Abstract
This chapter starts with kinematic concepts such as ‘motion’, ‘velocity’, ‘Eulerian and Lagrangean descriptions’, and then proceeds to describe ‘streamlines’, ‘trajectories’ and ‘streak lines’, illuminating these with illustrative examples. Next, the balance laws of mass and linear momentum are discussed both in global and local form and specialized to Eulerian fluids. The Bernoulli equation, defined as the path-integration of the scalar product of the momentum equation with the velocity field is given extensive space; it is discussed both when referred to non-inertial and inertial frames and when the integration is conducted along any path or along streamlines. Ample space is devoted to applications of the Bernoulli equation to typical examples, e.g., venturi pipes, Prandtl pipes, Torricelli flow out of a vessel, including clepsydra clocks. Global formulations of the momentum equation are equally touched and applied to the problem of Borda’s exit flows, impact of a jet on a wall, mixing processes of non-uniform velocities in plane conduits, hydraulic jumps and flow of a density preserving fluid through a periodic grid of wings. Aerodynamics is given a first glimpse by studying plane flow around infinitely long wings, specifically by deriving the Kutta-Joukowski condition of smooth flow off the wing’s trailing edge, which fixes the circulation around the wing. The chapter closes with a presentation of the balance of moment of momentum and its application to the Segner water wheel and Euler’s turbine equation.
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Notes
- 1.
The Eulerian description is also the preferential description in the mechanics of rigid bodies. For instance,
$$\begin{aligned} {\varvec{v}}({\varvec{x}}, t) = {\varvec{v}}_{s}(t) + {\varvec{\omega }}(t)\times \left( {\varvec{x}}-{\varvec{x}}_{s}(t)\right) , \end{aligned}$$where \({\varvec{v}}_{s}(t)\) and \({\varvec{\omega }}(t)\) are the velocity and angular velocity at the reference point \({\varvec{x}}_{s}(t)\).
- 2.
When using “convective” acceleration one usually tacitly assumes that a term \((\partial \phi /\partial z)w\) is present; by contrast, “advective” acceleration means that \((\partial \phi /\partial x)u+(\partial \phi /\partial y)v\) are dominant. However, this interpretation is not unanimously used, but very popular in geophysics.
- 3.
For a short biography of Leibniz see Fig. 3.12 .
- 4.
See Appendix 5.A, where the divergence theorem or Gauss law is derived.
- 5.
This term is actually ill-defined. A flow cannot be inviscid; it is the fluid, which might be inviscid.
- 6.
For a short biography of Newton see Fig. 3.17 .
- 7.
The difference of the two formulations is shown when Newton’s fundamental law is written down for the motion of a rocket with exhaustion of burning fuel.
- 8.
The derivation of this formula for an infinitesimal cube is best obtained, if one applies the formula
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\int \limits _{V}\rho {\varvec{v}}\,\mathrm {d}V =\int \limits _V\frac{\partial (\rho {\varvec{v}})}{\partial t}\,\mathrm {d}V +\int \limits _{\partial V}\rho {\varvec{v}} ({\varvec{v}}\cdot {\varvec{n}})\,\mathrm {d}A \end{aligned}$$to an infinitesimal cube fixed in space. If we apply this formula to the x-momentum and the cube of Fig. 3.18, we obtain
$$\begin{aligned}&\left\{ {\displaystyle \frac{\partial (\rho u)}{\partial t}}+{\displaystyle \frac{\partial }{\partial x}}\left( \rho u^2\right) +{\displaystyle \frac{\partial }{\partial y}}\left( \rho uv\right) +{\displaystyle \frac{\partial }{\partial z}}\left( \rho uw\right) \right\} \mathrm {d}x\mathrm {d}y\mathrm {d}z\\= & {} \bigg [ u \underbrace{\left\{ {\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial \rho u}{\partial x}}+{\displaystyle \frac{\partial \rho v}{\partial y}}+{\displaystyle \frac{\partial \rho w}{\partial z}} \right\} }_{\mathrm {balance~of~mass} =0} + \rho \underbrace{\left\{ {\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}+w{\displaystyle \frac{\partial u}{\partial z}} \right\} }_{\mathrm {d}u/\mathrm {d}t} \bigg ] \mathrm {d}x\mathrm {d}y\mathrm {d}z\\= & {} \rho {\displaystyle \frac{\mathrm{d}u}{\mathrm{d}t}} \mathrm {d}x\mathrm {d}y\mathrm {d}z. \end{aligned}$$ - 9.
For a short biographical sketch of Euler see Fig. 3.19 .
- 10.
Equation (3.86) can also be viewed as a special case of the so-called Reynolds transport theorem , according to which for a continuously differentiable scalar, vectorial and tensorial field \({\varvec{a}}\) the following expression
$$ \frac{\mathrm {d}}{\mathrm {d}t}\int \limits _{V(t)}\varvec{a}\,\mathrm {d}V= \int \limits _{V(t)}\frac{\partial \varvec{a}}{\partial t}\,\mathrm {d}V+ \oint \limits _{\partial V(t)}\varvec{a}(\varvec{v}\cdot \varvec{n})\,\mathrm {d}A $$holds true. In (3.86), \({\varvec{a}} = \rho {\varvec{v}}\).
- 11.
The tensor product \({\varvec{v}}\otimes {\varvec{v}}\) of the vector variable \({\varvec{v}}\) is that rank-2 tensor which in component representation has the form
$$ \varvec{v}\!\otimes \!\varvec{v}\, \hat{=} \left( \begin{array}{ccc} v_{1}v_{1} &{} v_{1}v_{2} &{} v_{1}v_{3} \\ v_{2}v_{1} &{} v_{2}v_{2} &{} v_{2}v_{3} \\ v_{3}v_{1} &{} v_{3}v_{2} &{} v_{3}v_{3} \\ \end{array} \right) . $$Its divergence is given by the column vector
$$ {\mathrm {div}\,}{(\varvec{v}\otimes \varvec{v})} = \left( \begin{array}{l} \displaystyle \frac{\partial v_{1}v_{1}}{\partial x} + \displaystyle \frac{\partial v_{1}v_{2}}{\partial y} + \displaystyle \frac{\partial v_{1}v_{3}}{\partial z} \\ \displaystyle \frac{\partial v_{2}v_{1}}{\partial x} + \displaystyle \frac{\partial v_{2}v_{2}}{\partial y} + \displaystyle \frac{\partial v_{2}v_{3}}{\partial z} \\ \displaystyle \frac{\partial v_{3}v_{1}}{\partial x} + \displaystyle \frac{\partial v_{3}v_{2}}{\partial y} + \displaystyle \frac{\partial v_{3}v_{3}}{\partial z} \\ \end{array} \right) . $$With these definitions the result (3.88) can easily be understood when written in component form. For a detailed explanation of the symbol “\(\hat{=}\)” see the footnote in connection with (3.275).
- 12.
For a short biography of Daniel Bernoulli see Fig. 3.22 .
- 13.
Atanh or Areatanh is the inverse function of tanh, sometimes also written as \(\mathrm {tanh}^{-1}\).
- 14.
For a short biography of Borda see Fig. 3.40 .
- 15.
- 16.
Nicolas Léonard Sadi Carnot (1796–1832) was a French military engineer and physicist. His work mainly deals with thermodynamics. We shall present his vita in Vol. 2, Chaps. 17 and 18.
- 17.
Of course, the opening angle \(\alpha \) in Fig. 3.47 must be sufficiently small to guarantee such conditions.
- 18.
A short vita of Prandtl is given in Vol. 2, Chap. 17.
- 19.
In the above normal stresses are positive when they are tensions and negative when they are pressures. For the shear stresses we used the following convention. The first index refers to the direction of the coordinate axis into which the vector of shear traction points. The second index refers to the direction of the unit normal vector of the areal element on which the shear traction acts. A shear traction vector is positive, if it points together with the unit normal vector into a positive or negative direction of the coordinates. Finally, Eq. (3.274) are the result of momentum balance; however, since one restricts attention to quantities that are small of order \(h^2\), (where h is a typical length of a tetrahedron edge), the balance of momentum reduces in this case to a force balance of surface forces.
- 20.
On the left-hand side \({\varvec{t}}\) represents the Cauchy stress tensor; on the right-hand side stands its matrix referred to Cartesian coordinates. The symbol \(\hat{=}\) is not an equality sign, because second rank tensors are not equal to matrices, but they are equivalent to tensors and the symbol \(\hat{=}\) states that the left and right hand-sides express the same ‘mathematical substance’. One also says that \({\varvec{t}}\) and its matrix are isomorphic to one another. For a short biographical sketch see Fig. 3.63 .
- 21.
Johann Andreas Segner (1704–1777), Professor at the University Göttingen. For a historical biography see [9].
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Hutter, K., Wang, Y. (2016). Hydrodynamics of Ideal Liquids. In: Fluid and Thermodynamics. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-33633-6_3
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