Chapter 4: Fluids Flashcards
Density, buoyancy equation, specific gravity
ρ=m/v
where ρ (rho) represents density, m is mass, and V is volume
SI unit: kg/m3 or g/mL or g/cm3
Density of water= 1 o/m3= 1000 kg/m3
The weight of any volume of a given substance with a known density can be calculated by multiplying the substance’s density by its volume and the acceleration due to gravity. This is a calculation that appears frequently when working through buoyancy problems on Test Day:
Fg = ρVg
Specific gravity: The density of a fluid is ofen compared to that of pure water at 1 atm and 4°C, a variable called specific gravity. It is at this combination of pressure and temperature that water has a density of exactly 1 m/cm3.
The specific gravity is given by: SG: ρ /1 g/m3
Pressure
Pressure is a ratio of the force per unit area. The equation for pressure is
P=F/A
where P is pressure, F is the magnitude of the normal force vector, and A is the area. The SI
unit of pressure is the pascal (Pa), which is equivalent to the 1 N/m2 newton per square meter
Absolute Pressure
Absolute (hydrostatic) pressure is the total pressure that is exerted on an object that is
submerged in a fluid. Remember that fluids include both liquids and gases. The equation for absolute pressure is
P = P0 + ρgz
where P is the absolute pressure,
P0 is the incident or ambient pressure (the pressure at the surface),
ρ is the density of the fluid,
g is acceleration due to gravity,
and z is the depth of the object.
Do not make the mistake of assuming that P0 always stands for atmospheric pressure.
In open air and most day-to-day situations P0 is equal to 1 atm, but in other fluid systems, the
surface pressure may be higher or lower than atmospheric pressure
Gauge pressure
Gauge pressure is the amount of pressure in a closed space above and beyond atmospheric pressure.
Pgauge = P – Patm = (P0 + ρgz) – Patm
Note that when P0 = Patm, then Pgauge = P – P0 = ρgz at a depth z.
Pascal’s principle and hydraulic systems
For fluids that are incompressible—that is, fluids with volumes that cannot be reduced by any significant degree through application of pressure—a change in pressure will be transmitted undiminished to every portion of the fluid and to the walls of the containing vessel.
W=P change in V= F1d1=F2d2
Remember when applying Pascal’s principle that the larger the area, the larger the
force, although this force will be exerted through a smaller distance.
Archimedes principle
When an object is placed in a fluid, it will sink into the fluid only to the point at which the
volume of displaced fluid exerts a force that is equal to the weight of the object.
Fbuoy = ρ fluid V fluid displaced g = ρ fluid V submerged g
The most common mistake students make using the buoyancy equation is to use the density of the object rather than the density of the fluid. Remember always to use the density of the fluid itself.
An object will float if its average density is less than the average density of the fluid it is
immersed in. It will sink if its average density is greater than that of the fluid.
Viscosity
Some fluids flow very easily, while others barely flow at all. The resistance of a fluid to flow is
called viscosity (η). Increased viscosity of a fluid increases its viscous drag, which is a
nonconservative force that is analogous to air resistance. Thin fluids, like gases, water, and
dilute aqueous solutions, have low viscosity and so they flow easily.
Low-viscosity fluids have low internal resistance to flow and behave like ideal fluids.
Assume conservation of energy in low-viscosity fluids with laminar flow.
Laminar flow
Smooth and orderly flow.
Poiseuille’s law:
Q= pi r4 Change P / 8 n L
where Q is the flow rate (volume flowing per time), r is the radius of the tube, ΔP is the pressure gradient, η (eta) is the viscosity of the fluid, and L is the length of the pipe.
Note that the relationship between the radius and pressure gradient is inverse exponential to the fourth power—even a very slight change in the radius of the tube has a significant effect on the pressure gradient,
assuming a constant flow rate.
Turbulent and Speed
Turbulent flow is rough and disorderly. Turbulence causes the formation of eddies, which are swirls of fluid of varying sizes occurring typically on the downstream side of an obstacle,
Calculations of energy conservation, such as Bernoulli’s equation, cannot be applied to turbulent flow systems. Luckily, the MCAT always assumes laminar (nonturbulent) flow for such questions.
For a fluid flowing through a tube of diameter D, the critical speed, vc, can be calculated as
Vc= Nrn/ pD
where vc is the critical speed, NR is a dimensionless constant called the Reynolds number, η is the viscosity of the fluid, and ρ is the density of the fluid.
Flow rate and Linear Speed
Linear speed is a measure of the linear displacement of fluid particles in a
given amount of time. Notably, the product of linear speed and cross-sectional area is equal to the flow rate. We’ve already said that the volumetric rate of flow for a fluid must be constant throughout a closed system.
Q = v1A1 = v2A2
where Q is the flow rate, v1 and v2 are the linear speeds of the fluid at points 1 and 2, respectively, and A1 and A2 are the cross-sectional areas at these points.
While flow rate is constant in a tube regardless of cross-sectional area, linear speed of a fluid will increase with decreasing cross-sectional area.
Bernoullis equation
First, the continuity equation arises from the conservation of mass of fluids. Liquids are essentially incompressible, so the flow rate within a closed space must be constant at all points. The continuity equation shows us that for a constant flow rate, there is an inverse relationship between the linear speed of the fluid and the cross-sectional area of the tube: fluids have higher speeds through narrower tubes.
Second, fluids that have low viscosity and demonstrate laminar flow can also be approximated to be conservative systems. The total mechanical energy of the system is constant if we discount the small viscous drag forces that occur in all real liquids.
P1+ 1/2 pv2+ pgh1 = P2 + 1/2 pv2+ pgh2
where P is the absolute pressure of the fluid, ρ is the density of the fluid, v is the linear speed, g is acceleration due to gravity, and h is the height of the fluid above some datum. Some of the terms of Bernoulli’s equation should look vaguely familiar
