A Composite Ultrasonic Transducer with a Fractal Architecture

To ensure the safe operation of many safety critical structures such as nuclear plants, aircraft and oil pipelines, non-destructive imaging is employed using piezoelectric ultrasonic transducers. These sensors typically operate at a single frequency due to the restrictions imposed on its resonant behaviour by the use of a single length scale in its design. To allow these transducers to transmit and receive more complex signals it would seem logical to use a range of length scales in the design so that a wide range of resonating frequencies will result. In this article we derive a mathematical model to predict the dynamics of an ultrasound transducer that achieves this range of length scales by adopting a fractal architecture. In fact, the device is modelled as a graph where the nodes represent segments of the piezoelectric and polymer materials. The electrical and mechanical fields that are contained within this graph are then expressed in terms of a finite element basis. The structure of the resulting discretised equations yields to a renormalisation methodology which is used to derive expressions for the non-dimensionalised electrical impedance and the transmission and reception sensitivities. A comparison with a homogenised (standard) design shows some benefits of these fractal designs

Analysis of a fractal ultrasonic transducer with a range of piezoelectric length scales

The transmission and reception sensitivities of most piezoelectric ultrasonic transducers are enhanced by their geometrical structures. This structure is normally a regular, periodic one with one principal length scale which, due to the resonant nature of the devices, determines the central operating frequency. There is engineering interest in building wide bandwidth devices, and so it follows that in their design, resonators that have a range of length scales should be used. This paper describes a mathematical model of a fractal ultrasound transducer whose piezoelectric components span a range of length scales. There have been many previous studies of wave propagation in the Sierpinski gasket but this paper is the first to study its complement. This is a critically important mathematical development as the complement is formed from a broad distribution of triangle sizes whereas the Sierpinski gasket is formed from triangles of equal size. Within this structure, the electrical and mechanical fields fluctuate in tune with the time dependent displacement of these substructures. A new set of basis functions is developed that allow us to express this displacement as part of a finite element methodology. A renormalisation approach is then used to develop a recursion scheme that analytically describes the key components from the discrete matrices that arise. Expressions for the transducer's operational characteristics are then derived and analysed as a function of the driving frequency. It transpires that the fractal device has a significantly higher reception sensitivity (18 dB) and a significantly wider bandwidth (3 MHz) than an equivalent Euclidean (standard) device

Investigating the performance of a fractal ultrasonic transducer under varying system conditions

As applications become more widespread there is an ever-increasing need to improve the accuracy of ultrasound transducers, in order to detect at much finer resolutions. In comparison with naturally occurring ultrasound systems the man-made systems have much poorer accuracy, and the scope for improvement has somewhat plateaued as existing transducer designs have been iteratively improved over many years. The desire to bridge the gap between the man-made and naturally occurring systems has led to recent investigation of transducers with a more complex geometry, in order to replicate the complex structure of the natural systems. These transducers have structures representing fractal geometries, and these have been shown to be capable of delivering improved performance in comparison with standard transducer designs. This paper undertakes a detailed investigation of the comparative performance of a standard transducer design, and a transducer based on a fractal geometry. By considering how these performances vary with respect to the key system parameters, a robust assessment of the fractal transducer performance is provided

Assessment of economic, thermal and hydraulic performances a corrugated helical heat exchanger filled with non-Newtonian nanofluid

Improved heat transfer efficiency with considering economic analysis in heating systems is an interesting topic for researchers and scientists in recent years. This research investigates the heat transfer rate (HTR) and flow of non-Newtonian water-Carboxyl methyl cellulose (CMC) based Al2O3 nanofluid in a helical heat exchanger equipped with common and novel turbulators using two-phase model. The requirements for dimensions and cost reduction and also energy saving in thermal systems are the main goal of this study. According to gained results usage of corrugated channel in helical heat exchanger has a considerable influence on thermal and hydraulic performance evaluation criteria (THPEC) index of helical heat exchanger and can improve the THPEC index. Thus, Re = 5000 is obtained as an optimum value, in which the maximum THPEC value is achieved. As it is found in this paper, in case of using novel heat exchanger instead of the basic smooth system, the thermal properties (by considering Nusselt number) increases about 210%, the hydraulic performance (friction factor) reduces about 28%, performance evaluation criteria index increases about 57% and the material consumption (in case of similar THPEC) decreases about 31%. In another word, with considering economic analysis for the basic and novel system which has same efficiencies, the novel one has lower length and consequently 31% lower material

Relation-theoretic almost ϕ-contractions with an application to elastic beam equations

In this article, we prove some results on existence and uniqueness of fixed points for an almost $\phi$-contraction mapping defined on a metric space endowed with an amorphous relation. Our results generalize and improve several well known fixed point theorems of the existing literature. To substantiate the credibility of our results, we construct some examples. We also apply our results to determine a unique solution of a boundary value problem associated with nonlinear elastic beam equations

A study of nonlocal fractional delay differential equations with hemivariational inequality

In this paper, we study an abstract system of fractional delay differential equations of order 1 < q < 2 with a hemivariational inequality in Banach spaces. To establish the existence of a solution to the abstract inequality, we employ the Rothe technique in conjunction with the surjectivity of multivalued pseudomonotone operators and features of the Clarke generalized gradient. Further, to show the existence of the fractional differential equation, we use the fractional cosine family and fixed point theorem. Finally, we include an example to elaborate the effectiveness of the findings

Convection in a Ferromagnetic Fluid Layer Influenced by Changeable Gravity and Viscosity

The motive of this work is to numerically evaluate the effect of changeable gravitational fields and varying viscosity on the beginning of convection in ferromagnetic fluid layer. The fluid layer is constrained by two free boundaries and varying gravitational fields that vary with distance across the layer. The authors hypothesized two categories of gravitational field variation, which can be subdivided into six distinct cases: (i) f(z)=z, (ii) f(z)=ez, (iii) f(z)=log(1+z), (iv) f(z)=−z, (v) f(z)=−z2, and (vi) f(z)=z2−2z. The normal mode method was applied, and the single term Galerkin approach was used to solve the ensuing eigenvalue problem. The results imply that, in the first three cases, the gravity variation parameter speeds up the commencement of convection, while, in the last three cases, the viscosity variation parameter and gravity variation parameter slow down the onset of convection. It was also observed that, in the absence of the viscosity variation parameter, the non-buoyancy magnetization parameter destabilizes the impact on the beginning of convection but, in the presence of the viscosity variation parameter, it destabilizes or stabilizes impact on the beginning of convection. In the case of oscillatory convection, the results illustrate that oscillatory modes are not permitted, suggesting the validity of the theory of exchange of stabilities. Additionally, it was also discovered that the system is more stable for case (vi) and more unstable for case (ii)

Mechanical characteristics of MHD of the non-Newtonian magnetohydrodynamic Maxwell fluid flow past a bi-directional convectively heated surface with mass flux conditions

In engineering and manufacturing industries, stretching flow phenomena have numerous real-world implementations. Real-world applications related to stretched flow models are metalworking, crystal growth processes, cooling of fibers, and plastics sheets. Therefore, in this work, the mechanical characteristics of the magnetohydrodynamics of the non-Newtonian Maxwell nanofluid flow through a bi-directional linearly stretching surface are explored. Brownian motion, thermophoresis, and chemical reaction impacts are considered in this analysis. Additionally, thermal convective and mass flux conditions are taken into consideration. The mathematical framework of the existing problem is constructed on highly non-linear partial differential equations (PDEs). Suitable similarity transformations are used for the conversion of partial differential equations into ordinary differential equations (ODEs). The flow problem is tackled with the homotopy analysis method, which is capable of solving higher-order non-linear differential equations. Different flow profiles against various flow parameters are discussed physically. Heat and mass transference mechanisms for distinct flow factors are analyzed in a tabular form. The outcomes showed that both primary and secondary velocities are the declining functions of magnetic and Maxwell fluid parameters. The heat transfer rate rises with the cumulative values of the Brownian motion and thermal Biot number. In addition, the mass transfer rate decreases with the rising Schmidt number, Brownian motion parameter, and chemical reaction parameter, while it increases with the augmenting thermophoresis parameter. It has been highlighted that streamlines in the current work for Maxwell and Newtonian models are in fact different from one another

A Numerical Computation for an Impulsive Fractional Differential Equation with a Deviated Argument

Symmetry analysis is an effective tool for understanding differential equations, particularly when analyzing equations derived from mathematical concepts. This paper is concerned with an impulsive fractional differential equation (IFDE) with a deviated argument. We implement two techniques, the Adomian decomposition method (ADM) and the fractional differential transform method (FDTM), for solving IFDEs. In these schemes, we obtain the solutions in the form of a convergent power series with easily computed components. This paper also discusses the existence and uniqueness of solutions using the Banach contraction principle. This paper presents a numerical comparison between the two methods for solving IFDEs. We illustrate the proposed methods with a few examples and find their numerical solutions. Moreover, we show the graph of numerical solutions via MATLAB. The numerical results demonstrate that the ADM approach is quite accurate and readily implemented

A numerical exploration of the comparative analysis on water and kerosene oil-based Cu–CuO/hybrid nanofluid flows over a convectively heated surface

Abstract The fluid flow over an extending sheet has many applications in different fields which include, manufacturing processes, coating, thin film decomposition, heat and mass transfer, biomedical applications, aerospace engineering, environmental science, energy production. Keeping in mind these applications, the non-Newtonian hybrid nanofluid flow comprising of Cu and CuO nanoparticles over an extending sheet is analyzed in this work. Two different base fluids called kerosene oil and water have been incorporated. The sheet is considered to be thermally convective along with zero mass flux condition. The main equations of modeled problem have been transformed to dimensionless form by using similarity variables. The designed problem is evaluated computationally by using bvp4c Matlab function. Validation of the present results is also performed. The impacts of magnetic, Brownian motion, chemical reaction, suction and thermophoresis factors are analyzed and discussed in details. The outcomes of the present investigation declare that the kerosene oil-based hybrid nanofluid flow has greater velocity and concentration profiles than that of the water-based hybrid nanofluid flow. The water-based hybrid nanofluid has greater temperature distribution than that of kerosene oil-based hybrid nanofluid flow. The streamlines of the kerosene oil-based Newtonian and non-Newtonian hybrid nanofluid flows are more stretched than water-based Newtonian and non-Newtonian hybrid nanofluid flows