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Internal Flows of Incompressible Fluids Subject to Stick-Slip Boundary Conditions

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Abstract

We study mathematical properties of internal three-dimensional flows of incompressible heat-conducting fluids with stick-slip boundary conditions, which state that the fluid adheres to the boundary until a certain criterion activates the slipping regime on the boundary. We look at this type of activated boundary condition as at an implicit constitutive equation on the boundary and establish the long-time and large-data existence of weak solutions for the incompressible three-dimensional Navier–Stokes–Fourier system with the viscosity and the heat conductivity depending on the temperature (internal energy). It is essential for our approach to know that the pressure, i.e., the quantity that is a consequence of the fact that the material is incompressible, is globally integrable. While this requirement is in the case of unsteady flows subject to a no-slip boundary condition open for most incompressible fluids, we show that this difficulty can be successfully overcome if one replaces the no-slip boundary condition by a stick-slip boundary condition. The result relies also on the approach developed in Bulíček et al. (Nonlinear Anal. Real World Appl. 10, 992–1015, 10).

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Notes

  1. In fact, the discussions whether the fluid should slip or stick to the boundary goes back to the founders of fluid mechanics and is discussed in length for example in the work of Stokes [53].

  2. The density of the fluid is uniform and equals to a positive constant ρ ∗. The governing equations (2) and (18) (considered below) are obtained from the general balance equations for linear momentum and energy by multiplying them by 1/ρ ∗ and denoting the quantities p, \(\mathbb {S}\) and q multiplied by 1/ρ ∗ again by p, \(\mathbb {S}\) and q.

  3. Note however that for internal flows characterized by (6), the conservation of the total energy of the system, i.e.,

    $$ {\int}_{\Omega} E(t, x) \, dx = {\int}_{\Omega} E(0, x) \, dx \quad \text{ for all }~ t>0 $$
    (21)

    might be required and one can observe that (21) holds if

    $$\begin{array}{@{}rcl@{}} \boldsymbol{q}\cdot \boldsymbol{n} &=& (\mathbb{S}\boldsymbol{v})\cdot \boldsymbol{n} = (\mathbb{S}\boldsymbol{n})\cdot \boldsymbol{v} = (\mathbb{S}\boldsymbol{n})_{\tau}\cdot \boldsymbol{v}_{\tau} =-\mathbf{s}\cdot \mathbf{v}_{\mathbf{\tau}}\\ &=&- \left\{\begin{array}{ll} 0 &\text{ if }~ |\boldsymbol{s}|\le \sigma_{*} \\ \sigma_{*} |\boldsymbol{v}_{\tau}| + \gamma_{*} |\boldsymbol{v}_{\boldsymbol{\tau}}|^{2} &\text{ if }~ |\boldsymbol{s}| > \sigma_{*} \end{array}\right. = -\frac{1}{\gamma_{*}} (|\boldsymbol{s}|-\sigma_{*})^{+}|\boldsymbol{s}|. \end{array} $$
    (22)

    The validity of (21) under the conditions (6) and (22) can be easily checked by integrating (18) over Ω and using the Gauss theorem.

  4. In fact, we use here the following inequality

    $${\int}_{\partial {\Omega}}|\boldsymbol{v}^{n}|^{\frac{8}{3}}\; dS \le C\|\boldsymbol{v}^{n}\|^{\frac{8}{3}}_{W^{\frac{3}{4},2}({\Omega})} \le C\|\boldsymbol{v}^{n}\|^{\frac{2}{3}}_{2} \|\boldsymbol{v}^{n}\|_{1,2}^{2}. $$

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Acknowledgments

The both authors acknowledge the support of the ERC-CZ project LL1202 financed by the Ministry of Education, Youth and Sports of the Czech Republic.

The authors acknowledge the membership to the Nečas Center for Mathematical Modeling (NCMM) and to the Charles University Center for Mathematical Modeling, Applied Analysis and Computational Mathematics (MathMAC).

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  1. Faculty of Mathematics and Physics, Mathematical Institute, Charles University in Prague, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Miroslav Bulíček & Josef Málek

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  1. Miroslav Bulíček
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Correspondence to Miroslav Bulíček.

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To Professor Willi Jäger on the occasion of his 75th birthday.

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Bulíček, M., Málek, J. Internal Flows of Incompressible Fluids Subject to Stick-Slip Boundary Conditions. Vietnam J. Math. 45, 207–220 (2017). https://doi.org/10.1007/s10013-016-0221-z

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