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

The thermal boundary layer on a flat plate is almost always computed by inserting the steady Blasius velocity field into a time-dependent energy equation, so that the temperature evolves along trajectories that do not. The inconsistency has been unavoidable: no exact solution of the two-dimensional unsteady laminar boundary layer was available until Sun [16] obtained one in terms of Kummer functions, using the diffusion time \(\tau=\nu t/\delta^{2}(x)\) as a similarity variable. We solve the corresponding thermal problem. The similarity transformation applied to the energy equation yields \[ \theta_{\tau}-\alpha f\theta_{\eta}+\gamma\tau(f_{\tau}\theta_{\eta} -f_{\eta}\theta_{\tau})=Pr^{-1}\theta_{\eta\eta}+Ec\,f_{\eta\eta}^{2}, \] the terms in \(\eta f_{\eta}\theta_{\eta}\) cancelling identically; the system closes only for a flat plate. Sun's solution, given through six auxiliary functions, collapses to three Kummer-\(U\) expressions and is invariant under the group generated by \(X=(\eta\tau+2)\partial_{\eta}+2\tau^{2}\partial_{\tau}\), with \(Xf=\tau f-\eta\). On group-invariant temperature fields the convective operator loses all dependence on the velocity field, and the energy equation reduces exactly to \[ \Theta''+2Pr\,\omega\Theta'+Pr\,Ec\,[F'(\omega)]^{2}=0 \] with \(\omega=(3\eta\tau+2)/2\sqrt{3}\tau^{3/2}\). Hence \[ \theta=\operatorname{erfc}(\sqrt{Pr}\,\omega)+Pr\,Ec\,\Phi(\omega), \] together with closed forms for the Nusselt number, a recovery factor \(r=2Pr\,P_{\infty}\), a Reynolds analogy factor \(0.7132Pr^{-1/2}\) and a thickness ratio \(1.0910Pr^{-1/2}\) exact at all $Pr$. Both fields satisfy the governing equations to below \(10^{-17}\). Boundary conditions are met to \(O(\tau^{-1})\), inherited from the parent solution.