Some helical flows of a Burgers’ fluid with fractional derivative

In this article, fractional calculus approach is used in the constitutive relationship of a Burgers’ fluid model. Integral transforms are used to calculate the velocity and the stress fields for some helical flows of a Burgers’ fluid with fractional derivative. Moreover, the behavior of different physical parameters involve in the Burgers’ fluid model is analyzed through several graphs.     

Exact solutions for generalized Burgers’ fluid in an annular pipe

This paper deals with some unsteady unidirectional transient flows of generalized Burgers’ fluid in an annular pipe. Exact solutions of some unsteady flows of generalized Burgers’ fluid in an annular pipe are obtained by using Hankel transform and Laplace transform. The following two problems have been studied: (1) Poiseuille flow due to a constant pressure gradient; (2) axial Couette flow in a annulus. The well known solutions for Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid, a second grade fluid and an Oldroyd-B fluid appear as limiting cases of our solutions.     

Hall and heat transfer effects on the steady flow of a generalized Burgers’ fluid induced by a sudden pull of eccentric rotating disks

This study looks at the magnetohydrodynamic (MHD) flow of a generalized Burgers’ fluid between two heated disks rotating about noncoaxial axes normal to the disks. The steady flow and heat transfer analysis is investigated by providing exact analytic solutions. The effect of Hall current is taken into consideration. Calculations are carried out for velocity, temperature, force, and torque exerted by the fluid on one of the disks. The physical interpretation for the emerging parameters is discussed with the help of graphs. The results are compared with those available in the existing literature.     

Group classification and exact solutions of a generalized Emden–Fowler equation

The aim of this work is to perform a complete symmetry classification of a generalized Emden-Fowler equation. The various forms of this equation are extensively studied in the literature and they have applications in astrophysical and physiological phenomena. The classical approach of group classification and the procedure based upon the Lie algebras of low dimension are employed for classification. Exact solutions of the invariant equations are derived.     

Pattern formation in a nonequilibrium phase transition for a generalized Burgers–Fisher equation

A nonequilibrium phase transition of a generalized Burgers–Fisher equation describing biological pattern formation with a periodic boundary condition is examined. In the presence of a weak external force, some approximate bifurcation solutions near a critical point and new spatially periodic patterns are obtained by using the perturbation method in an infinite-dimensional space. The result shows that the external force delays the bifurcation.     

Hybrid localized wave solutions for a generalized Calogero–Bogoyavlenskii–Konopelchenko–Schiff system in a fluid or plasma

Nonlinear Dynamics - Based on the long wave limit method and complex conjugate condition technique, we investigate hybrid localized wave solutions with different forms for the generalized...     

On one mechanism of formation of tornado‐like vortices in a rotating fluid

It is shown that during excitation of forced, resonant, inertial oscillations of large amplitude in a rigidly rotating fluid, the mechanism of formation of tornadolike vortices is primarily of a kinematic nature($advection of circulation of the azimuthal component velocity and stretching of vortex lines by the poloidal components of the velocity field that arise from excitation of inertial oscillations). The main parameters of the vortices are obtained by solutions of model problems. To excite such oscillations, it is necessary to deliver energy far exceeding the initial energy of the rotating fluid. Therefore, inertial oscillations by themselves cannot lead to the occurrence of intense atmospheric vortices. Nevertheless, such oscillations can apparently play the role of a trigger mechanism that activates more complex processes of vortex formation related to instability of the atmosphere.     

Debye-Hückel solution for steady electro-osmotic flow of micropolar fluid in cylindrical microcapillary

Analytic expressions for speed, flux, microrotation, stress, and couple stress in a micropolar fluid exhibiting a steady, symmetric, and one-dimensional electro-osmotic flow in a uniform cylindrical microcapillary were derived under the constraint of the Debye-Hiickel approximation, which is applicable when the cross-sectional radius of the microcapillary exceeds the Debye length, provided that the zeta potential is sufficiently small in magnitude. Since the aciculate particles in a micropolar fluid can rotate without translation, micropolarity affects the fluid speed, fluid flux, and one of the two non-zero components of the stress tensor. The axial speed in a micropolar fluid intensifies when the radius increases. The stress tensor is confined to the region near the wall of the mi- crocapillary, while the couple stress tensor is uniform across the cross-section.     

Debye-H¨uckel solution for steady electro-osmotic flow of micropolar fluid in cylindrical microcapillary

Analytic expressions for speed, flux, microrotation, stress, and couple stress in a micropolar fluid exhibiting a steady, symmetric, and one-dimensional electro-osmotic flow in a uniform cylindrical mi...     

Stability of plane Poiseuille–Couette flows of a piezo-viscous fluid

We examine stability of fully developed isothermal unidirectional plane Poiseuille–Couette flows of an incompressible fluid whose viscosity depends linearly on the pressure as previously considered in Hron et al. [J. Hron, J. Málek, K.R. Rajagopal, Simple flows of fluids with pressure-dependent viscosities, Proc. R. Soc. Lond. A 457 (2001) 1603–1622] and Suslov and Tran [S.A. Suslov, T.D. Tran, Revisiting plane Couette–Poiseuille flows of a piezo-viscous fluid, J. Non-Newtonian Fluid Mech. 154 (2008) 170–178]. Stability results for a piezo-viscous fluid are compared with those for a Newtonian fluid with constant viscosity. We show that piezo-viscous effects generally lead to stabilisation of a primary flow when the applied pressure gradient is increased. We also show that the flow becomes less stable as the pressure and therefore the fluid viscosity decrease downstream. These features drastically distinguish flows of a piezo-viscous fluid from those of its constant-viscosity counterpart. At the same time the increase in the boundary velocity results in a flow stabilisation which is similar to that observed in Newtonian fluids with constant viscosity.     

Investigation of generalized Fick’s and Fourier’s laws in the second-grade fluid flow

The second-grade fluid flow due to a rotating porous stretchable disk is modeled and analyzed. A porous medium is characterized by the Darcy relation. The heat and mass transport are characterized through Cattaneo-Christov double diffusions. The thermal and solutal stratifications at the surface are also accounted. The relevant nonlinear ordinary differential systems after using appropriate transformations are solved for the solutions with the homotopy analysis method (HAM). The effects of various involved variables on the temperature, velocity, concentration, skin friction, mass transfer rate, and heat transfer rate are discussed through graphs. From the obtained results, decreasing tendencies for the radial, axial, and tangential velocities are observed. Temperature is a decreasing function of the Reynolds number, thermal relaxation parameter, and Prandtl number. Moreover, the mass diffusivity decreases with the Schmidt number.     

Painlevé analysis,auto-Bäcklund transformation and analytic solutions of a (2 $$+$$ + 1)-dimensional generalized Burgers system with the variable coefficients in a fluid

Nonlinear Dynamics - Burgers-type equations are used to describe certain phenomena in gas dynamics, traffic flow, plasma astrophysics and ocean dynamics. In this paper, a (2 $$+$$ 1)-dimensional...     

On exact solutions of a class of fractional Euler–Lagrange equations

In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian L(x(t), where _a ^c D _t ^α x(t)) and 0<α<1, such that the following is the corresponding Euler–Lagrange

The Giesekus fluid in ω–D form for steady two-dimensional flows

Numerical results of the simulation of the Giesekus model in ω–D form, which has previously been introduced in Part I of this study, are presented. The model has been applied to the flow of a concentrated polymer solution through a planar 3.97:1 contraction. To obtain an accurate fit of the rheological properties of the fluid a four-mode model is used. The predictions of the numerical simulations are directly compared with the experimental results published by Quinzani et al. in 1994. For the velocity fields a good quantitative agreement is reached, especially in the upstream channel. Regarding the shear stress and first normal stress difference, qualitative predictions of the experimental profiles are obtained.     

A generalized algebraic method of new explicit and exact solutions of the nonlinear dispersive generalized Benjiamin–Bona–Mahony equations

Nonlinear dispersive generalized Benjiamin–Bona–Mahony equations are studied by using a generalized algebraic method. New abundant families of explicit and exact traveling wave solutions, including triangular periodic, solitary wave, periodic-like, soliton-like, rational and exponential solutions are constructed, which are in agreement with the results reported in other literatures, and some new results are obtained. These solutions will be helpful to the further study of the physical meaning and laws of motion of the nature and the realistic models. The proposed method in this paper can be further extended to the 2+1 dimensional and higher dimensional nonlinear evolution equations or systems of equations.     

Occurrence of tornado‐like vortices in a rotating fluid under forced inertial oscillations of large amplitude

Experimental estimates are obtained for the main parameters of tornadolike vortices that arise from excitation of forced axisymmetric inertial oscillations of large amplitude in a rigidly rotating fluid.     

High-order numerical methods of fractional-order Stokes’ first problem for heated generalized second grade fluid

The high-order implicit finite difference schemes for solving the fractionalorder Stokes’ first problem for a heated generalized second grade fluid with the Dirichlet boundary condition and the initial condition are given. The stability, solvability, and convergence of the numerical scheme are discussed via the Fourier analysis and the matrix analysis methods. An improved implicit scheme is also obtained. Finally, two numerical examples are given to demonstrate the effectiveness of the mentioned schemes.     

Various exact analytical solutions of a variable-coefficient Kadomtsev–Petviashvili equation

Nonlinear Dynamics - Under investigation is a generalized (3 + 1)-dimensional variable- coefficient Kadomtsev– Petviashvili equation in fluid mechanics. Various exact analytical solutions are...