Abstract
As it was shown in Sect. 3.2.5, the density variations in a fluid flow decrease with the square of the Mach number (the ratio of the fluid velocity to the sound speed). Hence, for many fluid flows, and especially for those of liquids, incompressibility is an excellent approximation. Moreover, it simplifies very much the equations of motion. This simplification provides us with the easiest context to study the effects of viscosity that we have neglected until now.
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Notes
- 1.
The reader may note that after an integration by part
$$\displaystyle{\int _{0}^{\ell} \frac{x(\ell-x)} {(h_{1} -\alpha x)^{2}}\mathit{dx} = -\frac{1} {\alpha } \int _{0}^{\ell} \frac{\ell-2x} {h_{1} -\alpha x}\mathit{dx}}$$while
$$\displaystyle{\int _{0}^{\ell} \frac{\ell-2x} {h_{1} -\alpha x}\mathit{dx} =\int _{ 0}^{\ell}\left \{\frac{\ell-2h_{1}/\alpha } {h_{1} -\alpha x} + \frac{2} {\alpha } \right \}\mathit{dx}}$$.
- 2.
Here is a very simple example of such a property: \(\sin (x/\varepsilon )\) is \(\mathcal{O}(1)\) but \(\frac{d^{2}} {\mathit{dx}^{2}} \sin (x/\varepsilon )\) is \(\mathcal{O}(1/\varepsilon ^{2})\).
- 3.
An example where we determine the solution of a differential equation using boundary layer theory is given in Sect. 12.4
- 4.
We can get an idea of the shape of the function f(η) by considering the asymptotic limits η ∼ 0 and η → ∞.
Near the origin, (4.46) and the boundary conditions impose that \(f(0) = f^{{\prime\prime}}(0) = f^{{\prime\prime\prime}}(0) = 0\); hence, a Taylor expansion yields
$$\displaystyle{f(\eta ) \approx a\eta -\frac{a^{2}} {48}\eta ^{4} + \mbox{ $\mathcal{O}(\eta ^{7})$}}$$where \(a = f^{{\prime}}(0) \simeq 0.332058\) (this value is determined by the boundary condition f(∞) = 1). This expression shows that the profile of the velocity is almost linear just before reaching the asymptotic value where f(η) ≃ 1. In this region (η ≫ 1), the function f verifies approximately \(f^{{\prime\prime}} = -\eta f^{{\prime}}/2\) whose solution for f ′ is Gauss function and thus for f the error function:
$$\displaystyle{f^{{\prime}}(\eta ) = Ae^{-\eta ^{2}/4}\qquad \;\Longrightarrow\;\qquad f(\eta ) \sim \mathrm{ erf}(\eta )\qquad \eta \rightarrow \infty }$$ - 5.
Sometimes called the Hagen–Poiseuille flow. Hagen studied it in 1839 and Poiseuille in 1840.
References
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Rieutord, M. (2015). Flows of Incompressible Viscous Fluids. In: Fluid Dynamics. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-09351-2_4
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