The magnetohydrodynamic wall jets: Techniques for rendering similar and perturbative non-similar solutions

Abstract

In the present work, it is given extensive analytic analysis on the magnetohydrodynamic (MHD) wall jets for the 1st time in the history. It is hopeful that this work provides a careful ground for further studies in this field. The main findings are summarized as: If u(x,0)=v(x,0)=u(x,y→∞)=0, then there will be no similarity solution of either algebraically or exponentially decaying type for a jet discharged over a plane surface in the presence of a transverse magnetic field. The proof is given via various analytic approaches together with a numeric validation. In particular, the conserved vector associated with the multiplier Λ=Λ(x,y,ψ,ψx,ψy)does not yield a conserved quantity accounting the above boundary conditions for the MHD wall jet. However, similarity solutions may exist in the context of Glauert type e-jets withu(x,0)∝x−12,v(x,0)∝x−34,u(x,y→∞)=0being connected to f′(0)=β,f(0)=α,f′(η→∞)=0 in similarity terms. The cases with α=0,β≠0 and β=0,α≠0are considered in the present paper. For the 1st case, it is found that a certain magnetic parameter is connected to dual solutions being representative of stretchable/shrinkable wall. It should be mentioned that for this case, some primary analysis have been recently performed by the present author in [27] and here, this part has been included mainly for the consistency purpose and the focus has been given to the stability analysis of the associated dual solutions. Spatial stability analysis revealed that the solutions associated with the stretchable wall are physically stable whilst the solutions associated with the shrinkable wall are stable only for very small magnetic parameters. For the 2nd case, it is given an explicit proof that upon the assumptions off′(η)≥0,f(η→∞)>0 and f″(0)>0 the Glauert type solutions exist if only−f(η→∞)<α<0, in particular implying that no suction assumption is allowed. As a consequence of the present findings, the MHD wall jet with u(x,0)=v(x,0)=u(x,y→∞)=0 was treated as non-similar flows. A perturbative treatment was accounted for the stream function, eventually leading to physically-acceptable solutions for this case. The following Table shows an overview of the original contributions of the present work to the existing literature. u(x,0)=v(x,0)=u(x,y→∞)=0 (The Original MHD Wall Jet)u(x,0)=0,v(x,0)∝x−34, u(x,y→∞)=0u(x,0)∝x−12,v(x,0)=0, u(x,y→∞)=0✓Proofs on which that No similarity solutions of the form of a-jet/e-jet exist.✓Similarity solutions: Explicit proof on which that only Injection is allowed upon setting some normal assumptions.✓Similarity Solutions: dual solutions exist associated with stretchable/shrinkable wall [27].✓Presenting Non-similar perturbative solution.✓Presenting an extensive stability analysis.