On the conditional stability of the rest state of a fluid of second grade in unbounded domains.

*(English)*Zbl 0697.76018In recent years there has been considerable interest in the study of the thermo-mechanics of the fluids of the differential type of complexity n. The Cauchy stress T in incompressible fluids of this kind has the representation \(T=-pl+f(L,\dot L,\ddot L,...,L^{(n)}),\) where \(L\equiv \text{grad} \nu\) and where \(-pI\) is the indeterminate part of the stress due to the constraint of incompressibility. Thus, in such fluids only a short part of the history of the deformation gradient affects the stress, through the first n material time derivatives of the gradient of the velocity.

The incompressible and homogeneous fluids of second grade form a subclass of fluids of complexity 2. In such fluids, the Cauchy stress T is given by \(T=-pI+\mu A_ 1+\alpha_ 1A_ 2+\alpha_ 2A^ 2_ 1,\) where \(\mu\) is the coefficient of viscosity, and \(\alpha_ 1\) and \(\alpha_ 2\) are material coefficients which are usually called the normal stress moduli.

We study the stability of flows of fluids of second grade in a half-space with boundaries free of stress. A detailed treatment of such problems within the context of the Navier-Stokes theory can be found in the lecture notes by G. P. Galdi and S. Rionero [Weighted energy methods in fluid dynamics and elasticity (1985; Zbl 0585.76001)]. In our work on fluids of second grade, we shall relax the requirement used by J. E. Dunn and R. L. Fosdick [Arch. Ration. Mech. Anal. 56, 191-252 (1974; Zbl 0324.76001)] and assume that \(\alpha_ 1+\alpha_ 2\) is arbitrary. We show that the rest state is conditionally stable, provided \(\mu >0\) and \(\alpha_ 1>0\). The results we establish hold for more general domains, essentially domains with flat boundaries, provided we satisfy the requirement that the boundary be free of stress.

When \(\alpha_ 1\) is assumed to be negative, other coefficients being the same as before, we can show that the flow under consideration is unstable in the usual Lyapunov sense. Our instability results are established by arguments different from those of Dunn and Fosdick. They showed that there are domains (small enough in size) wherein all disturbances blow up in time. Here we show that for any domain in question (which is unbounded) the basic state is Lyapunov unstable in a suitable norm. This method would also work in bounded domains, and thus the instability results of Dunn and Fosdick hold true for canisters of arbitrary size. It is worth observing that the equations of motion for an incompressible homogeneous fluid of second grade are generally third- order partial differential equations. Thus the usual adherence boundary conditions used in the Navier-Stokes theory might be insufficient to determine the solution fully. However, if we restrict attention to a certain subclass of flows, for example canister flows considered by Dunn and Fosdick, we are able to prove uniqueness.

In this paper we consider problems involving stress-free boundaries and we establish conditional stability for such a class of flows. To do this, we obtain certain energy estimates which can also be of use in proving the existence of solutions. It is important to note that the stress-free boundary conditions for a fluid of second grade do not appear, at first glance, to be the same as those for the Navier-Stokes fluid.

The incompressible and homogeneous fluids of second grade form a subclass of fluids of complexity 2. In such fluids, the Cauchy stress T is given by \(T=-pI+\mu A_ 1+\alpha_ 1A_ 2+\alpha_ 2A^ 2_ 1,\) where \(\mu\) is the coefficient of viscosity, and \(\alpha_ 1\) and \(\alpha_ 2\) are material coefficients which are usually called the normal stress moduli.

We study the stability of flows of fluids of second grade in a half-space with boundaries free of stress. A detailed treatment of such problems within the context of the Navier-Stokes theory can be found in the lecture notes by G. P. Galdi and S. Rionero [Weighted energy methods in fluid dynamics and elasticity (1985; Zbl 0585.76001)]. In our work on fluids of second grade, we shall relax the requirement used by J. E. Dunn and R. L. Fosdick [Arch. Ration. Mech. Anal. 56, 191-252 (1974; Zbl 0324.76001)] and assume that \(\alpha_ 1+\alpha_ 2\) is arbitrary. We show that the rest state is conditionally stable, provided \(\mu >0\) and \(\alpha_ 1>0\). The results we establish hold for more general domains, essentially domains with flat boundaries, provided we satisfy the requirement that the boundary be free of stress.

When \(\alpha_ 1\) is assumed to be negative, other coefficients being the same as before, we can show that the flow under consideration is unstable in the usual Lyapunov sense. Our instability results are established by arguments different from those of Dunn and Fosdick. They showed that there are domains (small enough in size) wherein all disturbances blow up in time. Here we show that for any domain in question (which is unbounded) the basic state is Lyapunov unstable in a suitable norm. This method would also work in bounded domains, and thus the instability results of Dunn and Fosdick hold true for canisters of arbitrary size. It is worth observing that the equations of motion for an incompressible homogeneous fluid of second grade are generally third- order partial differential equations. Thus the usual adherence boundary conditions used in the Navier-Stokes theory might be insufficient to determine the solution fully. However, if we restrict attention to a certain subclass of flows, for example canister flows considered by Dunn and Fosdick, we are able to prove uniqueness.

In this paper we consider problems involving stress-free boundaries and we establish conditional stability for such a class of flows. To do this, we obtain certain energy estimates which can also be of use in proving the existence of solutions. It is important to note that the stress-free boundary conditions for a fluid of second grade do not appear, at first glance, to be the same as those for the Navier-Stokes fluid.

##### MSC:

76A10 | Viscoelastic fluids |

##### Keywords:

fluid of second grade; thermo-mechanics of the fluids; Cauchy stress; stability of flows of fluids of second grade; half-space with boundaries free of stress; Navier-Stokes theory
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\textit{G. P. Galdi} et al., Arch. Ration. Mech. Anal. 109, No. 2, 173--182 (1990; Zbl 0697.76018)

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##### References:

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