A new formulation and analysis of a collapsing bubble with different content in a liquid induced during acoustic cavitation

2016 ◽  
Vol 26 (6) ◽  
pp. 1729-1746
Author(s):  
Ali Alhelfi ◽  
Bengt Ake Sunden
Keyword(s):  
Partial Differential Equations ◽  
Numerical Model ◽  
Differential Equations ◽  
Bubble Dynamics ◽  
Acoustic Cavitation ◽  
Ultrasonic Field ◽  
Bubble Surface ◽  
Correct Prediction ◽  
Partial Differential ◽  
Content Type

Purpose – The purpose of this paper is to present numerical investigation of the gas/vapor bubble dynamics under the influence of an ultrasonic field to give a more comprehensive understanding of the phenomenon and present new results Design/methodology/approach – In order to formulate the mathematical model, a set of governing equations for the gas inside the bubble and the liquid surrounding it are used. All hydrodynamics forces acting on the bubble are considered in the typical solution. The systems of equations required to be solved consist of ordinary and partial differential equations, which are both nonlinear and time dependent equations. A fourth order Runge-Kutta method is applied to solve the ordinary differential equations. On the other hand, the finite difference method is employed to solve the partial differential equations and a time-marching technique is applied. Findings – The numerical model which is developed in the current study permits a correct prediction of the bubble behavior and its characteristics in an acoustic field generated at this occasion. Originality/value – Previous studies considering numerical simulations of an acoustic bubble were performed based on the polytropic approximation or pressure uniformity models of the contents inside the bubble. In this study, an enhanced numerical model is developed to study the acoustic cavitation phenomenon and the enhancement concerns taking into account both the pressure and temperature gradients inside the bubble as well as heat transfer through the bubble surface into account which is very important to obtain the temperature of the liquid surrounding the bubble surface.

Differential quadrature method for nonlinear fractional partial differential equations

2018 ◽  
Vol 35 (6) ◽  
pp. 2349-2366 ◽  
Author(s):  
Umer Saeed ◽  
Mujeeb ur Rehman ◽  
Qamar Din
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
High Efficiency ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Operational Matrices ◽  
Fractional Partial Differential Equations ◽  
Infinite Domain ◽  
Partial Differential ◽  
Content Type

Purpose The purpose of this paper is to propose a method for solving nonlinear fractional partial differential equations on the semi-infinite domain and to get better and more accurate results. Design/methodology/approach The authors proposed a method by using the Chebyshev wavelets in conjunction with differential quadrature technique. The operational matrices for the method are derived, constructed and used for the solution of nonlinear fractional partial differential equations. Findings The operational matrices contain many zero entries, which lead to the high efficiency of the method and reasonable accuracy is achieved even with less number of grid points. The results are in good agreement with exact solutions and more accurate as compared to Haar wavelet method. Originality/value Many engineers can use the presented method for solving their nonlinear fractional models.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2620-2649 ◽  
Author(s):  
Rajni Rohila ◽  
R.C. Mittal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Spline Functions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

Reconstruction of the parameter in parabolic partial differential equations using Haar wavelet method

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Gopal Priyadarshi ◽  
B.V. Rathish Kumar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Computational Cost ◽  
Haar Wavelet ◽  
Haar Wavelets ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Wavelet Method ◽  
Content Type ◽  
Haar Wavelet Method

Purpose In the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose a wavelet collocation method based on Haar wavelets to identify a parameter in parabolic partial differential equations (PDEs). As Haar wavelet is defined in a very simple way, implementation of the Haar wavelet method becomes easier than the other numerical methods such as finite element method and spectral method. The computational time taken by this method is very less because Haar matrices and Haar integral matrices are stored once and used for each iteration. In the case of Haar wavelet method, Dirichlet boundary conditions are incorporated automatically. Apart from this property, Haar wavelets are compactly supported orthonormal functions. These properties lead to a huge reduction in the computational cost of the method. Design/methodology/approach The aim of this paper is to reconstruct the source control parameter arises in quasilinear parabolic partial differential equation using Haar wavelet-based numerical method. Haar wavelets possess various properties, for example, compact support, orthonormality and closed form expression. The main difficulty with the Haar wavelet is its discontinuity. Therefore, this paper cannot directly use the Haar wavelet to solve partial differential equations. To handle this difficulty, this paper represents the highest-order derivative in terms of Haar wavelet series and using successive integration this study obtains the required term appearing in the problem. Taylor series expansion is used to obtain the second-order partial derivatives at collocation points. Findings An efficient and accurate numerical method based on Haar wavelet has been proposed for parameter identification in quasilinear parabolic partial differential equations. Numerical results are obtained from the proposed method and compared with the existing results obtained from various finite difference methods including Saulyev method. It is shown that the proposed method is superior than the conventional finite difference methods including Saulyev method in terms of accuracy and CPU time. Convergence analysis is presented to show the accuracy of the proposed method. An efficient algorithm is proposed to find the wavelet coefficients at target time. Originality/value The outcome of the paper would have a valuable role in the scientific community for several reasons. In the current scenario, the parabolic inverse problem has emerged as very important problem because of its application in many diverse fields such as tomography, chemical diffusion, thermoelectricity and control theory. In this paper, higher-order derivative is represented in terms of Haar wavelet series. In other words, we represent the solution in multiscale framework. This would enable us to understand the solution at various resolution levels. In the case of Haar wavelet, this paper can achieve a very good accuracy at very less resolution levels, which ultimately leads to huge reduction in the computational cost.

Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm

2018 ◽  
Vol 28 (4) ◽  
pp. 828-856 ◽  
Author(s):  
Omar Abu Arqub
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Fluid Flows ◽  
Kernel Functions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Content Type

Purpose The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit. Design/methodology/approach The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions. Findings Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models. Research limitations/implications Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers. Practical implications The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability. Social implications Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest. Originality/value This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.

Optimization of coal handling system performability for a thermal power plant using PSO algorithm

2020 ◽  
Vol 10 (3) ◽  
pp. 359-376
Author(s):  
Subhash Malik ◽  
P.C. Tewari
Keyword(s):  
Partial Differential Equations ◽  
Power Plant ◽  
Differential Equations ◽  
Thermal Power Plant ◽  
Thermal Power ◽  
Performance Model ◽  
Transition Diagram ◽  
Partial Differential ◽  
Content Type ◽  
Handling System

PurposeThis paper deals with the optimization of coal handling system performability for a thermal power plant.Design/methodology/approachCoal handling system comprises of five subsystems, namely Wagon Tippler, Crusher, Bunker, Feeder and Coal Mill. The partial differential equations are derived on the behalf of transition diagram by using the Markov approach. These partial differential equations are further solved to obtain the performance model with the help of normalization condition. Numerous performability levels are achieved by putting the appropriate combinations of failure and repair rates (FRRs) in performance model. Performability optimization for coal handling system is obtained by varying the population and generation size.FindingsHighest performability level, that is, 93.33 at population size of 40 and 93.31 at generation size of 70, is observed.Originality/valueThe findings of this paper highlight the optimum value of performability level and FRRs for numerous subsystems. These findings are highly beneficial for plant administration to decide about the maintenance planning.

Fundamental solution of the system of equations of pseudo oscillations in the theory of thermoelastic diffusion materials with double porosity

2019 ◽  
Vol 15 (2) ◽  
pp. 317-336 ◽  
Author(s):  
Tarun Kansal
Keyword(s):  
Partial Differential Equations ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Volume Fraction ◽  
Displacement Vector ◽  
Double Porosity ◽  
Elementary Functions ◽  
Partial Differential ◽  
Thermoelastic Diffusion ◽  
Content Type

PurposeThe purpose of this paper to construct the fundamental solution of partial differential equations in the generalized theory of thermoelastic diffusion materials with double porosity.Design/methodology/approachThe paper deals with the study of pseudo oscillations in the generalized theory of thermoelastic diffusion materials with double porosity.FindingsThe paper finds the fundamental solution of partial differential equations in terms of elementary functions.Originality/valueAssuming the displacement vector, volume fraction fields, temperature change and chemical potential functions in terms of oscillation frequency in the governing equations, pseudo oscillations have been studied and finally the fundamental solution of partial differential equations in case of pseudo oscillations in terms of elementary functions has been constructed.

Solving the fractional heat-like and wave-like equations with variable coefficients utilizing the Laplace homotopy method

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Muhammad Nadeem ◽  
Shao-Wen Yao
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Perturbation Method ◽  
Variable Coefficients ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Content Type ◽  
The Laplace Transform ◽  
First Time

Purpose This paper aims to suggest the approximate solution of time fractional heat-like and wave-like (TFH-L and W-L) equations with variable coefficients. The proposed scheme shows that the results are very close to the exact solution. Design/methodology/approach First with the help of some basic properties of fractional derivatives, a scheme that has the capability to solve fractional partial differential equations is constructed. Then, TFH-L and W-L equations with variable coefficients are solved by this scheme, which yields results very close to the exact solution. The derived results demonstrate that this scheme is very effective. Finally, the convergence of this method is discussed. Findings A traditional method is combined with the Laplace transform to construct this scheme. To decompose the nonlinear terms, this paper introduces the homotopy perturbation method with He’s polynomials and thus the solution is provided in the form of a series that converges to the exact solution very quickly. Originality/value The proposed approach is original and very effective because this approach is, to the authors’ knowledge, used for the first time very successfully to tackle the fractional partial differential equations, which are of great interest.

Simulation of partial differential equations models in Java

2017 ◽  
Vol 34 (3) ◽  
pp. 800-813
Author(s):  
María José Cano ◽  
Eliseo Chacon-Vera ◽  
Francisco Esquembre
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
High Performance ◽  
Physical Models ◽  
Partial Differential ◽  
Content Type ◽  
Study Approach ◽  
The Creation ◽  
High Level ◽  
Interactive Simulations

Purpose Computer simulations improve the knowledge of physical models and are widely used in teaching and research. Key aspects are to understand their solutions and to make interactive changes to the models, observing their effects in real-time. The drawback of creating interactive simulations of physical models is the high level of programming expertise required. The purpose of this study is to facilitate this task. Design/methodology/approach Java is the perfect language for this task; it yields high-quality graphics and is widely spread in the scientific community. Because many important physical models are described by means of partial differential equations (PDEs), the combination of Java with FreeFem++, a C++ PDE solver based on the finite element method, is considered. Findings In this study, a Java library is introduced to numerically solve PDE equations via a run-time connection with FreeFem++. The solution is encapsulated into Java objects that are ready to be used in different programming tasks. The library also includes new Java visualization elements for solutions and meshes in the context of the Open Source Physics project library. Together, the connection features and the visualization elements facilitate the creation of Java simulations by programming researchers. For those with less programming capabilities, this work has been included into Easy Java Simulations, a tool to further ease the creation of interactive simulations. Originality/value The present study approach allows simulating models given PDEs. The equations are solved either in local or in remote mode (e.g. by a network accessible to a high-performance computer) and visualized locally, providing a high degree of interactivity to the end user.

The impact of impinging TiO2 nanoparticles in Prandtl nanofluid along with endoscopic and variable magnetic field effects on peristaltic blood flow

2018 ◽  
Vol 14 (3) ◽  
pp. 530-548 ◽  
Author(s):  
Sara I. Abdelsalam ◽  
Muhammad Mubashir Bhatti
Keyword(s):  
Magnetic Field ◽  
Blood Flow ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Temperature Profile ◽  
Variable Magnetic Field ◽  
Partial Differential ◽  
Content Type ◽  
Homotopy Perturbation ◽  
Linear Partial Differential Equations

Purpose The purpose of this paper is to study the variable magnetic field and endoscope effects on peristaltic blood flow of nanofluid containing TiO2 nanoparticles (NPs) through a porous annulus. The Prandtl fluid model is taken into account for the present flow. The mathematical modelling comprises the temperature, continuity, NP concentration, and equations of motion which are further simplified by taking a long peristaltic wave and creeping flow regime. Design/methodology/approach After using the long wavelength approximation, the obtained highly non-linear partial differential equations are solved using the homotopy perturbation scheme. The inclusion of the pertinent parameters is discussed mathematically and graphically for the pressure rise, friction forces, temperature profile, and concentration profile. The trapping phenomenon is also investigated with the help of contours. Findings Results show that the maximum velocity distribution exists near the centre of the annulus, whereas the average time flow boosts the velocity profile. It has also been shown that flow can pass readily without enormous pressure gradient imposed on the endoscope tube unlike the case of the slim section of the problem. Practical implications The nanofluids containing titanium NPs are increasingly utilised since such type of NPs is used by several manufacturers in sunscreen blockers and different types of endoscopy. In endoscopy, the variable magnetic field is used at the tip in order to detect or treat diseases. The NPs are used since they acquire specific thermal properties as compared with base fluids. The present study provides qualitative results showing the effect of inner tube of annulus on the fluid flow, the effect of variable magnetic field, and the change in the temperature profile on the flow field. Originality/value A new model is introduced that shows the utmost pressure that works against the positive peristaltic pump. It studies the blood flow that results in extremely non-linear partial differential equations that are solved by the homotopy perturbation method. The titanium NPs are being used in blocking the rays that penetrate the epidermis causing skin burns and short ultraviolet ageing rays that cause visible wrinkles, and thus are used in the manufacturing of sunscreens that are partially absorbed through the skin.

scholarly journals Numerical simulation method of lines together with a pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mushtaq Ali ◽  
Mohammed Almoaeet ◽  
Basim Karim Albuohimad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Ordinary Differential Equations ◽  
Method Of Lines ◽  
Pseudospectral Method ◽  
Partial Differential ◽  
Content Type ◽  
Jacobi Functions ◽  
Left And Right

PurposeThis study aims to use new formula derived based on the shifted Jacobi functions have been defined and some theorems of the left- and right-sided fractional derivative for them have been presented.Design/methodology/approachIn this article, the authors apply the method of lines (MOL) together with the pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative (SFPDEs). Then, using the collocation nodes to reduce the SFPDEs to the system of ordinary differential equations, which can be solved by the ode45 MATLAB toolbox.FindingsApplying the MOL method together with the pseudospectral discretization method converts the space-dependent on fractional partial differential equations to the system of ordinary differential equations.Originality/valueThis paper contributes to gain choosing the shifted Jacobi functions basis with special parameters a, b and give the authors this opportunity to obtain the left- and right-sided fractional differentiation matrices for this basis exactly. The results of the examples are presented in this article. The authors found that the method is efficient and provides accurate results, and the authors found significant implications for success in the science, technology, engineering and mathematics domain.

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