Green’s Function Solution of Dual-Phase-Lag Model

Lee, Yung-Ming; Lin, Pei-Chi; Tsai, Tsung-Wen

doi:10.1115/mnhmt2009-18425

Cited by 5 publications

(2 citation statements)

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“…Moreover, the analytical approaches provide the benchmark solutions to heat transfer problems. The separation of variables (SOVs), 34 Laplace transform (LT), 28 and Green's function 35,36 are the most used analytical approaches for DPL-based bioheat transfer-related problems. It is noted that the SOV is not applicable to time-dependent BCs.…”

Section: Introductionmentioning

confidence: 99%

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

Biswas,

Singh,

Srivastava

2023

Heat Trans

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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.

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Section: Introductionmentioning

confidence: 99%

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

Biswas,

Singh,

Srivastava

2023

Heat Trans

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“…Analytical solutions and numerical methods for the DPL equation with Dirichlet or Neumann boundary condition have been well studied. These analytical solution methods include the Laplace transform method , the separation of variables method , the Green's function method and the integral equation method . Conversely, the numerical methods for solving the DPL equation include the finite difference method , the finite element method , the lattice Boltzmann method , the Laplace transform method together with the control volume method , the high‐order TVD scheme with Roe's superbee limiter function , and the space‐time conservation element and solution element method .…”

Section: Introductionmentioning

confidence: 99%

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Sun

Dai

et al. 2015

Numerical Methods Partial

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Dual-phase-lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher-order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher-order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher-order accurate and unconditionally stable compact finite difference scheme for solving one-dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two-dimensional case and develop a fourth-order accurate compact finite difference method in space coupled with the Crank-Nicolson method in time, where the Robin's boundary condition is approximated using a third-order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth-order in space and second-order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis.

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Accurate numerical method for solving dual-phase-lagging equation with temperature jump boundary condition in nano heat conduction

Dai

Han

Sun

2013

International Journal of Heat and Mass Transfer

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Green’s Function Solution of Dual-Phase-Lag Model

Cited by 5 publications

References 0 publications

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

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

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Accurate numerical method for solving dual-phase-lagging equation with temperature jump boundary condition in nano heat conduction

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