Liquids and Gases in Motion; Fluid Dynamics

Appendices

Summary

• The motion of particles of a flowing medium (liquids or gases) is determined by the total force $\boldsymbol{F}=\boldsymbol{F}_{\mathrm{g}}+\boldsymbol{F}_{\mathrm{p}}+\boldsymbol{F}_{\mathrm{f}}$ which is the vector sum of gravity force, pressure force and friction force. The equation of motion is

  $$\displaystyle\boldsymbol{F}=\varrho\cdot\Delta V\cdot\frac{\,\mathrm{d}\boldsymbol{u}}{\,\mathrm{d}t}{\;},$$

  where u is the flow velocity of the volume element $\Delta V$ with the mass density $\varrho$.

• In a stationary flow $\boldsymbol{u}(\boldsymbol{r})$ is at every position r constant in time but can vary for different positions $\boldsymbol{r}_{i}$.

• Frictionless liquids $(\boldsymbol{F}_{\mathrm{f}}=\boldsymbol{0})$ are called ideal liquids. For them the Euler equation

  $$\displaystyle\frac{{\partial}\boldsymbol{u}}{{\partial}t}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}=\boldsymbol{g}-\frac{1}{\varrho}\mathrm{grad}\,p$$

  describes the motion of the liquid.

• The continuity equation

  $$\displaystyle\frac{{\partial}\varrho}{{\partial}t}+\mathop{\mathrm{div}}(\varrho\cdot\boldsymbol{u})=0$$

  describes the conservation of mass for a flowing medium. For incompressible media $(\varrho=\mathrm{const})$ the continuity equation reduces to $\mathop{\mathrm{div}}\boldsymbol{u}=0$.

• For frictionless incompressible flowing media the Bernoulli-equation

  $$\displaystyle p+\tfrac{1}{2}\varrho\cdot u^{2}=\mathrm{const}$$

  represents the energy conservation $E_{\mathrm{p}}+E_{\mathrm{kin}}=E=\mathrm{const}$. The pressure decreases with increasing flow velocity u.

  The Bernoulli equation is the basic equation for the explanation of the dynamical buoyancy and therefore also for aviation.

• For flow velocities u below a critical value $u_{\mathrm{c}}$ laminar flows are observed, while for $u> u_{\mathrm{c}}$ turbulent flows occur. This critical value $u_{\mathrm{c}}$ is determined by the Reynolds number $\mathrm{Re}=2E_{\mathrm{kin}}/W_{\mathrm{f}}$ which gives the ratio of kinetic energy to the friction energy of a volume element $\Delta V=L^{3}$ when $\Delta V$ is shifted by L.

• For laminar flows where the inertial forces are small compared to the friction forces no turbulence occurs and the stream lines are not swirled.

• For a laminar flow through a tube with circular cross section $\pi R^{2}$ the volumetric flow rate

  $$\displaystyle Q=\frac{\pi R^{4}}{8\eta}\mathrm{grad}\,p$$

  flowing per second through the tube is proportional to $R^{4}\cdot\mathrm{grad}\,p$ but inversely proportional to the viscosity η.

• A ball with radius r moving with the velocity u through a medium with viscosity η experiences a friction force

  $$\displaystyle\boldsymbol{F}_{\mathrm{f}}=-6\pi\eta r\cdot\boldsymbol{u}{\;},$$

  that is proportional to its velocity u.

• The complete description of a flowing medium is provided by the Navier–Stokes equation (8.36a) which reduces for ideal liquids $(\eta=0)$ to the Euler equation. The Navier–Stokes equation describes also turbulent flows, but for the general case no analytical solutions exist and the equation can be solved only numerically.

• For the generation and the decay of vortices friction is necessary. Vortices are generally generated at boundaries (walls and solid obstacles in the liquid flow).

• The flow resistance of a body in a streaming medium is described by the resisting force $F_{\mathrm{D}}=c_{\mathrm{D}}\cdot\varrho\cdot\tfrac{1}{2}u^{2}\cdot A$. It depends on the cross section A of the body and its drag coefficient $c_{\mathrm{D}}$ which is determined by the geometrical shape of the body. The force is proportional to the kinetic energy per volume element $\Delta V$ of the streaming medium. In laminar flows, $F_{\mathrm{D}}$ is much smaller than in turbulent flows.

• The aero-dynamical buoyancy is caused by the difference of the flow velocities above and below the body. This difference is influenced by the geometrical shape of the body and can be explained by the superposition of a laminar flow and turbulent effects (circulation).

Problems

8.1

Estimate the force that a horizontal wind with a velocity of $100\,\mathrm{k}\mathrm{m}/\mathrm{h}$ (density of air $=1.225\,\mathrm{k}\mathrm{g}/\mathrm{m}^{3}$) exerts ($\varrho=1.225\,\mathrm{k}\mathrm{g}/\mathrm{m}^{3}$; $c_{\mathrm{D}}=1.2$)

• a) on a vertical square wall of $100\,\mathrm{m}^{2}$ area

• b) on a saddle roof with $100\,\mathrm{m}^{2}$ area and length $L=6\,\mathrm{m}$ and a cross section that forms an isosceles triangle with $\alpha=150^{\circ}$.

8.2

Why can an airplane fly „on the head“ during flight shows, although it should experience according to Fig. 8.41 a negative buoyancy?

8.3

Why do the streamlines not intermix in a laminar flow although the molecules could penetrate a mean free path Λ into the adjacent layers?

Hint: Estimate the magnitude of Λ in a liquid.

8.4

Prove the relation (8.36b) using the component representation.

8.5

A cylinder is filled with a liquid up to the height H. The liquid can flow out through a pipe at height h (Fig. 8.54)

Fig. 8.54

To Probl. 8.5

• a) Calculate for an ideal liquid (no friction) the position $x(H)$ where the outflowing liquid hits the ground and the velocity $v_{x}(H)$ and $v_{z}(H)$ for z = 0. Compare this result with the velocity of a free falling body starting from $z=H$.

• b) What is the function $z(t)$ of the liquid surface in the cylinder with radius R for a liquid with the viscosity η streaming through a pipe with length L and radius $r\ll R$ at the height z = 0?

8.6

A pressure gauge as shown in Fig. 8.10c is placed into flowing water. The water in the stand pipe rises by $15\,\mathrm{c}\mathrm{m}$. The measurement with the device of Fig. 8.10a shows a pressure of $p=10\,\mathrm{m}\mathrm{bar}$. How large is the flow velocity?

8.7

A funnel with the opening angle $\alpha=60^{\circ}$ is filled with water up to the height H. The water can flow into a storage vessel with volume V through a horizontal pipe at the bottom of the funnel with length L and inner diameter d.

• a) What is the height $H(t)$ in the funnel as a function of time?

• b) What is the total flow mass $M(t)$?

• c) After which time is the funnel empty for $H=30\,\mathrm{c}\mathrm{m}$, $d=0.5\,\mathrm{c}\mathrm{m}$, $L=20\,\mathrm{c}\mathrm{m}$, and $\eta=1.002\,\mathrm{m}\mathrm{Pa}\cdot\mathrm{s}$?

• d) After which time is the storage vessel with a volume V = 4 litre full, if the water in the funnel is always kept at the height H by supplying continuously water?

8.8

A water reservoir has at $\Delta h$ below the water surface a drain pipe with inner diameter $d=0.5\,\mathrm{c}\mathrm{m}$ and length $L=1\,\mathrm{m}$ which is inclined by the angel α below the horizontal.

• a) How much water flows per second through the pipe for a laminar flow with $\eta=10^{-3}\,\mathrm{Pa}\cdot\mathrm{s}$ and $\Delta h=0.1\,\mathrm{m}$?

• b) Above which angle α the flow becomes turbulent if the critical Reynolds number is 2300?

8.9

What is the minimum diameter of a horizontal tube with $L=100\,\mathrm{m}$ to allow a laminar flow of water of $1\,\mathrm{l}\cdot\mathrm{s}^{-1}$ from a vessel with a water level $20\,\mathrm{m}$ above the horizontal tube?

8.10

What is the vertical path $z(t)$ of a ball with radius r falling through glycerine $(\eta=1480\,\mathrm{m}\mathrm{Pa}\cdot\mathrm{s})$ if it immerses at t = 0 and z = 0 into the glycerine with the initial velocity $v_{0}=2\,\mathrm{m}/\mathrm{s}$

• a) for $r=2\,\mathrm{m}\mathrm{m}$,

• b) for $r=10\,\mathrm{m}\mathrm{m}$?

8.11

Derive the Helmholtz equation (8.39) starting from (8.36a).