An Exact Solution of the Unsteady Laminar Thermal Boundary Layer on a Flat Plate
Abstract
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 $r=2Pr\,P_{\infty}$,…
Read the paper
Related studies
- eggNOG 5.0: a hierarchical, functionally and phylogenetically annotated orthology resource based on 5090 organisms and 2502 viruses
- Online Survival Analysis Software to Assess the Prognostic Value of Biomarkers Using Transcriptomic Data in Non-Small-Cell Lung Cancer
- OptiType: precision HLA typing from next-generation sequencing data
- Nicotinic Cholinergic System and COVID-19: In Silico Identification of an Interaction between SARS-CoV-2 and Nicotinic Receptors with Potential Therapeutic Targeting Implications
- Saccharomyces cerevisiae cells have three Omega class glutathione S-transferases acting as 1-Cys thiol transferases
- Hemolysis at low blood flow rates: in-vitro and in-silico evaluation of a centrifugal blood pump
Go deeper in Evidence Hub
Evidence Hub adds a study data sheet (design, sample size, evidence level, key numbers), an AI summary and a chat grounded in the paper's text.