Fluid Dynamics notes

MDCAT Physics

Fluid dynamics deals with the motion of liquids and gases, the forces acting on moving fluids, and the motion of objects through fluids. This chapter covers viscosity, drag, terminal velocity, types of fluid flow, the equation of continuity, and Bernoulli's equation with applications such as blood flow, Venturi metres and aeroplane wings.

Fluids and Fluid Flow

A fluid is a substance that can flow. Liquids and gases are fluids because they cannot resist a continuous shearing force. A liquid has a nearly fixed volume, while a gas expands to fill its container.

Fluid flow describes the motion of fluid particles. In steady flow, the velocity of the fluid at a particular point remains constant with time. The path followed by a fluid particle is called a streamline.

A liquid is nearly incompressible, while a gas can usually be compressed.
For steady flow, the velocity at any fixed point does not change with time.
A streamline is a curve whose tangent at any point gives the direction of fluid velocity.
Two streamlines cannot cross each other because a fluid particle cannot have two velocities at the same point.
In streamline flow, the paths of fluid particles can be tracked.
In turbulent flow, fluid motion is irregular and the paths of particles cannot be easily tracked.
Flow is more likely to be streamlined in a narrow pipe when the fluid has high viscosity and low density.

Viscosity and Its Effects

Viscosity is the internal friction between different layers of a fluid. When one layer moves relative to another, viscous force opposes their relative motion. A fluid with greater viscosity flows less easily.

For liquids, viscosity generally decreases when temperature increases. Heating reduces the internal attraction between liquid molecules, so the liquid flows more easily. The SI unit of coefficient of viscosity is kg m-1 s-1, also written as Pa s.

Viscosity is the resistance offered by a fluid to the relative motion of its layers.
The coefficient of viscosity is represented by eta, η.
SI unit of viscosity: kg m-1 s-1 or N s m-2.
For liquids, viscosity decreases with increase in temperature.
For gases, viscosity generally increases with increase in temperature.
A liquid of high viscosity and low density flowing through a pipe of small radius has more streamlined flow.
Honey has greater viscosity than water, so honey flows more slowly.

Fluid Drag and Stokes' Law

Fluid drag is the resistive force exerted by a fluid on an object moving through it. The drag force acts opposite to the direction of motion. It depends on the speed of the object, the viscosity and density of the fluid, and the shape and size of the object.

For a small spherical body moving slowly through a viscous fluid, Stokes' law gives the viscous drag. This law applies when the motion is smooth and the body is spherical.

Stokes' law: F = 6πηrv.
In Stokes' law, F is viscous drag, η is coefficient of viscosity, r is radius of the sphere and v is its speed.
Stokes' law is applicable to a spherical body moving slowly through a viscous fluid.
Drag acts in the direction opposite to the velocity of the object.
For the same fluid and speed, drag on a sphere is directly proportional to its radius.
For the same spherical body and fluid, viscous drag is directly proportional to speed.
A smooth and streamlined shape reduces drag compared with an irregular shape.

Terminal Velocity

When an object falls through a fluid, its weight acts downward and fluid drag acts upward. At first, the object accelerates. As its speed increases, drag also increases. Eventually, drag becomes equal to the effective downward force, so the resultant force becomes zero.

At this stage the object continues to move with constant speed called terminal velocity. For a small sphere falling through a viscous fluid, terminal velocity is obtained by balancing weight, upthrust and viscous drag.

At terminal velocity, acceleration is zero.
At terminal velocity, the resultant force on the falling body is zero.
The downward effective weight is balanced by upward viscous drag.
For a small sphere, terminal velocity is proportional to r2 when the fluid and material remain unchanged.
If the radii of two droplets are in the ratio 2:3, their terminal velocities are in the ratio 4:9.
When a large raindrop is divided into eight equal drops, each small drop has half the radius of the original drop and one-fourth its terminal velocity.
If the original terminal velocity is 32 m/s, the terminal velocity of each of the eight equal drops is 8 m/s.
Rain drops fall with nearly constant speed during the later part of their fall because air resistance balances their weight.

Falling Objects in Fluids

A falling object in air is affected by gravitational force, upthrust and air resistance. A hailstone and a raindrop with the same radius do not necessarily have the same terminal velocity because their densities and masses are different.

A hailstone is denser than a water drop of the same radius. It has a greater effective downward force and usually reaches the ground before the raindrop when both are released from the same height.

Weight acts downward on a falling body.
Buoyant force and air resistance act upward on a falling body.
Air resistance increases as the speed of a body increases.
At the beginning of a fall, acceleration is generally present because drag is small.
At terminal velocity, the body continues moving but its acceleration is zero.
A hailstone and a water drop of the same radius have different terminal velocities because their densities are different.
A hailstone released from the same height as a raindrop generally reaches the ground before the raindrop.

Equation of Continuity

The equation of continuity is based on the law of conservation of mass. In steady flow, the mass entering a section of a pipe per second must equal the mass leaving another section per second.

For an incompressible fluid, density remains constant. Therefore, the volume flow rate remains constant and the product of cross-sectional area and fluid speed has the same value at every section.

General continuity equation: rho1 A1 v1 = rho2 A2 v2.
For an incompressible fluid: A1v1 = A2v2.
A is cross-sectional area and v is fluid speed.
Volume flow rate: Q = Av.
The SI unit of volume flow rate is m3/s.
A fluid speeds up when it enters a narrower part of a pipe.
A fluid slows down when it enters a wider part of a pipe.
If the diameter of a pipe changes from 2 m to 4 m, the area becomes four times greater, so a speed of 16 m/s changes to 4 m/s.

Volume Flow Rate and Numerical Applications

Volume flow rate is the volume of fluid passing through a cross-section per unit time. It can also be calculated from mass flow rate when the density of the fluid is known.

For a circular pipe, the cross-sectional area is A = pi r2. Care must be taken to convert centimetres and millimetres into metres before using SI equations.

Volume flow rate: Q = volume/time = Av.
Mass flow rate: mass/time = rho Av.
For a circular pipe, A = pi r2.
For radius 1.5 cm and speed 7.0 m/s, Q is approximately 4.9 x 10-3 m3/s.
If 30 kg of water leaves in 60 s, the mass flow rate is 0.5 kg/s.
For water of density 1000 kg/m3 and outlet diameter 20 mm, the outlet speed is approximately 1.59 m/s.
The density of water is commonly taken as 1.0 x 103 kg/m3 in numerical problems.

Bernoulli's Equation

Bernoulli's equation is a statement of conservation of energy for a steadily flowing ideal fluid. It relates pressure, speed and elevation. If a fluid moves through a pipe, its pressure energy, kinetic energy and gravitational potential energy per unit volume remain constant.

The equation applies to steady, non-viscous, incompressible flow along a streamline. A rise in fluid speed is often accompanied by a fall in pressure when the height remains unchanged.

Bernoulli's equation: P + 1/2 rho v2 + rho gh = constant.
P represents pressure energy per unit volume.
1/2 rho v2 represents kinetic energy per unit volume.
rho gh represents gravitational potential energy per unit volume.
The equation relates pressure with speed and pressure with elevation.
At the same height, greater fluid speed generally means lower pressure.
At the same speed, greater elevation is associated with lower pressure if other conditions remain fixed.
Bernoulli's equation is not directly applicable when viscosity, turbulence or significant energy loss cannot be ignored.

Applications of Bernoulli's Equation

A Venturi metre has a wide section and a narrow section. As an incompressible liquid enters the narrow section, its speed increases according to the equation of continuity. Bernoulli's equation then shows that its pressure decreases.

The lift of an aeroplane wing is explained by a pressure difference between its upper and lower surfaces. Air moves faster over the curved upper surface, producing lower pressure above the wing than below it.

In a Venturi metre, fluid speed is greater in the narrow section.
Pressure is reduced in the narrow section of a Venturi metre.
The Venturi effect is the decrease in pressure caused by increased fluid speed in a narrow passage.
The lift on an aeroplane wing arises because air moves faster over the curved upper surface.
Lower pressure above the wing and higher pressure below it produce an upward force.
For air speeds of 120 m/s above and 90 m/s below a wing, with air density 1.3 kg/m3, the pressure difference is 4095 Pa.
Bernoulli's equation can also explain fluid flow through pipes of changing height and area.

Pressure in Fluids and Blood Flow

Pressure in a stationary liquid increases with depth. In a flowing liquid, pressure also depends on speed and elevation according to Bernoulli's equation. The pressure difference caused by height is called hydrostatic pressure difference.

For a person standing upright, blood must be pumped upward to the brain, and the pressure difference due to the height of the blood column is larger. When the person lies horizontally, the vertical height difference is small, so the maximum blood pressure in the body has the smallest value.

Pressure is force per unit area.
SI unit of pressure is pascal, Pa, which is N/m2.
Dimensions of pressure are M L-1 T-2.
Hydrostatic pressure difference is related to rho gh.
Blood pressure differences due to height are smaller when a person is lying horizontally.
A person standing upright has a greater pressure difference between lower and upper parts of the body.
Pressure, speed and elevation must be considered together in a moving fluid.

Surface Energy of Droplets

A liquid surface possesses surface energy because molecules at the surface experience unbalanced intermolecular forces. When small droplets merge to form a larger droplet, the total surface area decreases.

The decrease in surface area causes a decrease in surface energy. The excess energy is released, usually as heat or motion. Thus, energy is liberated when two droplets merge.

Surface energy is energy associated with the surface of a liquid.
A liquid tends to reduce its surface area.
When two droplets merge, the total surface area decreases.
When droplets merge, surface energy decreases and energy is liberated.
A large spherical droplet has less surface area than the combined surface areas of many smaller droplets of the same total volume.
The spherical shape is favoured because it gives minimum surface area for a given volume.

Key terms

Fluid: A substance that can flow and cannot resist a continuous shearing force.
Viscosity: The internal friction that opposes the relative motion of layers of a fluid.
Coefficient of viscosity: A measure of the internal resistance offered by a fluid to the relative motion of its layers.
Fluid drag: The resistive force exerted by a fluid on an object moving through it.
Stokes' law: The relation F = 6πηrv for viscous drag on a small sphere moving slowly through a fluid.
Terminal velocity: The constant maximum speed reached by a falling body when the resultant force becomes zero.
Streamline flow: Smooth flow in which fluid particles follow regular paths and the velocity at a point remains steady.
Turbulent flow: Irregular flow in which fluid particles move along disorderly paths.
Equation of continuity: An equation based on conservation of mass, written for an incompressible fluid as A1v1 = A2v2.
Volume flow rate: The volume of fluid passing through a cross-section per unit time, given by Q = Av.
Bernoulli's equation: An energy conservation equation relating pressure, speed and elevation in steady ideal fluid flow.
Venturi effect: The reduction in pressure that occurs when a fluid speeds up while passing through a narrow section.
Upthrust: The upward buoyant force exerted by a fluid on an immersed object.
Surface energy: The energy associated with the surface of a liquid due to intermolecular forces.

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Physics shortcuts

Comparing distance and displacement

Distance equals the magnitude of displacement only when the particle travels along a straight path without reversing direction.

Check whether the path is straight and one-directional.
If yes, distance = |displacement|.
Example: A particle moves 5 m east in a straight line. Distance = 5 m and displacement magnitude = 5 m.

This shortcut does not apply to a curved path or to motion involving a change of direction.

Projectile range and components

For a projectile launched and landing at the same level, use R = u² sin 2θ/g. Resolve the initial velocity into horizontal and vertical components when needed.

Write ux = u cos θ and uy = u sin θ.
For the same launch and landing level, R = u² sin 2θ/g.
Example: u = 20 m/s, θ = 30°, g = 10 m/s². R = 400 sin 60°/10 = 34.6 m.

The range formula does not apply directly when the projectile lands at a different height.

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