General Self-similar Solution Research Articles

Axisymmetric stagnation-point flow and heat transfer of nanofluid impinging on a cylinder with constant wall heat flux

The steady-state, viscous flow and heat transfer of nanofluid in the vicinity of an axisymmetric stagnation point of a stationary cylinder with constant wall heat flux is investigated. The impinging free-stream is steady and with a constant strain rate, k ?. Exact solution of the Navier-Stokes equations and energy equation are derived in this problem. A reduction of these equations is obtained by use of appropriate transformations introduced in this research. The general self-similar solution is obtained when the wall heat flux of the cylinder is constant. All the previous solutions are presented for Reynolds number Re = k ?a2/2n f ranging from 0.1 to 1000, selected values of heat flux and selected values of particle fractions where a is cylinder radius and n f is kinematic viscosity of the base fluid. For all Reynolds numbers, as the particle fraction increases, the depth of diffusion of the fluid velocity field in radial direction, the depth of the diffusion of the fluid velocity field in z-direction, shear-stresses and pressure function decreases. However, the depth of diffusion of the thermal boundary-layer increases. It is clear by adding nanoparticles to the base fluid there is a significant enhancement in Nusselt number and heat transfer.

General self-similar solution for expansion of non-Maxwellian plasmas

In this work, self-similar expansion of one-dimensional collisionless plasma into a vacuum has been calculated for non-Maxwellian particles. It is known that, during the expansion, the initial Maxwellian electron velocity distribution function (DF) is deformed. In this paper, this deformation is illustrated by the Vlasov simulation of plasma expansion. A part of deformation is associated with the appearance of high energy (super thermal) electrons. Therefore, a more complete and realistic plasma model has to include the effect of the super thermal electrons. It is found that the resulting DF of simulation has good consistency with the generalized Lorentzian (kappa) DF. Then the self-similar calculations were carried out for cold ions while the electrons having Lorentzian DF. Finally, it was found that, by increasing the population of energetic electrons, the expansion takes place faster and the ions are accelerated to higher energies.

Axisymmetric Stagnation Flow and Heat Transfer of a Compressible Fluid Impinging on a Cylinder Moving Axially

The steady-state viscous flow and also heat transfer in the vicinity of an axisymmetric stagnation point on a cylinder moving axially with a constant velocity are investigated. Here, fluid with temperature-dependent density is considered. The impinging freestream is steady and with a constant strain rate (strength) k¯. An exact solution of the Navier–Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by use of appropriate transformations. The general self-similar solution is obtained when the wall temperature of the cylinder or its wall heat flux is constant. All the solutions above are presented for Reynolds numbers, Re=k¯a2/2υ, ranging from 0.1 to 1000, low Mach number, selected values of compressibility factor, and different values of Prandtl numbers where a is cylinder radius and υ is kinematic viscosity of the fluid. Shear stress is presented as well. Axial movement of the cylinder does not have any effect on heat transfer but its increase increases the axial component of fluid velocity field and the shear stress.

Streamrate stagnation-point flow of a nanofluid on a stationary cylinder

The steady-state, viscous flow of Nanofluid in the vicinity of an axisymmetric stagnation point of a stationary cylinder is investigated. The impinging free-stream is steady and with a constant strain rate . Exact solution of the Navier–Stokes equations is derived in this problem. A reduction of these equations is obtained by use of appropriate transformations introduced in this research. The general self-similar solution is obtained when the wall temperature of the cylinder is constant. All the solutions above are presented for Reynolds numbers ranging from 0.1 to 1000 and selected values of particle. For all Reynolds numbers, as the particle fraction increases, the depth of diffusion of the fluid velocity field in radial direction, the depth of the diffusion of the fluid velocity field in -direction, shear-stresses and pressure function decreases.

Axisymmetric stagnation-point flow and heat transfer of a viscous, compressible fluid on a cylinder with constant heat flux

Existing solutions of the problem of axisymmetric stagnation-point flow and heat transfer on either a cylinder or flat plate are for incompressible fluid. Here, fluid with temperature dependent density is considered in the problem of axisymmetric stagnation-point flow and heat transfer on a cylinder with constant heat flux. The impinging free stream is steady and with a constant strain rate, k̄. An exact solution of the Navier–Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by use of appropriate transformations introduced for the first time. The general self-similar solution is obtained when the wall heat flux of the cylinder is constant. All the solutions above are presented for Reynolds numbers, Re=k̄a2/2υ, ranging from 0.01 to 1000, selected values of compressibility factors, and different values of Prandtl number, where a is cylinder radius and ν is the kinematic viscosity of the fluid. For all Reynolds numbers and surface heat flux, as the compressibility factor increases, both components of the velocity field, the heat transfer coefficient and the shear-stresses increase, and the pressure function decreases.

Axisymmetric stagnation-point flow and heat transfer of a viscous fluid on a moving plate with time-dependent axial velocity and uniform transpiration

The unsteady viscous flow and heat transfer in the vicinity of an axisymmetric stagnation point of an infinite moving plate with transpiration W0 are investigated when the axial velocity and wall temperature vary arbitrarily with time. The free stream is steady and with a strain rate of a. An exact solution of the Navier–Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by the use of appropriate transformations for the most general case when the transpiration rate is also time-dependent, but results are presented only for uniform values of this quantity. The general self-similar solution is obtained when the axial velocity of the plate and its wall temperature vary as specified time-dependent functions. For completeness, sample semi-similar solutions of the unsteady Navier–Stokes equations have been obtained numerically using a finite difference scheme. All the solutions above are presented for different values of dimensionless transpiration rate, S=W0/aυ, where υ is the kinematic viscosity of the fluid. The effects of the sundry parameters, including transpiration rate, Prandtl number, oscillation frequency and accelerating/decelerating parameter, on the velocity and temperature profiles, as well as surface shear stresses and heat transfer coefficient, are investigated and results are shown through graphs.

Mechanism of lightning discharge

A simple model describing lightning on the basis of the conventional equations of charged-gas hydrodynamics is proposed. In the 1D case, a general self-similar solution to the nonlinear equation is obtained. It is demonstrated that the uniform electric field and charge density distributions in a cloud are unstable. A virtually periodic structure of the field and charge densities is formed owing to their spatial redistribution. The theoretical results are in good agreement with the available data.

Axisymmetric Stagnation—Point Flow and Heat Transfer of a Viscous Fluid on a Rotating Cylinder With Time-Dependent Angular Velocity and Uniform Transpiration

The unsteady viscous flow and heat transfer in the vicinity of an axisymmetric stagnation point of an infinite rotating circular cylinder with transpiration U0 are investigated when the angular velocity and wall temperature or wall heat flux all vary arbitrarily with time. The free stream is steady and with a strain rate of Γ. An exact solution of the Navier-Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by the use of appropriate transformations for the most general case when the transpiration rate is also time-dependent but results are presented only for uniform values of this quantity. The general self-similar solution is obtained when the angular velocity of the cylinder and its wall temperature or its wall heat flux vary as specified time-dependent functions. In particular, the cylinder may rotate with constant speed, with exponentially increasing/decreasing angular velocity, with harmonically varying rotation speed, or with accelerating/decelerating oscillatory angular speed. For self-similar flow, the surface temperature or its surface heat flux must have the same types of behavior as the cylinder motion. For completeness, sample semi-similar solutions of the unsteady Navier-Stokes equations have been obtained numerically using a finite-difference scheme. Some of these solutions are presented for special cases when the time-dependent rotation velocity of the cylinder is, for example, a step-function. All the solutions above are presented for Reynolds numbers, Re=Γa2∕2υ, ranging from 0.1 to 1000 for different values of Prandtl number and for selected values of dimensionless transpiration rate, S=U0∕Γa, where a is cylinder radius and υ is kinematic viscosity of the fluid. Dimensionless shear stresses corresponding to all the cases increase with the increase of Reynolds number and suction rate. The maximum value of the shear stress increases with increasing oscillation frequency and amplitude. An interesting result is obtained in which a cylinder rotating with certain exponential angular velocity function and at particular value of Reynolds number is azimuthally stress-free. Heat transfer is independent of cylinder rotation and its coefficient increases with the increasing suction rate, Reynolds number, and Prandtl number. Interesting means of cooling and heating processes of cylinder surface are obtained using different rates of transpiration.

Self-similar Bianchi models: II. Class B models

In a companion article (referred to hereafter as paper I) a detailed study of the simply transitive spatially homogeneous (SH) models of class A concerning the existence of a simply transitive similarity group has been given. The present work (paper II) continues and completes the above study by considering the remaining set of class B models. Following the procedure of paper I we find all SH models of class B subjected only to the minimal geometric assumption to admit a proper homothetic vector field (HVF). The physical implications of the obtained geometric results are studied by specializing our considerations to the case of vacuum and γ-law perfect fluid models. As a result, we regain all the known exact solutions regarding vacuum and non-tilted perfect fluid models. In the case of tilted fluids, we find the general self-similar solution for the exceptional type VI−1/9 model and we identify it as an equilibrium point in the corresponding dynamical state space. It is found that this new exact solution belongs to the subclass of models nαα = 0, is defined for , and although it has a five-dimensional stable manifold there always exist two unstable modes in the restricted state space. Furthermore, the analysis of the remaining types guarantees that tilted perfect fluid models of types III, IV, V and VIIh cannot admit a proper HVF, strongly suggesting that these models either may not be asymptotically self-similar (type V) or may be extreme tilted at late times. Finally, for each Bianchi type, we give the extreme tilted equilibrium points of their state space.

Axisymmetric Stagnation-Point Flow and Heat Transfer of a Viscous Fluid on a Moving Cylinder With Time-Dependent Axial Velocity and Uniform Transpiration

The unsteady viscous flow and heat transfer in the vicinity of an axisymmetric stagnation point of an infinite moving cylinder with time-dependent axial velocity and with uniform normal transpiration Uo are investigated. The impinging free stream is steady and with a constant strain rate k¯. An exact solution of the Navier–Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by use of appropriate transformations for the most general case when the transpiration rate is also time-dependent but results are presented only for uniform values of this quantity. The general self-similar solution is obtained when the axial velocity of the cylinder and its wall temperature or its wall heat flux vary as specified time-dependent functions. In particular, the cylinder may move with constant speed, with exponentially increasing–decreasing axial velocity, with harmonically varying axial speed, or with accelerating–decelerating oscillatory axial speed. For self-similar flow, the surface temperature or its surface heat flux must have the same types of behavior as the cylinder motion. For completeness, sample semisimilar solutions of the unsteady Navier–Stokes and energy equations have been obtained numerically using a finite-difference scheme. Some of these solutions are presented for special cases when the time-dependent axial velocity of the cylinder is a step-function, and a ramp function. All the solutions above are presented for Reynolds numbers, Re=ka¯2/2υ, ranging from 0.1 to 100 for different values of dimensionless transpiration rate, S=Uo/ka¯, where a is cylinder radius and υ is kinematic viscosity of the fluid. Absolute value of the shear-stresses corresponding to all the cases increase with the increase of Reynolds number and suction rate. The maximum value of the shear- stress increases with increasing oscillation frequency and amplitude. An interesting result is obtained in which a cylinder moving with certain exponential axial velocity function at any particular value of Reynolds number and suction rate is axially stress-free. The heat transfer coefficient increases with the increasing suction rate, Reynolds number, Prandtl number, oscillation frequency and amplitude. Interesting means of cooling and heating processes of cylinder surface are obtained using different rates of transpiration. It is shown that a cylinder with certain type of exponential wall temperature exposed to a temperature difference has no heat transfer.

A model for quasi-spherical magnetized accretion flow

A model for axisymmetric magnetized accretion flow is proposed. The dominant mechanism of energy dissipation is assumed to be the magnetic diffusivity due to turbulence in the accretion flow. In analogy to the advection-dominated accretion flow (ADAF) solutions, a constant fraction of the resistively dissipated energy is stored in the accreting gas and the rest is radi- ated. We first introduce the general self-similar solutions which describe a resistive and nonrotating flow with purely poloidal magnetic field. The radial dependence of physical quantities is identical to that in viscous ADAF solutions. Although the main focus of this study is on nonrotating magnetized accretion flow, for rotating flow with both poloidal and toroidal components of magnetic field we find a radial scaling of solutions similar to the nonrotating case. We show that the accretion and the rotation velocities are both below the Keplerian rate, irrespective of the amount of cooling. We show that the set of equations is reduced to one second order differential equation for a nonrotating flow. The geometrical shape of the disk changes depending on the fraction of the resistively dissipated energy which is stored in the accreting gas. However, there is a hot low-density gas above the disk in almost all cases. The net accretion rate is calculated for a set of illustrative parameters.

Asymptotic properties of solutions of the viscous Hamilton-Jacobi equation

The purpose of the paper is to study properties of solutions of the Cauchy problem for the equation \( u_t-\Delta u + |\nabla u|^q=0 \) under the assumption \( (n+2)/(n+1)< q < 2 \). General selfsimilar solutions are constructed. Moreover, for initial data with some decay at infinity, we determine the leading term of the asymptotics of solutions in \( L^p(\mathbb{R}^n) \) which is described by either solutions of the linear heat equation or by particular selfsimilar solutions of the original equation.

Systems of coupled Riccati equations in discrete kinetic theory: New self-similar solutions, shock profiles, stability

Abstract In the kinetic theory of the discrete Boltzmann models (DBMs) the standard similarity shock waves solutions are, like the Broadwell shock waves, solutions of scalar Riccati equations. The corresponding microscopic densities, as well as the macroscopic conservative quantities: mass, momentum and energy, are monotonic. In the first part of the paper we construct new exact similarity shock waves solutions of systems of coupled Riccati equations, which arise for the DBMs with n conservation laws and n + k independent densities (here with n = 3, k = 2). The former standard shock waves are rational functions of two linear polynomials with an exponential variable while for the new waves the polynomials are quadratic and consequently nonmonotonic behaviours are possible. We stress that from the mathematical point of view we enter into a new domain of exact self-similar solutions associated to the systems of coupled Riccati equations. Due to the fact that, for the considered models, there exists a linear differential relation between two microscopic densities we obtain two different classes of solutions. For one class these two densities are necessarily of the former standard type while for the other all densities are of the new type of self-similar solutions. Furthermore we prove an important result: Let us consider microscopic densities associated to opposite velocities along the shock-axis. To any solution, there exists another one, called partner solution, such that those densities are exchanged with opposite similarity variable. This means that from any solution with mass, energy and pressure given, there exists a partner solution where these macroscopic quantities have opposite similarity variable. The new exact solutions are not the most general self-similar solutions of the considered models. In the second part we reduce the problem to the dynamical system of the coupled Riccati equations, with the limit conditions corresponding to the equilibrium states on both sides of the shocks. We analyse the structure of singularities corresponding to the equilibria, and obtain new nonmonotonic profiles of the microscopic densities by solving numerically the relevant two-point BVPs. For the new class of models studied in this paper we observe overshoots on one density for the exact solutions. In addition in the second part we find undershoots on one density and/or overshoot on one density accompanied by undershoots on another one. Contrary to the previously studied models, no nonmonotonic effects have been observed for the macroscopic conserved quantities: mass and energy.