# Fluid mechanics

The fluid mechanics , fluid mechanics and fluid mechanics is the science of the physical behavior of fluids . The knowledge gained in fluid mechanics are the laws of flow processes and serve to solve flow problems in the design of components through which flow is flowing as well as the monitoring of flows . It is used in mechanical engineering , chemical engineering , water and energy management , meteorology , astrophysics and medicine, among others . It finds its basis in continuum mechanics and thermodynamics , i.e. classical physics .

## Historical development

Fluid mechanics is based on continuum mechanics , physics and differential calculus , the historical development of which can be looked up there. At this point the specific fluid mechanical development should be sketched.

Archimedes (287–212 BC) dealt with fluid mechanics issues ( Archimedes principle , Archimedes screw ). Sextus Iulius Frontinus (approx. 35–103 AD) documented his knowledge of water supply in antiquity, over a thousand years before Leonardo da Vinci (1452–1519) dealt with flow processes.

Galileo Galilei (1564–1642) gave impetus to experimental hydrodynamics and revised the concept of vacuum introduced by Aristotle . Evangelista Torricelli (1608–1647) recognized the cause of the air pressure in the weight of the earth's atmosphere and connected the horizontally ejected jet of liquid with the laws of free fall ( Torricelli's law of discharge ). Blaise Pascal (1623–1662) dealt with, among other things, hydrostatics and formulated the principle of all-round pressure propagation. Edme Mariotte (1620–1684) made contributions to problems of liquids and gases and set up the first constitutive laws. Henri de Pitot (1695–1771) studied dynamic pressure in currents.

Isaac Newton published his three-volume Principia with the laws of motion in 1686 and also defined the viscosity of an ideal ( Newtonian ) liquid in the second book . Daniel Bernoulli (1700–1782) founded hydromechanics by combining pressure and speed in the energy equation named after him , and Leonhard Euler (1707–1783) formulated the equations of motion for ideal liquids . From now on, knowledge could also be gained by examining the mathematical equations. Jean-Baptiste le Rond d'Alembert (1717–1783) introduced Euler's point of view and complex numbers in potential theory , derived the local mass balance and formulated d'Alembert's paradox , accordingly none of the flow of ideal liquids on a body Force is exerted in the direction of the flow (which Euler already proved). Because of this and other paradoxes of frictionless flows it was clear that Euler's equations of motion had to be supplemented.

Claude Louis Marie Henri Navier (1785–1836) and George Gabriel Stokes (1819–1903) extended Euler's equations of motion to include viscous terms to the Navier-Stokes equations , which model flows realistically. Giovanni Battista Venturi (1746–1822), Gotthilf Heinrich Ludwig Hagen (1797–1884) and Jean Léonard Marie Poiseuille (1799–1869) carried out experimental studies in currents. William Froude (1810–1879) determined the swimming resistance of ships, Ernst Mach (1838–1916) carried out pioneering work in supersonic aerodynamics, Lord Rayleigh (1842–1919) examined hydrodynamic instabilities and Vincent Strouhal (1850–1922) examined the excitation of vibrations by shedding vortices . Hermann von Helmholtz (1821-1894) formulated the eponymous vortex theorems and established by mathematically worked out studies on hurricanes and storm scientific meteorology . Further groundbreaking work was presented by Osborne Reynolds (1832–1912, Reynolds equations , Reynolds number ) and Ludwig Prandtl (1875–1953, including on the hydrodynamic boundary layer ).

Andrei Nikolajewitsch Kolmogorow (1903–1987) expanded the theory of turbulent flow . From the middle of the 20th century, flow measurement technology and numerical fluid mechanics developed so far that solutions to practical problems can be found with their help.

## methodology

The subject of fluid mechanics is the movement of fluids, static, flowing or flowing media. The search for laws of motion and solutions to flow problems uses three methods:

Analytical methods
Laws are formulated in the form of equations that can be treated with the help of applied mathematics.
Experimental methods
The phenomenology of the flow processes is explored with the aim of finding out regularities.
numeric methods
With detailed insight into complicated and short-term flow processes, the calculations support and complement the analytical and experimental methods.

The complexity of the subject makes the combined use of all three methods necessary to solve practical flow problems.

## Sub-areas

### Fluid statics

Fluid statics consider fluids at rest , with hydrostatics assuming incompressibility, which water exhibits to a good approximation. The pressure distribution in stationary liquids and the resulting forces on container walls are of interest here, see picture. Floating bodies experience a static buoyancy and the question of the conditions under which the body is stable in swimming is of interest . Thermal effects are of secondary importance here.

The aerostatics considered the laws in calm atmosphere or the Earth's atmosphere and here density changes and thermal effects are crucial. For example, the stratification of the atmosphere and the pressure and temperature distribution over the altitude in the earth's atmosphere are considered.

### Similarity Theory

The similarity theory is concerned with drawing conclusions about an interesting but experimentally inaccessible (real) system from a known and accessible (model) system. B. bigger or smaller, faster or slower or only differs quantitatively from the model system in other dimensions, see picture. Two flows are kinematically similar if they perform similar spatial movements. The prerequisite for this is that there are similar boundary conditions ( geometric similarity ) and similar forces act on the fluid elements, which means dynamic similarity . The similarity considerations are also applied to heat transport problems with thermal similarity . The similarity theory was founded in 1883 by Osborne Reynolds in the form of Reynolds' law of similars, which states that the flows on the original and on the model are mechanically similar if the Reynolds numbers match.

### Stream filament theory

The streamline theory considers flows along an (infinitesimally) thin stream tube formed by streamlines , in which the state variables speed, pressure, density and temperature can be assumed to be constant over the cross section of the streamline . The integral forms of the basic equations can be applied to these volumes in order to develop further solutions to flow problems. A stationary flow area consists of stream filaments, so that it is possible to describe the global properties of the flow with the properties of the stream filaments. A prominent application is the flow through pipes and nozzles . The entirety of the one-dimensional flows of water are summarized under the collective name of hydraulics . The Fluid Power and Fluid contact the hydraulics and pneumatics to to transfer energy or process signals.

### Potential currents

In potential flows, the velocity field results from the derivation of a velocity potential , which is why such flows are fundamentally free of friction and rotation . A laminar flow at low Reynolds numbers follows a potential flow as a good approximation if the fluid dynamic boundary layer at the edges of the flow does not play an essential role. The potential theory is used in the layout and design of aircraft. Potential flows are relatively easy to calculate and allow analytical solutions to many flow problems.

Another idealization, which allows rotation but only considers incompressible media, allows the introduction of a stream function . However, this can only be used in plane or as a Stokes current function in three-dimensional, axially symmetric cases. The contour lines of the stream functions are streamlines.

In level, density-stable and rotation-free flows, the velocity field can be expressed with complex functions and thus their far-reaching properties can be used. With the help of this theory, the first lift-generating wing profiles could be developed at the beginning of the 20th century, see picture.

### Gas dynamics

The subject of gas dynamics is the rapid flow of density variable fluids found in aircraft and in nozzles . These currents are characterized by the Mach number M. Compressibility only becomes significant from Mach numbers greater than 0.2, so that high Reynolds numbers are then present and viscosity terms and gravitational forces are negligible. The currents are also faster than the heat transport, which is why adiabatic changes of state can be assumed. The laws are derived with the stream filament and similarity theory. A special phenomenon that can occur here is the shock wave and the compression shock , the most famous representative of which is the sound barrier .

### Fluid dynamics

Fluid dynamics is the area that deals with moving fluids. Analytical solutions can only be achieved by restricting them to one or two dimensions, to incompressibility, simple boundary conditions and to small Reynolds numbers, where the acceleration terms can be neglected compared to the viscosity terms. Although such solutions are of little practical relevance, they nevertheless deepen the understanding of flow processes.

With small Reynolds numbers, the viscosity of the fluid is able to dampen small fluctuations in the flow variables, so that a laminar flow, which may also be time-dependent, is then stable against small disturbances. With an increasing Reynolds number, this damping mechanism is overwhelmed and the laminar flow changes into an irregular turbulent flow . Turbulence research achieves insights into such flows through statistical considerations.

Conversely, with large Reynolds numbers, the viscosity terms are small compared to the acceleration terms and the influence of the boundary conditions on the flow is limited to areas close to the wall. The boundary layer theory founded by Ludwig Prandtl deals with these .

The aerodynamics studies the behavior of bodies in compressible fluids (eg, air) and determined forces and moments acting on flow around the body. Aerodynamics includes predicting wind forces on buildings, vehicles and ships.

The field of knowledge about wave movements in fluids deals with the temporal and spatial movements of a fluid around a mean position of rest. The aeroacoustics deals with the laws of such waves - sound waves - in the air. The hydromechanics distinguishes u. a. the gravity waves , the higher Stokes waves, see picture, the small capillary waves and the aperiodic solitons . In fluid dynamics, the causes, properties and the basic equations of these wave movements are examined.

Multiphase flows with solid, liquid and / or gaseous components are the flow forms that occur most frequently in nature and technology and are therefore of particular relevance. On the one hand, the mixture can already be represented in the continuum model, so that the mixture is present in every fluid element, which has advantages when considering large-scale movements. On the other hand, the flow of each phase can be described separately and the total flow then results from the interaction of the phases at their interfaces. Small-scale effects are in the foreground here.

Leachate flows through porous media are of interest in hydrogeology and filter technology . The surface tension , which is otherwise of secondary importance in currents, is decisive for the movement here. Because the pore shape of the solid phase is unknown, models are used that lead to the Richards equation .

### Linear stability theory

This subject investigates the extent to which the state of motion of a liquid is stable to small disturbances. The flow is considered at a boundary layer, which can be with a wall ( hydrodynamic boundary layer ) or with a liquid with other properties. In the case of instabilities, fluctuations in this boundary layer can lead to qualitatively different states, which often have clear structures (see Kelvin-Helmholtz instability in the picture).

### Flow measurement technology

Areas of application for flow measurement technology are research and development, where it is necessary to examine or optimize flow processes. However, flow measurement technology is also an essential component for process control in industrial plants in the chemical or energy industry. Reliable information about the properties of turbulent flows can only be obtained through flow measurement technology.

Of particular interest are the fundamental quantities of speed, pressure and temperature. Measurements can be taken with measuring probes placed in the flow. Pitot tubes measure the total pressure in the fluid, from which conclusions can be drawn indirectly about the speed. The thermal anemometer is another indirect speed measurement method. The downside to these indirect measurement methods is that the measured signal does not depend solely on the speed, but also by other state variables which must therefore be known.

Methods such as Particle Image Velocimetry and Laser Doppler Anemometry (see picture) allow direct and local speed measurement without probes. In aeroacoustics in particular, it is not the average values ​​that are of interest, but rather the fluctuation values ​​of the pressure, in particular the spectral power density , which is obtained through further signal processing .

### Numerical fluid mechanics

The performance of the computer allows to solve the basic equations in realistic boundary value problems and the achieved, realistic results have made numerical fluid mechanics an important tool in fluid mechanics. The numerical methods have established themselves in the aerodynamic design and optimization, because they allow a detailed insight into the flow processes, see picture, and investigation of model variants.

The methods known from applied mathematics for the solution of ordinary differential equations provide the flow area with a “numerical grid” in preparation. Potential flows require the least amount of effort, and the Euler equations also allow relatively coarse grids. The boundary layers and turbulence, which are important when applying the Navier-Stokes equations, require a high spatial resolution of the grid. In three dimensions, the number of degrees of freedom increases with the third power of the dimension, so that even in the 21st century the effort for direct numerical simulation in applications in vehicle development is not justifiable. Therefore, turbulence models are used that allow the necessary resolution to be reduced. Nevertheless, systems with tens of millions of equations for several thousand iteration or time steps have to be solved, which requires a computer network and efficient programming .

## Interdisciplinary fields of work

### Rheology

Rheology or flow science is an interdisciplinary science that deals with the deformation and flow behavior of matter and therefore also affects fluid mechanics. The phenomenological rheology is concerned with the formulation of material models that Strukturrheologie seeks to explain the macroscopic material behavior from the microscopic structure of the materials, and the rheometry provides techniques for the measurement of the rheological properties such. B. the viscosity , ready.

### Fluid energy machines

A discipline working together with mechanical engineering seeks to derive macroscopic values ​​of the flows with integral forms of the basic equations, such as volume flows, forces, work and performance. These quantities are of particular interest in engineering problems in fluid power machines. One of the first results in this area was formulated by Leonhard Euler in the Euler turbine equation named after him .

### Microfluidics

Microfluidics is the branch of microsystem technology that examines the flow around objects or flows through channels with dimensions smaller than one millimeter, see picture. The continuum mechanical treatment of flow and transport processes on this length scale is in many cases not easily possible. Corrections to the equations or even molecular dynamics simulations are necessary in order to correctly reproduce the flow processes. A prominent application is the print head of an inkjet printer . But the construction of a complete analysis laboratory on a chip ( English lab-on-a-chip for "laboratory on a chip" or micro-total-analysis system for "micro-complete analysis system") requires knowledge of the flow and transport processes on the microscale.

### Bio fluid mechanics

Bio-flow mechanics deals with the flow inside and around living bodies, whose characteristic feature is, among other things, that they are bordered by flexible and structured surfaces. The locomotion of protozoa, tadpoles and fish up to whales in the water is examined. When moving through the air, especially the flight of birds is explored. The transport of heat and substances in living beings during breathing, in the blood and lymphatic system and the water system are also of interest in medicine.

### Magnetohydrodynamics

Magnetohydrodynamics (MHD) takes into account the electrical and magnetic properties of liquids, gases and plasmas and also examines the movement under the effect of the fields generated by the medium itself and the movement in external fields. The equations of motion are the Euler equations extended by electrodynamic forces, the solution of which can be very complicated. However, making additional assumptions can simplify the equations to make them easier to solve. The assumption that the electrical conductivity of the plasma is infinitely high, therefore it has no electrical resistance, leads to the "ideal MHD" in contrast to the "resistive MHD" with finite conductivity. Typical applications of magnetohydrodynamics are the flow control and flow measurement in metallurgy and semiconductor - crystal breeding and the description of plasmas in stellar atmospheres and fusion reactors .

## Continuum mechanical fundamentals

From the perspective of statistical mechanics, flows can be viewed as particle flows or as continuum flows . The latter approach comes from continuum mechanics , in which the molecular structure of the fluids is disregarded and they are approximated as a continuum in which the physical properties are continuously smeared over space. This phenomenological approach enables realistic predictions to be formulated efficiently. The kinematic, physical and constitutive continuum mechanical equations relevant for fluid mechanics are summarized below.

### kinematics

Fluid mechanics uses Euler's approach , which examines the physical quantities present at a fixed point in space. Because the laws of physics relate to material points (here: fluid elements) and not to points in space, the substantial derivative must be used for the time derivative . This consists of a local and a convective part:

${\ displaystyle {\ dot {f}}: = {\ frac {\ mathrm {D} f} {\ mathrm {D} t}}: = {\ frac {\ partial f} {\ partial t}} + \ operatorname {grad} (f) \ cdot {\ vec {v}} = {\ frac {\ partial f} {\ partial t}} + ({\ vec {v}} \ cdot \ nabla) f \ ,.}$ The field f transported by the fluid can be scalar or vector valued and, like the speed, depends on the location and time. The partial derivative is the local derivative ; H. the rate of change observed at a fixed point in space, and the second term with the gradient grad or the Nabla operator is the convective part . In the case of a vector quantity , the notation with the vector gradient is preferred in fluid mechanics . ${\ displaystyle {\ tfrac {\ partial f} {\ partial t}}}$ ${\ displaystyle \ nabla}$ ${\ displaystyle {\ vec {f}}}$ ${\ displaystyle ({\ vec {v}} \ cdot \ nabla) {\ vec {f}}}$ In fluid mechanics, the primary unknown is the velocity and its gradient, the velocity gradient${\ displaystyle {\ vec {v}}}$ ${\ displaystyle \ mathbf {l}: = \ operatorname {grad} {\ vec {v}} = {\ begin {pmatrix} {\ frac {\ partial v_ {x}} {\ partial x}} & {\ frac {\ partial v_ {x}} {\ partial y}} & {\ frac {\ partial v_ {x}} {\ partial z}} \\ {\ frac {\ partial v_ {y}} {\ partial x} } & {\ frac {\ partial v_ {y}} {\ partial y}} & {\ frac {\ partial v_ {y}} {\ partial z}} \\ {\ frac {\ partial v_ {z}} {\ partial x}} & {\ frac {\ partial v_ {z}} {\ partial y}} & {\ frac {\ partial v_ {z}} {\ partial z}} \ end {pmatrix}}}$ is a key parameter when describing flow processes. The speed components relate to a Cartesian coordinate system with x, y and z coordinates. For a fluid element with (infinitesimally) small volume dv, the rate of change in volume results ${\ displaystyle v_ {x, y, z}}$ ${\ displaystyle {\ frac {\ mathrm {D}} {\ mathrm {D} t}} (\ mathrm {d} v) = \ operatorname {Sp} (\ mathbf {l}) \, \ mathrm {d} v \ ,.}$ The track Sp of the speed gradient is thus a measure of the rate of change in volume, which is associated with a change in density due to the mass balance below. The track is equal to the divergence div of the velocity field : The velocity gradient can be split additively into a symmetrical component d and a skew-symmetrical component w : ${\ displaystyle \ operatorname {Sp} (\ mathbf {l}) = \ operatorname {div} ({\ vec {v}}) \ ,.}$ ${\ displaystyle \ mathbf {l} = \ mathbf {d + w} \ quad {\ text {with}} \ quad \ mathbf {d}: = {\ frac {1} {2}} (\ mathbf {l + l} ^ {\ top}) \ quad {\ text {and}} \ quad \ mathbf {w}: = {\ frac {1} {2}} (\ mathbf {ll} ^ {\ top}) \, .}$ The superscript denotes the transposition . The symmetrical part d is the distortion speed tensor with which with ${\ displaystyle \ top}$ ${\ displaystyle {\ dot {\ varepsilon}} _ {1}: = {\ hat {e}} _ {1} \ cdot \ mathbf {d} \ cdot {\ hat {e}} _ {1} \ quad {\ text {and}} \ quad {\ dot {\ gamma}} _ {12}: = 2 {\ hat {e}} _ {1} \ cdot \ mathbf {d} \ cdot {\ hat {e} } _ {2}}$ the expansion rate in the direction and the shear rate in the 1-2 plane are calculated, which are spanned by mutually perpendicular unit vectors (with length one) . The skew-symmetric part w is the vortex tensor , which is above ${\ displaystyle {\ dot {\ varepsilon}} _ {1}}$ ${\ displaystyle {\ hat {e}} _ {1}}$ ${\ displaystyle {\ dot {\ gamma}} _ {12}}$ ${\ displaystyle {\ hat {e}} _ {1,2}}$ ${\ displaystyle \ mathbf {w} \ cdot {\ vec {u}} =: {\ vec {\ Omega}} \ times {\ vec {u}} \ quad \ forall {\ vec {u}}}$ a vector can be assigned, which in the case of the vortex tensor is called angular velocity and indicates the rotational speed of the fluid elements around themselves. According to the above definition is calculated ${\ displaystyle {\ vec {\ Omega}}}$ ${\ displaystyle {\ vec {\ Omega}} = {\ frac {1} {2}} \ operatorname {red} {\ vec {v}} \ ,.}$ The rotation red of the velocity field is called the vortex strength or vortex vector :

${\ displaystyle {\ vec {\ omega}}: = \ operatorname {red} {\ vec {v}} = 2 {\ vec {\ Omega}} \ ,.}$ Occasionally it is also defined what does not make a significant difference. ${\ displaystyle {\ vec {\ omega}} = {\ tfrac {1} {2}} \ operatorname {red} {\ vec {v}}}$ ### Laws of nature

The continuum mechanics formulated the following, applicable to each fluid element laws of nature:

1. Mass balance: ${\ displaystyle {\ frac {\ partial} {\ partial t}} \ rho + \ operatorname {div} (\ rho {\ vec {v}}) = {\ frac {\ partial \ rho} {\ partial t} } + \ operatorname {grad} (\ rho) \ cdot {\ vec {v}} + \ rho \, \ operatorname {div} ({\ vec {v}}) = {\ dot {\ rho}} + \ rho \ operatorname {div} ({\ vec {v}}) = 0}$ 2. Impulse balance: and${\ displaystyle \ rho {\ dot {\ vec {v}}} = \ rho \ left [{\ frac {\ partial} {\ partial t}} {\ vec {v}} + \ operatorname {grad} ({ \ vec {v}}) \ cdot {\ vec {v}} \ right] = \ rho \ left [{\ frac {\ partial} {\ partial t}} {\ vec {v}} + ({\ vec {v}} \ cdot \ nabla) {\ vec {v}} \ right] = \ rho \, {\ vec {k}} + \ operatorname {div} ({\ varvec {\ sigma}}) \ ,, }$ 3. Energy balance: ${\ displaystyle {\ dot {u}} = {\ frac {1} {\ rho}} {\ boldsymbol {\ sigma}}: \ mathbf {d} - {\ frac {1} {\ rho}} \ operatorname {div} \; {\ vec {q}} + r \ ,.}$ Here, ρ is the density , an acceleration due to gravity, the Cauchy's stress tensor , u the internal energy , the heat flow, internal heat sources e.g. B. from phase transitions, " " the Frobenius scalar product of vectors and ":" that of tensors. The angular momentum balance is reduced to the requirement for the symmetry of the stress tensor${\ displaystyle {\ vec {k}}}$ ${\ displaystyle {\ boldsymbol {\ sigma}}}$ ${\ displaystyle {\ vec {q}}}$ ${\ displaystyle r}$ ${\ displaystyle \ cdot}$ ${\ displaystyle ({\ varvec {\ sigma}} = {\ varvec {\ sigma}} ^ {\ top}) \ ,.}$ ### Material models

The system of kinematic and balance equations is completed by a material model of the fluid, which specifies the stress tensor as a function of the strain rate tensor, the density or other constitutive variables. The material model of the classical material theory for the linear viscous or Newtonian fluid

${\ displaystyle {\ boldsymbol {\ sigma}} = - p (\ rho) \ mathbf {I} +2 \ mu \ mathbf {d} + \ lambda \ operatorname {Sp} (\ mathbf {d}) \ mathbf { I}}$ is the material model mainly used in fluid mechanics. Here, p is the pressure generally dependent on the density ρ, λ and μ are the first and second Lamé constants and I is the unit tensor . The strain rate tensor is generally fully occupied and then rate-dependent shear stresses occur, which macroscopically make themselves noticeable as viscosity. In combination with the momentum balance, this model provides the Navier-Stokes equations . Because the pressure, the density and the strain rate tensor are objective (see Euclidean transformation ), the Navier-Stokes equations are invariant to a change in the reference system .

In the important special case of incompressibility , which can be assumed to a good approximation at flow velocities far below the wave propagation velocity in the fluid, this equation is simplified to

${\ displaystyle {\ boldsymbol {\ sigma}} = - p \ mathbf {I} +2 \ mu \ mathbf {d}}$ and the pressure p no longer results from a constitutive relationship, but only from the boundary conditions and the momentum balance. In the case of large Reynolds numbers or away from boundary layers, the viscous components can be neglected:

${\ displaystyle {\ boldsymbol {\ sigma}} = - p (\ rho) \ mathbf {I} \ ,.}$ A fluid with this stress tensor obeys the Euler equations of fluid mechanics . If the density is a one-to-one function of the pressure, then the fluid is Cauchy-elastic and conservative, and compression work in it is reversible.

In addition to these classical material models, fluid mechanics also considers any other flowing material, including plasma , non-Newtonian fluids or ductile materials with large deformations, where the elastic deformation can be neglected compared to the plastic .