The Westervelt equation with a causal propagation operator coupled to the bioheat equation.

Guy V.  Norton; Robert D.  Purrington

doi:10.3934/eect.2016013

Article Contents

2016, Volume 5, Issue 3: 449-461. Doi: 10.3934/eect.2016013

This issue Previous Article Nonlinear diffusion equations in fluid mixtures Next Article Recasting a Brinkman-based acoustic model as the damped Burgers equation

The Westervelt equation with a causal propagation operator coupled to the bioheat equation.

Guy V. Norton^1, and
Robert D. Purrington^1,

1.
Physics and Engineering Physics Department, Tulane University, New Orleans, LA 70118

Received: September 30, 2015

Revised: December 31, 2015

Published: August 2016

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Abstract

The Westervelt wave equation is frequently used to describe non-linear propagation of finite amplitude sound. If one assumes that the medium can be treated as a thermoviscous fluid, a loss mechanism can be incorporated. In this as in previous work the authors replaced the typical loss mechanism incorporated in the Westervelt equation with a causal Time Domain Propagation Factor (TDPF) which incorporates the full dispersive effects (both frequency dependent phase velocity and attenuation) in the numerical solution while remaining in the time-domain. In the present work we investigate heat deposition due to finite amplitude propagation through a dispersive medium (e.g., human tissue). To this end, the Westervelt equation with and without the TDPF is coupled to the Pennes bioheat equation and the coupled equations are solved using the method of finite differences to determine the resulting heat deposition. We show that non-linear effects are large and that proper treatment of dispersion results in significant changes as compared to modeling the medium as a thermoviscous fluid.

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Mathematics Subject Classification: Primary: 35Q; Secondary: 65M.

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