Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural
convection on a horizontal finite plate acting as a heat dipole, are considered at
large distances from the plate. It is shown that physically acceptable self-similar
solutions of the boundary-layer equations, which include buoyancy effects, exist in
certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for
plane, and Pr>1 for
axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions
are independent of the Prandtl number and can be determined from a momentum
balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl
number and are to be determined as part of the solution of the eigenvalue problem.
For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and
1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant
jet, for which self-similar solutions of the boundary-layer equations are also provided.
Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations
for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of
the Prandtl number.
Comparisons with finite-element solutions of the full Navier–Stokes equations show
that the self-similar boundary-layer solutions are asymptotically approached as the
plate Grashof number tends to infinity, whereas the self-similar solution to the full
Navier–Stokes equations is applicable, for a given value of the Prandtl number, only
to one particular, finite value of the Grashof number.
In the Appendices second-order boundary-layer solutions are given, and uniformly
valid composite expansions are constructed; asymptotic expansions for large values of
the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical
solutions of the Navier–Stokes equations.