Ordinary Fluid Flows Research Articles

Natural convection flow of a fluid using Atangana and Baleanu fractional model

A modified fractional model for the magnetohydrodynamic (MHD) flow of a fluid is developed utilizing Atangana–Baleanu fractional derivative (ABFD). Natural convection and wall oscillation instigate the flow over a vertical plate positioned in a porous medium. The partial differential equations (PDEs) are transmuted to ordinary differential equations (ODEs). The Laplace transform method with its inversion is employed to accomplish the exact solutions of momentum and heat equations. The final solution is expressed in terms of gamma function, modified Bessel function, and Mittag-Leffler function. The previous definitions Caputo fractional and Riemann–Liouville are rarely used by the researchers now due to their limitations. The newly introduced ABFD has got significance nowadays due to its nonlocal and nonsingular kernel. This work focuses on the oscillating boundary conditions for the viscous model in terms of ABFD. The influence of involved parameters is interpreted through plots. The velocity profile is an increasing function of fractional parameter and jumps for a higher Grashof number due to buoyancy push. Furthermore, the Atangana–Baleanu (AB) model is compared with the ordinary derivative model for limiting case and analyzed in detail. It is noted that the ordinary fluid flows faster compared to the fractional fluid.

Particle end loss in the edge plasma of a field-reversed configuration

Plasma parameters, particle end loss flux, flow velocity, and pressure are measured using a radial array of magnetic probes and directional electrostatic probes, in order to investigate particle loss processes in the edge layer of a field-reversed configuration (FRC). A plasma flow toward the end region is detected outside the separatrix between the axial midplane and the end region. The exhaust flow is also found in the end region. These results imply that particles are lost radially across the separatrix and then axially to the end. Measured flow velocity in the end region agrees within an error of 20% with the fluid-theory prediction, in which isentropy and axial momentum balance along magnetic flux tubes are assumed. The existence of the sonic condition in the end region is also suggested, analogous to ordinary fluid flow in a nozzle. The magnetic flux embedded in the edge layer of the confinement region and in the end region agrees within an error of 30%. These results indicate the applicability of the magnetohydrodynamics (MHD) theory for particle end loss. The end loss time along the open field agrees with the MHD prediction within an error of 20%. The measured particle loss flux from the end region is explained by the MHD theory within an error of 20%. The plasma outside the separatrix is considered to behave as hydrodynamic flow through the magnetic loss channel, contrary to the previous work [L. C. Steinhaur, Phys. Fluids 29, 3379 (1986)]. It seems that the magnetic mirror field improves the particle confinement in the edge plasma of the FRC and thus assist the FRC confinement as previously predicted [Slough et al., Nucl. Fusion 24, 1537 (1984)].

Solar wind low-frequency magnetohydrodynamic turbulence: extended self-similarity and scaling laws

Abstract. In this paper we review some of the work done in investigating the scaling properties of Magnetohydrodynamic turbulence, by using velocity fluctuations measurements performed in the interplanetary space plasma by the Helios spacecraft. The set of scaling exponents ξq for the q-th order velocity structure functions, have been determined by using the Extended Self-Similarity hypothesis. We have found that the q-th order velocity structure function, when plotted vs. the 4-th order structure function, displays a range of self-similarity which extends over all the lengths covered by measurements, thus allowing for a very good determination of ξq. Moreover the results seem to show that the scaling exponents are the same regardless the various observation periods considered. The obtained scaling exponents have been compared with the results of some intermittency models for Kraichnan's turbulence, derived in the framework of infinitely divisible fragmentation processes, showing the good agreement between these models and our observations. Finally, on the basis of the actually available data sets, we show that scaling laws in Solar Wind turbulence seem to be different from turbulent scaling laws in the ordinary fluid flows. This is true for high-order velocity structure functions, while low-order velocity structure functions show the same scaling laws. Since our measurements involve length scales which extend over many order of magnitude where dissipation is practically absent, our results show that Solar Wind turbulence can be regarded as a testing bench for the investigation of general scaling behaviour in turbulent flows. In particular our results strongly support the point of view which attributes a key role to the inertial range dynamics in determining the intermittency characteristics in fluid flows, in contrast with the point of view which attributes intermittency to a finite Reynolds number effect.

MFGA-IDT2 workshop: Astrophysical and geophysical fluid mechanics: the impact of data on turbulence theories

MFGA-IDT2 workshop: Astrophysical and geophysical fluid mechanics: the impact of data on turbulence theories

Experimental evidence for differences in the extended self-similarity scaling laws between fluid and magnetohydrodynamic turbulent flows.

It has been recently suggested that the various $q$th order velocity structure functions in turbulent flows are related to each other through well defined scaling laws, which extend outside the usual inertial domain. Even if theoretical models provide different scaling laws for fluid and magnetohydrodynamic turbulent flows, no attempt has been made up to now to furnish experimental evidence for these differences. By using measurements from the solar wind turbulence and from turbulence in ordinary fluid flows, we show that the differences can be observed only by looking at the high-order velocity structure functions.

A Thermodynamic Derivation of Equations for Dielectric Relaxation Phenomena in Anisotropie Polarizable Media

Using the general methods of non-equilibrium thermodynamics, a theory for anisotropic polarizable media in which dielectric relaxation phenomena occur is developed. Assuming that ii microscopic phenomena give rise to dielectric relaxation, the contributions of these phenomena to the macroscopic polarization are introduced as vectorial internal degrees of freedom in the Gibbs relation. Moreover, it is assumed that a viscous flow phenomenon occurs which is analogous to the viscous flow of ordinary fluids. The phenomenological equations connected with irreversible changes in the polarization and the generalized laws of Ohm, Fourier and Newton are derived, and the Onsager-Casimir reciprocity relations are formulated. For the case that the equations of state can be linearized explicit expressions for the free energy, the internal energy and the entropy are given and it is shown that a complete set of equations describing the mechanical, electromagnetic and thermodynamic behaviour of the media studied in the present paper can be obtained.

On the propagation of linear transverse acoustic waves in isotropic media with mechanical relaxation phenomena due to viscosity and a tensorial internal variable: I. General formalism

In this paper we consider the propagation of linear transverse acoustic waves in isotropic media in which mechanical relaxation phenomena occur. It is assumed that the irreversible mechanical processes in the medium are due to viscosity and to changes in a tensorial internal variable and that these processes can be described with the aid of non-equilibrium thermodynamics. The viscous flow phenomenon is analogous to the viscous flow of ordinary fluids. In particular we investigate the velocity and attenuation of the waves and we consider the limiting cases of waves high and low frequencies. For high frequencies we obtain expressions for the phase velocity and attenuation which are analogous to those for viscous fluids and it is seen that the tensorial internal variables does not influence the propagation of waves. If the frequencies are sufficiently low, the expression for the phase velocity is analogous to the expression for the phase velocity in purely elastic media, while the damping is influenced by both irreversible phenomena and the imaginary part of the complex wave number is proportional to the square of the frequency.

A thermodynamic derivation of the stress-strain relations for burgers media and related substances

A generalization is given of the author's thermodynamic theory for mechanical phenomena in continuous media. The developments are based on the general methods of non-equilibrium thermodynamics. Temperature effects are fully taken into account. It is assumed that several microscopic phenomena occur which give rise to inelastic strains (for instance, slip, dislocations, etc.). The contributions of these phenomena to the inelastic strain tensor are introduced as internal degrees of freedom in the Gibbs relation. Moreover, it is assumed that a viscous flow phenomenon occurs which is analogous to the viscous flow of ordinary fluids. An explicit form for the entropy production is derived. The phenomenological equations (Fourier's law and generalizations of Lévy's law and of Newton's law for viscous fluid flow) are given, and the Onsager-Casimir reciprocity relations are formulated. It follows from the theory that several types of (macroscopic) stress fields may occur in a medium: A stress field τ eq αβ which is of a thermoelastic nature, a stress fields τ vi αβ which is analogous to the viscous stresses in ordinary fluids, and stress fields τ k mαβ which are probably connected with the microscopic stress fields surrounding imperfections in the medium. The stress field τ eq αβ + τ vi αβ is the mechanical stress field which occurs in the equations of motion and in the first law of thermodynamics, and stress fields of the type τ eq αβ + τ k mαβ play the role of thermodynamic affinities in the phenomenological equations which are generalizations of Lévy's law. If the equations of state may be linearized (for example, Hooke's law and the Duhamel-Neumann law), and if the phenomenological coefficients may be regarded as constants, an explicit form for the stress-strain relation may be derived. In this case the relation for distortional phenomena in isotropic media has the form of a linear relation among the deviators of the mechanical stress tensor, the first n derivatives with respect to time of this tensor, the tensor of total strain (the sun of the elastic and inelastic strains), and the first n + 1 derivatives with respect to time of the tensor of total strain, where n is the number of phenomena that give rise to inelastic deformations. The well-known Burgers equation is a special case of this relation if n = 2. Moreover, the stress-strain relations for ordinary viscous fluids, for thermoelastic media, and for Maxwell, Kelvin, Jeffreys, and Poynting-Thomson media are also special cases of the more general relation mentioned above. In case the equations of state may be linearized explicit expressions are given for the free energy, the internal energy, and the entropy, both for isotropic and anisotropic media. If it is not permissible to linearize the equations of state and/or to regard the phenomenological coefficients as constants, the stress-strain relation is of a very complicated nature. Plasticity phenomena are left out of consideration.

Certain reducible helical hydromagnetic flows

Intrinsic formulations are obtained of the equations governing three-dimensional steady flows of an inviscid magnetic fluid with infinite electrical conductivity both when the streamlines of the magnetic field H and the fluid velocity v everywhere have the same unit tangent, and when H lies in the normal plane to the fluid streamlines at every point, so that v×H=0 and v·H=0 respectively. Using these intrinsic equations certain results concerning hydromagnetic helical flows are derived, and particular classes of such flows are reduced (in the sense of Power and Walker1)) to associated non-magnetic two-dimensional flows. Finally, reciprocal relations are used to link classes of hydromagnetic helical flows to four-parameter classes of ordinary fluid flows.

Flow of plasma around the Earth

The interplanetary plasma flow measured by means of Explorer 10 is supersonic in the sense that the flow speed is greater than the Alfven speed. It is shown that this has some of the same implications as supersonic flow does in ordinary fluid flow theory, and possible consequences are discussed. The position of the shock that would develop if the interplanetary plasma were a perfect gas with a short mean free path is calculated and is shown to be well outside the orbit of Explorer 10, so that it would not have been seen. Speculations on the possible structure of the shock in collisionless plasma are given, and it is shown that this structure may account for the turbulent character of the distant magnetic field. (auth)

Experiments on laminar and turbulent flow of He II in wide capillaries

Laminar and turbulent flow of pure 4He through wide capillaries of circular cross section, respectively 82μ, 174 μ and 255 μ in diameter, have been studied in the He II region above 1°K. The viscosity of the normal fluid has been derived from the laminar flow behaviour using a special technique. It was found that the Reynolds number Re = d ϱυ n η n , containing the total density ϱ, the normal fluid velocity υ n, the viscosity η n and the diameter of the capillary d, is a parameter characterising the flow in the whole velocity range. One of the fundamental equations of motion in the high velocity range appeared to be the Navier Stokes equation containing the total density, the normal fluid velocity and the viscosity of the normal fluid. The dependence of the pressure gradient on the normal fluid velocity is then represented by an equation analogous to the classical empirical formula for turbulent flow of ordinary fluids as given by Blasius. It was found that within the experimental accuracy direction and magnitude of the superfluid velocity did not affect the pressure gradient. In connection with the phenomena of turbulence, the various results on gravitational flow are discussed. Finally an experiment on turbulent flow of 3He- 4He mixtures in the He II region, using a capillary 156μ in diameter, is communicated.

Pressure Loss in the Pipe Bends, in which Granular Solids are Transported

The authors have examined experimentally the pressure loss in the pipe bends, in which granular solids are transported. In order to investigate the pressure losses an assumption has been made that the total pressure loss consists of the losees in the flow of ordinary fluids through a pipe bend and those due to solids. It has been shown that the former is presented by the known formula and the authors have obtained the tact that the pressure losses due to solids increases in proportion to the weight ratio of solids to air in flow. Moreover the flow of solids in glass pipe bends has been observed. The observation has helped us in taking account of the effect of solids flow upon the pressure losses due to solids.

Pressure Loss in the Pneumatic Conveyance of Granular Solids

The authors examined experimentally the pressure losses in the flow of suspensions of granular solids through a straight pipe. In order to investigate the pressure losses an assumption was made that the total pressure drop consisted of friction losses in the flow of ordinary fluids through a pipe and those due to solids. It was shown that the former was presented by the Blasius formula λ=0.316 4/Rε1/4 and the latter was expressed by the new formula that the authors had obtained here. It is given as λ8=8/(√(8/λ)-1.72μ)2-λ and is independent of the characteristics of solids. By this formula the losses due to solids are easily obtained, if only μ and λ are known. The effect of solids upon the pressure losses is considered by introducing both the weight ratio of solids to air in flow and the friction velocity, in which the characteristic of the method of present analysis lies.