Narrow Results

Incompressible fluids are studied in such disciplines as ocean engineering, astrophysics and aerodynamics. Under investigation in this paper is a $(3+1)$-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation in an incompressible fluid. Based on the known bilinear form, BLMP hybrid solutions comprising a lump wave, a periodic wave and two kink waves, and hybrid solutions comprising a breather wave and multi-kink waves are derived. We observe the interaction among a lump wave, a periodic wave and two kink waves. Fission of a kink wave is observed: A kink wave divides into a breather wave and three kink waves. On the contrary, we see the fusion among a breather wave and three kink waves: The breather wave and three kink waves merge into a kink wave. Finally, we observe the interaction among a breather wave and four kink waves.

In this paper, the investigation is conducted on a (2 + 1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation for an incompressible fluid. Via the Riemann theta function, periodic-wave solutions are derived, and breather-wave solutions are constructed with the aid of the extended homoclinic test approach. Based on the polynomial expansion method, several traveling-wave solutions are derived. Besides, we observe that the amplitude of the breather keeps unchanged during the propagation and the traveling wave which is kink shaped propagates stably. Furthermore, we analyze the transition between the periodic-wave and soliton solutions, which implies that the periodic-wave solutions tend to the soliton solutions via a limiting procedure.

We are concerned with a system of partial differential equations (PDEs) describing internal flows of homogeneous incompressible fluids of Bingham type in which the value of activation (the so-called yield) stress depends on the internal pore pressure governed by an advection–diffusion equation. After providing the physical background of the considered model, paying attention to the assumptions involved in its derivation, we focus on the PDE analysis of the initial and boundary value problems. We give several equivalent descriptions for the considered class of fluids of Bingham type. In particular, we exploit the possibility to write such a response as an implicit tensorial constitutive equation, involving the pore pressure, the deviatoric part of the Cauchy stress and the velocity gradient. Interestingly, this tensorial response can be characterized by two scalar constraints. We employ a similar approach to treat stick-slip boundary conditions. Within such a setting we prove long-time and large-data existence of weak solutions to the evolutionary problem in three dimensions.

We study the free boundary problem for two-dimensional current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. For this amplitude equation, a local-in-time existence theory for smooth solutions to the Cauchy problem was established earlier by the authors under a suitable stability condition. However, the solution found therein enjoyed a loss of regularity (of order two) in comparison to the regularity of the initial data. In this work, we are able to prove an existence result with optimal regularity, in the sense that the regularity of the initial data is preserved in the evolution for positive times.

The paper is concerned with the free boundary problem for two-dimensional current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo [On the weakly nonlinear Kelvin–Helmholtz instability of tangential discontinuities in MHD, J. Hyperbolic Differ. Equations8(4) (2011) 691–726] have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. The local-in-time existence of smooth solutions to the Cauchy problem for the amplitude equation was shown in [Approximate current-vortex sheets near the onset of instability, J. Math. Pures Appl.105(4) (2016) 490–536; Existence of approximate current-vortex sheets near the onset of instability, J. Hyperbolic Differ. Equations]. In the present paper, we prove the continuous dependence in strong norm of solutions on the initial data. This completes the proof of the well-posedness of the problem in the classical sense of Hadamard.

The present paper deals with the study of viscous contributions to the pressure for the viscous potential flow analysis of Kelvin–Helmholtz instability with tangential magnetic field at the interface of two viscous fluids. Viscosity enters through normal stress balance in the viscous potential flow theory and tangential stresses for two fluids are not continuous at the interface. Here, we have considered viscous pressure in the normal stress balance along with the irrotational pressure and it is assumed that the addition of this viscous pressure will resolve the discontinuity between the tangential stresses and the tangential velocities at the interface. The viscous pressure is derived by mechanical energy equation and this pressure correction applied to compute the growth rate of magnetohydrodynamic Kelvin–Helmholtz instability. A dispersion relation is obtained and stability criterion is given in the terms of critical value of relative velocity. It has been observed that the inclusion of irrotational shear stresses have stabilizing effect on the stability of the system.

This paper studies the characteristics of a micro-beam interacting with an incompressible fluid in a fluid chamber with an opening in its bottom face for fluid flow. The Euler–Bernoulli equation for transverse deformation of an elastic beam is coupled with the fundamental hydrodynamic equation, which is solved by Galerkin and separation of variables method. The 2D fluid flow assumption in Cartesian coordinate has been used. Natural frequencies and mode shapes of wet beam are calculated and compared with the dry beam. The effects of geometrical parameter changes are also computed as a benchmark for the design of the micro-pump. It is observed that fluid coupling causes a decrease for beam’s natural frequencies, especially in higher modes. Furthermore, since the results of the dry and wet beam show a small discrepancy in lower modes, the mode related to the dry beam was employed as the trial function in the forced vibration analysis of the coupled system.

The fluid equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term — hence describing viscous flow. The form of the Navier–Stokes equations means they can be transformed to full/partial inhomogeneous parabolic differential equations: differential equations in respect of space variables and the full differential equation in respect of time variable and time dependent inhomogeneous part. Orthogonal polynomials as the partial solutions of obtained Helmholtz equations were used for derivation of analytical solution of incompressible fluid equations in 1D, 2D and 3D space for rectangular boundary. New one anti-curl method was proposed for derivation of velocities in incompressible fluid and was shown how this method works with rectangular boundaries. Finally, solution in 3D space for any shaped boundary was expressed in term of 3D general solution of 3D Helmholtz equation accordantly.