1. MECHANICAL PROPERTIES OF FLUIDS Mohamed Sherif K, HSST Physics, GHSS Athavanad

2. FLUIDS  Fluid is a substance which can flow (air & liquid)  Unlike a solid, a fluid has no definite shape of its own.  Solids and liquids have a fixed volume, whereas a gas fills the entire volume of its container.  Solids and liquids have lower compressibility compared to gases

3. 10.2 PRESSURE Average pressure (P av ) is normal force (F) acting per unit area (A). Units : N/m 2, Pascal, Atm (atmospheric pressure),Psi,bar, torr 1 atm =1.013×10 5 Pascal Pressure is a scalar quantity

4. Density (ρ) Density is defined as mass per unit volume. =, V = Its unit is kg/m 3. Density of water at 4 0 C is 10 3 kg/m 3. The relative density of a substance is the ratio of its density to the density of water at 4°C.

5. 10.2.1 Pascal’s Law Pressure inside a fluid at rest is same at all points if they are at the same height

6. 10.2.2 Variation of Pressure with Depth Consider, Pressure at point 1 = P 1 Pressure at point 2 = P 2 Mass of fluid inside cylinder = m Area of the base of cylinder = A Height of the cylinder = h Density of fluid =ρ m = ρhA ∴ P 2 − P 1 = ρgh When point 1 is open to atmosphere, P 1 = Atmospheric pressure (P a ) P 2 = P (absolute pressure) ∴ P = P a + ρgh Gauge pressure = P − P a = ρgh

7. Hydrostatic paradox

8. 10.2.3 Atmospheric Pressure and Gauge Pressure  The pressure of the atmosphere at any point is equal to the weight of a column of air of unit cross sectional area extending from that point to the top of the atmosphere.  At sea level it is 1.013 × 10 5 Pa (1 atm)  P a = ρgh  76 cm at sea level equivalent to one atmosphere (1 atm)  A pressure equivalent of 1 mm is called a torr (after Torricelli). 1 torr = 133 Pa.  In meteorology, a common unit is the bar and millibar. 1 bar = 105 Pa

9. 10.2.3 Atmospheric Pressure and Gauge Pressure

10. 10.2.4 Hydraulic Machines Pascal’s law for transmission of fluid pressure Whenever external pressure is applied on any part of a fluid contained in a vessel, it is transmitted undiminished and equally in all directions. Hydraulic lift and hydraulic brakes are based on the Pascal’s law. F 2 = PA 2 = F 1 A 1 /A 2

11. 10.3 STREAMLINE FLOW Streamline Flow is a steady flow of liquid, in which each particle of liquid follows the same path and the same velocity as that of its predecessor. (The path taken by a fluid particle under a steady flow is called a streamline) Critical velocity : Streamline flow is possible only when, velocity of flow is less than a limiting value. This velocity is called critical velocity.

12. Equation of continuity Consider streamline flow of a liquid of density ‘ρ’ through a pipe of different area of cross section. [ Note: Mass = Volume × ρ = (Area × Length) × ρ ] Mass of fluid flowing in through the large area ‘A 1 ’ in a time ‘Δt’ is given by M 1 = A 1 × (v 1 Δt ) × ρ Mass of fluid flowing out through the small area ‘A 2 ’ in a time ‘Δt’ is given by M 2 = A 2 × ( v 2 Δt ) × ρ Fluid mass flowing in = Fluid mass flowing out A 1 × (v 1 Δt ) × ρ = A 2 × ( v 2 Δt ) × ρ A 1 v 1 = A 2 v 2 Av = constant This equation is called Equation of Continuity

13. Turbulent flow when velocity of flow is greater than the critical velocity, the liquid flow becomes disorderly and zigzag and is called turbulent flow.

14. 10.4 BERNOULLI’S PRINCIPLE The work done on the fluid at left end (BC) is W 1 = P 1 A 1 (v 1 Δt) = P 1 ΔV The work done by the fluid at the other end (DE) is W 2 = P 2 A 2 (v 2 Δt) = P 2 ΔV The total work done on the fluid is W 1 – W 2 = (P 1 − P 2 ) ΔV Part of this work goes into changing the kinetic energy of the fluid, and part goes into changing the gravitational potential energy

15. 10.4 BERNOULLI’S PRINCIPLE change in gravitational potential energy is ΔU = ρgΔV (h 2 − h 1 ) The change in its kinetic energy is employing the work – energy theorem We now divide each term by ΔV to obtain This is Bernoulli’s equation

16. BERNOULLI’S THEOREM The sum of pressure energy, kinetic energy, and potential energy per unit mass is always constant for the streamline flow of a non-viscous and incompressible fluid

17. 10.4.1 Speed of Efflux: Torricelli’s Law Torricelli ‘s Law The speed of efflux (fluid outflow) from an open tank is given by a formula identical to that of a freely falling body According to equation of continuity Since A 2 >> A 1, v 2 = 0 (fluid at rest) Applying Bernoulli’s theorem

18. 10.4.1 Speed of Efflux: Torricelli’s Law Applying Bernoulli’s theorem P 1 =P a (atmospheric pressure) y 2 − y 1 = h (shown in the figure)

19. 10.4.2 Venturi-meter It is a gauge used to measure the rate of flow of fluid when fluid is steady Speed at the constriction (Using equation of continuity) According to Bernoulli’s equation Speed of the fluid at the wide neck

20. 10.4.3 B LOOD F LOW AND H EART A TTACK Accumulation of plaque constricts the artery. To drive blood through it, the activity of heart increases. Speed of blood in the region increases, lowering the inside pressure of artery. Artery may collapse due to high external pressure. Heart exerts further pressure and opens the artery to force the blood through it. Blood rushes through the opening and the internal pressure of artery drops. This leads to repeat collapse and results in heart attack

21. 10.4.4 D YNAMIC L IFT Aerofoil or lift on aircraft wing Wings of aeroplane look similar to an aerofoil. Aerofoil moving against the wind causes the streamline to crowd more above the wing than below it. Therefore, the speed on top is more than it is below it. Upward force resulting in a dynamic lift of wings balances the weight of the plane

22. 10.5 VISCOSITY It is the resistance of the fluid motion. This force exists when there is relative motion between the layers of liquid Laminar − For any layer of liquid, its upper layer pulls it forward while lower layer pulls it backward. This results in force between the layers. This type of flow is known as laminar

23. 10.5 VISCOSITY Co-efficient of viscosity, Shearing stress = Strain rate = Unit of viscosity is poiseiulle (Pl) or Nsm −2 or Pa s. Thin liquids are less viscous than thick liquids. Viscosity of liquids decreases with temperature while it increases in case of gases.

24. Stoke’s Law An object moving through a fluid drags the liquid in contact. This force between the layers of the fluid makes the body experience a retarding force. Retarding force (F) depends on velocity of the object (v) viscosity of the fluid (η) radius of the sphere (a) ∴ F = 6πηav This is known as Strokes’ law

25. Terminal Velocity (v t ) When a spherical body falls through a viscous fluid, it experiences a viscous force. The magnitude of viscous force increases with the increase in velocity of the falling body under the action of its weight. As a result, the viscous force soon balances the driving force (weight of the body) and the body starts moving with a constant velocity known as its terminal velocity Using Strokes’ law Where, ρ − Mass density of sphere σ − Mass density of fluid Terminal velocity

26. 10.6 REYNOLDS NUMBER When the rate of flow of a fluid is large, the flow becomes turbulent. An obstacle placed in the path of a fast moving fluid causes turbulence Reynolds (R e ) number implies if the flow would be turbulent or not R e = Where, ρ − Density of fluid d − Dimension of pipe v − Speed of fluid flow η − Viscosity of the fluid

27. 10.6 REYNOLDS NUMBER R e is dimensionless. R e 2000 [Turbulent flow] R e is ratio of inertial force to viscous force. Use − Turbulence promotes mixing; increases the transfer rate of mass, momentum, and energy.

28. 10.7 SURFACE TENSION

29.  The Surface tension is the property by virtue of which the free surface of a liquid behaves like elastic stretched membrane tending to contract.  Surface tension (S) is measured as the tangential force (due to the surface molecules) per unit length.  Surface Tension(S) = Force / Length.  Its unit is N/m.

30. 10.7.1 Surface Energy Surface of a liquid acts as a stretched membrane. So molecules in this layer posses’ elastic potential energy. This elastic potential energy is called surface energy. Also Surface Energy = Work done / surface Area

31. 10.7.2 Surface Energy and Surface Tension Consider a rectangle; the side ‘AB’ is movable. Dip the frame in a soap solution; then due to surface tension ‘AB’ moves through a distance ‘b’..: Work done = Tangential Force × b But, Tangential Force = Surface Tension (S) × 2l. This is because; surface tension is acting on the upper surface and lower surface of the soap film. Tangential Force = S ×2 l Work done = S × 2l ×b Surface Energy = Work done /Area Numerically, Surface Energy = Surface Tension

32. 10.7.3 Angle of Contact Angle between tangent to the liquid surface at the point of contact and solid surface inside the liquid is called angle of contact.

33. 10.7.4 Drops and Bubbles r − Radius of drop P 0 − Pressure outside the bubble P i − Pressure inside the bubble S − Surface tension of the bubble Surface energy = 4πr 2 S Let radius increase by Δ r. Then, extra surface energy = [4 π (r + Δr) 2 − 4πr 2 ] S = 8πrΔrS  (1) Energy gain in the pressure difference P i − P 0 ∴ Work done, W = (P i − P 0 ) 4πr 2 Δr  (2) At equilibrium, the energy used is balanced by the energy gained. From equations (1) and (2), (P i − P 0 ) = 2S/r For Bubbles, having two interfaces (P i − P 0 ) = 2S/r

34. 10.7.5 Capillary Rise

35. Angle of contact between water and glass is acute. Surface of water in the capillary is concave. Pressure difference between two sides of top surface, P a − P 0 = 2S/r Considering points A and B, they must be at same pressure i.e., P 0 + hρg = P a Therefore, capillary rise is due to surface tension. Height of water rise, (θ=0 0 )

36. 10.7.6 Detergents and Surface Tension Ordinary water will not remove greasy dirt. This is because water could not wet greasy dirt. When we add detergent, small glob of dirt can be captured by detergent molecule by surrounding it. This ‘new system molecules’ can be captured or wet by water molecules. So they can be removed