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This investigation deals with the time-dependent flow of an incompressible viscous fluid bounded by an infinite plate. The fluid is electrically conducting under the influence of a transverse magnetic field. The plate moves with a time dependent velocity in its own plane. Both fluid and plate exhibit rigid body rotation with a constant angular velocity. The solutions for arbitrary velocity and magnetic field is presented through similarity and numerical approaches. It is found that rotation induces oscillations in the flow.

Although the flapping oscillatory motion is organized, its effect on the mean flows of turbulent plane jets is similar to the effect of turbulence. Therefore, the close relation for the stress due to flapping-induced velocity components is assumed to be similar to the relation for the turbulent Reynolds stress, and the similarity theory applicable for mean flows of turbulent plane jets is extended to that of unstable flapping turbulent plane jets. The results show the effect of flapping motion on mean flows can be regarded as two folds: one is to enhance the mixing by flapping-induced Reynolds stress and the other is to modify the effective advection. The results from similarity analyses are in good agreement with the experimental data.

A generalized nonlinear heat equation with the fractional derivative is proposed, whose similarity solutions are derived from a type of special scalar transformation with two parameters. With the help of separated variable method, two special series solutions of the standard heat equation are obtained. Finally, through computation of the left Riemann–Liouville fractional derivative, we obtain two approximated computation formulas of the factional-order ordinary differential equation which could be used to calculate the numerical solutions of the generalized nonlinear heat conduction equation.

We introduce isospectral and non-isospectral Lax pairs, then apply the Tu scheme to generate the isospectral integrable hierarchy and the non-isospectral hierarchy, whose Hamiltonian structure, hereditary operator, symmetries are followed to obtain. In addition, a kind of Bäcklund transformation of the long-water wave hierarchy for the isospectral hierarchy is constructed. Through reductions of the isospectral hierarchy, we again get the long-water wave system whose similarity solutions, nonlinear self-adjointness and the non-invariant solutions are investigated, respectively, by the use of symmetry analysis. Finally, we make use of the weight method of the variables appearing in the long-water wave system to analyze the conservation laws of the system.

In this paper, we investigate some symmetries and Lie-group transformations of an integrable system by using the symmetry analysis method. It follows that the resulting similarity solutions are obtained by applying the characteristic equations of the symmetries. By applying the software Maple,we work out some exact solutions of the generalized KdV system, including the rational solutions, the periodic solutions, the dark soliton solutions, and so on. Finally, we make use of the self-adjoint operators to investigate the nonlinear self-adjointness and the conservation laws of the generalized KdV integrable system.

We investigate in the present paper the magnetic Rayleigh problem, where a semi-infinite flat plate is moving with a power-law velocity, in a non-Newtonian power-law fluid (Ostwald–de Wael model). The non-stationary flow of this electrically conducting fluid in a transverse magnetic field is then analyzed. The solutions of this problem are obtained by means of similarity techniques. The main goal, is to investigate existence, uniqueness and behavior of such solutions, according to the values of the physical parameters.

In this paper, group analysis of the fourth-order time-fractional Burgers–Korteweg–de Vries (KdV) equation is considered. Geometric vector fields of Lie point symmetries of the equation are investigated and the corresponding optimal system is found. Similarity solutions of the equation are presented by using the obtained optimal system. Finally, a useful method called invariant subspaces is applied in order to find another solutions.

In this paper, we have studied the propagation of cylindrical shock waves in a self-gravitating perfect gas under the influence of azimuthal magnetic field. The method of Lie group invariance is used to construct some special class of self-similar solutions in the presence of the azimuthal magnetic field. The different cases of solutions with a power law and exponential law shock paths are obtained with the choice of arbitrary constants appearing in the expressions for the infinitesimal generators. The similarity solution for cylindrical shock wave with power law shock path is discussed in detail. The effects of variation of Alfven-Mach number, gravitation parameter, initial density variation index and adiabatic exponent on the flow variables are analyzed graphically. It is obtained that the increase in the values of Alfven-Mach number, gravitation parameter and adiabatic exponent have decaying effect on the shock strength. Also, the shock strength increases with an increase in the values of initial density variation index. A comparison is also made between the solutions in gravitating and non-gravitating cases in the presence of magnetic field.

This paper concerns with the similarity solutions using Lie group theoretic method for one-dimensional unsteady isothermal flow behind the shock wave in the mixture of non-ideal gas and small solid particles in rotating medium. Group theoretic technique brings the different possible cases of potential solutions with the power law, particular case of power law and exponential law shock paths with the different choices of constants present in the generators of the Lie group of transformations. Similarity solutions for non-ideal dusty gas (a mixture of non-ideal gas and small solid dust particles) exist if the density of the ambient medium is constant. Solutions are obtained for both exponential law and power law shock paths. The effects of the variation of the physical parameters on the flow variables distribution behind the shock wave front, and on the shock are discussed. The shock strength increases with an increase in the ratio of the specific heat of the solid particles to the specific heat of the gas at constant volume ${\beta }^{\prime }$, the initial azimuthal velocity variation index ${𝜃}_1$, and the ratio of the density of solid particles to the initial density of the gas $G_0$. The shock wave decays with an increase in the value of gas non-idealness parameter $\overline{b}$.

The tidal flushing of an estuary or inlet is often used to assist with the dispersal of waste water on a falling tide. During inflow, water is generally drawn radially towards the estuary mouth. The nature of the inflow and outflow depends on the topography of the sea floor, particularly if the floor is shelving into deeper water or if there is an off-shore bar.

The analytic model of Wolanski and Imberger (1987) is examined in detail and a full set of similarity solutions are derived for the inflow regime. These are applicable to a range of bottom profiles which may be used to model a limited set of practical examples or to provide verification for a numerical scheme.

A numerical solution is also developed for more general depth profiles. It uses a five point finite-difference stencil to derive an algebraic system representing the elliptic equation satisfied by the stream function. MATLAB programming is used to implement and display the solution on an 80486 AT machine (IBM PC).

A variety of bottom profiles were investigated to assess the effects of the slope of the sea floor, and of a sand bar along the beach and across the inlet mouth. Implications of the bar on the inflowing movement of water were also investigated, with special reference to the Tweed River mouth and neighbouring coastline.