Frequency response function of burning solid propellants

Summary Burning stability analyses of solid propellants are performed, within the framework of a thermal theory of monodimensional heterogeneous deflagration waves, from a linearized point of view. The frequency response function is determined both for pressure and radiation driven combustion waves. Two limiting schemes for the quasi-steady gas phase are implemented: distributed and sharp flame models. Work available from the literature in this field is reviewed, corrected, and extended. Particular attention is given to the connection between pressure and radiation driven admittance functions. Burning stability properties predicted from linear and nonlinear theories are compared. A way to rate experimentally the dynamic burning stability of condensed substances with optically opaque condensed phase and transparent flame is suggested.

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Sommario Vengono svolte analisi lineari della stabilità di combustione di propellenti solidi per endoreattori, nell'ambito di una teoria termica di onde eterogenee di deflagrazione monodimensionale. Si determina la funzione di risposta in frequenza, o ammettenza acustica, per onde di combustione pilotate sia dalla pressione sia da un fascio di luce monocromatica. Sono discussi due casi limite di modelli di fiamma per la fase gassosa supposta quasi-stazionaria: fiamma distribuita e fiamma concentrata. I lavori svolti in precedenza nel settore in esame sono rivisti, corretti ed estesi. Particolare attenzione viene dedicata alla corrispondenza tra la funzione di ammettenza pilotata dalla pressione e la funzione di ammettenza pilotata da radiazione. Sono confrontate le proprietà di stabilità della combustione predette dalle teorie lineari e dalle teorie non lineari. Viene suggerito un possibile criterio per classificare sperimentalmente la stabilità dinamica di combustione di sostanze condensate con fase condensata otticamente opaca e fiamma trasparente.

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Paper presented at the V Congresso Nazionale dell'Associazione Italiana Di Aeronautica ed Astronautica (AIDAA), Milano, 22 – 26 October 1979. Project supported by Centro di Studio per Ricerche sulla Propulsione e sulla Energetica. Milano, 24 September 1979.     

Effects of magnetohydrodynamic flow past a vertical plate with variable surface temperature

An analysis is performed to study the magnetohydrodynamic flow of an electrically conducting, viscous incompressible fluid past a semi-infinite vertical plate with variable surface temperature under the action of transversely applied magnetic field. The heat due to viscous dissipation and the induced magnetic field are assumed to be negligible. The dimensionless governing equations are unsteady, two-dimensional, coupled and non-linear governing equations. It is found that the magnetic field parameter has a retarding effect on the velocities of air and water.     

Effect of variable viscosity on boundary layer flow along a continuously moving plate with variable surface temperature

Flow of an incompressible viscous fluid past a continuously moving semi-infinite plate is studied by taking into account variable viscosity and variable temperature. Velocity and temperature profiles are shown graphically whereas the numerical values of the skin-friction and the rate of heat transfer are listed in a table. The effect of different parameters on the flow field is discussed.     

Heat transfer to a continuous moving flat surface with variable wall temperature

Transient heat transfer from a continuous moving flat surface with varying wall temperature is studied. Numerical results are presented for the transient temperature profiles and heat transfer rates from the wall for Prandtl numbers varying from 0.01 to 1000. Asymptotic solutions for steady state heat transfer rates for large Prandtl number are also presented.     

Mixed convection flow of second grade fluid along a vertical stretching flat surface with variable surface temperature

In the present study we have explored the effects of thermal buoyancy on flow of a viscoelastic second grade fluid past a vertical, continuous stretching sheet of which the velocity and temperature distributions are assumed to vary according to a power-law form. The governing differential equations are transformed into dimensionless form using appropriate transformations and then solved numerically. The methods here employed are (1) the perturbation method together with the Shanks transformation, (2) the local non-similarity method with second level of truncation and (3) the implicit finite difference method for values of ξ ( = Gr _x /Re _x ², defined as local mixed convection parameter) ranging in [0, 10]. The comparison between the solutions obtained by the aforementioned methods found in excellent agreement. Effects of the elasticity parameter λ on the skin-friction and heat transfer coefficients have been shown graphically for the fluids having the values of the Prandtl number equal to 0.72, 7.03 and 15.0. Effects of the viscoelastic parameter and the mixed convection parameter, ξ, on the temperature and velocity fields have also been studied. We notice that with the increase in visco-elastic parameter λ, velocity decreases whereas temperature increases and that velocity gradient is higher than that of temperature. On leave of absence from the Department of Mathematics, University of Dhaka, Bangladesh.     

Transient natural convection along a vertical clyinder with variable surface temperature and mass diffusion

A numerical study is carried out for thermal and concentration driven transient natural convection adjacent to a vertical cylinder. The temperature and concentration level at the cylinder surface are assumed to vary as power-law type functions, with exponents n and m respectively in the streamwise co-ordinate. The governing boundary layer equations are converted into a non-dimensional form. A Crank-Nicolson type of implicit finite-difference method is used to solve the governing non-linear set of equations. Numerical results are obtained and presented with various thermal and mass Grashof numbers and power law variations. Transient effects of velocity, temperature and concentration are analyzed. Local and average skin-friction, Nusselt number and Sherwood number are shown graphically.     

Study of the extinction of gunpowder in the combustion model with a variable surface temperature

The results of a calculation of the rate of transient combustion of gunpowder during a fall in pressure are presented; these are obtained by the numerical integration of the equations of transient-combustion theory, allowing for the variable surface temperature of the k phase. For rapid and severe pressure drops extinction always occurs, no introduction of special extinction conditions being required. The change in the rate of burning during the extinction process is of a smooth nature.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 92–100, May–June, 1973.     

Flow past an impulsively started vertical plate with variable surface temperature and mass flux

Finite-difference solution of the transient natural convection flow of an incompressible viscous fluid past an impulsively started semi-infinite vertical plate with variable surface temperature and mass flux is presented here. The Velocity profiles are compared with exact solution and are found to be in good agreement. The steady-state velocity, temperature and concentration profiles are shown graphically. It is observed that there is a rise in the velocity due to the presence of a mass diffusion. The local as well as average skin-friction, Nusselt number and Sherwood number are shown graphically. Received on 27 May 1998     

Viscous dissipative fluid flow past a semi‐infinite vertical plate with variable surface temperature

An analysis of the effect of viscous dissipative heat on two‐dimensional viscous incompressible fluid flow past a semi‐infinite vertical plate with variable surface temperature is carried out. The dimensionless governing equations are unsteady, two‐dimensional, coupled, and non‐linear governing equations. A most accurate, unconditionally stable and fast converging implicit finite‐difference scheme is used to solve the non‐dimensional governing equations. Velocity and temperature of the flow have been presented graphically for various parameters occurring in the problem. The local and average skin friction and Nusselt number are also shown graphically. It is observed that greater viscous dissipative heat causes a rise in the temperature. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite difference solution of unsteady natural convection boundary layer flow over an inclined plate with variable surface temperature

The unsteady natural convection boundary layer flow over a semi-infinite inclined plate is considered with the wall temperatureT _w ,(x) (=T +ax ⁿ )varying as the power of the axial coordinate. The governing equations are solved by an implicit finite difference scheme of Crank-Nicolson type. Numerical results are obtained for different values of Prandtl number, Grashof number and n for different angles of inclination. The steadystate velocity and temperature profiles, local and average skin frictions and Nusselt numbers are shown graphically. The effects of the angle of inclination and exponent n on velocity and temperature profiles, skin friction and Nusselt number have been discussed. The velocity, temperature and Nusselt number of the present study are compared with the available results and a good agreement is found to exist between the two.

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Differenzlösung für nichtstationäre natürliche Grenzschicht-Konvektionsströmung an einer geneigten Platte mit veränderlicher Oberflächentemperatur

Zusammenfassung Die nichtstationäre natürliche Grenzschicht-Konvektionsströmung an einer halbunendlichen geneigten Platte wird unter Zugrundelegung der GesetzmäßigkeitT _w ,(x) (=T +ax ⁿ für die Wandtemperatur als Funktion der Achsialkoordinate untersucht, und zwar mit Hilfe eines impliziten Differenzverfahrens vom Crank-Nicolson Typ und bei Variation der Prandtl- und Grashof-Zahlen, des Exponenten n und des Neigungswinkels. Graphisch dargestellt sind die Geschwindigkeits-und Temperaturprofile im stationären Zustand, die örtlichen und gemittelten Reibungsbeiwerte und der Nusselt-Zahlen. Der Einfluß des Neigungswinkels und des Exponenten n auf diese Größe wird diskutiert. Im Vergleich mit den Ergebnissen aus anderen Arbeiten konnte gute Übereinstimmung festgestellt werden.

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Nomenclature g acceleration due to gravity - Gr _L Grashof number at x=L - L length of the plate - n exponent in the power law variation of the wall temperature - Nu _x local Nusselt number - Nu _X dimensionless local Nusselt number - average Nusselt number - dimensionless average Nusselt number - p pressure - Pr Prandtl number - t time - t dimensionless time - T temperature - Tw temperature on the plate - T dimensionless temperature - u x-velocity component - U dimensionlessX-velocity component - v y-velocity component - V dimensionlessY-velocity component - x spatial coordinate along the plate - X dimensionless spatial coordinate along the plate - y spatial coordinate normal to the plate - Y dimensionless spatial coordinate normal to the plate Greek symbols thermal diffusivity - ß volumetric coefficient of thermal expansion - t dimensionless time-step - X dimensionless finite difference grid spacing in theX-direction - Y dimensionless finite difference grid spacing in theY-direction - angle of inclination of plate with horizontal - kinematic viscosity - density - _x local skin friction - _X dimensionless local skin friction - average skin friction - dimensionless average skin friction     

Steady nonlinear hydromagnetic flow and heat transfer over a stretching surface of variable temperature

The steady nonlinear hydromagnetic flow of an incompressible, viscous and electrically conducting fluid with heat transfer over a surface of variable temperature stretching with a power-law velocity in the presence of variable transverse magnetic field is analysed. Utilizing similarity transformation, governing nonlinear partial differential equations are transformed to nonlinear ordinary differential equations and they are numerically solved using fourth-order Runge–Kutta shooting method. Numerical solutions are illustrated graphically by means of graphs. The effects of magnetic field, stretching parameter and Prandtl number on velocity, skin friction, temperature distribution and rate of heat transfer are discussed.     

Finite difference analysis of unsteady natural convection flow along an inclined plate with variable surface temperature and mass diffusion

An analysis is performed to study the transient laminar natural convection flows along an inclined semi-infinite flat plate in which the wall temperatureT _w ^′ and species concentration on the wallC _w ^′ vary as the power of the axial co-ordinate in the formT _w ^′ (x)=T _∞ ^′ +ax ⁿ andC _w ^′ =C _∞ ^′ +bx ^m respectively. The dimensionless governing equations considered here are unsteady, two-dimensional, coupled and non-linear integro-differential equations. A finite difference scheme of Crank-Nicolson type is employed to solve the problem. The velocity, temperature, concentration, skin friction, Nusselt number and Sherwood number are studied in detail for various sets of values of parameters. Correlation equations are also established for Nusselt number and Sherwood number in terms of parameters.     

On thermal boundary layer of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection

 Heat transfer characteristics of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection were investigated. Similarity solutions of the laminar boundary layer equations describing heat transfer and fluid flow in a quiescent fluid were obtained and solved numerically. Velocity and temperature profiles as well as the Nusselt number, Nu, were studied for two thermal boundary conditions; uniform surface temperature and variable surface temperature, for different parameters; Prandtl number Pr, temperature exponent b, velocity exponent m, injection parameter d and power-law index n. It was found that decreasing injection parameter d, and power-law index n and increasing Prandtl number Pr and surface temperature exponent b enhance the heat transfer coefficient. Received on 27 April 2000     

Nonstationary motion of a shell on the surface of a heavy fluid

A three-dimensional nonstationary problem of vibrations of a flexible shell moving on the surface of an ideal heavy fluid. The forces due to surface tension are ignored. The problem is formulated in the space of the acceleration potential. The potential of the pulsating source is found by solving the Euler equation and the continuity equation taking into account the free-surface conditions (linear theory of small waves) and the conditions at infinity. The density distribution function of the dipole layer is determined from the boundary conditions on the surface of the shell. Formulas for determining the shape of gravity waves on the fluid surface and the natural frequencies of vibrations of the shell are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 66–75, July–August, 2009.