The receptivity problem of plane Bingham–Poiseuille flow with respect to weak perturbations is addressed. The relevance of this study is highlighted by the linear stability analysis results (spectra and pseudospectra). The first part of the present paper thus deals with the classical normal-mode approach in which the resulting eigenvalue problem is solved using the Chebychev collocation method. Within the range of parameters considered, the Poiseuille flow of Bingham fluid is found to be linearly stable. The second part investigates the most amplified perturbations using the non-modal approach. At a very low Bingham number (B ≪ 1), the optimal disturbance consists of almost streamwise vortices, whereas at moderate or large B the optimal disturbance becomes oblique. The evolution of the obliqueness as function of B is determined. The linear analysis presented also indicates, as a first stage of a theoretical investigation, the principal challenges of a more complete nonlinear study.
The Darcy model with the Boussinesq approximations is used to study double-diffusive
instability in a horizontal rectangular porous enclosure subject to two sources of
buoyancy. The two vertical walls of the cavity are impermeable and adiabatic while
Dirichlet or Neumann boundary conditions on temperature and solute are imposed on
the horizontal walls. The onset and development of convection are first investigated
using the linear and nonlinear perturbation theories. Depending on the governing
parameters of the problem, four different regimes are found to exist, namely the
stable diffusive, the subcritical convective, the oscillatory and the augmenting direct
regimes. The governing parameters are the thermal Rayleigh number, RT, buoyancy
ratio, N, Lewis number, Le, normalized porosity of the porous medium,
ε, aspect ratio of the enclosure, A, and the thermal and solutal
boundary condition type, κ,
applied on the horizontal walls. On the basis of the nonlinear perturbation theory
and the parallel flow approximation (for slender or shallow enclosures), analytical
solutions are derived to predict the flow behaviour. A finite element numerical method
is introduced to solve the full governing equations. The results indicate that steady
convection can arise at Rayleigh numbers below the supercritical value, indicating
the development of subcritical flows. At the vicinity of the threshold of supercritical
convection the nonlinear perturbation theory and the parallel flow approximation
results are found to agree well with the numerical solution. In the overstable regime,
the existence of multiple solutions, for a given set of the governing parameters, is
demonstrated. Also, numerical results indicate the possible occurrence of travelling
waves in an infinite horizontal enclosure.
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