Translation groups of the boundary-layer flows induced by continuous moving surfaces

Abstract

The steady plane boundary-layer flows of velocity field {u(x, y), v(x, y)} induced by continuous moving surfaces are revisited in this paper. It is shown that the governing balance equations, as well as the asymptotic condition u(x, ∞) = 0 at the outer edge of the boundary layer are invariant under arbitrary translations y → y + y0(x) of the transverse coordinate y. The wall conditions, i.e. the prescribed stretching velocity u(x, 0) ≡ Uw(x) and the transpiration velocity v(x, 0) ≡ Vw(x) distributions, however, undergo in general substantial changes. The consequences of this basic symmetry property on the structure of the solution space are investigated. It is found that starting with a primary solution which describes the boundary-layer flow induced by an impermeable surface, infinitely many translated solutions can be generated which form a continuous group, the translation group of the given primary solution. The elements of this group describe boundary-layer flows induced by permeable surfaces stretching under transformed wall conditions, Uw(x) → Ũw(x) = u[x, y0(x)] and Vw(x) → Ṽw(x) = v[x, y0(x)] − y′0(x)u[x, y0(x)], respectively. In this way, starting with a known solution {u(x, y), v(x, y)} so that the inverse y0(x) = u−1(x, Ũw) of u[x, y0(x)] exists, a new solution {ũ(x, y), ṽ(x, y)} corresponding to any desired stretching velocity distribution Ũw(x) can be prepared. It also turns out that several exact solutions discovered during the latter decades are not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.