Azimuthal Fluid Velocity Research Articles

Lie group transformation method for shock wave in rotating non-ideal gas with or without magnetic field, and interaction of characteristic shock with weak discontinuity

An analytical solution for power-law shock paths and a numerical solution for exponential-law shock paths to the system of equations that describes a cylindrical shock wave in a rotating non-ideal gas with or without an axial magnetic field is determined by utilizing the Lie group invariance method. In an undisturbed medium, the axial magnetic field and azimuthal fluid velocity are meant to be variable; however, the density is taken to be constant. The liberty to choose the value of arbitrary constants that are in the equation for an infinitesimal generator gives rise to three different cases, i.e., the power law, a particular case of the power law, and the exponential-law shock paths. In the power-law case, a particular solution in an analytical form is obtained, while for an exponential-law case, a numerical solution is obtained. By considering this analytical solution, the development of the characteristic shock and its interaction with a weak discontinuity are also discussed. The effects of the rotational and non-idealness parameters on the characteristic shock and on the acceleration wave's amplitude are discussed. The expressions for the jump in shock acceleration and the amplitude of the transmitted and reflected wave are obtained.

The propagation of strong cylindrical shock wave in a rotating axisymmetric non‐ideal gas with radiation heat flux

In this manuscript, we studied a quasi‐linear hyperbolic system of partial differential equations which describes the one‐dimensional adiabatic unsteady flow behind a strong cylindrical shock wave propagating in a rotating non‐ideal gas, which has a varying azimuthal fluid velocity together with a varying axial fluid velocity, with radiation heat flux, and similarity solutions are obtained by using the Lie group‐theoretic method. The basic idea of this method is that it changes the system of PDEs representing the one‐dimensional flow through the similarity variable to the system of ODEs. The flow profiles are drawn behind the shock followed by a brief discussion on the behavior of the solutions via graphs. The effects of variation in non‐ideal parameter, adiabatic index of the gas, Alfven–Mach number and ambient azimuthal velocity exponent on the flow variables are described. All numerical calculations have been performed using the “Mathematica” software.

Propagation of cylindrical shock waves in rotational axisymmetric dusty gas with magnetic field: Isothermal flow

In this article, the effect of the dust particles is studied on the propagation of a cylindrical shock wave in rotational axisymmetric ideal gas under isothermal flow conditions with the magnetic field. Here, magnetic pressure, azimuthal fluid velocity, and axial fluid velocity are supposed to vary according to a power law in the undisturbed medium. With the help of Sakurai's technique, we obtain approximate solutions analytically by expanding the flow parameters in the form of a power series in ϕ=(CV)2. The power series method is used to derive the zeroth and the first-order approximations. The solutions for the zeroth-order approximation are constructed in analytical form. Distributions of the hydrodynamical quantities are analyzed graphically behind the shock front. Also, the effects of shock Cowling number (co), mass fraction of the solid particles in the mixture (kp), adiabatic exponent (γ), and rotational parameter (L) on the flow variables are discussed. It is investigated that the density and pressure near the line of symmetry in the case of isothermal flow become zero, and hence a vacuum is formed at the axis of symmetry when the flow is isothermal. The present work may be used to verify the correctness of the solution obtained by self-similarity and numerical methods. Furthermore, the results obtained in the present work are found to be in good agreement with those obtained from the study by Nath and Singh [Can. J. 98, 1077 (2020)].

Similarity solutions for cylindrical shock wave in rotating non-ideal gas using Lie group theoretic method

In this paper, the propagation of cylindrical shock wave in rotating non-ideal gas under adiabatic flow condition using Lie group of transformation method is investigated. The density is assumed to be constant and azimuthal fluid velocity is assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the Local Lie group of transformations bring about two different cases of solutions i.e. with a power-law and exponential-law shock paths. Numerical solutions are obtained for both the cases. Distribution of gasdynamical quantities is illustrated through figures. It is obtained that the reduced flow variables pressure and azimuthal fluid velocity decrease in general, whereas density and radial fluid velocity increase in case of power-law shock path. The reduced azimuthal fluid velocity decreases, whereas reduced density, pressure and radial fluid velocity increase in case of exponential-law shock path. Also, it is obtained that shock strength decreases with increase in value of adiabatic exponent or gas non-idealness parameter, whereas it increases due to increase in ambient azimuthal fluid velocity exponent.

A self-similar solution for unsteady adiabatic and isothermal flows behind the shock wave in a non-ideal gas using Lie group analysis method with azimuthal or axial magnetic field in rotating medium

The similarity solutions using Lie group analysis for shock wave propagation in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field in the case of isothermal and adiabatic flows are obtained. All possible cases of similarity solutions are discussed using the Lie group analysis for the isothermal and adiabatic flows. The arbitrary constants involved in the generators of local Lie groups bring various possible cases of solutions with exponential law and power law shock paths. Similarity solution for isothermal and adiabatic flows with power law shock path is discussed in detail. The density of ambient medium is taken to be constant. The axial and azimuthal fluid velocities and magnetic field in the ambient medium are assumed to be varying according to power law. The effect on shock wave strength and that on the flow variables due to variation of the Alfven Mach number, adiabatic index of the gas, non-idealness parameter, rotational parameter and initial magnetic field variation exponent are investigated. It is found that these parameters have decaying effects on shock wave. The obtained results in the case of isothermal and adiabatic flows are also compared with each other.

Approximate Analytical Solution for Ionizing Cylindrical Magnetogasdynamic Shock Wave in Rotational Axisymmetric Self-Gravitating Perfect Gas: Isothermal Flow

The propagation of ionizing cylindrical magnetogasdynamic shock wave in rotational axisymmetric self-gravitating perfect gas under isothermal flow condition is investigated. Mathematical model for the considered problem using system of PDEs is presented. The density, magnetic pressure, azimuthal fluid velocity and axial fluid velocity are assumed to be varying according to power law with distance from the axis of symmetry in the undisturbed medium. The flow variables are expanded in power series and using that the zeroth and first order approximations are discussed. Solutions for zeroth order approximation are constructed in approximate analytical form. The effect of flow parameters namely: gravitational parameter $$G_{0}$$ , shock Cowling number $$C_{0}$$ , rotational parameter L and ambient density variation index q are studied on the flow variables and total energy of disturbance. Distribution of gasdynamical quantities are discussed. Radial fluid velocity and mass tends to zero near the axis of symmetry in general; but magnetic pressure, axial fluid velocity, non-dimensional components of vorticity vector $$l_{\theta }^{(0)}$$ and $$l_{z}^{(0)}$$ tend to positive infinity near the axis of symmetry. Azimuthal fluid velocity decreases as we move inwards from the shock to the axis of symmetry. Density and pressure vanish near the axis of symmetry thus forming a vacuum there. $$J_{0}$$ decreases with increase in value of ambient density variation index q or gravitational parameter $$G_{0}$$ ; whereas it increases with increase in value of rotational parameter L or shock Cowling number $$C_{0}$$ .

Similarity solutions for magnetogasdynamic shock waves in a rotating ideal gas using the Lie group-theoretic method

The propagation of a cylindrical shock wave under the influence of an azimuthal magnetic field in a rotating medium for adiabatic flow conditions is investigated using the Lie group transformation method. The density, magnetic field, and azimuthal and axial fluid velocities are assumed to vary in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the local Lie group of transformations bring about different cases of solutions, i.e., with a power-law shock path, with an exponential-law shock path, and a particular case of a power-law shock path. Numerical solutions are obtained in the case of a power-law shock path and exponential-law shock path. The distributions of gasdynamical quantities are discussed based on figures. The effects of varying the values of the adiabatic exponent $$\gamma $$ , Alfven–Mach number $$M_\mathrm{A}^{-2}$$ , ambient azimuthal fluid velocity variation index $$\lambda _{1}$$ , and ambient density variation index $$\phi $$ on the flow variables and shock strength are studied. With an increase in the adiabatic exponent or the strength of the magnetic field, the shock strength decreases. However, an increase in the ambient density variation index or ambient azimuthal fluid velocity variation index results in an increase in the shock strength. The numerical calculations are carried out using Mathematica software.

Similarity solutions for cylindrical shock wave in rotating ideal gas with or without magnetic field using Lie group theoretic method

The propagation of a cylindrical shock wave in rotating ideal gas under adiabatic flow condition is investigated using Lie group of transformation method. Both the cases of shock without magnetic field and under the influence of axial magnetic field are considered separately. The density and azimuthal fluid velocity in the case of shock without magnetic field and density, azimuthal fluid velocity and axial magnetic field in the case of shock under the influence of magnetic field are assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for the infinitesimals of the local Lie group of transformations bring about two different cases of solutions, i.e. with a power law and exponential law shock paths. Exact solutions are obtained in the case of power law shock path for both the cases of cylindrical shock with and without magnetic field. It is not possible to obtain the exact solution in the case of exponential law shock path. In this case, the numerical solutions can be obtained by using the respective boundary conditions. Distribution of gasdynamical and magnetogasdynamical flow quantities are illustrated through figures.

Approximate analytical solution for ionizing cylindrical shock wave in rotational axisymmetric non-ideal gas: isothermal flow

The propagation of an ionizing cylindrical shock wave in rotational axisymmetric non-ideal gas under isothermal flow condition with an azimuthal magnetic field is investigated. The electrical conductivity is assumed to be negligible in the medium ahead of the shock wave, which after the passage of the shock wave becomes infinitely large. The magnetic pressure, azimuthal fluid velocity, and axial fluid velocity are assumed to be varying according to the power law with distance from the axis of symmetry in the undisturbed medium. The zeroth and first-order approximations are discussed by the aid of the power series method. Solutions for the zeroth-order approximation are constructed in analytical form. Distributions of hydrodynamical quantities are discussed. The effect of flow parameters, namely, shock wave Cowling number c∗, adiabatic exponent γ, rotational parameter L, and gas non-idealness parameter [Formula: see text] are studied on the flow variables. Due to the consideration of a rotating medium or due to the presence of magnetic field, the total energy of the disturbance increases, while with an increase in adiabatic exponent γ the total energy of the disturbance decreases. Density and pressure vanish near the axis of symmetry, thus forming a vacuum there.

Approximate analytical solution for shock wave in rotational axisymmetric perfect gas with azimuthal magnetic field: Isothermal flow

The propagation of cylindrical shock wave in rotational axisymmetric perfect gas under isothermal flow condition with azimuthal magnetic field is investigated. Distributions of gas dynamical quantities are discussed. The density, magnetic pressure, azimuthal fluid velocity and axial fluid velocity are assumed to be varying according to power law with distance from the axis of symmetry in the undisturbed medium. Approximate analytical solutions are obtained by expanding flow variables in power series. Zeroth-order and first-order approximations are discussed with the aid of power series method. Solutions for zeroth-order approximation are constructed in approximate analytical form. The effect of flow parameters namely: shock Cowling number $$C_{0}$$, ambient density variation index q and adiabatic exponent $$\gamma $$ are studied on the flow variables. Consideration of magnetic pressure increases the total energy of disturbance of zeroth order while with increase in ambient density variation index or adiabatic exponent, the total energy of disturbance decreases.

Numerical investigation of electroconvection induced by strong unipolar injection between two rotating coaxial cylinders

In this paper, the interaction between a Couette shear flow and an electroconvective motion induced by an unipolar injection between two-coaxial cylinders is numerically investigated. A flow is generated by two counter-rotating coaxial cylinders inducing a shear flow. Space charges are injected in the flow through a metallic electrode placed on the inner cylinder and brought to a given potential. Transient numerical simulations have been carried out to investigate the structure of the induced flow. The entire set of the coupled Navier-Stokes and EHD equations is solved using an efficient finite volume technique. The behaviour of the flow subjected to an applied voltage between the two electrodes is analyzed and time evolution of the charge density distributions is presented. The interaction between the convective motion induced by space charge injection and the mainstream flow, emphasizes the appearance of periodic counter-rotating electroconvective cells. The electroconvective cells are convected in the annular space by the azimuthal fluid velocity. From the stability point of view the bifurcation diagram is very similar to the one obtained in the case when Re = 0. We observe a threshold value Tc of the instability parameters T above which the electroconvective instability initiates. A non-linear criterion Tf under which the electro-convective motion is suppressed is also found. When increasing the Reynolds number the flow induced by the two rotating cylinders has a sweeping effect on the charge density distribution. Consequently the instability parameter T must be drastically increased to allow the electroconvective instability to develop. For Re = 10 a subcritical instability characterized by an hysteresis loop and therefore a linear and non-linear criteria, Tc and Tf respectively, are determined. While for Re = 0, Tc = 122.42 and Tf = 86.5, for Re = 10 we numerically found that Tc = 802 and Tf = 722. The magnitude of the linear and non-linear criteria are directly linked to the value of the Reynolds number.

Erratum to: Self-similar solution of cylindrical shock wave propagation in a rotational axisymmetric mixture of a non-ideal gas and small solid particles

Similarity solutions are obtained for one- dimensional isothermal and adiabatic unsteady flow behind a strong cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has a variable azimuthal fluid velocity together with a variable axial fluid velocity. The shock is assumed to be driven out by a moving piston and the dusty gas to be a mixture of non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-condition is maintained and variable energy input is continuously supplied by the piston. The shock Mach number is not infinite, but has a finite value. The azimuthal and axial component of the fluid velocity in the ambient medium are assumed to be vary and obey power laws, and the density of the ambient medium is taken to be constant. In order to obtain the similarity solutions the angular velocity of the ambient medium is assumed to be decreasing as the distance from the axis increases. Effects of the variation of the parameter of non-idealness of the gas in the mixture, the mass concentration of solid particles and the ratio of the density of solid particles to the initial density of the gas are investigated.

Liquid sodium model of geophysical core convection

Convective motions in Earth’s outer core are responsible for the generation of the geomagnetic field. We present liquid sodium convection experiments in a spherical vessel, designed to model the convective state of planetary cores such as the Earth’s. Heat transfer, azimuthal fluid velocities, and properties of temperature fluctuations were measured for different rotation rates and temperature drops across the convecting sodium. We observed small-scale convective motions with strong, large-scale azimuthal winds and developed turbulence despite the fact that convective heat transport was weak and the temperature profile was close to diffusive. In the context of Earth’s outer core, our observations suggest models which imply a thermal Rayleigh number R a ≈ 6 × 1 0 23 and a convective velocity near 2 × 1 0 − 4 m/s. Also, the energy spectrum of outer core may exhibit important structure down to length and time scales of 1 km and 30 days. Furthermore, we calculate an estimate of Ohmic dissipation, 0.1 TW, in the core based on the shape of experimentally observed power spectra.

Реакция вязкой жидкости на торсиально осциллирующий эллицсоид: Решение методом «слабого разделения» линеаризованной проблемы стационарного состояния

The steady-state linearized problem of a prolate (oblate) spheroid torsionally oscillating in an infinite, viscous, incompressible fluid is considered. The use of spheroidal co-ordinates results in a separable partial differential equation whose solutions are orthogonal spheroidal functions. The solution of the boundary-value problem is then expressible as an infinite series of spheroidal functions. However, it is quite difficult to convert a spheroidal function series into a series in powers of the eccentricity. Here it is shown that with the use of a special «modified spheroidal» co-ordinate system the solution of the linearized Navier-Stokes equations can be obtained directly in the form of a perturbation expansion in powers of the square of the eccentricity. Both the azimuthal fluid velocity and resulting torsional reaction of the fluid on the spheroid are evaluated to all orders in the perturbation expansion. These results illustrate the usefulness of the «weak separation» method of solving certain partial-differential-equation boundary-value problems.