<abstract><p>In this paper, we find the solution of the time-fractional Newell-Whitehead-Segel equation with the help of two different methods. The newell-Whitehead-Segel equation plays an efficient role in nonlinear systems, describing the stripe patterns' appearance in two-dimensional systems. Four case study problems of Newell-Whitehead-Segel are solved by the proposed methods with the aid of the Antagana-Baleanu fractional derivative operator and the Laplace transform. The numerical results obtained by suggested techniques are compared with an exact solution. To show the effectiveness of the proposed methods, we show exact and analytical results compared with the help of graphs and tables, which are in strong agreement with each other. Also, the results obtained by implementing the suggested methods at various fractional orders are compared, which confirms that the solution gets closer to the exact solution as the value tends from fractional-order towards integer order. Moreover, proposed methods are interesting, easy and highly accurate in solving various nonlinear fractional-order partial differential equations.</p></abstract>
In this paper, we introduce a modified method which is constructed by mixing the residual power series method and the Elzaki transformation. Precisely, we provide the details of implementing the suggested technique to investigate the fractional-order nonlinear models. Second, we test the efficiency and the validity of the technique on the fractional-order Navier-Stokes models. Then, we apply this new method to analyze the fractional-order nonlinear system of Navier-Stokes models. Finally, we provide 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.
We present a lattice-gas (generalised Ising) model for liquid droplets on solid surfaces. The time evolution in the model involves two processes: (i) Single-particle moves which are determined by a kinetic Monte Carlo algorithm. These incorporate into the model particle diffusion over the surface and within the droplets and also evaporation and condensation, i.e. the exchange of particles between droplets and the surrounding vapour. (ii) Larger-scale collective moves, modelling advective hydrodynamic fluid motion, determined by considering the dynamics predicted by a thin-film equation. The model enables us to relate how macroscopic quantities such as the contact angle and the surface tension depend on the microscopic interaction parameters between the particles and with the solid surface. We present results for droplets joining, spreading, sliding under gravity, dewetting, the effects of evaporation, the interplay of diffusive and advective dynamics, and how all this behaviour depends on the temperature and other parameters. * Electronic address: A.J. Archer@lboro.ac.uk arXiv:1906.08121v1 [cond-mat.soft] 19 Jun 2019 is to use Monte Carlo methods to simulate the dynamic properties of Hamiltonian systems [18]. However, it is not clear to us that one should enforce detailed balance when constructing coarse grained models of such non-equilibrium systems, since droplets on surfaces are highly dissipative systems, due to the evaporation, the strong coupling to the substrate, etc. Of course, at equilibrium, there should be detailed balance in the model.To model collective hydrodynamic motion in our model, we generate large-scale collective particle moves by considering the dynamics that is predicted by a thin-film equation. This is a time evolution equation for the liquid film thickness profile h, i.e. the height of the liquid-vapour interface above the surface [2,[19][20][21][22]. This equation is derived from the Navier-Stokes equations for a film of liquid on a surface by making a long-wave approximation, which greatly simplifies the analysis. When generating these 'thin-film moves', we first determine the liquid film height profile from the configuration of occupied lattice sites, we then evolve the thin-film equation for a short amount of time, and then map back to the lattice. In our model, an important quantity is the parameter λ, defined below in Eq. (13), which is proportional to the ratio of the number of KMC attempted particle moves to the number of thin-film moves. When λ = 0, our model reduces to just evolving the thin-film equation, whilst in the limit λ → ∞, our model is a pure KMC model. Therefore, this parameter λ ∝ D/η, where D is a single-particle diffusion coefficient and η is the viscosity.Key quantities that characterises how a liquid wets a surface are the spreading parameter S = γ sv − (γ sl + γ lv ), where γ sv , γ sl , and γ lv are the solid-vapour, solid-liquid and liquid-vapour interfacial tensions, respectively [2], and also the contact angle θ, which is given by Young's equation [2]:The...
In biological systems, the MHD boundary layer bioconvection flow through permeable surface has several applications, including electronic gadgets, heating systems, building thermal insulation, geological systems, renewable energy, electromagnetism and nuclear waste. The bioconvection caused by the hydromagnetic flow of a special form of water-based nanoliquid including motile microorganisms and nanoparticles across a porous upright moving surface is investigated in this report. The combination of motile microbes and nanoparticles causes nanofluid bioconvection is studied under the cumulative impact of magnetic fields and buoyancy forces. The Brownian motion, thermophoresis effects, heat absorption/generation, chemical reaction and Darcy Forchhemier impact are also unified into the nonlinear model of differential equations. The modeled boundary value problem is numerically computed by employing a suitable similarity operation and the parametric continuation procedure. The parametric study of the flow physical parameters is evaluated versus the velocity, energy, volume fraction of nanoparticles, motile microorganisms’ density, skin friction, Sherwood number and Nusselt number. It has been observed that the velocity profile reduces with the effect of porosity parameter k1, inertial parameter k2, Hartmann number and buoyancy ratio. While the energy transition profile significantly enhances with the flourishing values of Eckert number Ec, heat absorption/generation Q and Hartmann number respectively.
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