MECHANICAL PROPERTIES OF FLUIDS

Mohamed Sherif K, HSST Physics, GHSS Athavanad

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

10.2 PRESSURE

Average pressure (Pav) is normal force (F) acting per unit area (A).

Units : N/m2, Pascal , Atm (atmospheric pressure),Psi ,bar, torr1 atm =1.013×105 PascalPressure is a scalar quantity

Density (ρ)

Density is defined as mass per unit volume.

𝜌 = , V = 𝑀𝑎𝑠𝑠 𝑉𝑜𝑙𝑢𝑚𝑒Its unit is kg/m3.

Density of water at 40C is 103kg/m3. The relative density of a substance is the ratio of its density to the density of water at 4°C.

10.2.1 Pascal’s Law

Pressure inside a fluid at rest is same at all points if they are at the same height

10.2.2 Variation of Pressure with Depth

Consider,Pressure at point 1 = P1

Pressure at point 2 = P2

Mass of fluid inside cylinder = mArea of the base of cylinder = AHeight of the cylinder = hDensity of fluid =ρ

m = ρhA∴P2 − P1 = ρghWhen point 1 is open to atmosphere,P1 = Atmospheric pressure (Pa)P2 = P (absolute pressure)

∴ P = Pa + ρghGauge pressure = P − Pa = ρgh

Hydrostatic paradox

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 × 105 Pa (1 atm)

Pa = 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

10.2.3 Atmospheric Pressure and Gauge Pressure

10.2.4 Hydraulic Machines

Pascal’s law for transmission of fluid pressureWhenever 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.

F2 = PA2 = F1A1/A2

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.

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 ‘A1’ in a time ‘Δt’ is given byM1 = A1 × (v1 Δt ) × ρMass of fluid flowing out through the small area ‘A2’ in a time ‘Δt’ is given byM2 = A2 × ( v2 Δt ) × ρFluid mass flowing in = Fluid mass flowing outA1 × (v1 Δt ) × ρ = A2 × ( v2 Δt ) × ρA1v1 = A2v2

Av = constantThis equation is called Equation of Continuity

Turbulent flow

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

10.4 BERNOULLI’S PRINCIPLE

The work done on the fluid at left end (BC) is W1 = P1A1(v1Δt) = P1ΔVThe work done by the fluid at the other end (DE) is W2 = P2A2(v2Δt) = P2ΔVThe total work done on the fluid is W1 – W2 = (P1− P2) ΔV

Part of this work goes into changing the kinetic energy of the fluid, and part goes into changing the gravitational potential energy

10.4 BERNOULLI’S PRINCIPLE

change in gravitational potential energy is ΔU = ρgΔV (h2 − h1)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

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

10.4.1 Speed of Efflux: Torricelli’s Law

Torricelli ‘s LawThe 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 A2 >> A1,v2 = 0 (fluid at rest) Applying Bernoulli’s theorem

10.4.1 Speed of Efflux: Torricelli’s Law

Applying Bernoulli’s theorem

P1 =Pa (atmospheric pressure)y2 − y1 = h (shown in the figure)

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

10.4.3 BLOOD FLOW AND HEART ATTACK

•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

10.4.4 DYNAMIC LIFT

•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

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

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.

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 onvelocity of the object (v)viscosity of the fluid (η)radius of the sphere (a)

∴ F = 6πηavThis is known as Strokes’ law

Terminal Velocity (vt)

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 sphereσ − Mass density of fluidTerminal velocity

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 (Re) number implies if the flow would be turbulent or not

Re =

Where,ρ − Density of fluidd − Dimension of pipev − Speed of fluid flowη − Viscosity of the fluid

10.6 REYNOLDS NUMBER

Re is dimensionless.Re < 1000 [Streamline or laminar flow]2000 ≥ Re ≥ 1000 [Unsteady flow]Re > 2000 [Turbulent flow]

Re is ratio of inertial force to viscous force.

Use − Turbulence promotes mixing; increases the transfer rate of mass, momentum, and energy.

10.7 SURFACE TENSION

10.7 SURFACE TENSION

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.

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

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 × bBut, 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 lWork done = S × 2l ×bSurface Energy = Work done /Area

Numerically, Surface Energy = Surface Tension

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.

10.7.4 Drops and Bubbles

r − Radius of dropP0 − Pressure outside the bubblePi − Pressure inside the bubbleS − Surface tension of the bubble

Surface energy = 4πr2SLet radius increase by Δ r.Then, extra surface energy = [4 π (r + Δr)2 − 4πr2] S

= 8πrΔrS (1)

Energy gain in the pressure difference Pi − P0

Work done, ∴ W = (Pi − P0) 4πr2Δr (2)At equilibrium, the energy used is balanced by the energy gained.From equations (1) and (2),(Pi − P0) = 2S/r

For Bubbles, having two interfaces(Pi − P0) = 2S/r

10.7.5 Capillary Rise

10.7.5 Capillary Rise

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, Pa − P0 = 2S/r

Considering points A and B, they must be at same pressure i.e.,P0 + hρg = Pa

Therefore, capillary rise is due to surface tension.

Height of water rise, (θ=00)

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