A closed-form solution of DPL bioheat transfer problem with time-periodic boundary conditions

A closed-form (long-time) solution of one-dimensional dual-phase lag bioheat transfer problem with consistent time-periodic boundary conditions (BCs) is presented in this paper for planar, cylindrical, and spherical skin tissue for a newly developed solution methodology. The steady-periodic solution is composed of a steady-state part and an oscillating part; corresponding to the constant and oscillating parts of BCs, respectively. Using the superposition principle, these two parts are split into two problems, which are solved separately. The steady-state part is fairly straightforward to obtain, while for the oscillating part, an alternate Laplace transform (LT) approach is proposed in this work. It is demonstrated that for sinusoidal BCs, a closed-form solution in the time domain can be obtained by evaluating an approximate convolution integral, which emulates the effect of the inverse LT. The obtained closed-form solution is free of any series summation or numerical inversion, thereby, making it computationally very efficient compared with conventional LT and eigenfunctions-based approaches. The current methodology is verified with the established eigenfunctions expansion-based methodology. It can be seen that the long-time solutions obtained by these two approaches are almost identical. The verified methodology is further extended for the time-periodic nonsinusoidal BCs. The ease of implementation and simplicity of the new methodology for both sinusoidal and nonsinusoidal BCs is demonstrated using a few test cases. It is evident from the results that the developed methodology leads to an efficient and accurate solution.