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Geodesic Loewner paths with varying boundary conditions.

ABSTRACT: Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner's equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped 'mathematical' plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of π/5.

SUBMITTER: McDonald R

PROVIDER: S-EPMC7655755 | biostudies-literature |

REPOSITORIES: biostudies-literature

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Project description:Piezocrystals, especially the relaxor-based ferroelectric crystals, have been subject to intense investigation and development within the past three decades, motivated by the performance advantages offered by their ultrahigh piezoelectric coefficients and higher electromechanical coupling coefficients than piezoceramics. Structural anisotropy of piezocrystals also provides opportunities for devices to operate in novel vibration modes, such as the d36 face shear mode, with domain engineering and special crystal cuts. These piezocrystal characteristics contribute to their potential usage in a wide range of low- and high-power ultrasound applications. In such applications, conventional piezoelectric materials are presently subject to varying mechanical stress/pressure, temperature and electric field conditions. However, as observed previously, piezocrystal properties are significantly affected by a single such condition or a combination of conditions. Laboratory characterisation of the piezocrystal properties under these conditions is therefore essential to fully understand these materials and to allow electroacoustic transducer design in realistic scenarios. This will help to establish the extent to which these high performance piezocrystals can replace conventional piezoceramics in demanding applications. However, such characterisation requires specific experimental arrangements, examples of which are reported here, along with relevant results. The measurements include high frequency-resolution impedance spectroscopy with the piezocrystal material under mechanical stress 0-60 MPa, temperature 20-200 °C, high electric AC drive and DC bias. A laser Doppler vibrometer and infrared thermal camera are also integrated into the measurement system for vibration mode shape scanning and thermal conditioning with high AC drive. Three generations of piezocrystal have been tested: (I) binary, PMN-PT; (II) ternary, PIN-PMN-PT; and (III) doped ternary, Mn:PIN-PMN-PT. Utilising resonant mode analysis, variations in elastic, dielectric and piezoelectric constants and coupling coefficients have been analysed, and tests with thermal conditioning have been carried out to assess the stability of the piezocrystals under high power conditions.

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Geodesic Loewner paths with varying boundary conditions.

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