Mechanical Properties of Fluids — Formula Sheet

$P$: Pressure (Pa) · $F$: Normal force (N) · $A$: Area (m²)

Normal force per unit area. Pressure is a scalar. SI unit: pascal (1 Pa = 1 N m⁻²). 1 atm ≈ 1.013 × 10⁵ Pa.

$h$: Depth below the free surface (m) · $\rho$: Density of the liquid (kg m⁻³) · $g$: Acceleration due to gravity (m s⁻²)

Depends only on the depth h, not on the shape or cross-section of the vessel (hydrostatic paradox).

$P$: Absolute pressure at depth h (Pa) · $P_0$: Atmospheric pressure (Pa) · $P_{gauge}$: Gauge pressure (Pa)

Absolute pressure includes atmospheric pressure P₀. Gauge pressure is the excess over atmospheric, which is what a tyre gauge reads.

$F_1,A_1$: Force (N) and area (m²) of the small piston · $F_2,A_2$: Force (N) and area (m²) of the large piston

Pressure applied to an enclosed fluid is transmitted undiminished. A small force on a small piston gives a large force on a large piston. Energy is not multiplied, because the large piston moves through a smaller distance.

$T$: Surface tension (N m⁻¹) · $F$: Force due to the surface film (N) · $l$: Length of the line on which the force acts (m)

Force per unit length acting along the surface, perpendicular to a line drawn on it. For a film on a wire frame, l = 2 × wire length (two surfaces). SI unit: N m⁻¹ (= J m⁻²). Decreases as temperature rises.

$W$: Work done = increase in surface energy (J) · $\Delta A$: Total increase in surface area (m²)

Work done in increasing the surface area isothermally is stored as surface energy. Surface energy per unit area is numerically equal to surface tension.

$R$: Radius of the bubble (m) · $T$: Surface tension of the soap solution (N m⁻¹)

A bubble of radius R has two surfaces of area 4πR² each, so ΔA = 8πR² when it is blown from nothing. For enlarging from R₁ to R₂: W = 8πT(R₂² − R₁²).

$R$: Radius of the big drop (m) · $n$: Number of identical droplets formed

Volume is conserved: nr³ = R³, so r = R/n^{1/3}. The total surface area increases, so work must be done. When droplets merge into one drop, the same amount of energy is released.

$P_i-P_o$: Excess pressure, inside minus outside (Pa) · $T$: Surface tension (N m⁻¹) · $R$: Radius of the drop (m)

Pressure inside a curved liquid surface is greater on the concave side. Smaller drops have larger excess pressure. For an air bubble inside a liquid there is also only one surface, so the result is 2T/R as well.

$R$: Radius of the soap bubble (m)

Twice the value for a drop, because a soap bubble in air has two surfaces (inner and outer).

$h$: Height of capillary rise (m) · $\theta$: Angle of contact · $r$: Radius of the capillary bore (m) · $\rho$: Density of the liquid (kg m⁻³)

θ is the angle of contact. For θ < 90° (water in glass) the liquid rises; for θ > 90° (mercury in glass) cos θ < 0 and the liquid is depressed. h ∝ 1/r (Jurin's law).

$\eta$: Coefficient of viscosity (Pa s) · $A$: Area of the layer (m²) · $dv/dx$: Velocity gradient (s⁻¹)

The viscous force between layers is proportional to the area and to the velocity gradient. It opposes relative motion. SI unit of η: N s m⁻² = Pa s. CGS unit: poise (1 Pa s = 10 poise).

$F$: Viscous force (N) · $r$: Radius of the sphere (m) · $v$: Speed of the sphere relative to the fluid (m s⁻¹)

Viscous drag on a small sphere moving slowly through a fluid (streamline flow). The force grows with speed, which is why a terminal velocity exists.

$v_t$: Terminal velocity (m s⁻¹) · $\rho$: Density of the material of the sphere (kg m⁻³) · $\sigma$: Density of the fluid (kg m⁻³) · $\eta$: Coefficient of viscosity of the fluid (Pa s)

ρ is the density of the sphere and σ the density of the fluid. v_t ∝ r². If σ > ρ, v_t is negative, meaning the body rises (as an air bubble in water does).

$R_n$: Reynolds number (no unit) · $\rho$: Density of the fluid (kg m⁻³) · $v_c$: Critical velocity of the fluid (m s⁻¹) · $d$: Diameter of the pipe (m)

A pure number (no unit) that indicates the type of flow. Textbook values: streamline for R_n < 1000, turbulent for R_n > 2000, unsteady in between. Rearranged, the critical velocity is v_c = R_n·η/(ρd).

$A_1,A_2$: Cross-sectional areas at two sections (m²) · $v_1,v_2$: Speeds of flow at those sections (m s⁻¹)

For an incompressible fluid in steady flow, the volume flow rate Av is constant. A narrower section means faster flow. For circular pipes, v ∝ 1/r², not 1/r.

$P$: Pressure (Pa) · $\rho$: Density of the fluid (kg m⁻³) · $v$: Speed of flow (m s⁻¹) · $h$: Height above a reference level (m)

Applies along a streamline for steady, incompressible, non-viscous flow. It is energy conservation per unit volume: pressure energy + kinetic energy + potential energy. Where speed is high, pressure is low.

$v$: Speed of efflux (m s⁻¹) · $h$: Depth of the hole below the free surface (m)

Speed of liquid coming out of a small hole at depth h below the free surface of an open tank. It equals the speed of a body falling freely through height h.