Existence of Turning Points for the Response Diagram of the Poiseuille Flow with Prescribed Flow-rate

We study the stationary Poiseuille flow in a cylindrical channel of arbitrary cross-section with temperature dependent viscosity and internal dissipation. We assume the flow-rate Φ given and the axial pressure gradient μ unknown. This leads to a nonlocal problem. We show the existence in the response diagram, the plane (Φ, μ), of two turning points. 1. In this paper we study the stationary Poiseuille flow in a pipe, assuming the viscosity η and the thermal conductivity κ to be given functions of the temperature u. Let Ω be an open and bounded subset of R with a regular boundary Γ representing the section of the pipe. If the axial pressure gradient μ is a given constant for the determination of the z-component v(x, y) of the velocity and of the temperature u, we have (see [4]) the following Problem A: find v(x, y) and u(x, y) such that −∇ · (η(u)∇v) = μ in Ω, v = 0 on Γ, (1.1) −∇ · (κ(u)∇u) = η(u)|∇v| in Ω, u = 0 on Γ. (1.2) The Navier-Stokes system, for the problem at hand, reduces to (1.1), whereas (1.2) is the energy equation. In the right hand side of equation (1.2) we have the heating source corresponding to the viscous forces. We suppose, on physical grounds, η(u) ∈ C(R), κ(u) ∈ C(R), η(u) > 0, κ(u) > 0 for all u ∈ R. (1.3) If μ = 0 it is easily seen that the only solution of Problem A is v(x, y) = 0, u(x, y) = 0 (see Lemma 1.2). Moreover, if |μ| is sufficiently small a branch of small solutions Received September 28, 2006. 2000 Mathematics Subject Classification. Primary 76D03, 76D05.