Stability of harmonic fields

Incompressible flows of an ideal two-dimensional fluid on a closed, orientable surface with positive genus are studied. Linear stability of the harmonic, i.e. irrotational and incompressible, solutions to the Euler equations is shown using the Hodge-Helmholtz decomposition. We also demonstrate that any surface Euler flow is stable with respect to small harmonic velocity perturbations.

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Global static stability

The global static stability of a Starling Resistor conveying non-Newtonian fluid is considered. The Starling Resistor consists of two rigid circular tubes and axisymmetric collapsible tube mounted between them. Upstream and downstream pressures are the boundary condition as well as external to the collapsible tube pressure. Quasi one-dimensional model has been proposed and a boundary value problem in terms of nondimensional parameters is considered. Nonuniqueness of the boundary value problem is regarded as static instability. The analytical condition of instability which defines a surface in parameter space has been studied numerically. The influence of fluid rheology on stability of collapsible tube is established. The external pressure was chosen to be of special form giving uniform reference solution. Mechanical response of the tube resembles generalised ”string” model. The steady equations of motion and incom- pressibility lead to a boundary value problem which has a nontrivial solution only for specific values of nondimensional parameters. Thus the static instability is associated with nonuniqueness of boundary value problem solution. The analytical condition of nonuniqueness is investigated numerically for different values of parameters including the automodel case.

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Local dynamical instability

A dynamical model of flow of a nonlinear viscous power-law hardening medium in a cylindrical deformable channel is considered. The flow and deformation are axisymmetric. $$ \frac{\partial{Q}}{\partial{t}}+\frac{\partial{}}{\partial{z}}\left(\frac{(3n+1)Q^2}{(2n+1)\pi{}R^2}\right)+\frac{\mu{}}{\rho{}}\frac{(3n+1)^n 2^{\frac{3-n}{2}}Q^n}{n^n\pi{}^{n-1}R^{3n-1}}=$$ $$=-\frac{\pi{}R^2}{\rho{}}\frac{\partial{P}}{\partial{z}} $$ $$ \frac{\partial{Q}}{\partial{z}}+\frac{\partial{\left(\pi{}R^2\right)}}{\partial{t}}=0 $$ $$\beta\left(R-R_0\right)=P$$ Stationary solutions and their stability with respect to small perturbations are studied. Stability domains are found in the space of dimensionless parameters. In this paper it is shown that the stability conditions depend on the power-law exponent \(n\). The pseudoplastic model with a small value of \(n\) describes the rigid-plastic behavior of the medium. In the framework of this approximation, the critical parameters of stability of a plastic medium differ from those of a Newtonian fluid. In the latter case, the instability of flow is observed only for \(\chi\geq 1\), whereas in the first case we have an additional instability domain for \(\chi\leq 0.06\). Here $$\chi{}=\frac{2\rho{}q^2(3n+1)}{\beta{}R_0^5\pi{}^2(2n+1)}$$

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Experimental study

"Starling resistor" facility models various biological flows. Experiments on an elastic, thin (Penrose) tubes subjected to non-Newtonian flows held in Laboratory of Experimental Hydrodynamics, Institute of Mechanics of Lomonosov Moscow State University.



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Viscoplastic squeezing

The problem of quasi-static compression and spreading (s queezing) of a thin viscoplastic layer between approaching absolutely rigid parallel-arranged plates is solved using asymptotic integration methods rapidly developed in recent years in the mechanics of deformable thin bodies. A solution symmetric about the coordinate axes is sought in the same region of the layer as in the classical Prandtl problem. The layer material is characterized by a yield point and a hardening function relating the intensities of the stress and strain rate tensors. The conditions of no-flow and reaching certain values by tangential stresses are imposed on the plate surfaces. The coefficients at the terms of the asymptotic expansions corresponding to the minus first and zero powers of the small geometrical parameter are obtained. An approximate analytical solution in the case of power hardening and large Saint-Venant numbers is given. The physical meaning of the roughness coefficient characterizing the cohesion between the plates and viscoplastic material is discussed.

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