Zhiliang Lin

Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls

This paper focuses on the theoretical treatment of the laminar, incompressible, and time-dependent flow of a viscous fluid in a porous channel with orthogonally moving walls. Assuming uniform injection or suction at the porous walls, two cases are considered for which the opposing walls undergo either uniform or nonuniform motions. For the first case, we follow Dauenhauer and Majdalani Phys. Fluids 15, 1485 2003 by taking the wall expansion ratio to be time invariant and then proceed to reduce the Navier‐Stokes equations into a fourth order ordinary differential equation with four boundary conditions. Using the homotopy analysis method HAM, an optimized analytical procedure is developed that enables us to obtain highly accurate series approximations for each of the multiple solutions associated with this problem. By exploring wide ranges of the control parameters, our procedure allows us to identify dual or triple solutions that correspond to those reported by Zaturska et al. Fluid Dyn. Res. 4, 151 1988. Specifically, two new profiles are captured that are complementary to the type I solutions explored by Dauenhauer and Majdalani. In comparison to the type I motion, the so-called types II and III profiles involve steeper flow turning streamline curvatures and internal flow recirculation. The second and more general case that we consider allows the wall expansion ratio to vary with time. Under this assumption, the Navier‐ Stokes equations are transformed into an exact nonlinear partial differential equation that is solved analytically using the HAM procedure. In the process, both algebraic and exponential models are considered to describe the evolution of t from an initial 0 to a final state 1. In either case, we find the time-dependent solutions to decay very rapidly to the extent of recovering the steady state behavior associated with the use of a constant wall expansion ratio. We then conclude that the time-dependent variation of the wall expansion ratio plays a secondary role that may be justifiably ignored.

The improved homotopy analysis method for the Thomas–Fermi equation

Abstract The homotopy analysis method (HAM) is sharpened to solve the Thomas–Fermi equation. Some techniques are employed, including the use of asymptotic analysis to introduce proper transformation, and the use of optimal initial guess and optimal auxiliary linear operator to accelerate the convergence of homotopy approximations. The optimal convergence-control parameters are determined by the minimum of the squared residual error. As a result, the initial slop is provided with more-than-10-digit accuracy, which is far more accurate than the results obtained by other authors using the same method. It demonstrates the flexibility and power of the HAM equipped with these techniques.

An iterative HAM approach for nonlinear boundary value problems in a semi-infinite domain

a b s t r a c t In this paper, we propose an iterative approach to increase the computation efficiency of the homotopy analysis method (HAM), a analytic technique for highly nonlinear problems. By means of the Schmidt-Gram process (Arfken et al., 1985) (15), we approximate the right-hand side terms of high-order linear sub-equations by a finite set of orthonormal bases. Based on this truncation technique, we introduce the Mth-order iterative HAM by using each Mth-order approximation as a new initial guess. It is found that the iterative HAM is much more efficient than the standard HAM without truncation, as illustrated by three nonlinear differential equations defined in an infinite domain as examples. This work might greatly improve the computational efficiency of the HAM and also the Mathematica package BVPh for nonlinear BVPs.

Nonlinear Wave–Current Interaction in Water of Finite Depth

AbstractThe interaction of nonlinear progressive waves and a uniform current in water of finite depth is investigated analytically by means of the homotopy analysis method (HAM). With HAM, the velocity potential of the flow and the surface elevation are expressed by the Fourier series, and the nonlinear free surface boundary conditions are satisfied by continuous mapping. Unlike a perturbation method, the present approach does not depend on any small parameters; thus, the solutions are suitable for steep waves and strong currents. To verify the HAM solutions, experiments are conducted in the wave–current flume of the Education Ministry Key Laboratory of Hydrodynamics at Shanghai Jiao Tong University (SJTU) in Shanghai, China. It is found that the HAM solutions are in good agreement with experimental measurements. Based on the series solutions of the validated analytical model, the influence of water depth, wave steepness, and current velocity on the physical properties of the coexisting wave–current field...

Mass, momentum, and energy flux conservation for nonlinear wave-wave interaction

A fully nonlinear solution for bi-chromatic progressive waves in water of finite depth in the framework of the homotopy analysis method (HAM) is derived. The bi-chromatic wave field is assumed to be obtained by the nonlinear interaction of two monochromatic wave trains that propagate independently in the same direction before encountering. The equations for the mass, momentum, and energy fluxes based on the accurate high-order homotopy series solutions are obtained using a discrete integration and a Fourier series-based fitting. The conservation equations for the mean rates of the mass, momentum, and energy fluxes before and after the interaction of the two nonlinear monochromatic wave trains are proposed to establish the relationship between the steady-state bi-chromatic wave field and the two nonlinear monochromatic wave trains. The parametric analysis on e1 and e2, representing the nonlinearity of the bi-chromatic wave field, is performed to obtain a sufficiently small standard deviation Sd, which is applied to describe the deviation from the conservation state (Sd = 0) in terms of the mean rates of the mass, momentum, and energy fluxes before and after the interaction. It is demonstrated that very small standard deviation from the conservation state can be achieved. After the interaction, the amplitude of the primary wave with a lower circular frequency is found to decrease; while the one with a higher circular frequency is found to increase. Moreover, the highest horizontal velocity of the water particles underneath the largest wave crest, which is obtained by the nonlinear interaction between the two monochromatic waves, is found to be significantly higher than the linear superposition value of the corresponding velocity of the two monochromatic waves. The present study is helpful to enrich and deepen the understanding with insight to steady-state wave-wave interactions.

HAM Solution of Nonlinear Wave Interaction With Uniform Current

The interaction of nonlinear progressive waves and uniform currents in water of finite depth is investigated analytically by means of the homotopy analysis method (HAM). In the HAM, the velocity potential of the wave is expressed by Fourier series and the nonlinear free surface boundary conditions are satisfied by continuous mapping. The present approach does not depend on any small parameters, thus the solutions are suitable for steep waves and strong currents. To verify the HAM solutions, experiments are conducted in wave-current flume of Education Ministry Key Laboratory of Hydrodynamics at SJTU. It is shown that the HAM solutions are in good agreement with experimental measurements. The present study demonstrated that the great potential of the HAM to solve more complex wave-current interaction problems leading to engineering applications in traditional offshore industry and marine renewable energy sector.Copyright

On the quartet resonance of gravity waves in water of finite depth

The interaction among a quartet of resonant progressive waves in water of finite depth is considered by the Homotopy Analysis Method (HAM). The problem is governed by a linear PDE with a group of nonlinear boundary conditions on the unknown free surface. Convergent multiple steady-state solutions have been gained by means of the HAM. Its worth noting that, in the most cases of the quartet resonance, wave energy is often exchanged periodically among different wave modes. However, our computations indicate for the first time that there exist some cases in which energy exchange disappears and besides the resonant wave component may contain much small percentage of the total wave energy. This work verifies that the HAM is valid even for some rather complicated nonlinear PDEs, and can be used as a powerful tool to understand some nonlinear phenomena.

On the dispersion relation of nonlinear wave current interaction by means of the HAM

The influence of exponentially sheared currents on unidirectional bichromatic waves in deep water is investigated by the HAM. The governing equations contain four coupled PDEs, including a nonlinear vorticity transport equation and two nonlinear free-surface conditions on the unknown wave elevation. No constrain is made for the primary wave amplitudes, and the current owns a exponential type profile along the vertical line. Convergent solutions are obtained with the help of convergence-control parameter. It is found that a critical characteristic current profile slope exists for each parts of phase velocity caused by nonlinear interaction, under/above which the mean flow vorticity increases/decreases the corresponding part of phase velocity. This work indicates that the HAM is a powerful tool for complicated coupled nonlinear PDEs, which deserves more attention for further development.

Exact HAM Solutions for the Viscous Rotational Flowfield in Channels with Regressing and Injecting Sidewalls

This paper focuses on the theoretical treatment of the laminar, incompressible, and time-dependent flow of a viscous fluid in a porous channel with orthogonally moving walls. Assuming uniform injection or suction at theporous walls, two cases are considered for which the opposing walls undergo either uniform or non-uniform motions. For the first case, we follow Dauenhauer and Majdalani 1 by taking the wall expansion ratioα to be time invariant and then proceed to reduce the Navier-Stokes equations into a fourth order ordinary differential equation (ODE) with four boundary conditions. Using the Homotopy Analysis Method (HAM), an optimized analytical procedure is developed that enables us to obtain highly accurate series approximations for each of the multiple solutions associated with this problem. By exploring wide ranges of the control parameters, our procedure allows us to identify dual or triple solutions that correspond to those reported by Zaturska, Drazin and Banks. 2 Specifically, two new profiles are captured that are complementary to the type I solutions explored by Dauenhauer and Majdalani. 1 In comparison to the type I motion, the so-called types II and III profiles involve steeper flow turning streamline curvatures and internal flow recirculation. The second and more general case that we consider allows the wall expansion ratio to vary with time. Under this assumption, the Navier-Stokes equations are transformed into an exact nonlinear partial differential equation that is solved analytically using the HAM procedure. In the process, both algebraic and exponential models are considered to describe the evolution ofα(t) from an initialα0 to a final stateα1. In either case, we find the time-dependent solutions to decay very rapidly to the extent of recovering the steady-state behavior associated with the use of a constant wall expansion ratio. We then conclude that the time-dependent variation of the wall expansion ratio plays a secondary role that may be justifiably ignored.