Differential Equations In Fluid Mechanics Research Articles

MHD boundary layer micropolar fluid flow over a stretching wedge surface: Thermophoresis and brownian motion effect

To investigate the consequence of thermophoresis and Brownian diffusion on convective boundary layer micropolar fluid flow over a stretching wedge-shaped surface. The effects of 
 non-dimensional parameters namely coupling constant parameter (0.01 ≤ B1 ≤ 0.05), magnetic parameter (1.0 ≤ M ≤ 15.0), Grashof number (0.3 ≤ Gr ≤ 0.9), modified Grashof number 
 (0.3 ≤ Gm ≤ 0.8), micropolar parameter (2.0 ≤ G2 ≤ 7.5), vortex viscosity constraint (0.02 ≤ G1 ≤ 0.2), Prandtl number (7.0 ≤ Pr ≤ 15.0), thermal radiation parameter (0.25 ≤ R ≤ 0.50), Brownian motion parameter (0.2 ≤ Nb ≤ 0.62), thermophoresis parameter (0.04 ≤ Nt ≤ 0.10), heat generation parameter (0.1 ≤ Q ≤ 0.5), Biot number (0.65 ≤ Bi ≤ 1.0), stretching parameter (0.2 ≤ A ≤ 0.5), Lewis number (3.0 ≤ Le ≤ 7.0), and chemical reaction parameter (0.2 ≤ K ≤ 0.7) on the steady MHD heat and mass transfer is investigated in the present study. The coupled non-linear partial differential equations are reduced into a set of non-linear ordinary differential equations employing similarity transformation. Furthermore, by using the Runge-Kutta method followed by the shooting technique, the transformed equations are solved. The main goal of this study is to investigate the numerical analysis of nanofluid flow within the boundary layer region with the effects of the microrotation parameter and velocity ratio parameter. The novelty of this paper is to propose a numerical method for solving third-order ordinary differential equations that include both linear and nonlinear terms. To understand the physical significance of this work, numerical analyses and tabular displays of the skin friction coefficient, Nusselt number, and Sherwood number are shown. The new approach of the present study contributes significantly to the understanding of numerical solutions to non-linear differential equations in fluid mechanics and micropolar fluid flow. Micropolar fluids are becoming even more of a focus due to the desire for engineering applications in various fields of medical, mechanical engineering, and chemical processing.

On the space-time analyticity of the inhomogeneous heat equation on the half space with Neumann boundary conditions

We consider the inhomogeneous heat equation on the half-space R + d with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol [A direct approach to Gevrey regularity on the half-space. In: Partial differential equations in fluid mechanics. Cambridge: Cambridge Univ. Press; 2018. p. 268–288. (London math. soc. lecture note ser.; vol. 452).].

Approximate deconvolution discretisation

A new strategy is presented for the construction of high-order spatial discretisations extracted from a lower-order basic discretisation. The key consideration is that any spatial discretisation of a derivative of a solution can be expressed as the exact differentiation of a corresponding ‘filtered’ solution. Hence, each numerical discretisation method may be directly linked to a unique spatial filter, expressing the truncation error of the basic method. By approximately deconvolving the implied filter of the basic numerical discretisation an augmented high-order method can be obtained. In fact, adopting a deconvolution of the implied filter of suitable higher order enables the formulation of a new spatial discretisation method of correspondingly higher order. This construction is illustrated for finite difference (FD) discretisation schemes, solving partial differential equations in fluid mechanics. Knowing the implied filter of the basic discretisation, one can derive a corresponding higher order method by approximately eliminating the implied spatial filter to a certain desired order. We use deconvolution to compensate for the implied filter. This corresponds to a ‘sharpening’ of numerical solution features before the application of the basic FD method. The combination will be referred to as Approximate Deconvolution Discretisation (ADD). The accuracy of the deconvolved FD scheme depends on the order of approximation of the deconvolution filter. We present the ‘sharpening’ of several well-known FD operators for first- and second-order derivatives and quantify the achieved accuracy in terms of the modified wavenumber spectrum. Examples include high-order extensions up to new schemes with spectral accuracy. The practicality of the deconvolved FD schemes is illustrated in various ways: (i) by investigation of exactly solvable advection and diffusion problems, (ii) by tracking the evolution of the numerical solution to the Taylor-Green vortex problem and (iii) by showing that ADD yields spectral accuracy for the Burgers equation and for double-jet flow of an incompressible fluid.

MAGNETOHYDRODYNAMICS AND POWER-LAW FLUIDS IN DOUBLE LID-DRIVEN CAVITY WITH SEMI-CIRCULAR BODIES

The use of complex fluids is one of the modern techniques used in small devices in order to enhance their thermal performance. This paper is a numerical study of a complex fluid imprisoned in a chamber with two hot bodies exposed to a magnetic field of constant intensity. The upper and lower walls move horizontally at a constant velocity, while the lateral sides are thermally insulated. The numerical simulation of the system was achieved based on the finite volume method that solves the differential equations of fluid mechanics and heat transfer. Simulations were carried out under the following conditions: Re = 1 to 40, Ri = 0 to 100, n = 0.6 to 1.4 and Ha = 0 to 100. The study showed that the thermal activity of the two bodies is different and related to initial condition. Also, the effect of the magnetic field is strong in the case of shear-thinning fluids, while its effect is diminished in the case of shear-thickening fluids.

Application of the Variational Iteration Method for the time-fractional Kaup-Kupershmidt Equation and the Boussinesq-Burger equation

The variational iteration method is used to deal with linear and nonlinear differential equations. The main characteristics of the method lie in its flexibility and ability to accurately and easily solve nonlinear equations. In this work, a general framework is presented for a variational iteration method for the analytical treatment of partial differential equations in fluid mechanics. The Caputo sense is used to describe fractional derivatives. The time-fractional Kaup-Kupershmidt (KK) equation is investigated, as it is the solution of the system of partial differential equations via the Boussinesq-Burger equation. By comparing the results that are obtained by the variational iteration method with those obtained by the two-dimensional Legendre multiwavelet, the optimal homotopy asymptotic method (OHAM), the q-homotopy analysis transform method, the Laplace Adomian Decomposition Method, and the homotopy perturbation method, the first method proved to be very effective and convenient. The main methodology in this work is anticipated to be applied to various fractional calculus, linear, and nonlinear problems.

The Introductory Analysis on Fourier Series

Numerical analysis benefits from spectral methods. The Fourier series and Chebyshev polynomials were initially used to solve ordinary differential equations by Cornelius Lanczos. For the first time, spectral approaches were employed by Kreiss, Oliger, Orszag, and other scholars in the 1970s to resolve the partial differential equations of fluid mechanics. In Fourier analysis, general functions are expressed as the sums of a variety of fundamental function options, such as sines and cosines.This paper gives an introductory analysis on Fourier Series. Starting with the basic definition of Fourier coefficients and Fourier Series, the paper introduces the concept and properties of convolution. It also talks about different kernels and good kernel family, analyzing the relationship between these kernels and the original function. Then the paper will combine these ideas and prove the uniqueness and convergence of Fourier Series.

On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs

AbstractWe examine the power series solutions of two classical nonlinear ordinary differential equations of fluid mechanics that are mathematically related by their large-distance asymptotic behaviours in semi-infinite domains. The first problem is that of the ‘Sakiadis’ boundary layer over a moving flat wall, for which no exact analytic solution has been put forward. The second problem is that of a static air–liquid meniscus with surface tension that intersects a flat wall at a given contact angle and limits to a flat pool away from the wall. For the latter problem, the exact analytic solution—given as distance from the wall as a function of meniscus height—has long been known (Batchelor, G. K. (1967) An Introduction to Fluid Dynamics, chapter 1: The physical properties of fluids. Cambridge). Here, we provide an explicit solution as meniscus height versus distance from the wall to elucidate structural similarities to the Sakiadis boundary layer. Although power series solutions are readily obtainable to the governing nonlinear ordinary differential equations, we show that—in both problems—the series diverge due to non-physical complex or negative real-valued singularities. In both cases, these singularities can be moved by expanding in exponential gauge functions motivated by their respective large distance asymptotic behaviours to enable series convergence over their full semi-infinite domains. For the Sakiadis problem, this not only provides a convergent Taylor series (and conjectured exact) solution to the ODE, but also a means to evaluate the wall shear parameter (and other properties) to within any desired precision. Although the nature of nonlinear ODEs precludes general conclusions, our results indicate that asymptotic behaviours can be useful when proposing variable transformations to overcome power series divergence.

On theories of natural decomposition method applied to system of nonlinear differential equations in fluid mechanics

In shallow waters, the Wu-Zhang (WZ) system describes the (1+1)-dimensional dispersive long wave in two horizontal directions, which is important for the engineering community. This paper presents proofs for various theorems and shows that the natural decomposition method (NDM) solves systems of linear and nonlinear ordinary and partial differential equations under proper initial conditions, such as the Wu-Zhang system. We use a combination of two methods, namely the natural transform method to deal with the linear terms and the Adomian decomposition method to deal with the nonlinear terms. Several examples of linear and nonlinear systems (ODEs and PDEs) are given, including the Wu-Zhang (WZ) system. The present approach, which has numerous applications in the science and engineering fields, is a great alternative to the many existing methods for solving systems of differential equations. It also holds great promise for additional real-world applications.

An efficient hybrid numerical method for multi-term time fractional partial differential equations in fluid mechanics with convergence and error analysis

The fundamental purpose of this paper is to study the numerical solution of multi-term time fractional nonlinear Klein–Gordon equation, using regularized beta functions and fractional order Bernoulli wavelets. First, the exact formulas for the fractional integrals of the fractional order Bernoulli wavelets were obtained. Using properties of the regularized beta functions and their operational matrices the operational matrices of the fractional order Bernoulli wavelets were calculated. Through new operational matrices and appropriate collocation points, the time fractional nonlinear Klein–Gordon equation were transformed to a system of nonlinear algebraic equations. The convergence analysis and error bound of the proposed method were then performed. A sufficient number of numerical simulations were considered to show the effectiveness and validity of the presented numerical method and its theoretical analysis.

New Iterative Method of Solving Nonlinear Equations in Fluid Mechanics

This paper presents the results of applying a new iterative method to linear and nonlinear fractional partial differential equations in fluid mechanics. A numerical analysis was performed to find an exact solution of the fractional wave equation and fractional Burgers’ equation, as well as an approximate solution of fractional KdV equation and fractional Boussinesq equation. Fractional derivatives of the order α are described using Caputo's definition with <i>0</i> < α ≤ <i>1</i> or <i>1</i> < α ≤ <i>2</i>. A comparative analysis of the results obtained using a new iterative method with those obtained by the Adomian decomposition method showed the first method to be more efficient and simple, providing accurate results in fewer computational operations. Given its flexibility and ability to solve nonlinear equations, the iterative method can be used to solve more complex linear and nonlinear fractional partial differential equations.

Existence, uniqueness, Ulam–Hyers stability and numerical simulation of solutions for variable order fractional differential equations in fluid mechanics

In this paper, a numerical approximation is shown to solve solutions of time fractional linear differential equations of variable order in fluid mechanics where the considered fractional derivatives of variable order are in the Caputo sense. Existence, uniqueness of solutions and Ulam–Hyers stability results are displayed. To solve the considered equations a numerical approximation based on the shifted Legendre polynomials are proposed. To perform the method, an operational matrix of fractional derivative with variable-order is derived for the shifted Legendre polynomials to be applied for developing the unknown function. By substituting the aforesaid operational matrix into the considered equations and using the properties of the shifted Legendre polynomials together with the collocation points, the main equations are reduced to a system of algebraic equations. The approximate solution is calculated by solving the obtained system which is technically easier for checking. We also study the error analysis for the approximate solution yielded by the introduced method. Finally, the accuracy and performance of the proposed method are checked by some illustrative examples. The illustrative examples results establish the applicability and usefulness of the proposed method.

Relations between skin friction and other surface quantities in viscous flows

This paper presents the derivations of the exact relations between skin friction and other important dynamical and kinematical quantities on a stationary curved surface in a viscous flow by applying the standard methods of differential geometry to the governing partial differential equations in fluid mechanics. In particular, the mathematical structures of the effects of the surface curvature are explicitly expressed, which extend the previous results on a flat surface. These relations reveal that skin friction is intrinsically coupled with surface pressure, temperature, and scalar concentration through the boundary enstrophy flux, heat flux, and mass flux, respectively. As an example, the relation between skin friction and surface pressure is examined in the Oseen flow over a sphere to elucidate the significant effect of the surface curvature at a very small Reynolds number. Two other validation examples are a gravity-driven creeping liquid film flow over a wavy surface and the Falkner-Skan flow over a wedge. Furthermore, the relation is applied to a simulated turbulent channel flow to explore the local near-wall coherent structure and understand its dynamical roles in turbulence production.

Application of the Kudryasov method with characteristic set algorithm to solve some partial differential equations in fluid mechanics

In this paper, we pay attention the analytical method named, the Kudryashov method combined with characteristic set algorithm for finding the exact travelling solutions of two non-linear PDE in fluid mechanics, which named surface wave equation and the generalized Kuramoto-Sivashinsky equation. The solution procedure of the Kudryashov method can be reduced to solve a large system of algebraic equations, which is hard to solve, then we use characteristic set algorithm to solve this problem. The obtained results show that the Kudryashov method combined with characteristic set algorithm is effective.

Two examples on the property of the noise in the systems of equations of fluid mechanics

In this manuscript, we discuss two examples of the results on the systems of partial differential equations in fluid mechanics obtained via stochastic analysis, for which deterministic case has no counterpart. Firstly, it is well-known that for a deterministic system of equations in fluid mechanics, a lack of diffusion creates a significant obstacle in proving its global well-posedness for a sufficiently small initial data. Nevertheless, it may be proven that for the Hall-magnetohydrodynamics system forced by L e ́ vy noise, such a result may be obtained even with zero viscous diffusion. Secondly, we may extend the classical Kelvin’s conservation of circulation flow, as well as the Cauchy’s vorticity transport formula, on the Euler equations to show that the solution to the damped Euler equations enjoys the exact same identities except with an exponential decay in time. Such identities are known to completely fail once we add diffusion and consider the damped Navier–Stokes equations. Nevertheless, discovering a stochastic Lagrangian formulation, we are able to show that such identities indeed do continue to hold informally on average, i.e. via a mathematical expectation, over the random ensemble at earlier time. The stochastic Lagrangian formulation and these identities have also been obtained for the solution to the Boussinesq system. These results are rigorously proven in Mohan and Yamazaki (0000) and Yamazaki (0000). In this manuscript we will not pursue complete proofs, which were in particular mostly inspired by Kim (2011) and Iyer (2006), but rather elaborate on motivations, difficulty and remaining open problems for future works.

The balance principle and the Reynolds transport theorem in introductory fluid mechanics

This paper traces the evolution of the use of the Reynolds transport theorem to derive finite control volume equations for mass, momentum, and energy in introductory engineering fluid mechanics textbooks used in the United States. The paper demonstrates the superior merits of an alternative approach using the balance principle. The balance principle is easier to teach, to understand, and to apply in more complex situations. It better prepares students to understand the derivation of the partial differential equations of fluid mechanics and the finite volume equations of computational fluid dynamics. For these reasons, instructors should consider using the balance principle instead of (or at minimum in addition to) the Reynolds transport theorem in introductory fluid mechanics courses.

A Lagrangian Approach to Modeling and Analysis of a Crowd Dynamics

Modeling of crowd and pedestrian dynamics has intrigued engineers, physicists, and sociologists alike, even in recent times. Scientists have long sought to model the collective motion of large groups of individuals and study the mathematical basis of what seems to be apparently random behavior. A large number of macroscopic models have been proposed that describe crowd motion as a whole, much like the partial differential equations of fluid mechanics. This paper proposes a Lagrangian approach to the modeling of crowd dynamics by taking into consideration the various forces that act between the members of a crowd while they are in motion in a 2-D field. We attempt a realistic modeling of the attractive and repulsive forces between the members and seek to give a definite mathematical backbone to the terms “panic” and “evacuation.” That the dynamics is stable is demonstrated by constructing an appropriate Lyapunov energy function. We then linearize the dynamics to obtain mathematical expressions for the small perturbations about an equilibrium point. Through machine simulations and by tracking the motion of actual crowd systems, we show the validity of the mathematics of group formation and evacuation that we have proposed.

Lie Symmetry Method for Solutions of Differential Equations with Applications in Physics

Abstract. A mathematical method in pure mathematics (differential geometry) for finding solutions of differential equations is considered. The method is based on constructing a Lie algebra associated to a given system of differential equation, called Lie algebra of the symmetries of the given system. This Lie algebra is a vector space which maps a given solution, such as a constant solution, to another solution, it is a significant tool for finding new solution for system of differential equation specially partial differential equations. Then we will apply it to some differential equations in fluid mechanics and physics.

Variational Homotopy Perturbation Method An Efficient Scheme For Solving Partial Differential Equations In Fluid Mechanics

In this paper, an application of variationalhomotopy perturbation method is applied to solve Korteweg-de Vries (KdV)and Burgers equations. The study reveals that the method is very effective and simple.

Solution of Fractional Partial Differential Equations in Fluid Mechanics by Extension of Some Iterative Method

An extension of the so-called new iterative method (NIM) has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM) and the variational iteration method (VIM) reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.

Computational Fluid Dynamics in the Design and Analysis of Thermal Processes: A Review of Recent Advances

The design of thermal processes in the food industry has undergone great developments in the last two decades due to the availability of cheap computer power alongside advanced modelling techniques such as computational fluid dynamics (CFD). CFD uses numerical algorithms to solve the non-linear partial differential equations of fluid mechanics and heat transfer so that the complex mechanisms that govern many food-processing systems can be resolved. In thermal processing applications, CFD can be used to build three-dimensional models that are both spatially and temporally representative of a physical system to produce solutions with high levels of physical realism without the heavy costs associated with experimental analyses. Therefore, CFD is playing an ever growing role in the development of optimization of conventional as well as the development of new thermal processes in the food industry. This paper discusses the fundamental aspects involved in developing CFD solutions and forms a state-of-the-art review on various CFD applications in conventional as well as novel thermal processes. The challenges facing CFD modellers of thermal processes are also discussed. From this review it is evident that present-day CFD software, with its rich tapestries of mathematical physics, numerical methods and visualization techniques, is currently recognized as a formidable and pervasive technology which can permit comprehensive analyses of thermal processing.