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## 1. INTRODUCTION The subject of applied fluid mechanics

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**1. INTRODUCTIONThe subject of applied fluid mechanics**• Applied fluid mechanics is constructed on the ground of mechanics, mathematics and physics. It is related also to such branches of science as strength of materials, physical chemistry. Analyzed laws of fluid mechanics are realized in hydraulic engineering, hydraulic machinery and hydraulic systems. • Fluid mechanics as a science developed along two different paths. The first purely theoretical path of a precise mathematical analysis is based on the laws of mechanics. It led to the development of so-called hydromechanics. • The next purely experimental path based on the ground of observations of a stream and pipe flow also on the data of field and laboratory investigations was directed to the solution of engineering problems. At the beginning of the 20th century this scientific activity was separated from mechanics and was named hydraulics. • Today both of them hydromechanics and hydraulics are employed together. Theoretical solutions of hydromechanics are verified by hydraulic experiments, the results of experimental investigations are generalized by the laws of hydromechanics.**HISTORICAL BACKGROUND**THEORETICAL HYDROMECHANICS HYDROMECHANICS EXPERIMENTAL HYDROMECHANICS (hydraulics) HYDROSTATICS HYDRAULICS HYDRODINAMICS HYDROSTATICSis deal with fluids under staties equilibrium, conditions HYDRODYNAMICSis researrch fluid motion laws**APPLIED**Hydraulics of open canal Pipeline hydraulics Riverine flows Flood hydraulics HYDRAULICS Drainage hydraulics Sea hydraulics Non-Newton liquids hydraulics Aerohydraulics Scepage (Ground water flow)**HISTORY**GRACE ARCHIMEES – Author of the first low of statics of fluid 287 B.C. mechanics GRACE LEONARDO da VINCH – Earliest e[perimentalist in hydraulics (1452-1519) GALILEO – To study the motion of remote (1564-1642) celestial bodies than that of a stream running at one’s feet J.NEWTON – Friction law (1642-1724) S. STEVIN – Theorical fundation (1548-1620) TORRICELLI – Declared that it is easier (1608-1647) PASCALI – Laid theorical fundation of fluid mechanics (1623-1662)**Switezerland**DANILA BERNOULI – Laid theorical fundation of fluid fluid (1700-1782) (Bernoulli equation) Germany LEONARDAS EULERIS – Euler equations Equilibrium between surface (1707-1783) (Pressure) and volumetric (gravity and inertial) forces L. NAVJE-J. STOKS – Equations of real fluid motion (open chanal flow hydraulics) ŠEZY, BAZEN – Engineers (1885) Great Britain O. REINOLDS (1842-1912) – Laid fundamentals of present day fluid mechanics L. PRANTL**Conception of a fluid**• Material which under equilibrium conditions is not capable to resist the action of tension and shear stress is called a fluid. • Both liquid and gas satisfy this definition, therefore they are called within this subject by a common name – fluids. • Beside indicated properties of a fluid there are some significant differences between a liquid and a gas. The liquid is more compact, it keeps definite volume and forms a free surface, if its volume is less than a vessel in which a liquid is contained. • Due to inability of a fluid to resist shear and tension, under the action of gravity forces located in a vessel a liquid takes its shape and forms a free surface.**Gas easily expands (and compresses also) and occupies a**vessel of any volume completely without empty space left and without forming a free surface. Volume of gas is reversely proportional to pressure. • There is a group of liquids which do not satisfy the description of a fluid. They are called non-Newtonian liquids. Under definite conditions their behavior differs from that of usual liquids. Non-Newtonian liquid dynamics is a subject of a special branch of mechanics. Only Newtonian liquid is a subject of applied fluid mechanics. • Due to the conception of a fluid the most analyzed laws are valid for both liquids and gases. The validity of gas motion laws is limited by velocity v£10m/s and pressure p£ 1 Mpa. The limits divine liquid and gas dynamics. Gas dynamics laws should be applied for gas system computation above the indicated limits of velocity and pressure only.**Forces acting a fluid**• All fluid acting forces are divided in two groups: surface and volumetric forces. • Surface forces act at any point of surface under consideration and appears in shape of distributed load. Intensity of it is characterized by stress measured in N/m2. Pressure, shear, surface tension belong to surface forces. • Surface force which is proportional to intensity of surface force stress and area of the surface A, i.e.The force is applied to definite point, which does not coincide with geometrical centre Fs = A.**Volumetric forces act at any point of volume under**consideration. Intensity of the force is estimated by acceleration characterizing the force. Gravity, inertial and centrifugal forces belong to this group of them. • Fv = ma. • The force Fv is applied to the gravity centre of fluid volume, direction of it coincides with direction of acceleration.**2. MECHANICAL PROPERTIES OF FLUIDSDensity**• It is mass of fluid volume unit. Density is denoted by and expressed as mass m and volume V ratio: • Density depends on fluid type, pressure and temperature. Density of some fluids is given in Table 2.1. kg/m3.**Compressibility**• It is property of fluid to change volume under action of pressure. Liquidcompressibility characterizes by compressibility coefficient • Compressibility coefficient p of liquid depends on liquid type, temperature and pressure. Sometimes compressibility of liquid and always of gas characterizes by bulk modules of elasticity**Even large changes of pressure cause small changes of**volume. Increment of pressure in 100 kPa , for example, results in decrement water volume in 1/20000 part only. Such small changes may be neglected in the case of usual engineering computations, therefore they are neglected and liquid is considered incompressible. • Gas is highly compressible material therefore changes of volume and density of gas must be taken in all computations. According to Clapeyron equation : where R is universal constant of gas, pressure p is directly proportional to density . For the case of isothermal process of gas (T=const) according to (2.1) and (2.5) pV=const. It means that presureof gas is reversal to the volume and is linearly related with it. Thus E=p. Compressibility coefficient p is not applied for gas computation.**Thermal expansion**• It is property of fluid to change volume from change of temperature. The property is characterized by thermal expansion coefficient : • The coefficient depends on temperature also on pressure. Thermal expansion of a fluid is taken into account in most engineering computations, especially when closed hydraulic systems are designed. Expansion of liquid in such system may cause increment of pressure above admissible limits and damage the system. • Density, compressibility and thermal expansion are related through volume V, which takes part in expression of parameters, characterizing these properties. It is seen from Table 2.1.**Table 2.1. Density , kg/m3 as function on fluid type,**temperature T and pressure p**Viscosity**• It is property of a fluid to resist shear stress appearing in fluid flow, when layers of it move with different velocities (see Fig. 1). Shear stress in dy thickness layer may be expressed as: • Proportionality between shear stress and velocity u gradient • coefficient expresses resistance of the fluid to shear stress and is called dynamic viscosity.**u=f(y)**y y u Viscosity illustration scheme • In engineering computations they use so-called kinematics viscosity which is related with dynamic viscosity through density q**Unit of the viscosity m2/s is too large for practical**application, therefore for the sake of convenience smaller units Stock 1St = 1 cm2/s and centi Stock 1cSt = 1 mm2/s. • Viscosity of a fluid great extent depends on temperature and to much smaller extent to pressure (Table 2.2). • It should be noted that change of with T is different for liquid and for gas: increment of temperature leads to decrement of liquid viscosity and increment of gas viscosity. • The phenomenon may be explained by different origin of the viscosity. Molecules of liquid are packed densely, therefore friction forces are prevalent here. Thermal expansion increases distance between molecules, friction farces and related with then viscosity decreases.**Table 2.2. Kinematics viscosity of some fluids, Cst**Molecules of gas are passkeys less densely, therefore dynamic strike type interaction between them is prevalent. Increment of temperature leads to increment of strokes type molecule interaction frequency, what results in increment of viscosity.**3**5 4 1 2 Fig. 2.4 Capillary phenomena Capillarity • It is phenomenon of difference between liquid free surface level in narrow channel 3, 4 and in open space 5 (Fig. 2.4). • The reasons of the phenomena are intermolecular forces cohesion or adhesion.**Drop of water on glass surface (Fig.2.4, 1) due to cohesion**phenomena expands and forms thin layer, expanding in radial direction. In vertical glass tube water rises up and forms free surface of concave meniscus shape above level of water outside the tube. • Due to adhesion mercury on glass surface tends to gather into compact drop, in tube – to for free surface of convex shape below level of mercury outside the tube. • Capillary phenomena are important for agriculture, also civil engineering. Capillary belt above ground water free sulface is available for plants. • The same belt when reaches excavation bottom makes a lot of problems for constructing of fundament of building. In sand capillary belt may have 5 – 10 cm height, in clay – up to 3 m.**Difference of liquid levels in tube and in surrounding area**may be computed by formula Here A is constant in area units depending on liquid type and tube material. For water and glass tube A 30 mm2.**Evaporation**• It is phenomenon of molecules division from face surface of a liquid. • Increment temperature makes motion of molecules faster and evaporation more intensive. At boiling temperature Tb forms inside mass of liquid bubbles of vapor. Evaporation becomes more intensive, energy received from external heat source equalizes with vaporization energy. • Boiling temperature depends on liquid type and pressure . • Boiling of liquid in hydraulic system is undesirable. To avoid boiling pressure at any point of the system must be larger boiling pressure, read from Tb – pbgraph according to possible temperature of the liquid.**100oC**Tb Fig. 2.5 Water boiling temperatute – pressure relationship pb pat**Ideal fluid conception**• Due to presence of many factors solution of fluid mechanics problems are rather complex. To simplify the problems introduction of assumptions is rather often. • Ideal is imaginary incompressible (p= 0) fluid having no viscosity ( = 0). • It was mentioned above that changes of volume and density of a liquid are rather small, therefore neglecting of them introduce into results of computations very small errors and do not influence on reliability of them. • Compression of gas is much more significant. When changes of pressure are not large and requirements to accuracy of computations are not high, even gas may be assumed in compressible.**Viscosity influences hydraulic loss of energy. When minor**loss, dependency of which on viscosity is very slight, is prevalent, viscosity may be neglected also and solution of a problem simplified essentially. • Conception of ideal fluid is applied in the case of constant pressure and temperature, when overflow the dams or outflow through orifice is investigated. The phenomena are often in hydraulic structures and fluid machinery fields.**pm**T Fig. 2.1 Scheme to illustrative problem 2.1 Illustrative problem 1 Strength and hermeticallity of rigid tank is tested by heating of water inside of it . Given: initial temperature T1=20oC, pressure p1=0 MPa. Coefficients of compressibility and thermal expansion of water p=0.498*10-9 m2/N and T=0.15*10-3 1/K. Find temperature T2 at which pressure in the tank reaches magnitude p2 reaches magnitude 3.0 MPa. Computation. Expansion of water due to heating goe s on simultaneously with compression due to increment of pressure. Thus change of volume V in expressions (2.2) and (2.5) are of the same magnitude.**Equalizing these expressions gives**Vpp= VTT from where T=pp/T=3*106*0.489*10-9/0.15*10-3=9.78 C. Final temperature corresponding pressure p2=3.0 MPa T2=T1+T=20.0+9.78=29.8 oC.**h**F l 1 3 Fs 2 v Fig. 2.3 Scheme to problem 2.1:1-glass plate; 2-slid surface; 3-layer of greasing oil Fg Illustrative problem 2 Glass plate is placed on inclined plane surface, which is greased by layer of an oil .**Given: dimensions of glass plate 1000x700x6 mm; density of**glass g=2700 kg/m3; thickness of greasing layer =1 mm; greasing oil kinematical viscosity =120 cSt and density =890 kg/m3; slid surface inclination angle =30o. Find glass plate slid velocity v. Computation. Projection forces acting the plate to the plane of slid surface gives equation : Sliding force Fs=Fgsin. Gravity force Fg=mpg=gVpg=glbhg, where mp and Vp are mass and volume of the plate.**Resisting to slid friction force F=Ap= Ap. Dynamic**viscosity of greasing oil let as express through kinematical viscosity and density of oil as =, plate area – through dimensions of it as Ap=lbh. Change of velocity on greasing oil layer du=v, thickness of the layer dy=. Equality between sliding and resisting to slid forces now may be recorded asglbhgsin=lbh from where