ vanishes as $\eta \to \infty$, so no Cauchy data are prescribed there. Consequently more than one exact solution satisfies the same asymptotic conditions; at $\mathit{Pr}=1$ both $erfc\left(\omega \right)$ and $f_{\eta }$ do. Within the group-invariant class the solution (41) is unique, which is the natural similarity-theoretic statement. The practical spread is quantifiable: (48) gives $\mathit{St}/(c_f/2)=1$ exactly at $\mathit{Pr}=1$, against $0.7132$ from (53), and the difference may be taken as a measure of the residual indeterminacy.

Figure 10. (a) Approach of the wall and outer values to their prescribed limits; the shaded band is $\pm 5\%$. (b) The exact thickness ratio ${\delta }_T/{\delta }_u=1.091{\mathit{Pr}}^{-1/2}$, Equation (54), compared with the steady ${\mathit{Pr}}^{-1/3}$ estimate.

Figure 10. (a) Approach of the wall and outer values to their prescribed limits; the shaded band is $\pm 5\%$. (b) The exact thickness ratio ${\delta }_T/{\delta }_u=1.091{\mathit{Pr}}^{-1/2}$, Equation (54), compared with the steady ${\mathit{Pr}}^{-1/3}$ estimate.

10. Conclusions

The diffusion-time similarity transformation $\eta =y/\delta \left(x\right)$, $\tau =\nu t/{\delta }^2\left(x\right)$ transforms the unsteady energy equation into (19), with the same coefficients $\alpha ,\beta ,\gamma$ as the momentum equation. The terms in $\eta f_{\eta }{\theta }_{\eta }$ arising from $u{T}_x$ and $v{T}_y$ cancel identically. The coupled system closes only for $U=const$, so the flat plate is the distinguished member of the family $U=C{x}^m$.

Sun’s exact velocity solution collapses to the three expressions (25)–() and is invariant under the one-parameter group generated by $X=(\eta \tau +2){\partial }_{\eta }+2{\tau }^2{\partial }_{\tau }$, with invariant $\omega =(3\eta \tau +2)/2\sqrt{3}{\tau }^{3/2}$ and $Xf=\tau f-\eta$. In particular $u/U_0=1/2\tau +F\left(\omega \right)$.

On group-invariant temperature fields the convective operator of the energy equation loses all dependence on the velocity field, and the unsteady thermal boundary layer reduces exactly to ${\Theta }^{\prime \prime }+2\mathit{Pr}\omega {\Theta }^{\prime }+\mathit{Pr}\mathit{Ec}{\left[F^{\prime }\left(\omega \right)\right]}^2=0$, the unsteady counterpart of the Pohlhausen equation. Its solutions are elementary: $\theta =erfc\left(\sqrt{\mathit{Pr}}\omega \right)+\mathit{Pr}\mathit{Ec}\Phi \left(\omega \right)$ for an isothermal wall, and a recovery factor $r=2\mathit{Pr}{P}_{\infty }$ for an adiabatic wall. The engineering results are ${\mathit{Nu}}_x=0.6910(1-\mathit{Pr}\mathit{Ec}{P}_{\infty })\sqrt{\mathit{Pr}{\mathit{Re}}_x/\tau }$, $\mathit{St}/(c_f/2)=0.7132{\mathit{Pr}}^{-1/2}(1-\mathit{Pr}\mathit{Ec}{P}_{\infty })$, ${\delta }_T/{\delta }_u=1.0910{\mathit{Pr}}^{-1/2}$, ${\mathit{Ec}}^{*}=2/r$, and h independent of x.

The prevailing practice of substituting the steady Blasius field into the unsteady energy equation misestimates the wall heat flux by $2.081{\mathit{Pr}}^{1/6}{\tau }^{-1/2}$; the two agree only near $\tau =4.33{\mathit{Pr}}^{1/3}$, and over the whole early transient the discrepancy is of order unity. A consistent unsteady treatment of the velocity field is therefore not a refinement but a requirement.

Two extensions suggest themselves. A temperature field of the form $\theta ={\Theta }_0\left(\omega \right)+{\tau }^{-1}{\Theta }_1\left(\omega \right)+\dots$ would relax the large-$\tau$ restriction of Section 9 systematically, each order inheriting the same linear operator and requiring only a quadrature; the leading correction would in particular address the $O(1/2\tau )$ defect in the boundary conditions. More broadly, the mechanism identified in (37) is not special to this flow: whenever a boundary-layer solution is invariant under a one-parameter group, the convective operator restricted to invariant scalars becomes universal, and any passive scalar reduces to an ordinary differential equation with the velocity field entering only through a source term.