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Abstract

The paper constitutes a first step toward a fuller investigation of the interactions of two electrically conducting streams of different velocities in the presence of magnetic fields. These studies have a theoretical and practical importance in relation with problems connected with jets discharged in the ionosphere, cooling schemes for hypersonic vehicles, processes requiring the possibility of controlling mixing rates, and so on.

Conditions under which the magneto-fluid-dynamics equations can be linearized with respect to the magnetic field are first derived. Subsequently it is shown that, within the limits imposed by the linearization processes, the flow fields of an ideal, electrically conducting fluid admit of a discontinuity surface for the tangential velocity (shear discontinuity) even when the applied magnetic field is normal to the discontinuity surface itself.

Upon the assumptions of fluid with constant properties and of laminar flow the pertinent boundary-layer-type equations are derived for the following two cases of interaction of two semi-infinite, electrically conducting streams in the presence of a magnetic field: a) the field is stationary with respect to a fixed frame of reference; b) the field is stationary with respect to the slower stream. Power series solutions are used in terms of the magnetic parameter . Solutions for the first approximation are found for several values of the initial velocity ratio k=u 20/u 10. To determine the limits of validity of the first approximation, the second approximation has been solved for one particular case and it has been concluded that results are valid to within terms of order m for m less than about 0.5. Results of the analysis are: the magnetic field influences both the inviscid and the viscous regions.

In case a) the magnetic field decelerates the two inviscid streams at the common constant rate . In case b) the magnetic field has no effect on the slower stream and decelerates the faster stream until it reaches asymptotically the same velocity u 2 as the slower stream. For values of m<1 the inviscid solution for the faster stream can be expressed as a power series in terms of the magnetic parameter m.

The effects of the magnetic field on the viscous regions depend upon both the magnetic parameter m and the initial velocity ratio k. These effects generally increase as m increases and as k decreases. In case a) they are more largely felt in the regions of lower velocity.

To a first approximation in m, the overall action of the magnetic field results, in both cases a) and b), in a flattening of the velocity profiles and in an increasing of the mixing region height. In both cases the layers closer to the faster stream are generally decelerated more than the inviscid faster stream itself. In case a) the layers closer to the slower stream are decelerated less than the inviscid stream.

Extensions of the analysis to less simple cases which can describe the physical phenomenon more closely are suggested.

Riassunto

Il lavoro rappresenta un primo passo avanti verso lo studio dell’interazione di due correnti elettricamente conduttive aventi diverse velocità in presenza di un campo magnetico. Questi studi hanno interesse sia teorico che pratico per i problemi connessi con getti scaricanti nella ionosfera, schemi di raffreddamento per velivoli ipersonici, processi per controllare la velocità di mescolamento e cosi via.

Si studiano le condizioni per cui le equazioni della magnetofluidodinamica possono essere linearizzate rispetto al campo magnetico.

Si mostra successivamente che, nei limiti imposti dai processi di linearizzazione, il campo de flusso di un fluido ideale elettricamente conduttivo ammette una superficie di discontinuità per la velocità tangenziale (discontinuità tangenziale) anche quando il campo magnetico applicato è normale alla superficie di discontinuità.

Si derivano, con le assunzioni di fluido con proprietà costanti e flusso laminare, le equazioni relative allo strato limite per i seguenti due casi di interazione di due correnti seminfinite elettricamente conduttive in presenza di un campo magnetico: a) campo stazionario rispetto ad un determinato sistema di riferimento; b) campo stazionario rispetto alla corrente più lenta.

Si danno soluzioni in serie di potenze in termini del parametro magnetico e si passa ad applicazioni numeriche relative alla prima approssimazione per alcuni valori del rapporto tra le velocità iniziali k=u 20/u 10. Per determinare i limiti di validità della prima approssimazione si risolve la seconda approssimazione per un caso particulare e si conclude che è lecito fermarsi fino ai termini in m per m minore di circa 0,5.

Risultati dell’analisi sono:

Il campo magnetico influenza sia le regioni non viscose che quelle viscose.

Nel caso a) il campo magnetico decelera le due correnti non viscose al comune valore costante . Nel caso b) il campo magnetico non ha effetto sulla corrente più lenta e decelera la più veloce finchè questa raggiunge asintoticamente la stessa velocità u 2 della corrente più lenta. Per valori di m < 1 la soluzione non viscosa per la corrente più veloce può essere espressa in serie di potenze in termini del parametro magnetico m.

Gli effetti dei campi magnetici sulle regioni viscose dipendono sia dal parametro magnetico m che dal rapporto tra le velocità iniziali k. Questi effetti generalmente aumentano con m e diminuiscono con k. Nel caso a) si avvertono maggiormente nelle regioni di più basse velocità.

Dalla prima approssimazione in m l’azione del campo magnetico risulta essere sia nel caso a) che nel caso b) un appiattimento di profili di velocità ed un aumento dello spessore della regione di interazione. In entrambi i casi gli strati più vicini alla corrente più veloce sono generalmente decelerati più che la stessa corrente non viscosa più veloce. Nel caso a) gli strati più vicini alla corrente più lenta sono decelerati meno della corrente non viscosa.

Si suggeriscono estensioni dell’analisi a casi meno semplici che possano descrivere più da vicino il fenomeno fisico.

Résumé

Un premier pas vers une investigation plus complète de l’interaction de deux écoulements conducteurs à vitesses différentes en présence d’un champ magnétique. L’importance théorique et pratique de telles études est liée aux problèmes des jets débouchant dans l’ionosphère, des méthodes de refrigération aux vitesses hypersoniques, des méthodes de contrôle de mélanges, etc.

La possibilité de linéariser les équations par rapport au champ magnétique est d’abord établie. Dans les limites de la théorie linéarisée, l’écoulement d’un fluide idéal, électriquement conducteur, admet une discontinuité tangentielle de vitesse, même en présence d’un champ magnétique normal à la surface de discontinuité.

Les deux cas suivants d’interaction d’écoulements laminaires et homogènes semi-infinis sont examinés:
  1. (a)

    champ magnétique stationnaire par rapport à un repère fixe;

     
  2. (b)

    champ stationnaire relativement à l’écoulement lent.

     

Les solutions sont obtenues en séries de puissances d’un paramètre magnétique . La première approximation est obtenue pour plusieurs valeurs du rapport de vitesse initial k=u 20/u 10. Pour déterminer les limites de validité de la première approximation, la seconde a été résolue dans un cas particulier et montre que les résultats contenant les termes d’ordre m seulement sont valables jusque m= 0.5 environ. L’analyse démontre que le champ magnétique influence la couche limite aussi bien que l’écoulement potentiel.

Dans le cas (a) le champ applique aux deux écoulements potentiels une décélération commune , dans le cas (b) le champ est sans effet sur l’écoulement lent mais retarde l’autre jusqu’à ce qu’il atteigne asymptotiquement la vitesse du premier. Pour m<1, la solution potentielle de l’écoulement rapide peut être exprimée sous forme d’une série dans les puissances de m.

Les effets dans la couche limite dépendent à la fois de m et de k. Ils augmentent en général quand m croît et k décroît. Dans le cas (a) ils sont plus importants dans les régions de faible vitesse.

Pour la première approximation en m, le champ magnétique induit dans les deux cas un aplatissement du profil des vitesses et un accroissement de la hauteur de la couche d’interaction, tandis que les couches proches de l’écoulement rapide subissent une décélération plus forte que l’écoulement potentiel. Dans le cas (a) les couches proches de l’écoulement lent subissent une décélération inférieure à l’écoulement potentiel. Il est suggéré d’étendre l’analyse à des situations qui s’approchent davantage de la réalité physique.

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Copyright information

© Springer-Verlag Wien 1959

Authors and Affiliations

  • Luigi G. Napolitano
    • 1
  1. 1.Department of AeronauticsUniversity of NaplesNapoliItalia

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