PHYSICAL PROPERTIES

1. PHYSICAL PROPERTIES The main distinction between fluids is according to their aggregate state : • Liquids (possess the property of free surface) • Gases (expand to completely fill available space limited by impermeable boundaries)

2. PHYSICAL PROPERTIES In fluid mechanics we imagine continuously distributed mass of arbitrary substance within space - continuum hypothesis (Mass exists in each point of space). It is possible to divide continuum into infinitesimal volumes, and not to lose its physical properties. Unlike solids, fluids will deform independent of extent of applied force.

3. PHYSICAL PROPERTIES Fluids are also divided intohomogeneous and heterogeneous. Homogeneous are those that in each point of space have the same value of a given physical quantity. Liquids even at very high pressures achieve only very little change in volume. In most practical engineering cases liquids are treated as incompressible (changes in pressure of 1 bar at room temperature results in a change of water volume of only 0.005%). In some particular engineering problems (e.g.. “water hammer”) it plays an important role and can not be ignored.

4. PHYSICAL PROPERTIES Generally, gases should be treated as compressible (e.g.. air). In some engineering problems (air flow under atmospheric conditions and room temperature, with the speed < 50 m/s) gases can be observed as incompressible. Density is by definition related to “mass” density  , meaning uniformly distributed mass of fluid m involume V. (dimension M/L3 kg/m3) (Generally  depends on pressure p (N/m2) and temperature T (K), writing  = f(p,T)).

5. PHYSICAL PROPERTIES The total change in the density of liquids can be described with partial changes at constant temperature and constant pressure: T – isothermal compressibility coefficient(1/Pa) P – thermal elongation coefficient(1/K) For incompressible fluids the relationship T = P = 0 is valid.

6. PHYSICAL PROPERTIES Between elasticity modulus of liquid EF and compressibilitycoefficient relation EF = 1/T is also valid. In solving engineering problems one uses EF 2*109 Pa for water. Barotropic fluids - compressible fluids with the relation given by  = f(p). Adiabatic process – exactly defined mass of fluid that is insulated from the environment so there is no heat exchange with environment.

7. PHYSICAL PROPERTIES It is known from experience that any solid body immersed in fluid flow experiences force ; equally any solid body moving through fluid in repose. That force is a consequence of viscosity, or internal friction, as basic properties of the real fluids. Due to interaction of neighboring fluid particles deformation takes place as a result of stresses caused by friction.

8. PHYSICAL PROPERTIES Friction is the cause of mechanical energy losses. In some cases of real fluid flow, energy losses can be ignored due to their minor influence (ideal or inviscid fluids). Real liquids are divided into Newtonian fluids and anomalous viscous liquids. In fluid mechanics, Rheology is the study of the relationship between stresses in the fluid and the speed of deformation (caused by friction).

9. PHYSICAL PROPERTIES Rheological diagram defines the relationship between shear stress and velocity changes perpendicular to solid boundary. Newtonian liquids (1a), Bingman liquids (1b), structurally viscous fluids (2a,b), dilatation liquids (3a,b).

10. PHYSICAL PROPERTIES During the movement of fluid there is internal friction (viscosity) between adjacent particles of liquid. That friction is approximately independent on existing normal stresses and proportional to the velocity difference between two adjacent fluid particles. Velocity profile between two infinitely long parallel plates separated by constant and small vertical distance h.The lower plate is at rest, while the one above moves with the velocity v0 (pressure is constant everywhere, fluid particle velocity at the contact with lower and upper plate are v = 0 and v = v0).

11. PHYSICAL PROPERTIES Due to friction (internal resistance) upper “faster” fluid particle is decelerated and lower “slower” fluid particle is accelerated. The corresponding tangential stress is defined with constitutive Newtonian equation for fluids: Proportionality coefficient  is called dynamic viscosity and has the unit Pas. Dividing the coefficient of dynamic viscosity with the density gives coefficient of kinematic viscosity (independent on density)  =  / m2/s.

12. PHYSICAL PROPERTIES Attractive molecular forces of both liquids act on the contact surface of two immiscible liquids, or a liquid and a gas. If a drop of lower density fluid is set above the higher density liquid, the drop will retain its shape, or will be spilled in a thin film over the surface of the denser fluid (water drop on mercury or oil drop on water).

13. PHYSICAL PROPERTIES Molecules of liquid in rest are exposed to mutually attractive force with the influence radius rM =10-7 cm. Attraction forces of gas molecules on liquid molecules are negligibly small.

14. PHYSICAL PROPERTIES Net force on liquid molecule at the distance a > rM from the contact surface with gas is zero (Forces acting with the same magnitude in all direction FM = 0). On the other hand, a liquid molecule at distance a < rM experiences net force FM 0 FM increase as a decrease. Finally, at the contact surface between liquid and gas remain only minimum number of molecules required for the formation of free surface.

15. PHYSICAL PROPERTIES Minimum surface required to envelop some volume of liquid is achieved in the form of a droplet. Retention of the droplet form is possible only if there is a certain state of stress in the contact surface with the gas (surface tension). The stresses at the contact surface are called capillary stresses, labeled with symbol  and having the unit N/m. Thin tubes in which the effect of capillarity is highly expressed are called capillaries.

16. PHYSICAL PROPERTIES Stresses  on a curved segment of contact surface dA have resultant force dFnthat is perpendicular to that surface. Resultant dFn is proportional to the surface curvature. Resultant pressure pKPa is defined with relation pK = dFn /dA and equation:

17. PHYSICAL PROPERTIES If a liquid is in contact with a solid boundary its molecules are under the influence of both fluids (gas and liquids), as well as under the influence of solid boundary (adhesive force). If the attractive force between solid wall and fluid molecules is much stronger than the attractive forces between the molecules of the liquid, the liquid near the solid walls has a tendency to spread on it. a) water and glass b) mercury and glass

18. PHYSICAL PROPERTIES

19. HYDROSTATIC The state of equilibrium is related to fluid at absolute or relative rest. According to fluid definition, it is possible only if shear stresses are absent and normal stresses (pressure) are present. Pressure is a scalar with the magnitude dependent on position p(x,y,z). As a first step, we analyze pressure distribution on fluid particle at rest (equilibrium of external forces).

20. HYDROSTATIC On a fluid particle in z axis direction acts mass force (weight) and surface force (pressure) : Equilibrium is achieved if all external forces cancel out: Adopting that pressure difference between “upper” and “lower” surface is given by p one gets:

21. HYDROSTATIC After the transition z0 one gets the equality : In hydrostatic condition there is a pressure gradient in vertical direction and it acts against the direction of gravity. Therefore, pressure increases with increasing depth, linearly dependent on fluid density . Mass force is not present in horizontal direction, so pressure change in that direction does not exist. To obtain the absolute amount of pressure it is necessary to integrate the above expression by variable z.

22. HYDROSTATIC For the constant of integration one can choose the absolute zero pressure p0= 0 Pa (vacuum) or relative zero pressure patm(standard atmospheric pressure) that is generally used in technical application (prel= paps- patm ). Standard atmospheric pressure is defined at 15 0C and zero elevation (sea surface) paps= 1,013 bar = 1,013*105 Pa. In many engineering problems the fluid layer is so thin that density can be accepted as constant along the vertical. Consequently, pressure increase is linear :

23. HYDROSTATIC Pressure distribution diagram with adoption of atmospheric pressure at free surface (integration constant p0= patm) can be drawn for horizontal and vertical components.

24. HYDROSTATIC Dividing p with g one gets the so-called pressure head . In the sum with geodetic datum z (from some referential point) one gets the so-called piezometric head. Piezometric head is constant for arbitrary point within the fluid domain as long as fluid is at rest.

25. HYDROSTATIC If liquid density is not uniform along the vertical column it is necessary to carry out the integration as given below:

26. HYDROSTATIC Pressure is - according to definition - infinitesimal force dF that acts on infinitesimal surface dA. The total pressure force is obtained by integration over entire surface A (made up of infinitesimals dA). In addition to the intensity of hydrostatic pressure force we are also interested in position of force vertex. Let’s analyze the general case of arbitrary surface area A in a plane at an angle  to the free surface horizontal plane. With h we label depth (vertical distance) from free surface up to some point, and with  the coordinate in the plane where observed surface is situated.

27. HYDROSTATIC Water depth can be defined as a function of coordinate : Total force is calculated with the aim of integration: Moment of the surface is expressed by integral:

28. HYDROSTATIC This gives an expression for the total pressure force : where: hT = Tsinthe depth of observed surface A centroid, pTpressure in the point of observed surface Acentroid at the depth hT. The acting point of total force FP is derived from condition of moment balance .

29. HYDROSTATIC The sum of infinitesimal moments dM (pressure forces dF multiplied with corresponding arms) is equal to resultant moment (total pressure force FP multiplied with resultant arm H):

30. HYDROSTATIC Applying the Steiner rule: one gets: where: Imoment of inertia for surface A around  axes (through the origin of coordinate system) ITmoment of inertia for surface A around  axes (through the centroid of surface A)

31. HYDROSTATIC In solving some practical problems one would benefit from using the force components instead of total force: Total pressure force in horizontal direction FPx is obtained multiplying the pressure pTx = ghTx at the point of surface projection AX centroid and surface projection area AX :

32. HYDROSTATIC On the upper side of the immersed body the pressure acts with intensity gh, while on the lower side a pressure has intensity g(h + h). Pressure difference at the immersed body surface is present everywhere , so after the integration over the entire surface one gets the so-called buoyancy force:

33. HYDROSTATIC Vertical component of total pressure force is equal to the weight of water column above the observed surface (up to the water free surface): The division of total pressure force on horizontal and vertical components is an engineering adaptation in solving problems with pressure action on curved surfaces.

34. HYDROSTATIC - relative equilibrium If a fluid is contained in a vessel which is at rest, or moving with constant linear velocity, it is not affected by the motion of the vessel; but if the container is given a continuous acceleration, this will be transmitted to the fluid and affect the pressure distribution within. Since the fluid remains at rest relative to the container, there is no relative motion of the particles of the fluid and, therefore, no shear stresses, fluid pressure being everywhere normal to the surface on which it acts. Under these conditions the fluid is said to be in relative equilibrium.

35. HYDROSTATIC - relative equilibrium Although it sounds paradoxical, analysis of the systems in relative equilibrium belongs thematically to hydrostatic chapter. An observer who travels with the liquid in the relative equilibrium observes the fluid as if it were in rest. Accordingly, external forces on the liquid are in equilibrium. External forces again consist of pressure force and weight. Novelty is the existence of pressure gradient in horizontal Direction.

36. HYDROSTATIC - relative equilibrium Partial change of pressure in x-direction is given by: Analog, for 3-D valid notation is: If we are moving through the space along the line of the same pressure (isobar), total change in pressure is equal zero:

37. HYDROSTATIC - relative equilibrium For the moving system at relative equilibrium holds : Free surface (water table) always represents the surface of the same pressure (p=konst.=p0 = patm). Acceleration vector is always perpendicular to that surface. We should remind ourselves that liquid at absolute rest (non movable cane filled with liquid) has horizontal water table due to the presence of only one acceleration vector (gravity) acting in vertical direction.

38. HYDROSTATIC - relative equilibrium At a constant change of vehicle speed in time and direction beside the mass force in the vertical direction (gravitational acceleration aZ = g ) coexist another mass force in horizontal direction (aX0). For the free surface one has to apply: which after integration gives the equation of the free surface:

39. HYDROSTATIC - relative equilibrium By setting the coordinate system origin at free surface in the middle of the vehicle (intersection of free surface lines in total and relative equilibrium) equation of water table reads: To determine the pressure in an arbitrary point of liquid domain, at vertical distance h from the free surface, one can use equations:

40. HYDROSTATIC - relative equilibrium Next example of relative equilibrium is the case of vessel that rotates around its axis at a constant angular velocity . The function of free surface is one more time obtained from the condition that the resultant vector of mass forces is perpendicular to it.

41. HYDROSTATIC - relative equilibrium After the integration we get the equation: After dividing by g and adoption ofz2 as integration constant: Constant z2 is defined according to adopted position of coordinate system origin (at intersection of free surface and vessel axis before the rotation). It means that after the onset of rotation volume integral holds:

42. HYDROSTATIC - relative equilibrium Adopting the relation z1 – z2 = z one gets: Finally, equation of free surface for the liquid in vessel that rotates with constant angular velocity  around its axis is given by: