Viscosity Effects on the Propagation of Acoustic Transients

2000 ◽  
Author(s):  
G. C. Gaunaurd ◽  
G. C. Everstine
Keyword(s):  
Analytic Solution ◽  
Propagation Speed ◽  
Viscosity Coefficient ◽  
Viscous Medium ◽  
Initial Value ◽  
One Dimensional ◽  
Impulsive Excitation ◽  
Viscosity Effects ◽  
Parabolic Pde ◽  
Acoustic Transients

Abstract The propagation of an impulsive excitation applied at the origin of a lossy viscous medium is studied by operational techniques as the excitation advances through the medium. The solution of the governing partial differential equation (PDE) for such transient propagation problems has been elusive. Such solution is found and quantitatively examined here using a one-dimensional model in space and time. As expected, as the transient advances through space, its amplitude decreases, and its width broadens. Such is the damping effect of viscosity that one would anticipate from elementary considerations in related disciplines such as electrodynamics. Such is also the smoothing-out effect of dispersion. We also obtain an approximate solution of the present boundary-initial value problem based on the method of steepest descents. This approximation agrees with the first term of the complete analytic solution given here. The pertinent dispersion relation associated with the governing parabolic PDE is shown to impose a restrictive condition on the allowable values of the propagation speed and the kinematic viscosity coefficient, thus assuring that propagation with attenuation does take place. Various numerical results illustrate and quantitatively describe the propagation of the transient pulse in several nondimensional graphs.

Viscosity Effects on the Propagation of Acoustic Transients

2001 ◽  
Vol 124 (1) ◽  
pp. 19-25 ◽  
Author(s):  
G. C. Gaunaurd ◽  
G. C. Everstine
Keyword(s):  
Error Function ◽  
Propagation Speed ◽  
Viscosity Coefficient ◽  
Viscous Medium ◽  
Impulsive Excitation ◽  
Viscosity Effects ◽  
Complementary Error Function ◽  
Parabolic Pde ◽  
Acoustic Transients ◽  
Infinite Sum

The propagation of an impulsive excitation applied at the origin of a lossy viscous medium is studied by operational techniques as the excitation advances through the medium. The solution of the governing partial differential equation (PDE) for such transient propagation problems has been elusive. Such solution is found as an infinite sum of properly weighted successive integrals of the complementary error function, and it is quantitatively examined here using a one-dimensional model in space and time. As expected, as the transient advances through space, its amplitude decreases, and its width broadens. Such is the damping effect of viscosity that one would anticipate from elementary considerations in related disciplines such as electrodynamics. Such is also the smoothing-out effect of dispersion. We also obtain an approximate solution of the present boundary-initial value problem based on the method of steepest descents. This approximation agrees with the first term of the complete analytic solution given here. The pertinent dispersion relation associated with the governing parabolic PDE is shown to impose a restrictive condition on the allowable values of the propagation speed and the kinematic viscosity coefficient, thus assuring that propagation with attenuation does take place. Various numerical results illustrate and quantitatively describe the propagation of the transient pulse in several nondimensional graphs.

On the theory of electrohydrodynamically driven capillary jets

1997 ◽  
Vol 335 ◽  
pp. 165-188 ◽  
Author(s):  
ALFONSO M. GAÑÁN-CALVO
Keyword(s):  
Surface Charge ◽  
Electric Fields ◽  
Gas Flow ◽  
Wave Speed ◽  
Propagation Speed ◽  
Field Equations ◽  
Steady Regime ◽  
One Dimensional ◽  
Relaxation Effects ◽  
Capillary Jets

Electrohydrodynamically (EHD) driven capillary jets are analysed in this work in the parametrical limit of negligible charge relaxation effects, i.e. when the electric relaxation time of the liquid is small compared to the hydrodynamic times. This regime can be found in the electrospraying of liquids when Taylor's charged capillary jets are formed in a steady regime. A quasi-one-dimensional EHD model comprising temporal balance equations of mass, momentum, charge, the capillary balance across the surface, and the inner and outer electric fields equations is presented. The steady forms of the temporal equations take into account surface charge convection as well as Ohmic bulk conduction, inner and outer electric field equations, momentum and pressure balances. Other existing models are also compared. The propagation speed of surface disturbances is obtained using classical techniques. It is shown here that, in contrast with previous models, surface charge convection provokes a difference between the upstream and the downstream wave speed values, the upstream wave speed, to some extent, being delayed. Subcritical, supercritical and convectively unstable regions are then identified. The supercritical nature of the microjets emitted from Taylor's cones is highlighted, and the point where the jet switches from a stable to a convectively unstable regime (i.e. where the propagation speed of perturbations become zero) is identified. The electric current carried by those jets is an eigenvalue of the problem, almost independent of the boundary conditions downstream, in an analogous way to the gas flow in convergent–divergent nozzles exiting into very low pressure. The EHD model is applied to an experiment and the relevant physical quantities of the phenomenon are obtained. The EHD hypotheses of the model are then checked and confirmed within the limits of the one-dimensional assumptions.

scholarly journals AN ANALYTIC SOLUTION FOR ONE-DIMENSIONAL DISSIPATIONAL STRAIN-GRADIENT PLASTICITY

2009 ◽  
Vol 50 (3) ◽  
pp. 407-420
Author(s):  
ROGER YOUNG
Keyword(s):  
Boundary Condition ◽  
Strain Gradient ◽  
Analytic Solution ◽  
Strain Gradient Plasticity ◽  
Numerical Results ◽  
Gradient Plasticity ◽  
One Dimensional ◽  
Numerical Result ◽  
Formulation Analysis ◽  
The One

AbstractAn analytic solution is developed for the one-dimensional dissipational slip gradient equation first described by Gurtin [“On the plasticity of single crystals: free energy, microforces, plastic strain-gradients”, J. Mech. Phys. Solids48 (2000) 989–1036] and then investigated numerically by Anand et al. [“A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results”, J. Mech. Phys. Solids53 (2005) 1798–1826]. However we find that the analytic solution is incompatible with the zero-sliprate boundary condition (“clamped boundary condition”) postulated by these authors, and is in fact excluded by the theory. As a consequence the analytic solution agrees with the numerical results except near the boundary. The equation also admits a series of higher mode solutions where the numerical result corresponds to (a particular case of) the fundamental mode. Anand et al. also established that the one-dimensional dissipational gradients strengthen the material, but this proposition only holds if zero-sliprate boundary conditions can be imposed, which we have shown cannot be done. Hence the possibility remains open that dissipational gradient weakening may also occur.

scholarly journals A graphical solution of a differential equation with application to hill’s treatment of nerve excitation

1937 ◽  
Vol 123 (832) ◽  
pp. 382-395 ◽  
Keyword(s):  
Viscous Medium ◽  
Graphical Solution ◽  
Initial Value ◽  
First Order ◽  
Rate Dependent ◽  
Monomolecular Reaction ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Physical And Chemical ◽  
Nerve Excitation

Linear differential equations with constant coefficients are very common in physical and chemical science, and of these, the simplest and most frequently met is the first-order equation a dy / dt + y = f(t) , (1) where a is a constant, and f(t) a single-valued function of t . The equation signifies that the quantity y is removed at a rate proportional to the amount present at each instant, and is simultaneously restored at a rate dependent only upon the instant in question. Familiar examples of this equation are the charging of a condenser, the course of a monomolecular reaction, the movement of a light body in a viscous medium, etc. The solution of this equation is easily shown to be y = e - t / a { y 0 = 1 / a ∫ t 0 e t /a f(t) dt , (2) where y 0 is the initial value of y . In the case where f(t) = 0, this reduces to the well-known exponential decay of y .

Multi-Parameter Optimization of Damped Linear Continuous Systems

1972 ◽  
Vol 94 (1) ◽  
pp. 1-7 ◽  
Author(s):  
O. B. Dale ◽  
R. Cohen
Keyword(s):  
Optimal Parameter ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
System Response ◽  
Continuous Systems ◽  
Frequency Interval ◽  
State Variables ◽  
Initial Value ◽  
Unknown Parameters ◽  
One Dimensional

A method is presented for obtaining and optimizing the frequency response of one-dimensional damped linear continuous systems. The systems considered are assumed to contain unknown constant parameters in the boundary conditions and equations of motion which the designer can vary to obtain a minimum resonant response in some selected frequency interval. The unknown parameters need not be strictly dissipative nor unconstrained. No analytic solutions, either exact or approximate, are required for the system response and only initial value numerical integrations of the state and adjoint differential equations are required to obtain the optimal parameter set. The combinations of state variables comprising the response and the response locations are arbitrary.

scholarly journals How Hyperpolarization and the Recovery of Excitability Affect Propagation through a Virtual Anode in the Heart

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Nicholas P. Charteris ◽  
Bradley J. Roth
Keyword(s):  
Wave Front ◽  
Time Constant ◽  
Computer Model ◽  
Propagation Speed ◽  
One Dimensional ◽  
Upper Limit ◽  
Rapid Propagation ◽  
Upper Limit Of Vulnerability ◽  
Key Factor ◽  
Sodium Inactivation

Researchers have suggested that the fate of a shock-induced wave front at the edge of a “virtual anode” (a region hyperpolarized by the shock) is a key factor determining success or failure during defibrillation of the heart. In this paper, we use a simple one-dimensional computer model to examine propagation speed through a hyperpolarized region. Our goal is to test the hypothesis that rapid propagation through a virtual anode can cause failure of propagation at the edge of the virtual anode. The calculations support this hypothesis and suggest that the time constant of the sodium inactivation gate is an important parameter. These results may be significant in understanding the mechanism of the upper limit of vulnerability.

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