Spherical Coefficients of Slice Regular Functions

Given a quaternionic slice regular function f, we give a direct and effective way to compute the coefficients of its spherical expansion at any point. Such coefficients are obtained in terms of spherical and slice derivatives of the function itself. Afterwards, we compare the coefficients of f with those of its slice derivative $$\partial _{c}f$$ ∂ c f obtaining a countable family of differential equations satisfied by any slice regular function. The results are proved in all details and are accompanied to several examples. For some of the results, we also give alternative proofs.

On the spherical expansion for calculating the sound radiated by a baffled circular piston

An efficient and accurate method for calculating the sound radiated by a baffled circular rigid piston is using spherical harmonics, and the solution is a series containing the integral of spherical Bessel functions. The integral is usually calculated with the generalized hypergeometric functions in existing literatures, which shows poor convergence at middle and high frequencies due to the overflow and the loss of significant figures. A rigorous and closed form solution of the integral is derived in this paper based on the recurrence method, which is accurate in the whole frequency range and thousands of times faster than the existing methods. It is shown that the proposed method can be extended for the calculation of the sound radiated by a baffled piston and an unbaffled resilient disk with axisymmetric velocity and pressure profiles, respectively, and some baffled rotating sources where the velocity profile is asymmetric.

On the real differential of a slice regular function

In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (calledspherical expansion), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces on which it results to be constant.

Energy of Laser Induced Shockwaves

High throughput experimentation is a possibility to develop new materials in a short time in order to meet the demands of efficient characterisation of compositions. Thus, fundamentals of a new hardness measurement method are investigated based on laser-induced shockwaves. In this study, plasma is created with a nanosecond pulsed TEA CO2 laser on top of an indenter. Further interactions of the plasma with the high intensity laser beam result in a shockwave. The pressure of the shockwave is used to push an indenter inside a material surface. So far, the energy transfer of the shockwave on indenters is not fully understood. Therefore, pendulum experiments are conducted to calculate how much energy can be transferred from the shockwave into the indenter. For these experiments, a bob, which geometry is equal to the indenter geometry, is connected to a thread pendulum and maximum deflection angles are recorded with a high-speed camera. Under standard conditions and the assumption of a spherical expansion of the shockwave, the experiments show that with a 6 J pulse energy a shockwave energy of up to 9 μJ can be used for indentation tests.

Modeling of Drug Delivery by A Pump Driven Micro-Needle Array System

Background: Micro-needles were proposed as one of the alternatives to deliver drugs painlessly passing through stratum corneum in recent years. In this work, a mathematical model is presented to characterize the in fusion flow of a hollow micro-needle array driven by a micro-pump. Methods: By assuming the injection of each micro-needle undergoes a spherical expansion and diffusion, the model is able to calculate the time-varying expansion radius, and the diffusion boundary, provided that the material properties and the micro-needle system parameters are known. Results and Conclusion: The calculation results show that the expansion caused by the infusion of micro-needles stops and the flow rate drops to zero in a short time. However, the diffusion boundary is much bigger than the expansion and the infusion continues if the surrounding material is absorptive. The experimental results of jet infusion through a single needle in silicon rubber and polyacrylamide gel agree with the calculation results qualitatively.

Contact Law and Coefficient of Restitution in Elastoplastic Spheres

A complete contact cycle of an elastoplastic sphere consists of loading and unloading phases. The loading phase may fall into three sequential regimes: elastic, mixed elastic–plastic, and fully plastic. In this paper, we distinguish the transition points among the three regimes via the material hardness and a dimensionless geometric parameter corresponding to the onset of the fully plastic regime. Based on Johnson’s simplified spherical expansion model, together with the well-supported force–indentation relationships in the elastic and fully plastic regimes, we build an analytical approximation for the mixed elastic–plastic regime by enforcing the C1 continuity of a loading force–indentation curve. Unloading responses of the elastoplastic sphere are characterized by an elastic force–indentation relation, which has a Hertzian-type form but takes into account the effects of the strain hardening that occurs in the mixed elastic–plastic regime. We validate the model by comparing with existing quasi-static and impact experiments and show that the model can precisely capture the force–indentation responses. Further validation is performed by employing the proposed compliance model to investigate the coefficient of restitution (COR). We achieve agreement between our numerical results and the experimental data reported in other studies. Particularly, we find that the COR is inversely proportional to the impacting velocity with an exponent equal to 1/6, instead of 1/4 reported by many other models.