Fluid Dynamics

Fluid is a substance that offers no permanent resistance to deforming forces.

Statics and Dynamics

The study of fluids has two parts:

Fluid statics: Concentrates on fluids at rest.
Fluid dynamics: Concerns with fluid in motion.

Fluid statics includes:

Atmospheric pressure.
Liquid pressure.
Archimedes' principle.
Pascal's law.

Motion of fluids is an important application in our daily life, such as:

River flow.
Water distribution systems.
Gas pipelines.
Aviation (travels in air).
Motion underneath the surface of water.
Hydrodynamics: The study of the flow of liquid.
Hydrostatics: The study of liquid at rest.

Laminar and Turbulent Flow

Fluid can be either liquid or gas.
The volume of a gas depends on pressure and temperature, obeying Boyle's law and Charles' law.
The volume of a liquid does not depend on pressure if the temperature is constant.
Liquid is incompressible, meaning its density is constant for any pressure, provided the temperature is kept constant.

Streamline

In the flow of fluids, the path or the flow of the fluid particles is called streamline.

Types of Fluid Motion

Motion of fluids can be classified into two types:

Laminar Flow
Turbulent Flow

Laminar Flow

If fluid particles are moving steadily in smooth paths in layers, with each layer moving smoothly past the adjacent layers with no mixing, such a flow is called laminar flow.
In this steady laminar flow, streamlines do not cross each other, and every fluid particle arriving at a given point has the same velocity.

Turbulent Flow

If the flow or path of the fluid particles is irregular, their direction is always changing or whirling, this fluid movement is called turbulent flow.
In turbulent flow, the speed of the fluid at a point is continuously undergoing changes in both magnitude and direction.

Ideal Laminar Fluid Flow

Ideal laminar fluid flow considers the following facts:

The fluid is incompressible.
The flow has no friction, or friction may be neglected (zero viscosity).
The fluid flow is laminar (steady), and turbulent flow is not taken into account.
- Sea breeze is a laminar flow; a storm is a turbulent flow.
The liquid entering one end of the pipe leaves from the other end with a constant volume flow rate.
The equation A1v1 = A2v2 is known as the equation of continuity for fluid flow.
The larger the area of the pipe, the slower is the speed of the fluid.
The smaller the area of the pipe, the faster is the speed of the fluid.
The speed of water spraying from a hose increases by making the hose area smaller.

Bernoulli's Equation

Bernoulli's equation, in fluid dynamics, relates the pressure, flow speed, and height for the steady (laminar) flow of an ideal incompressible fluid.
The internal friction of the fluid flow is negligible (i.e., no viscosity).
This equation was first derived in 1783 by the Swiss mathematician Daniel Bernoulli.
Bernoulli's equation can be derived using the work-energy principle to a fluid flowing in a tube.

Derivation

Consider a section of the fluid flowing in the tube with an uneven cross-section.
This fluid section is initially between two cross-sections a and c.
After a small time interval, the fluid at a moves to b, and the fluid at c moves to d. [The fluid between a and b becomes the fluid between c and d.]
The cross-sectional area, speed, and pressure at the two ends of the fluid section are A1, v1, P1 and A2, v2, P2 respectively.
According to the work-energy principle, the work done on this section of fluid must equal the change in total mechanical energy.
Total mechanical energy = kinetic energy + potential energy
Since the fluid is incompressible, A1\Delta x1 = A2\Delta x2 = \Delta V
The work done on the fluid in this case is due to the pressure of the fluid.
The work done \Delta W = F1\Delta x1 + (-F2)\Delta x2 = P1A1\Delta x1 - P2A2\Delta x2 = P1\Delta V - P2\Delta V = (P1 - P2)\Delta V
- (F₂ has a negative sign because the force at c opposes the force at a.)
The mass of the fluid between a and b = m1 = \rho A1\Delta x_1 = \rho \Delta V
The mass of the fluid between c and d = m2 = \rho A2\Delta x_2 = \rho \Delta V
The change in kinetic energy \Delta KE = \frac{1}{2} m2 v2^2 - \frac{1}{2} m1 v1^2 = \frac{1}{2} \rho \Delta V (v2^2 - v1^2)
The change in potential energy \Delta PE = \rho \Delta V g (h2 - h1)
Using the work-energy principle, \Delta W = \Delta KE + \Delta PE
(P1 - P2)\Delta V = \frac{1}{2} \rho \Delta V (v2^2 - v1^2) + \rho \Delta V g (h2 - h1)
(P1 - P2) = \frac{1}{2} \rho (v2^2 - v1^2) + \rho g (h2 - h1)
P1 + \frac{1}{2} \rho v1^2 + \rho gh1 = P2 + \frac{1}{2} \rho v2^2 + \rho gh2
P + \frac{1}{2} \rho v^2 + \rho gh = constant
This equation is known as Bernoulli's equation.
In words, Bernoulli's theorem states that the sum of pressure energy, kinetic energy, and potential energy per unit volume of an incompressible, non-viscous fluid in a streamlined laminar flow is constant.

Manifestations of Bernoulli's Theorem

Lift on the Wings of an Aeroplane
- The airflow moving above the top curved surface of the wing has a longer distance to travel and needs to go faster to have the same transit time as the air traveling along the lower flat surface (horizontal position).
- Hence, the velocity of airflow is high along the top surface and low along the bottom surface. This causes a low pressure above the wing and a high pressure below the wing, which provides a lift on the wing.
Insect Sprayer
- When the piston of the sprayer is pressed, the fast-moving air causes a low pressure at the tip of the nozzle.
- The liquid inside the container is at a high pressure and therefore rushes up through the stem and out the nozzle.
Magnus Effect on a Spinning Ball
- The Magnus effect is a particular manifestation of Bernoulli's theorem that is commonly associated with a spinning object moving through a fluid.
- In the case of a ball spinning through the air, the turning ball drags some of the air around with it.
- This causes the difference in pressure of the air on opposite sides of the spinning ball and produces the deflection of the ball.
- The effect is named after the German physicist Heinrich Gustav Magnus, who described the effect in 1852.
- When the velocity of air and velocity of the ball are in opposite directions, it reduces the velocity of airflow, resulting in high pressure.
- When the velocity of air and velocity of the ball are in the same direction, it increases the velocity of airflow, resulting in low pressure.
The Roofs of Houses Can Fly Away During Cyclones
- During cyclones, moving air gains speed above the roof of a house, causing reduced pressure on the roof.
- Therefore, the pressure inside the house is greater, which can raise the roof.

Viscosity (Fluid Friction)

Most fluids are not ideal ones and offer some resistance to motion.
This resistance to fluid motion is like an internal friction analogous to friction when a solid moves on a surface.

What is viscosity?

The resistance to fluid motion (or internal resistance of fluid) is called viscosity.
Viscosity arises when there is relative motion between layers of the fluid.
More precisely, it measures resistance to flow arising due to the internal friction between the fluid layers as they slip past one another when fluid flows.
Viscosity can also be thought of as a measure of a fluid's thickness (stickiness) or its resistance to objects passing through it.
A fluid with large viscosity resists motion because its strong intermolecular forces give it a lot of internal friction, resisting the movement of layers past one another.
A fluid with low viscosity flows easily because its molecular makeup results in very little friction when it is in motion.
Gases also exhibit viscosity, but it is harder to notice in ordinary circumstances.
- The viscosity of liquids decreases rapidly with an increase in temperature, and the viscosity of gases increases with an increase in temperature.
- Thus, upon heating, liquids flow more easily, whereas gases flow more slowly.
Viscosity does not change as the amount of matter changes; therefore, it is an intensive property.
The SI unit of viscosity is pascal second (Pa·s), which is equivalent to newton second per meter squared (N·s/m²).
It is sometimes referred to as the poiseuille (Pl).
There are two ways to measure a fluid's viscosity:
- Measure fluid's resistance to flow when an external force is applied (dynamic viscosity).
- Measure the resistive flow of a fluid under the weight of gravity (kinematic viscosity), which is more useful than dynamic viscosity (absolute viscosity).

Newton's Law of Viscosity

Newton's law of viscosity states that the shear stress between two adjacent layers of fluid is directly proportional to the negative value of the velocity gradient between the same two adjacent fluid layers.
In symbols, shear stress \tau \propto - \frac{dv}{dy}
\tau = -\eta \frac{dv}{dy}, where \eta = the viscosity (coefficient of viscosity), \frac{dv}{dy} = velocity gradient

Define shear stress

Shear stress is the shearing force per unit area.
That is, \tau = \frac{F}{A}, where F = the shearing force acting between two adjacent layers of a fluid, A = the area of the fluid layer.

Define viscosity

Viscosity is defined as the ratio of shearing stress to the velocity gradient.
\eta = - \frac{\tau}{\frac{dv}{dy}} = \frac{F/A}{\frac{dv}{dy}}

Determination of Kinematic Viscosity using Stokes' Law

Stokes' law, derived by the British scientist Sir George G. Stokes in 1851, expresses the drag force resisting the fall of small spherical objects through a fluid medium under the influence of gravity.
It states that the drag force F acting upward in resistance to the fall of a spherical object in a viscous liquid is F = 6 \pi \eta r v
where \eta = the coefficient of viscosity of the liquid, r = the radius of the sphere, v = the constant velocity of fall called the terminal velocity.
A simple way of measuring viscosity is that a small sphere ball is dropped through the fluid, and the time of fall and distance of fall are measured. From the distance and time of fall, the velocity can be determined.
Consider the fall of a sphere ball. The forces acting on the object are:
- Weight of the sphere ball: W = \frac{4}{3} \pi r^3 \rho_1 g
- Upward thrust: F1 = \frac{4}{3} \pi r^3 \rho2 g
- Drag force: F = 6 \pi \eta r v
where \rho1 = the density of the sphere, \rho2 = the density of the liquid, g = the acceleration due to gravity.
At a constant velocity of fall (terminal velocity), the upward and downward forces are in balance. Therefore,
- F = W - F_1
- 6 \pi \eta r v = \frac{4}{3} \pi r^3 \rho1 g - \frac{4}{3} \pi r^3 \rho2 g
- 6 \pi \eta r v = \frac{4}{3} \pi r^3 (\rho1 - \rho2) g
- \eta = \frac{2 g r^2 (\rho1 - \rho2)}{9 v}

Surface Tension and Capillarity

Surface Tension

Surface tension is the tendency of the surface of a liquid to behave like a stretched elastic skin (or membrane).
It is due to the attractive forces between liquid molecules called cohesion.
Cohesion refers to forces between molecules of the same substance; for example, cohesion makes water molecules pull together, making the water surface stretch like an elastic skin.
Surface tension is the reason a water strider and it is able to stride along water surface without sinking.
Paperclips, razor blades, and light coins are also able to rest on water without sinking, although their density is higher than that of water.
Surface tension is defined as the force per unit length that acts across any line on a surface tending to pull the surface apart.
\gamma = \frac{F}{l}
where F = force, l = length, \gamma = surface tension. The SI unit of surface tension is newton per meter (N/m).
Surface tension can also be referred to as a force because it is the tension between liquid molecules.