Search Results (12)

In this paper, we investigate the existence and uniqueness of solutions for the viscous Burgers’ equation for the isothermal flow of power-law non-Newtonian fluids $\rho ({\partial }_{t}u+u{\partial }_{x}u)=\mu {\partial }_x\left({\left|{\partial }_{x}u\right|}^{p-2}{\partial }_{x}u\right),$ augmented with the initial condition $u(0,x)=u_0$, $0<x<L$, and the boundary condition $u(t,0)=u(t,L)=0$, where $\rho$ is the density, $\mu$ the viscosity, u the velocity of the fluid, $1<p<2$, $L>0$, and $T>0$. We show that this initial boundary problem has an unique solution in the Buchner space $L^2\left(0,T;{W_0}^{1,p}(0,1)\right)$ for the given set of conditions. Moreover, numerical solutions to the problem are constructed by applying the modeling and simulation package COMSOL Multiphysics 6.0 at small and large Reynolds numbers to show the images of the solutions. Full article

In this paper, we develop a hybrid approach to solve the viscous Burgers’ equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity dominates. PINNs, on the other hand, offer a data-driven framework that can address complex boundary and initial conditions more flexibly. We demonstrate that PINNs capture the key dynamics of the Burgers’ equation, such as shock wave formation and the smoothing effects of viscosity, and show how the combination of these methods provides a powerful tool for solving nonlinear partial differential equations. Full article

The second order Burger’s equation model is used to study the turbulent fluids, suspensions, shock waves, and the propagation of shallow water waves. In the present research, we investigate a numerical solution to the time fractional coupled-Burgers equation (TFCBE) using Crank–Nicolson and the cubic B-spline (CBS) approaches. The time derivative is addressed using Caputo’s formula, while the CBS technique with the help of a $\theta$-weighted scheme is utilized to discretize the first- and second-order spatial derivatives. The quasi-linearization technique is used to linearize the non-linear terms. The suggested scheme demonstrates unconditionally stable. Some numerical tests are utilized to evaluate the accuracy and feasibility of the current technique. Full article

In this paper, we study the Kelvinons, which are monopole ring solutions to the Euler equations, regularized as the Burgers vortex in the viscous core. There is finite anomalous dissipation in the inviscid limit. However, in the anomalous Hamiltonian, some terms are growing as logarithms of Reynolds number; these terms come from the core of the Burgers vortex. In our theory, the turbulent multifractal phenomenon is similar to asymptotic freedom in QCD, with these logarithmic terms summed up by an RG equation. The small effective coupling does not imply small velocity; on the contrary, velocity is large compared to its fluctuations, which opens the way for a quantitative theory. In the leading order in the perturbation theory in this effective coupling constant, we compute running multifractal dimensions for high moments of velocity circulation, which is in good agreement with the data for quantum Turbulence and available data for classical Turbulence. The logarithmic dependence of fractal dimensions on the loop size comes from the running coupling in anomalous dimensions. This slow logarithmic drift of fractal dimensions would be barely observable at Reynolds numbers achievable at modern DNS. Full article

In the current work, the stochastic Burgers’ equation is studied using the Dynamically Orthogonal (DO) method. The DO presents a low-dimensional representation for the stochastic fields. Unlike many other methods, it has a time-dependent property on both the spatial basis and stochastic coefficients, with flexible representation especially in the strongly transient and nonstationary problems. We consider a computational study and application of the DO method and compare it with the Polynomial Chaos (PC) method. For comparison, both the stochastic viscous and inviscid Burgers’ equations are considered. A hybrid approach, combining the DO and PC is proposed in case of deterministic initial conditions to overcome the singularities that occur in the DO method. The results are verified with the stochastic collocation method. Overall, we observe that the DO method has a higher rate of convergence as the number of modes increases. The DO method is found to be more efficient than PC for the same level of accuracy, especially for the case of high-dimensional parametric spaces. The inviscid Burgers’ equation is analyzed to study the shock wave formation when using the DO after suitable handling of the convective term. The results show that the sinusoidal wave shape is distorted and sharpened as the time evolves till the shock wave occurs. Full article

Due to issues such as heat accumulation, the site area, and project investment, the reasonable determination of the hole spacing for heat exchangers has become one of the key design points of the ground-coupled heat pump system. Based on the definition of heat penetration in heat transfer and the research method of the inverse problem, a direct algorithm of the heat penetration distance in the aquifer was proposed using the analytical solution to the mathematical model for one-dimensional heat convection–conduction problems. Taking a vertical ground-coupled heat pump project in Hefei, Anhui Province, China, as an example, a three-dimensional hydro-thermal coupling numerical simulation model was established, and the influence radius during the refrigeration and heating periods under the action of a single borehole heat exchanger was determined. Comparing the heat penetration distance with the influence radius, the results show that the relative errors of the results obtained by the two methods are less than 10%, which verifies the rationality and effectiveness of the calculated penetration distance in the aquifer. At the end of the cooling or heating period, the heat penetration distance in the aquifer is calculated to be 7.59 m. Therefore, the proposed method is straightforward and efficient, which can provide a convenient approach to determining the reasonable hole spacing of the heat pump system. Full article

The effect of water temperature variation in a river channel on groundwater temperature in the confined aquifer it cuts can be generalized to a one-dimensional thermal convection-conduction problem in which the boundary water temperature rises instantaneously and then remains constant. The basic equation of thermal transport for such a problem is the viscous Burgers equation, which is difficult to solve analytically. To solve this problem, the Cole–Hopf transform was used to convert the second-order nonlinear thermal convection-conduction equation into a heat conduction equation with exponential function-type boundary conditions. Considering the difficulty of calculating the inverse of the image function of the boundary function, the characteristics and properties of the Laplace transform were used to derive the theoretical solution of the model without relying on the transformation of the boundary function, and the analytical solution was obtained by substituting the boundary condition into the theoretical solution. The analytical solution was used to analyze the temperature response laws of aquifers to parameter variation. Subsequently, a 40-day numerical simulation was conducted to analyze the boundary influence range and the results from the analytical method were compared to those from the numerical method. The study shows that: (1) the greater the distance from the river canal and the lower the aquifer flow velocity, the slower the aquifer temperature changes; (2) the influence range of the river canal boundary increases from 18.19 m to 23.19 m at the end of simulation period as the groundwater seepage velocity v increases from 0.08 m/d to 0.12 m/d; (3) the relative errors of the analytical and numerical methods are mostly less than 5%, confirming the rationality of the analytical solution. Full article

The basic characteristics of cylindrical as well as spherical solitary and shock waves in degenerate electron-nucleus plasmas are theoretically investigated. The electron species is assumed to be cold, ultra-relativistically degenerate, negatively charged gas, whereas the nucleus species is considered a cold, non-degenerate, positively charged, viscous fluid. The reductive perturbation technique is utilized in order to reduce the basic equations (governing the degenerate electron-nucleus plasmas under consideration) to the modified Korteweg-de Vries and Burgers equations. The latter are numerically solved and analyzed to detect the basic characteristics of solitary and shock waves in such electron-nucleus plasmas. The nonlinear nucleus-acoustic waves are found to be propagated in the form of solitary as well as shock waves in such degenerate electron-nucleus plasmas. Their basic properties as well as their time evolution are significantly modified by the effects of cylindrical as well as spherical geometries. The results of this study is expected to be applicable not only to astrophysical compact objects, but also to ultra-cold dense plasmas produced in laboratory. Full article

A boundary-layer is a thin fluid layer near a solid surface, and viscous effects dominate it. The laminar boundary-layer calculations appear in many aerodynamics problems, including skin friction drag, flow separation, and aerodynamic heating. A student must understand the flow physics and the numerical implementation to conduct successful simulations in advanced undergraduate- and graduate-level fluid dynamics/aerodynamics courses. Numerical simulations require writing computer codes. Therefore, choosing a fast and user-friendly programming language is essential to reduce code development and simulation times. Julia is a new programming language that combines performance and productivity. The present study derived the compressible Blasius equations from Navier–Stokes equations and numerically solved the resulting equations using the Julia programming language. The fourth-order Runge–Kutta method is used for the numerical discretization, and Newton’s iteration method is employed to calculate the missing boundary condition. In addition, Burgers’, heat, and compressible Blasius equations are solved both in Julia and MATLAB. The runtime comparison showed that Julia with $for$ loops is 2.5 to 120 times faster than MATLAB. We also released the Julia codes on our GitHub page to shorten the learning curve for interested readers. Full article

We provide a lower bound for the blow up time of the $H^2$ norm of the entropy solutions of the inviscid Burgers equation in terms of the $H^2$ norm of the initial datum. This shows an interesting symmetry of the Burgers equation: the invariance of the space $H^2$ under the action of such nonlinear equation. The argument is based on a priori estimates of energy and stability type for the (viscous) Burgers equation. Full article

The benefits and properties of iterative splitting methods, which are based on serial versions, have been studied in recent years, this work, we extend the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations. For such interesting partial differential examples with higher dimensional, fractional, and nonlinear terms, we could apply the parallel iterative splitting methods, which allow for accelerating the solver methods and reduce the computational time. Here, we could apply the benefits of the higher accuracy of the iterative splitting methods. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, The flexibilisation with multisplitting methods allows for decomposing large scale operator equations. In combination with iterative splitting methods, which use characteristics of waveform-relaxation (WR) methods, we could embed the relaxation behavior and deal better with the nonlinearities of the operators. We consider the convergence results of the parallel iterative splitting methods, reformulating the underlying methods with a summation of the individual convergence results of the WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods with respect to the synchronous and asynchronous treatments. Furthermore, we present different numerical applications of fluid and phase field problems in order to validate the benefit of the parallel versions. Full article

In this paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-differential equations with functional derivatives. We continue the work of Oberlack and Wacławczyk (2006, Arch. Mech. 58, 597), (2013, J. Math. Phys. 54, 072901), where the extended Lie symmetry analysis is performed in the Fourier space. Here, we introduce a method to perform the extended Lie symmetry analysis in the physical space where we have to deal with the transformation of the integration variable in the appearing integral terms. The method is based on the transformation of the product y(x)dx appearing in the integral terms and applied to the functional formulation of the viscous Burgers equation. The extended Lie symmetry analysis furnishes all known symmetries of the viscous Burgers equation and is able to provide new symmetries associated with the Hopf formulation of the viscous Burgers equation. Hence, it can be employed as an important tool for applications in continuum mechanics. Full article