scholarly journals Analysis of the self-similar solutions of a generalized Burger's equation with nonlinear damping

2001 ◽  
Vol 7 (3) ◽  
pp. 253-282 ◽  
Author(s):  
Ch. Srinivasa Rao ◽  
P. L. Sachdev ◽  
Mythily Ramaswamy
Keyword(s):  
Initial Conditions ◽  
Nonlinear Damping ◽  
Nonlinear Ordinary Differential Equation ◽  
The Self ◽  
Similar Reduction ◽  
Generalized Burgers Equation ◽  
Different Types ◽  
Self Similar ◽  
Parameter Ranges ◽  
Similar Solutions

The nonlinear ordinary differential equation resulting from the self-similar reduction of a generalized Burgers equation with nonlinear damping is studied in some detail. Assuming initial conditions at the origin we observe a wide variety of solutions – (positive) single hump, unbounded or those with a finite zero. The existence and nonexistence of positive bounded solutions with different types of decay (exponential or algebraic) to zero at infinity for specific parameter ranges are proved.

Self-similar cylindrical expansion of impurity particles in a plasma

1994 ◽  
Vol 52 (3) ◽  
pp. 345-352 ◽  
Author(s):  
Ming Y. Yu ◽  
R. Bharuthram
Keyword(s):  
Cylindrical Geometry ◽  
The Self ◽  
Dust Particles ◽  
Charged Impurity ◽  
Physical Interest ◽  
Different Types ◽  
Fluid Theory ◽  
Self Similar ◽  
Similar Solutions

The radial self-similar expansion of negatively charged impurity or dust particles in a cylindrical geometry is investigated using the multi-fluid theory. Three different types of self-similar solutions are considered. Various asymptotic limits of physical interest in the self-similar space are shown to occur.

Annular self-similar solutions in ideal magnetogasdynamics

2008 ◽  
Vol 74 (4) ◽  
pp. 531-554 ◽  
Author(s):  
R. M. LOCK ◽  
A. J. MESTEL
Keyword(s):  
Magnetic Field ◽  
Initial Value Problem ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
The Self ◽  
Initial Value ◽  
Azimuthal Magnetic Field ◽  
Self Similar ◽  
Similar Solutions ◽  
Z Pinch

AbstractWe consider the possibility of self-similar solutions describing the implosion of hollow cylindrical annuli driven by an azimuthal magnetic field, in essence a self-similar imploding liner z-pinch. We construct such solutions for gasdynamics, for ideal ‘β=0’ plasma and for ideal magnetogasdynamics (MGD). In the latter two cases some quantities are singular at the annular boundaries. Numerical solutions of the full ideal MGD initial value problem indicate that the self-similar solutions are not attractive for arbitrary initial conditions, possibly as a result of flux-freezing.

Freely draining gravity currents in porous media: Dipole self-similar solutions with and without capillary retention

2007 ◽  
Vol 18 (3) ◽  
pp. 337-362 ◽  
Author(s):  
JOCHONIA S. MATHUNJWA ◽  
ANDREW J. HOGG
Keyword(s):  
Unsaturated Flow ◽  
Multiple Scales ◽  
Initial Conditions ◽  
The Self ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Theoretical Predictions ◽  
Analytic Expressions ◽  
Self Similar ◽  
Similar Solutions

We analyse the two-dimensional, gravitationally-driven spreading of fluid through a porous medium overlying a horizontal impermeable boundary from which fluid can drain freely at one end. Under the assumption that none of the intruding fluid is retained within the pores in the trail of the current, the motion of the current is described by the dipole self-similar solution of the first kind derived by Barenblatt and Zel'dovich (1957). We show that small perturbations of arbitrary shape imposed on this solution decay in time, indicating that the self-similar solution is linearly stable. We use the connection between the perturbation eigenfunctions and symmetry transformations of the self-similar solution to demonstrate that variables can always be specified in terms of which the rate of decay of the perturbations is maximised. Unsaturated flow can be modelled by assuming that a constant fraction of the fluid is retained within the pores by capillary action in the trail of the current. It has been shown (Barenblatt and Zel'dovich, 1998; Ingerman and Shvets, 1999) that in this case, the motion of the current is described by a self-similar solution of the second kind characterised by an anomalous exponent. We derive leading-order analytic expressions for the anomalous exponent and the self-similar quantities valid for small values of the fraction of fluid retained using direct asymptotic analysis and by using a novel application of the method of multiple scales. The latter offers a number of advantages and permits the evolution of the current to be clearly connected with its initial conditions in a way not possible with conventional approaches. We demonstrate that the theoretical predictions provided by these expressions are in excellent agreement with results from the numerical integration of the governing equations.

On a self-similar solution for the decay of turbulent bursts

1992 ◽  
Vol 3 (4) ◽  
pp. 319-341 ◽  
Author(s):  
S. P. Hastings ◽  
L. A. Peletier
Keyword(s):  
Asymptotic Behaviour ◽  
Physical Parameter ◽  
Dimensional Analysis ◽  
Existence And Uniqueness ◽  
The Self ◽  
Similar Solution ◽  
Self Similar Solution ◽  
Eigenvalue Parameter ◽  
Self Similar ◽  
Similar Solutions

We discuss the self-similar solutions of the second kind associated with the propagation of turbulent bursts in a fluid at rest. Such solutions involve an eigenvalue parameter μ, which cannot be determined from dimensional analysis. Existence and uniqueness are established and the dependence of μ on a physical parameter λ in the problem is studied: estimates are obtained and the asymptotic behaviour as λ → ∞ is established.

Natural convection flow far from a horizontal plate

1999 ◽  
Vol 387 ◽  
pp. 227-254 ◽  
Author(s):  
VALOD NOSHADI ◽  
WILHELM SCHNEIDER
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
The Self ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Self Similar ◽  
Similar Solutions

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

scholarly journals Theoretical interpretation of fetch limited wind-drivensea observations

2005 ◽  
Vol 12 (6) ◽  
pp. 1011-1020 ◽  
Author(s):  
V. E. Zakharov
Keyword(s):  
Gravity Waves ◽  
Wave Breaking ◽  
Field Studies ◽  
The Self ◽  
Theoretical Interpretation ◽  
Simple Analytical Model ◽  
Surface Gravity Waves ◽  
Interaction Terms ◽  
Self Similar ◽  
Similar Solutions

Abstract. We show that the results of major fetch limited field studies of wind-generated surface gravity waves on deep water can be explained in the framework of simple analytical model. The spectra measured in these experiments are described by self-similar solutions of ``conservative" Hasselmann equation that includes only advective and nonlinear interaction terms. Interaction with the wind and dissipation due to the wave breaking indirectly defines parameters of the self-similar solutions.

scholarly journals ACCRETION AND PLASMA OUTFLOW FROM DISSIPATIONLESS DISCS

2010 ◽  
Vol 19 (03) ◽  
pp. 339-365 ◽  
Author(s):  
S. V. BOGOVALOV ◽  
S. R. KELNER
Keyword(s):  
Angular Momentum ◽  
Similar Case ◽  
Gravitational Energy ◽  
Accretion Disc ◽  
The Self ◽  
Low Viscosity ◽  
Variable Polarity ◽  
Self Similar ◽  
Similar Solutions ◽  
High Electric Conductivity

We consider the specific case of disc accretion for negligibly low viscosity and infinitely high electric conductivity. The key component in this model is the outflowing magnetized wind from the accretion disc, since this wind effectively carries away angular momentum of the accreting matter. Assuming magnetic field has variable polarity in the disc (to avoid magnetic flux and energy accumulation at the gravitational center), this leads to radiatively inefficient accretion of the disc matter onto the gravitational center. In such a case, the wind forms an outflow, which carries away all the energy and angular momentum of the accreted matter. Interestingly, in this framework, the basic properties of the outflow (as well as angular momentum and energy flux per particle in the outflow) do not depend on the structure of accretion disc. The self-similar solutions obtained prove the existence of such an accreting regime. In the self-similar case, the disc accretion rate (Ṁ) depends on the distance to the gravitational center, r, as [Formula: see text], where λ is the dimensionless Alfvenic radius. Thus, the outflow predominantly occurs from the very central part of the disc provided that λ ≫ 1 (it follows from the conservation of matter). The accretion/outflow mechanism provides transformation of the gravitational energy from the accreted matter into the energy of the outflowing wind with efficiency close to 100%. The flow velocity can essentially exceed the Kepler velocity at the site of the wind launch.

scholarly journals Tree crowns grow into self-similar shapes controlled by gravity and light sensing

2018 ◽  
Vol 15 (142) ◽  
pp. 20170976 ◽  
Author(s):  
Laurent Duchemin ◽  
Christophe Eloy ◽  
Eric Badel ◽  
Bruno Moulia
Keyword(s):  
Tree Growth ◽  
Initial Conditions ◽  
Tree Crown ◽  
Tree Crowns ◽  
Light Sensing ◽  
The Core ◽  
Long Period ◽  
Self Similar ◽  
Similar Solutions ◽  
Front Model

Plants have developed different tropisms: in particular, they reorient the growth of their branches towards the light (phototropism) or upwards (gravitropism). How these tropisms affect the shape of a tree crown remains unanswered. We address this question by developing a propagating front model of tree growth. Being length-free, this model leads to self-similar solutions after a long period of time, which are independent of the initial conditions. Varying the intensities of each tropism, different self-similar shapes emerge, including singular ones. Interestingly, these shapes bear similarities to existing tree species. It is concluded that the core of specific crown shapes in trees relies on the balance between tropisms.

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