Important Forces before starting:

Reference : Internet

Reynolds number :

Ratio of inertial forces to viscous forces.

(dimensionless)

Reynold number = (density)*(velocity)*(Diameter)/(Dynamic viscosity)

The ratio ν = µ/ρ is termed the kinematic viscosity.

Reynold number is used to categorize the fluids systems in which the effect of viscosity is important in controlling the velocities or the flow pattern of a fluid.

Flow in a pipe is laminar if the Reynolds Number (based on diameter of the pipe) is less than 2100 and is turbulent if it is greater than 4000.

Transitional Flow prevails between these two limits.

Capillary number :

Ratio of viscous force to interfacial force.

It is used while flooding or EOR.

More the capillary number, higher the oil recovery.

Capillary number = (Dynamic viscosity)*(Velocity)/(interfacial tension)

For a flowing liquid, if N_C >>1, then viscous forces dominate over interfacial forces; however

if N_C <<1, then viscous forces are negligible compared with interfacial forces.

Capillary numbers are usually large for high-speed flows and low for low-speed flows; thus, typically for flow through pores in the reservoir N_C is ~10⁻⁶, and for flow in production tubulars N_C is ~1.

Froude number :

Ratio of inertia force to the gravitational force.

Froude number = (velocity of flow)/square root of [(length of flow)*(acceleration due to gravity)]

Froude number is useful to describe the flow in open channels, flow over notches and weirs, the motion of a ship in turbulent sea conditions (ship resistance), flow over spillways, etc.

When:
Fr = 1, critical flow,
Fr > 1, supercritical flow (fast rapid flow),

Fr < 1, subcritical flow (slow / tranquil flow)

Nusselt number :

Ratio of convective to conductive heat transfer at a boundary in a fluid. (dimensionless)

Nusselt Number : Nu = hL/k.

Where, h = Convection Heat Transfer Coefficient, L = Characteristic Length, k = Thermal Conductivity of the Fluid.

Nusselt number can never be less than 1.

Used in calculations of heat transfer between a moving fluid and a solid body.

Mach number :

Ratio of inertia force to the elastic force.

Mach number = velocity of flow/velocity of sound in that fluid.

Pic : From NASA

If the Mach number is one, then the flow velocity is equal to the velocity of sound in the fluid.

If it is less than one, then the flow is called subsonic flow, and

If it is greater than one the flow is called supersonic flow.

Euler’s number :

Ratio of inertia force to the pressure force.

Euler's number = (velocity of fluid)/under root of{(pressure)/(density of fluid)}

Euler’s number is significant in cases where pressure gradient exists such as flow through pipes, water hammer pressure in penstocks, discharge through orifices and mouthpieces, etc.

Bond number or

Eötvös number :

Ratio of the gravitational force (mg, where m is the mass and g is the acceleration of gravity) to the force due to surface tension γL (where γ is the surface tension and L is some characteristic length such as the radius for a sphere)

Bo = mg/γL

For spherical bubbles,

Bond number = (ρ - ρ′) L ² g /σ,

where ρ is the density of a bubble or drop, ρ′ is the density of the surrounding medium, L is a characteristic dimension, g is the acceleration of gravity, and σ is the surface tension of the bubble or drop.

A value of N_b <<1 implies the flow is only weakly dependent on gravitational forces,

Nb >>1 implies gravitational forces dominate over interfacial forces.

Stokes Number :

Stoke Number = (relaxation time)*(fluid velocity)/(characteristic dimension of the obstacle (typically its diameter))

Note: Commonly used in particles suspended in fluid.
For Stk << 1, the particle negotiates the obstacle.
For Stk >> 1, the particle travels in straight line and eventually collides with obstacle.

Wave number :

reciprocal of wavelength, so the number of wave cycles per unit of distance, abbreviated as k.

Weber number :

The Weber number is defined as

We=ρg *Vr2 * L/σ

where ρg is the gas mass density (kg/m3), Vr is the relative velocity between gas and liquid (m/s), L the drop or liquid jet diameter (m), and σ (N/m) the surface tension of the liquid.

Useful in analysing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces.

prandtl number :

prandtl number = cp* v / k,

where c _p is the heat capacity per unit volume of the fluid,

v its kinematic viscosity, and

k its thermal conductivity.

Used in calculations of heat transfer between a moving fluid and a solid body.

grashof number :

Ratio of the buoyant to viscous force acting on a fluid in the velocity boundary layer.