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2020 | OriginalPaper | Buchkapitel

5. Flows of Fluids with Pressure Dependent Material Coefficients

verfasst von : Miroslav Bulíček

Erschienen in: Fluids Under Pressure

Verlag: Springer International Publishing

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Abstract

It has been well documented in many studies that the material parameters of a fluid may essentially depend on the pressure and that they can vary by several orders of magnitude. The material parameters, for which this dependence is observed, are mainly the viscosity (due to the internal forces in the fluid) and the friction (due to fluid–(rigid) solid interaction). In addition, these large variations with respect to the pressure in the material parameters occur although the variations of the density are almost negligible (in comparison with other parameters). Therefore it is still reasonable to describe the above mentioned phenomena in many fluids by incompressible models. Likewise, the viscosity and the drag of many fluids vary with the shear rate and such shear (rate)-dependent viscosity and friction are extensively used, ranging from geophysics, chemical engineering, and bio-material science up to the food industry, enhanced oil recovery, carbon dioxide sequestration, or extraction of unconventional oil deposits, etc.

The aim of this study is to present an overview of available results for models with very complicated rheological laws used in engineering praxes. As particular examples that fit into the class of models studied here, we refer to the Darcy model, to the Brinkman models, and to the Bingham models. Nevertheless, the aim of this study is much more ambitious and we go much beyond these standard models and present a kind of unifying theory, which is based on the use of the so-called maximal monotone graphs, which seems to be very appropriate from the point of view of mathematical analysis of the problem.

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Vorheriges Kapitel The Role of Pressure in the Theory of Weak Solutions to the Navier-Stokes Equations

Nächstes Kapitel Finite Element Pressure Stabilizations for Incompressible Flow Problems

We can consider the normal component of v since we control the divergence of v.

Whenever we write f on Γ, we are automatically employing the trace operator.

The space $L^{r}_{ \operatorname {\mathrm {div}},\Gamma _1}(\Omega )$ can be equivalently defined as

$$\displaystyle \begin{aligned}L^{r}_{\operatorname{\mathrm{div}},\Gamma_1}(\Omega)\mathrel{\mathop{:}}= \left\{ \boldsymbol{\varphi} \in L^{r}(\Omega) \mid \mathrm{ for all} u\in W^{1,r'}_{\Gamma_2}(\Omega) \quad \int_{\Omega} \nabla u \cdot \boldsymbol{\varphi} \, dx =0\right\}. \end{aligned}$$

We say that the graph $\mathcal {A}$ is strictly monotone with respect to v if for all $(\boldsymbol {m}_i,\boldsymbol {v}_i)\in \mathcal {A}$, there holds

$$\displaystyle \begin{aligned}(\boldsymbol{m}_1-\boldsymbol{m}_2)\cdot (\boldsymbol{v}_1-\boldsymbol{v}_2)= 0 \quad \implies \quad \boldsymbol{v}_1=\boldsymbol{v}_2. \end{aligned}$$

Similarly, we say that the graph is strictly monotone with respect to m if there holds

$$\displaystyle \begin{aligned}(\boldsymbol{m}_1-\boldsymbol{m}_2)\cdot (\boldsymbol{v}_1-\boldsymbol{v}_2)= 0 \quad \implies \quad \boldsymbol{m}_1=\boldsymbol{m}_2. \end{aligned}$$

The generalization of (N1) was already studied in the preceding section.

Ideally, one would like to fix the pressure at one point. But since we deal only with Lebesgue integrable functions, such procedure cannot be applied. Nevertheless, we can try to mimic the fixing the value of p at some x ₀ just by fixing the integral

$$\displaystyle \begin{aligned}\int_{B(x_0)} p\, dx \end{aligned}$$

over some very small ball B centered at x ₀.

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Titel: Flows of Fluids with Pressure Dependent Material Coefficients
verfasst von: Miroslav Bulíček
Verlag: Springer International Publishing
Buch: Fluids Under Pressure
Print ISBN: 978-3-030-39638-1

Electronic ISBN: 978-3-030-39639-8

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-39639-8_5

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