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Mathematical derivation of the power law describing polymer flow through a thin slab. (English) Zbl 0819.76007

Summary: We consider the polymer flow through a slab of thickness \(\varepsilon\). The flow is described by \(3D\) incompressible Navier-Stokes system with a nonlinear viscosity, being a power of a norm of the shear rate (power law). We consider the limit when \(\varepsilon \to 0\) and prove that the limit averaged velocity, averaged over the thickness, satisfies a nonlinear two-dimensional Poiseuille’s law, with nonlinear viscosity depending on the power of the length of the gradient of the pressure. It is found out that the powers in the starting law and in the limit law are conjugate. Furthermore, we prove a convergence theorem for velocity and pressure in appropriate functional spaces.

MSC:

76A05 Non-Newtonian fluids
35Q35 PDEs in connection with fluid mechanics
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References:

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