Formulas for orthogonal projectors generated by the problem on small motions of three viscoelastic fluids in a stationary container

In the paper, we consider the problem on small motions of three viscoelastic fluids in a stationary container. One of models of such fluids is Oldroid's model. It is described, for example, in the [1]. It should be noted that the present paper based on the previous N. D. Kopachevsky and his co-authors works [2, 3, 4]. Namely, problem on small movements of two viscoelastic fluids has already investigated in [2].

We can apply an operator approach of mentioned work to the initial-boundary-value problem generated by the problem of small motions of three(or more) viscoelastic fluids in a stationary container.

Studying of this problem shows us that some complications appear when we use the method of orthogonal projection. This complications arise for the reason that fluid with two (lower and upper) free boundaries appears in the case of problem for three fluids. Existence of such fluid lead us to new more complicated auxiliary problems.

The aim of this paper is to get the formulas for orthogonal projectors on the spaces generated by the problem.

This paper is organized as follows. After introduction in section 2 we formulate mathematical statement of the problem: linearized equations of movements, stickiness conditions, kinematic and dynamic conditions. Further, in this section, we receive the law of full energy balance and choose the functional spaces generated by the problem. To apply of orthogonal projection method we need to get orthogonal projectors on corresponding spaces. The law of action of the first of them we receive in section 3. To get this formula we need to solve the auxiliary transmission problem. This problem lead us to three Zaremba problems (a separate problem for each fluid). Let us remark that Zaremba problem corresponding to the fluid with two free boundaries is more complicated then others two. To solve this problem we need to search solution as the sum of two functions. In other words, we have to solve two Zaremba problems instead of one. In section 4 we obtain the law of action of the second orthogonal projector. The three second type auxiliary S.G. Krein problems appear in the process of reasoning. It is obvious that the problem for the fluid with two free boundaries is more complicated as before. In section 5 we conclude that operator approach lets us to realize the transition from the initial-boundary-value problem to an operator differential equation in sum of Hilbert spaces. Note that properties of main operator of this problem are exactly the same that the properties of main operator arising for problem of two fluids. This fact lets us to consider that all solvability statements proved for the problem of two fluids are true for the problem of three fluids, too.

Finally, note also that this results can be extend to the case of any number of viscoelastic fluids in a stationary container.