Mechanical Properties of Fluids Flashcards
Fluids
Substance that can flow
Characteristics of a fluid
- Atoms are arranged in a random manner
- Fluid cannot withstand tangential stress for an indefinite period
- Fluid has no definite shape of its own
- Can exert / withstand a force in a direction perpendicular to its surface
Fluid statics
Branch of physics that deals with the study of fluids at rest
Fluid dynamics
Branch of physics that deals with study of fluids in motion
Thrust
Total force exerted by a liquid on any surface in contact with it
Show that a liquid at rest exerts force perpendicular to surface of container at every point
- Suppose the liquid exerts a force F on the bottom surface in an inclined direction OA
- Tangential component: OC = Rcos theta
- Normal component: OD = Rsin theta
However, liquid is at rest, hence Rcos theta = 0; theta = 90 degree
Pressure
Thrust acting normally per unit area around that point
Fluid pressure
p = delta F / delta A
Density
Mass per unit volume
ro = M / V
Specific gravity
Ratio of density of substance to density of water at 4 degree celsius
Density formula (in terms of specific gravity)
Density = Specific gravity * Density of water at 4 degree Celsius
Pascal’s Law
- Pressure exerted at any point on an enclosed liquid is transmitted equally in all directions
- A change in pressure applied to an enclosed incompressible fluid is tranmitted undiminished to every point of the fluid and the walls of the containing vessel
- The pressure in a fluid at rest is same at all points if we ignore gravity
Derivation of Pascal’s Law
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Applications of Pascal’s Law
- Hydraulic Lift: Force multiplier; Small area force applied –> Larger area larger force
- Hydraulic Brakes: Piston attached to brake pedal through a lever system
Pressure exerted by a liquid column
W = Mass * g = Ah * ro * g
P = W / A = h * ro * g
Effect of Gravity on Fluid Pressure
derive
P2 - P1 = h * ro * g
* Liquid pressure is same at all points at the same horizontal level
* Pressure at any point depends upon h
* Absolute pressure P, at a depth h below the liquid surface open to atmosphere is greater than atmospheric pressure by amount h ro g
Gauge Pressure
Gauge pressure is the difference between the pressure being measured and the atmospheric pressure
Atmospheric Pressure
Pressure exerted by atmosphere
1.013 * 10^5 Nm^-2 or Pa or 1atm
Hydrostatic Paradox
Pressure exerted by a liquid column depends only on the height of the liquid column and not on the shape of the containing vessel
Mercury Barometer
- Measures atmospheric pressure
- 1m long glass tube closed at one end is filled with clean and dry mercury
- After closing the end of the tube with the thumb, the tube is inverted into a dish of mercury
Open - tube manometer
- U-tube containing some liquid
- One end of the tube is open to atmosphere and other end connected to vessel
- Total pressure P = pressure at A
Height of atmosphere
- Value of g does not change; temp remains uniform; assume density to be uniform
h ro g = pressure = 1.013 * 10^5 / 1.3 * 9.8 = around 8km
Systolic blood pressure is the top number and refers to the amount of pressure experienced by the arteries while the heart is beating.
Diastolic blood pressure is the bottom number and refers to the amount of pressure in the arteries while the heart is resting in between heartbeats.
Buoyancy
Upward force acting on a body immersed in a fluid is called upthrust / buoyancy
Centre of Buoyancy
Force of buoyancy acts through the centre of gravity of displaced fluid
Archemedes’ Principle
When a body is partially or wholly immersed in a fluid it experiences upward thrust equal to weight of fluid displaced by it and its upthrust acts through the centre of gravity of the displaced liquid
Streamline flow
When a liquid flows such that each particle of the liquid passing a given point moves along the same path and has the same velocity as its predecessor
Properties of streamlines
- No two streamlines can cross each other
- Tangent at any point on the streamline gives the direction of velocity of fluid particle
- Greater number of streamlines = Larger fluid velocty
- Fluid velocity remains constant at any point but may be different at different points of same streamline
Critical velocity
Critical velocity is the speed and direction at which a fluid flows through a conduit without becoming turbulent.
Tube of Flow
Bundle of streamlines forming a tubular region
Turbulent Flow
When liquid vel > critical vel = liquid flow becomes zig-zag
Factors affecting Critical Velocity
- Coefficient of viscosity of liquid
- Density of liquid
- Diameter of tube
vc = k (viscosity)/ (ro) (D) – do diminsional analysis - Higher viscosity and lower density = streamlined
- Lower viscosity and higher density = turbulent
Viscosity
Property of fluid by virtue of which an internal force of friction comes into play when a fluid is in motion and which opposes the relative motion between its different layers
Velocity gradient
Rate of change of velocity with distance in the direction of increasing distance (dv / dx)
Formula of force of viscosity
F = - coeff of viscosity * A * (dv / dx)
dv / dx is also the velocity gradient
Why is there a negative sign in the coeff of viscosity formula
Viscous force acts in a direction opposite to the direction of motion of the liquid
Coefficient of viscosity
Tangential viscous force required to maintain a unit velocity gradient between its two parallel layers each of unit area
Viscosity of a liquid with temp
Increase in temperature = KE increases = weaker intermolecular forces of attraction = lower viscosity
Viscosity of a gas with temp
- Due to diffusion of molecules from one moving layer to another
- Rate of diffusion of a gas directly proportional to square root of absolute temperature
- Viscosity of a gas increases with temp
Pressure on viscosity
- Increase with increase in pressure - liquids
- For water decreases with increase in pressure
- Viscosity of gases is independent of pressure
Poiseuille’s Formula
Volume of a liquid flowing out per second through a horizontal capillary tube:
Q = V / t = (pi * r^4 * pressure) / (8 * coeff. of viscosity * l)
Stokes Law
Backward viscous force acting on a small spherical body of radius r moving with uniform velocity v through fluid of viscosity is given by:
F = 6 * pi * viscosity * r * v
Terminal Velocity
Maximum constant velocity acquired by a body while falling through a viscous medium
Terminal Velocity expression
Derive
v = 2/9 * r^2 * (density of body - density of liquid) * g / viscosity
Explain how different densities of fluid affect terminal velocity
Take ro = density of body
sigma = density of liquid
1. ro > sigma = downward direction
2. ro < sigma = upward direction
3. ro = sigma = suspension in air
What is terminal velocity independent of
Height
Laminar Flow
- Velocity of flow of a liquid is less than its critical velocity, liquid flows steadily
- Each layer of liquid slides over the other layer - different lamina are sliding over one another
Velocity profile of a non-viscous liquid
Velocity of all particles at any section are same hence plane velocity profile
Velocity profile
Surface obtained by joining heads of velocity vecotrs for particles in a section of a flowing liquid
Velocity profile of a viscous liquid
Velocity of layers at axis is maximum; velocity decreases as we go towards wall of pipe and becomes zero for layer in contact with pipe
Parabolic velocity profile
Reynold’s Number
Valur decides nature of flow of a liquid through a pipe
R = ro * v * D / viscosity
Significance of Reynold’s Number
0 < Re < 2000: streamlined / laminar
Re > 3000: turbulent
2000 < Re < 3000: unstable (may change from streamlined to turbulent)
Ideal Fluid
Fluid which is non-viscous, incompressible, and flow is steady and irrotational
Critical Reynold’s Number
Exacty value at which turbulence sets in a fluid
Equation of Continuity
During streamlined flow of the non-viscous and incompressible fluid through a pipe of varying cross-section the produc of area of cross section and normal fluid velocity remains constant
av = constant
Equation of continuity proof
m1 = a1v1 delta t ro1
m2 = a2v2 delta t ro2
m1 = m2
and since it is incompressible, ro1 = ro2
a1v1 = a2v2
Atomizer
Spray liquids based on Bernoulli’s principle
Horizontal tube B decreasing pressure less than atm Pressure in container = liquid rising up and clollinding with high speed air breaks up into a fine spray
Velocity Head
Potential Head
Pressure head
Kinetic energy per unit weight of the liquid = v^2 / 2g
Potential energy per unit weight of the liquid = h
Pressure energy per unit weight of the liquid = P / (ro * g)
Bernoulli’s Principle
Bernoulli’s Principle states that the sum of pressure energy, kinetic energy and potential energy per unit volume of an incompressible, non-viscous fluid in a streamlined irrotational flow remains constant along streamline
Prove bernoulli’s principle
P + 1/2 ro v^2 + ro g h = constant
Toricelli’s Law of Efflux
Torricelli’s law of efflux, also known as Torricelli’s theorem or Torricelli’s principle, describes the speed of a fluid flowing out of a hole in a tank
Derivation of toricelli’s law of efflux
v1 = root (2gh + 2(P - Pa)/ ro)
When tank open to atmosphere P = Pa
Dynamic Lift
Force that acts on a body by virtue of its motion through a fluid
Magnus Effect
Difference in lateral pressure, which causes a spinning ball to take a curved path which is convex towards the greater pressure side
Ball moving without spin
- Velocity of fluid (air) above and below the ball at corresponding points is the same resulting in zero pressure difference
- Air exerts no upward or downward force on the ball
Ball moving with spin
- Drags air along with it
- Rough surface = more air dragged
- Velocity of air above the ball relative to the ball is larger and below it is smaller
Aerofoil
Name given to a solid object shaped to provide an upward vertical force as it moves horizontally through air
How does aircraft balance
- The cross-section of an airplane’s wings resembles an aerofoil
- As the aerofoil moves against the wind, the streamlines of air crowd together more above the wing than below it, causing the air to flow faster on top
- This difference in flow speed creates an upward force, known as dynamic lift, which helps balance the weight of the plane.
Cohesive vs Adhesive forces
Cohesive: Force of attraction between molecules of same substance
Adhesive: Force of attraction between molecules of two different substances
Molecular theory of surface tension
- Molecule A: Well inside the liquid; attracted equally in all directions; net force is zero
- Molecule B: Lying inside the surface film; sphere of influence lies partly outside; molecule experiences less force upward and more force downward by molecules
- Molecule C: Half its sphere of influence lies above the surface; resultant downward force on such a molecule is maximum; PE of molecules of surface film higher than those well inside and since PE must be minimum, surface film needs to have minimum surface area - free surface behaves like an elastic stretched membrane
Surface tension
Property by virtue of which the free surface of a liquid at rest behaves like an elastic stretched membrane tending to contract so as to occupy minimum surface area
Surface tension = Force / Length
Surface energy
Extra energy posessed by the molecules of surface film of unit area compared to the molecules in the interior
Work done / increase in surface area = surface energy
Surface tension of liquid-air interface
S = W / 2l = mg / 2l
Angle of contact
The angle between tangent to the liquid surface at
the point of contact and solid surface inside the
liquid
Pressure difference between two sides of a curved liquid surface
- Surface is plane: Molecule is attracted on both sides the same (Pl = Pv)
- Surface is convex: Resultant downward force F on molecule (Pressure on liquid side > Pressure on vapour side –> Pl > Pv)
- Surface is concave: Resultant upward force F due to surface tension on molecule A (Pv > Pl)
Pressure on its concave side is greater than pressure on convex side
Excess pressure inside a
1. liquid drop
2. Soap bubble
3. Air bubble
- 8pi R dR sigma; p = 2 sigma / R
- p = 4 sigma / R (two free surfaces)
- p = 2 sigma / R
