Fluid Mechanics for MAP/Introduction

The importance of flow phenomena is out of question. Natural phenomena or technological applications are completely or partially involves flow phenomena. It can be met in a diverse range of length of time scales. Atmospheric flows and blood flows are two examples for this diversity. As a tool making specie, humankind learned also how to utilize flow phenomena. Hence, those, who deal with flowing matter, should be better equipped with theoretical understanding and capability to use experimental and numerical investigation tools.

Fluid mechanics have played an important role in human life. Therefore, it also attracted many curious people. Even in the ancient Greek history, systematic theoretical works have been done. The devlopement of governing equations of fluid flow started already in the 16th century. In the 18th and 19th century, the conservation laws for mass, momentum and energy was already known in its most general form. In the 20th century, developments were in theoretical, experimental and recently numerical. In the theoretical field, mostly solutions of the governing equations for special cases were provided. Experimental methods have been developed to measure flow velocities and fluid properties. By the developement of computers , the numerical treatment of fluid mechanical problems opened new perspectives in research. It is the common blief that in the 21th century, the activities would be most intensive in the development new experimental and numerical tools and application of those for developing new technologies.

• 
  Aquaduct (Pont Du Gard), 7th cent. BC
• 
  Greek ship (Trireme), 7th cent. BC
• 
  Helicopter design of Leonardo da Vinci

Besides theoretical considerations, exepriments and simulations are heavily used in research. If possible, most productive and accurate approach is the combination of all three methods. However, sometimes environmental conditions can be so harsh for any experimental technique that only theoretical or numerical methods can be used. For example, it is very hard or almost impossible to obtain the velocity or temperature distribution in the die casting mold or in the crucible used for crystal growth, because of very high temperatures.

Force applied on a matter creates stresses on it. Stress is simply force per unit area:

${\displaystyle \displaystyle \tau ={\frac {F}{A}}\left[{\frac {N}{m^{2}}}=Pa\right]}$

Hence the unit of stress is ${\displaystyle Pa}$. There can be normal and shear stresses in and on the matter.

Shear stress is proportional to the deformation rate of the matter, i.e. strain rate:

${\displaystyle \displaystyle \tau \propto {\frac {\delta \theta }{\delta t}}}$

${\displaystyle \displaystyle \tan {\delta \theta }={\frac {\delta u\delta t}{\delta y}}}$

${\displaystyle \displaystyle u}$ is the deformation speed. For very small deformation angles

${\displaystyle \displaystyle \delta \theta ={\frac {\delta u\delta t}{\delta y}}\ \rightarrow \ {\frac {\delta \theta }{\delta t}}={\frac {\delta u}{\delta y}}\ \rightarrow \ \tau \propto {\frac {\delta u}{\delta y}}\ \rightarrow \ \tau =\mu {\frac {\delta \theta }{\delta t}}=\mu {\frac {\delta u}{\delta y}}}$

${\displaystyle \displaystyle \mu }$ is the dynamic viscosity of the fluid.

For the same ${\displaystyle \displaystyle \tau }$ and fluid having higher viscosity ${\displaystyle \displaystyle \mu }$, the deformation rate, i.e. velocity gradient is smaller.

Dynamic viscosity is a thermodynamic property of the material and it depends on temperature and pressure. In general, viscosity of liquids drop by increasing temperature, whereas that of gases increases. The viscosities of liquids and gases increase with increasing pressure.

${\displaystyle \displaystyle \mu =f\left(T,P\right)\left[Pa\cdot s\right]}$

Often dynamic viscosity is normalized by the density of the fluid and this quantity is called “kinematic viscosity”:

${\displaystyle \displaystyle \upsilon ={\frac {\mu }{\rho }}\left[{\frac {m^{2}}{s}}\right]}$

One can judge the dominance of inertial effects to viscous effects by using a dimensionless number, namely Reynolds number:

${\displaystyle \displaystyle Re={\frac {\rho U_{c}l_{c}}{\mu }}={\frac {U_{c}l_{c}}{\upsilon }}}$

${\displaystyle \displaystyle U_{c}}$ and ${\displaystyle \displaystyle l_{c}}$ are characteristic velocity and length scales of the flow.

When one tries to deform a piece of material, some of the above properties appear depending on the amplitude and duration of the applied stress.

• Long time application of weak stress: Solids initially deform and then resist to deform. Fluids deform (flow) continuously.

• Short time application of weak stress: If deformation follows the stress, material is elastic!!!. If deformation rate follows the stress, material is viscous.

• Application of high stress: After a certain stress (yield stress), some solids start to deform irreversibly. These are called plastic solids. There are also yield stress fluids, whose threshold stress is much lower than plastic solids.

In many technical applications, the distance between the molecules (mean free path) (${\displaystyle \displaystyle \lambda }$) are much larger than the molecular diameter. For air, ${\displaystyle \lambda }$ is around ${\displaystyle 5\times 10^{-8}m}$.

The molecules are not fixed in a lattice but move about freely relative to each other. Thus fluid density, or mass per unit volume, has no precise meaning because the number of molecules occupying a given volume continually changes. If the selected unit volume ${\displaystyle \displaystyle \delta V}$ is smaller than the cube of the mean free path between the molecules, there will be large scatter in the determination of density, since the molecules move freely relative to each other, i.e. at one instant the number of molecules in the unit volume is not constant. This effect becomes unimportant if the unit volume is large compared with, say, the cube of the molecular spacing, when the number of molecules within the volume will remain nearly constant in spite of the enormous interchange of particles across the boundaries. In other words, when ${\displaystyle \displaystyle \delta V}$ is selected such that the selected volume contains in average the number of molecules, the density converges to a level. The acceptable size of the unit volume for many liquids and gases is about ${\displaystyle \displaystyle 1\mu m^{3}}$. Over this value, the medium can be accepted as continuum , such that the variations in space and time can be accepted to be smooth and differential equations can be written to describe the fluid motion. If, however, the chosen unit volume is too large, there could be a noticeable variation within the selected volume owing to the non-uniform bulk distribution of molecules caused by temperature and/or pressure variations in the flow field.

Surface tension phenomena occur at the interface of one liquid and another liquid, gas or a solid wall. The cohesive forces between molecules down into a liquid are shared with all neighboring atoms. Those on the surface have no neighboring atoms above, and exhibit stronger attractive forces upon their nearest neighbors on the surface. This enhancement of the intermolecular attractive forces at the surface is called surface tension.

Evaporation occurs at the liquid gas interface. When the vapor pressure of liquid is less than the liquid's saturation pressure at the given liquid temperature, the evaporation and condensation occurs at the same time on the interface.At the liquid solid interface, at a given temperature, liquids starts to boil at saturation pressure.

Instead of increasing the temperature of the liquid, one can decrease the pressure of the liquid so that it starts to boil, or so to say cavitates.

One can meet cavitation in nature and in technical application. One known example is the cavitation damage on ship propellers. An interesting natural occurrence of cavitation was observed while the snapping shrimp hunts ^[2].

Four basic types of line patterns are used to visualize flows:

• A streamline is a line everywhere tangent to the velocity vector at a given instant.
• A pathline is the actual path traversed by a given fluid particle.
• A streakline is the locus of particles which have earlier passed through a prescribed point.
• A timeline is a set of fluid particles that form a line at a given instant.

In a steady flow streamlines, streaklines and pathlines are identical.

Laminar flows are:

• smooth,
• the disturbances are damped via viscous effects,
• and they are in general deterministic.

In turbulent flows:

• flow and fluid variables show random fluctuations in time and space, i.e. the flow is stochastic
• there are eddies of velocity and length scales over a very wide range

Laminar to turbulent transition occurs when the disturbances in the flow can not be damped anymore by viscous forces. This happens when the inertia of the flow is increased and/or the flow configuration (boundaries, states of the fluid(s)) causes the generation and/or amplification of very small disturbances. As Reynolds number (Re) is the ratio of the inertial forces to viscous forces, for different types of flows, over a critical Reynolds number, transition to turbulence takes place. Below a list of simple but still technically interesting flow cases and critical Reynolds numbers are listed:

• Pipe flow: ${\displaystyle Re={\frac {U_{b}D_{pipe}}{\nu }}>2200}$
• Jet flow: ${\displaystyle Re={\frac {U_{b}D_{jet}}{\nu }}>1000}$
• Flow over a flat plate: ${\displaystyle Re={\frac {U_{\infty }\delta _{l}}{\nu }}>950}$

where ${\displaystyle U_{b}}$, ${\displaystyle U_{\infty }}$ are the bulk velocity of the fluid or the velocity of fluid approaching to the plate. ${\displaystyle D_{pipe}}$, ${\displaystyle D_{jet}}$ and ${\displaystyle l}$ are the pipe diameter, jet diameter or the length of the plate. ${\displaystyle \delta _{l}=1.72{\sqrt {\nu l/U_{\infty }}}}$ is the displacement thickness.

Density and volume change:

${\displaystyle \displaystyle \rho ={\frac {m}{V}}}$

${\displaystyle \displaystyle {\frac {d\rho }{dV}}=-{\frac {m}{V^{2}}}=-{\frac {m}{V}}{\frac {1}{V}}=-\rho {\frac {1}{V}}}$

${\displaystyle \displaystyle {\frac {d\rho }{\rho }}=-{\frac {dV}{V}}}$

The Bulk Modulus:

${\displaystyle \displaystyle E_{v}=-{\frac {dp}{\frac {dV}{V}}}={\frac {dp}{\frac {d\rho }{\rho }}}\ [Pa]}$

Large values of Ev means that the fluid is relatively incompressible.

Under standard atmospheric conditions:

${\displaystyle E_{v}=2.15\times 10^{9}\ Pa=21500\ bar\ }$

for water and

${\displaystyle E_{v}=1.42\times 10^{5}\ Pa=1.4\ bar\ }$

for air. Therefore air is 15000 times more compressible than water.

Liquids can be accepted to be incompressible in many applications. Air can be compressible, especially when there are large changes of pressure in the flow.

${\displaystyle {\frac {d\rho }{dt}}=0}$ and ${\displaystyle {\frac {d\rho }{dx_{i}}}=0}$

Where i = 1,2,3.

Following chart covers most of the flow phenomena, which might occur in a flow problem. When one deals with a flow problem, first task is to classify the flow. Correct classification helps to choose the correct and most efficient methods to deal with this problem.