Similarity solutions for liquid metal systems near a sharply cornered conductive region

We study the boundary value problem ( P m , a ) : f ‴ + [ ( m + 1 ) / 2 ] f f ″ − m f ′ 2 = 0 on ( 0 , + ∞ ) , subject to the boundary conditions f ( 0 ) = a ∈ R , f ′ ( 0 ) = − 1 and f ′ ( + ∞ ) = 0 . The problem arises in the study of similarity solutions for high frequency excitation of liquid metal systems in an antisymmetric magnetic field. We give a complete picture of solutions of ( P m , a ) for the physical interesting case: m < − 1 and a ⩾ 0 .