Review of potential flow solutions for velocity and shape of long isolated bubbles in vertical pipes

The motion of elongated gas bubbles in vertical pipes has been studied extensively over the past century. A number of empirical and numerical correlations have emerged out of this curiosity; amongst them, analytical solutions have been proposed. A review of the major results and resolution methods based on a potential flow theory approach is presented in this article. The governing equations of a single elongated gas bubble rising in a stagnant or moving liquid are given in the potential flow formalism. Two different resolution methods (the power series method and the total derivative method) are studied in detail. The results (velocity and shape) are investigated with respect to the surface tension effect. The use of a new multi-objective solver coupled with the total derivative method improves the research of solutions and demonstrates its validity for determining the bubble velocity. This review aims to highlight the power of analytical tools, resolution methods and their associated limitations behind often well-known and wide-spread results in the literature.

Prediction of nonlinear dynamic coefficients of tilting-pad journal bearings with pad perturbation

In this paper, a modified partial derivative method is developed to predict the linear and nonlinear dynamic coefficients of tilting-pad journal bearings with journal and pad perturbation. To this end, Reynolds equation and its boundary conditions along with equilibrium equations of the pad are used. Finite difference, partial derivative method, and perturbation technique have been employed simultaneously for solving these equations. The accuracy of the results is investigated by comparing the linear dynamic coefficients of three types of tilting-pad journal bearings with those published the literature. It is shown that the nonlinear dynamic coefficients depend on Sommerfeld number, eccentricity ratio, and length to diameter ratio. Similar to the case of linear dynamic coefficients of TPJB, it is observed that the eccentricity ratio effects on nonlinear dynamic coefficients are more notable when the eccentricity ratio is higher than 0.8 or less than 0.2.

Method Development and Validation for the Estimation of Etizolam and Propranolol Hydrochloride in bulk and Combined Dosage Forms by Simultaneous and Derivative Spectroscopic Methods

Accurate, simple, sensitive and rapid economic UV spectroscopic methods were developed for the estimation of Etizolam and Propranolol Hydrochloride in bulk and combined dosage form. The present study deals with the UV spectroscopic method development and validation for the Simultaneous Equation method and First Derivative method of Etizolam and Propranolol Hydrochloride in bulk and combined dosage form at determined wavelength of Etizolam and Propranolol Hydrochloride at 244nm and 288nm for Simultaneous Equation method and 234nm and 289nm for First Derivative Method. The linearity range for Etizolam and Propranolol Hydrochloride was 1-5µg/ml and 10-50µg/ml, and exhibit good correlation coefficient of Etizolam and Propranolol Hydrochloride was 0.9877 and 0.9977 for Simultaneous Equation method and 0.9872 and 0.9977 for First Derivative method, respectively and excellent mean recovery (98-102%). The precision was found to be within limit (%RSD <2). Comparatively First Derivative method is more sensitive than Simultaneous Equation method. The methods were validated statistically and parameters like linearity, precision, accuracy, specificity and assay was studied according to ICH guidelines and can be applicable in determination of both drugs in routine quality control analysis of drugs in bulk and combined dosage form.

A noniterative reconstruction method for solving a time-fractional inverse source problem from partial boundary measurements

In this paper, a noniterative method for solving an inverse source problem governed by the two-dimensional time-fractional diffusion equation is proposed. The basic idea consists in reconstructing the geometrical support of the unknown source from partial boundary measurements of the associated potential. A Kohn-Vogelius type shape functional is considered together with a regularization term penalizing the relative perimeter of the unknown set of anomalies. Identifiability result is derived and uniqueness of a minimizer is ensured. The shape functional measuring the misfit between the solutions of two auxiliary problems containing information about the boundary measurements is minimized with respect to a finite number of ball-shaped trial anomalies by using the topological derivative method. In particular, the second-order topological gradient is exploited to devise an efficient and fast noniterative reconstruction algorithm. Finally, some numerical experiments are presented, showing different features of the proposed approach in reconstructing multiple anomalies of varying shapes and sizes by taking noisy data into account .

Topology optimization of structures subject to self-weight loading under stress constraints

PurposeThe purpose of this paper is to present an approach for structural weight minimization under von Mises stress constraints and self-weight loading based on the topological derivative method. Although self-weight loading topology has been the subject of intense research, mainly compliance minimization has been addressed.Design/methodology/approachThe resulting minimization problem is solved with the help of the topological derivative method, which allows the development of efficient and robust topology optimization algorithms. Then, the derived result is used together with a level-set domain representation method to devise a topology design algorithm.FindingsNumerical examples are presented, showing the effectiveness of the proposed approach in solving a structural topology optimization problem under self-weight loading and stress constraint. When the self-weight loading is dominant, the presence of the regularizing term in the formulation is crucial for the design process.Originality/valueThe novelty of this research work lies in the use of a regularized formulation to deal with the presence of the self-weight loading combined with a penalization function to treat the von Mises stress constraint.

Brittle fracture on plates governed by topological derivatives

PurposeIn the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.Design/methodology/approachThe Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.FindingsNumerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.Originality/valueThis is the first work concerning brittle fracture modeling of plate structures based on the topological derivative method.

An optimal finite-difference method based on the elongated stencil for 2D frequency-domain acoustic wave modeling

Frequency-domain finite-difference (FDFD) modeling plays an important role in exploration seismology. However, a major disadvantage of FDFD modeling is the computational cost, especially for large-scale models. By compactly distributing nonzero strips, the elongated stencil helps to generate a narrow-bandwidth impedance matrix, improving computational efficiency without sacrificing numerical accuracy. To further improve the accuracy and efficiency of modeling, we have developed an optimal FDFD method with an elongated stencil for 2D acoustic-wave modeling. The Laplacian term is approximated using the directional-derivative method and the average-derivative method. The dispersion analysis indicates that this elongated-stencil-based method (ESM) achieves higher accuracy than other finite-difference methods with the elongated stencil, and it is more suitable for large grid-spacing ratios. To keep the phase-velocity error within 1%, 15-point and 21-point schemes in the ESM only require approximately 2.28 and 2.19 grid points per wavelength, respectively, when the grid-spacing ratio, namely, the ratio of directional sampling intervals, is not less than 1.5. Moreover, we also adopt a variable-stencil-length scheme, in which the stencil length varies with the velocity, to further reduce the computational cost in frequency-domain modeling. Several numerical examples are presented to demonstrate the effectiveness of our ESM.