Non-dimensional numbers are commonly used in fluid mechanics to categorize the p

Question

Non-dimensional numbers are commonly used in fluid mechanics to categorize the problem, e.g. the ratio between to liquid densities is a non-dimensional number. Try to build these numbers for the following quantities:
a) Speed of sound and object velocity
b) Stopping distance, relaxation time, and fluid velocity
c) Dynamic viscosity [kg m-1s-1], density, velocity, and characteristic length of an object.
d) Density difference, gravitational acceleration, characteristic length, coefficient of surface tension [kg s-2].

Explanation / Answer

a) Mach number

Mach number is defined as the ratio of object velocity and Speed of sound
It is dimensionless

b) Stoke's number

Stoke's number is defined as the ratio of stopping distance of a particle to a characteristic dimension of the obstacle (Stopping distance is the product of relaxation time and fluid velocity)
It is dimensionless

c) Reynolds Number

Reynolds Number is defined as the ratio of dynamic pressure and shearing stress
dynamic pressure = rho u^2 (Product of density and square of the velocity) shearing stress = mu*u/L (ratio of product of dynamic viscosity and velocity to the charactaristic length) It is dimensionless d) Bond Number Bond Number is defined as the ratio of of body forces to surface tension forces Bd = (rho'-rho)L^2*g/sigma (rho'-rho) is the Density difference g is the gravitational acceleration L is the characteristic length sigma is the coefficient of surface tension It is dimensionless

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