ME 310 Chap.1 Fluid Properties: Dimensions, Viscosity and Surface Tension
Fluid Properties: Dimensions, Viscosity and Surface Tension
The organising idea of the course
ME 310 opens by framing all of fluid mechanics as Newton's second law applied to a chunk of fluid. On one side sits inertia, mass times acceleration. On the other side sit the forces a fluid can exert: pressure, gravity, viscosity and surface tension.
The early homework asks students to state this idea and then decide which parts of it matter for a particular situation, such as a swimmer hanging motionless underwater, who mostly feels pressure created by gravity acting on the water above.
That habit of choosing forces before choosing equations runs through every later chapter.
Dimensions as a checking tool
The first week also sets up dimensional reasoning with force, mass, length and time. Ratios of like quantities, such as the circle constant or an angle in radians, have no dimensions at all.
Viscosity, by contrast, is a stress divided by a rate of strain and carries force times time per length squared. Rearranging a formula to isolate a coefficient, for example the drag coefficient in the wind-load equation, reveals whether it is dimensionless.
A dimensionless number keeps the same value in SI and in US customary units, which the homework demonstrates with the Froude number.
Viscosity, no slip and the Newtonian idea
Viscosity acts like friction inside the fluid, opposing any sliding of one layer over another.
Two facts drive the calculations: fluid touching a wall moves with the wall, and in a Newtonian fluid the shear stress is proportional to the velocity gradient. For a thin gap between surfaces the velocity profile is nearly linear, so the shear stress is simply viscosity times the relative speed divided by the gap.
Many real materials break the linear rule in different ways: they thin under shear, need a yield stress before flowing, remember past deformation, or change viscosity over time while being sheared.
Surface tension as a line force
Surface tension acts along an interface and enters a free-body diagram as a force per unit length of cut edge.
Cutting a droplet, a jet or a bubble in half and balancing pressure against that edge force gives the pressure jump across the curved surface.
The same balance, with gravity added, explains how far water climbs in a narrow tube.
What the early homework practises
The first two problem sets combine short reading questions, such as how the textbook defines Newtonian and non-Newtonian fluids, with calculations on dimensions, a shaft sliding through a lubricated bearing, the pressure inside a small jet and a water strider supported by surface tension.
What this chapter covers
- 01
Inertia balanced by pressure, gravity, viscosity and surface tension
- 02
Dimensions in force, length and time units
- 03
Dimensionless numbers and unit independence
- 04
Newton's law of viscosity and the no-slip condition
- 05
Non-Newtonian behaviour: thinning, yield stress, memory
- 06
Pressure jumps across droplets, jets and bubbles
- 07
Capillary rise from a force balance
Worked example · free
Terminal speed of a cylinder sliding down an oiled tube
- 2Shear in the gap: τ = μV/h, acting on the side area 2πRL, so the viscous force is (2πμRL/h)V = 20.1V newtons.
- 2Newton's law downward: M dV/dt = Mg − 20.1V. At terminal speed dV/dt = 0.
- 2Terminal speed V = Mg/20.1 = 4.91/20.1 = 0.244 m/s.
- 1The solution approaches that value as 1 − exp(−t/τ) with τ = M/20.1 = 0.025 s.
Key terms
- No-Slip Condition
- The no-slip condition is the boundary rule that fluid in contact with a solid surface moves at the surface's own velocity.
- Dynamic Viscosity
- Dynamic viscosity is the property that relates shear stress to the rate of shear strain in a fluid; its SI unit is the pascal-second.
- Newtonian Fluid
- A Newtonian fluid is one whose shear stress is linearly proportional to its rate of shear strain, as for water, air and most oils.
- Surface Tension
- Surface tension is the force per unit length acting along the interface between two immiscible fluids, tending to reduce the interface area.
- Dimensional Homogeneity
- Dimensional homogeneity is the requirement that every additive term in a physical equation has the same dimensions.
Fluid Properties: Dimensions, Viscosity and Surface Tension FAQ
Why does ME 310 describe fluid mechanics as F = ma?
Because every problem balances inertia against some subset of pressure, gravity, viscosity and surface tension. Naming which of those forces act, and which can be neglected, tells you whether you need hydrostatics, Bernoulli, a viscous solution or a surface-tension balance.
How do I find the dimensions of a coefficient in an equation?
Rearrange the equation so the coefficient stands alone, substitute the dimensions of every other quantity, and simplify using force equals mass times length over time squared. If everything cancels, the coefficient is dimensionless and has the same value in any consistent unit system.
What makes a fluid non-Newtonian?
Its shear stress is not simply proportional to shear rate. It may thin or thicken as it is sheared, require a threshold stress before it flows, store elastic energy, or change viscosity with time under steady shearing.
Why is the pressure inside a small droplet higher than outside?
Surface tension pulls the curved interface inward along its whole rim. For the droplet to stay in equilibrium the inside pressure must exceed the outside by twice the surface tension divided by the radius, so smaller droplets carry larger excess pressure.
Exam move
Practise the free-body step: for any setup, list the four fluid forces and cross out those you can justify ignoring. Then check dimensions of the equation you choose before substituting numbers.
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