Fluid Mechanics Exam 1

What is fluid mechanics?

A branch of classical physics concerned with the behavior and response of matter that continuously deforms (or flows) under the actions of shear stress.

What is shear stress?

A force acting parallel to the surface of a material (parallel force / area); tries to slide or tear the internal layers apart.

What is a “free surface”?

Interface between a liquid and the air (or some other gas), where the liquid is effectively “free” to move, not constrained by a lid, etc.

Skipped states of matter

Gases _____ expand unless ______

continuously expand unless confined by a container

What is compressibility? How does it separate liquids and gases?

Compressibility is the ability to make a volume smaller (change density) with the application of pressure. This is the major difference between liquids and gases; gases are compressible while liquids are effectively incompressible.

What does it mean for a fluid to be incompressible?

A fluid is considered incompressible if its density remains nearly constant as pressure changes. This is typical of liquids. In natural systems where pressure changes are very minimal, the density variations in gases are so small that their flow can also be approximated to be incompressible (i.e. not true for rocketry, aerodynamics, aircrafts).

What is the definition of a fluid as it relates to shear stress? Why does this happen?

A fluid is a state of matter that deforms continuously and permanently under the action of a shear stress, no matter how small this stress. This occurs because the intermolecular forces (IMFs) within a fluid are not strong enough to maintain a definite shape, making fluids deform/flow under the action of stress for as long as the force is applied.

If you apply a force to a surface, what are its components called?

Component parallel/tangential to the surface is shear stress. The component perpendicular to the surface is normal stress, AKA pressure.

What are the fundamental dimensions and units of the International System (SI) system? What is this system alternatively referred to as?

Fundamental dimensions: Mass (M), length (L), time (T); units: kg, meters, seconds; Often referred to as the MLT system.

What is the expression for force in the MLT system?

F = MLT-2

1 Newton is…

the force needed to accelerate a body of mass 1kg at a rate of 1m/s2

What are the fundamental dimensions and units of the British Gravitational (BG) system? What is this system alternatively referred to as?

Fundamental dimensions: Force (F), length (L), time (T); units: lbf (pound-force), ft, seconds. Alternative name: FLT system

What is the expression for mass in the FLT system?

Mass = FL-1T2

1 slug is …

. . . the mass of an object that accelerates at 1 ft/s2 under the action of a 1 lbf force. ( 1 slug = 1 lbf s2 / ft)

What is density?

Mass per unit volume (m/∀)

What is specific weight? What are two formulas for it?

The force exerted by a fluid due to gravity (its weight) on a unit volume. ɣ = (weight/volume) = ρg

What is specific gravity?

The ratio of the density of a fluid to the density of liquid water at standard temperature (15.56℃). s_liquid = ρ_liquid / ρ_water = ɣ_liquid / ɣ_water.

What is specific volume? What are two formulas for it?

The volume occupied by a unit mass of the fluid, i.e. how many m_³ (or ft³) are occupied by a 1 kg (or 1 slug) of fluid. ∀_s = volume / mass = 1 / ρ

What is water density (in both SI and BG units) at standard conditions?

1000 kg/m³ or 1.94 slug/ft³

What is the specific weight of water (in both SI and BG units) at standard conditions?

9.81 kN/m³ or 62.43 lbf/ft³ (a.k.a pcf, pounds per cubic foot)

What is the significance of the continuum hypothesis?

It allows us to treat fluids as continuous media such that we can apply the usual rules of calculus to small (infinitesimal) elements of them; lets us treat a tiny proportion of our fluid (ex: a 1 mm water droplet) as our smallest fluid particle, without being concerned with the molecular interactions that make up this “particle”.

What is the volume modulus of elasticity (bulk modulus)? What are the equations for it? What are its dimensions?

It is a mathematical way to describe compressibility in fluids. Ev = -∀ dp/d∀ = - dp / (d∀ / ∀) = dp / (dρ / ρ). E_v has the same dimensions as pressure (units Pa or psi)

A higher bulk modulus means a liquid is ____(easier/harder) to compress.

harder

If a fluid is at rest, can it experience shear stress?

No, it can’t both be at rest and experiencing shear stresses. A fluid continuously deforms/flows under the action of shear stresses. It can only experience normal stress (pressure).

What is (dynamic) viscosity (μ)?

Viscosity is a fluid property that quantifies the resistance of a fluid to deformation when acted upon by shear stresses. It measures the frictional force between different layers of the fluid that are in relative motion.

What is kinematic viscosity? What is its formula?

Kinematic viscosity describes the ability of a fluid to diffuse momentum, i.e. it’s a measure of how easily momentum is exchanged between different fluid layers. Higher kinematic viscosity means it’s easier for momentum to be exchanged. Formula: 🇻 = μ/ρ

What are the formulas for (viscous) shear stress?

τ = μ (du/dy) = force/area, where du/dy is the velocity gradient (also called shear strain?)

What are the units of (dynamic) viscosity?

N s / m2 or Pa s or kg/(m s)

What are Newtonian fluids?

Fluids that have a constant viscosity. (obey the linear relationship between shear stress and strain where the constant of proportionality is the dynamic viscosity μ.

Lecture 3, slide 12 → has stuff about a poise, not sure if it’s needed

What does it mean if a fluid’s specific gravity is greater than 1?

It means the fluid is more dense than water.

Does high density correlate with high viscosity?

No

For liquids, as temperature increases, dynamic viscosity _____ (increases/decreases). For gases, it ______(increases/decreases). Why?

For liquids, as temp increases, dynamic viscosity decreases; molecules gain more energy to overcome cohesive forces and can move with less resistance/friction, decreasing μ. For gases, as temp increases, dynamic viscosity increases. This is because for gases, viscosity is caused by the exchange of momentum during molecular collisions; IMFs are negligible. Heating increases the likelihood of collisions, thus increasing viscosity.

What is a cohesive force? What is an adhesive force?

Cohesive→ attraction between liquid molecules (of the same substance?) ; adhesive → attraction between liquid molecules and the molecules of a different substance (ex: the air or the walls of their container)

Surface tension occurs at…. Why?

At the free surface between a liquid and a gas (i.e. air) due to differences in cohesive and adhesive forces. A molecule at the free surface will be pulled inward by cohesive forces, outward by adhesive forces. There are fewer air/gas molecules to be attracted to, so the inward force is larger than the outward force. So, liquids try to minimize their surface area → makes a spherical surface, as spheres have the least surface area.

Surface tension is the property of…

an interface, it requires contact with something; it’s not a property of a specific liquid or gas

What is surface tension (σs)?

It is the tensile force per unit length of the surface with common units N/m or lbf/ft.

What is a meniscus and what causes its formation?

It’s the curved surface that forms between the free surface of a liquid (interface between liquid and air) and the walls of a solid container. For water, (concave meniscus) its adhesive force with the glass is stronger than its cohesive forces with itself. Water molecules on the sides are pulled up, cohesive forces pull the middle down (bc water is more attracted to itself than air). On the other hand, mercury’s cohesive forces are stronger than its adhesive forces, creaing a convex meniscus.

What is the angle of contact (ø)?

It’s the angle between the solid wall of the tube and the curved surface of the meniscus.

Will surface tension increase or decrease as temperature increases?

Decrease; higher temp = higher energy = weaker cohesive forces.

When does capillary rise stop?

It will halt when the upward force (from when adhesive forces with wall exceed cohesive forces) is balanced by the weight of the water/liquid column (weight = mg = ρ∀g = ρgπR2h). (i.e. when the adhesive forces pulling it up, the cohesive forces pulling it down, and gravitational force are all in equilibrium).

If you place a glass tube (two open ends) into a bath of liquid, what will happen if the liquid in the bath is water? What if the liquid is mercury? Respond in terms of meniscus shape and relative water level.

If it’s water, the meniscus will be concave (u-shaped), and the water level in the tube will rise above that of the water level in the bath. If it’s mercury, it will form a convex (n-shaped) meniscus and the mercury level in the tube will be lower than that of the mercury level in the bath.

What are the two scenarios where a fluid is in static equilibrium?

1. If it’s at rest or 2. if it’s moving with a constant velocity where different fluid layers are not moving relative to each other (rigid body motion). In either case, there is no acceleration and the net force acting on the fluid must be zero. No tangential stresses exist in the fluid.

The pressure at a point in a fluid at rest is ______ of the direction of action. This means pressure is _____.

independent; pressure is isotropic (same in all directions).

Is pressure a scalar or vector quantity?

Scalar

What is the hydrostatic equation?

p = patm + ρgh = patm + ɣh

Pressure in an ______ fluid increases ______ with depth.

incompressible; linearly (true as long as density is constant (which is true for an incompressible fluid) and gravity is constant)

What does psia mean?

pound per square inch absolute; it is a measure of absolute pressure. This is the total pressure; the sum of atmospheric pressure and the pressure due to the weight of the fluid).

What is gage pressure?

The pressure due to the weight of the fluid. Can also be described as the pressure relative to atmospheric pressure. Most pressure measurement devices are calibrated to read zero in the atmosphere, and so they indicate gage pressure.

What is the basic formula for absolute pressure?

p = p_atm + ɣh

What is the basic formula for gage pressure?

p = ɣh

What is a pressure head? What does it measure?

PRessure head expresses pressure in terms of height of fluid. h is the height of fluid needed to create pressure p. Formula: h = p / ɣ

What does Pascal’s law state? How can we apply this in pressure problems?

States that a pressure change at any point in a confined incompressible fluid is transmitted, undiminished, throughout the fluid. Effectively, this means that pressure is constant along a horizontal plane in an interconnected, continuous fluid.

What is a barometer used for? What is its setup?

It’s used to measure the absolute pressure of the atmosphere (this is why atmospheric pressure is usually referred to as the barometric pressure. A test tube is flipped upside down, the open end is submerged in liquid (usually mercury ) bath. Inside the tube is a vacuum. Pressure head from bath level to fluid level in tube is measured, this is equal to p_atm

What are notable things about the tube used for a barometer?

Length and cross-section of the tube have no effect on the height of the liquid column. All you need is for the diameter of the tube to be large enough for effects of surface tension (capillarity) to be negligible.

What is a u-shaped manometer used for? What does it contain?

It’s commonly used to measure small and moderate pressure differences. A manometer contains one or more fluids such as mercury, water, alcohol, or oil

What is the force acting on the bottom of a liquid-filled container of depth H, specific weight ɣ? What is the point of application? Why do we use gage pressure?

Pressure = ɣH = force/area → Force = ɣH(Area). Point of application is the centroid, (force applied to the surface is uniform) which is located in the center of the bottom of the container. We use gage pressure because the container will experience a normal force -p_atmA that counteracts the p_atmA felt at the bottom of the container.

In hydrostatic force problems, what is the center of pressure?

The point of application of the resultant force, F_R

For hydrostatic force on a plane surface, what is the formula for F_R and y_p?

FR = ɣh_cA and yp = yc + I_cx / y_cA

What happens to y_c and y_p as depth immersion increases?

As you get submerged deeper and deeper in a fluid, the center of pressure (point of application of FR) approaches yc (the location of the centroid)