Dimension less numbers in applied fluid mechanics

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. It is also known as a bare number or pure number or a quantity of dimension one[1] and the corresponding unit of measurement in the SI is one (or 1) unit[2][3] and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities, to which dimensions are regularly assigned, are length, time, and speed, which are measured in dimensional units, such as meter , second and meter per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.

1. SARDAR PATEL COLLEGE OF ENGINEERING,BAKROL

2. SUB:- ADVANCE FLUID MECHANICS ( 2160602) GUIDED BY: CHIRAG SIR SR. NO. NAME ENROLLMENT NO. 1 NILESH PRAJAPATI 151240106065 2 TIRATH PRAJAPATI 151240106066 TOPIC : DIMENSION LESS NUMBERS

3. Dimensionless number These are numbers which are obtained by dividing the inertia force by viscous force or gravity force or pressure force or surface tension force or elastic force . As this is ratio of once force to other, it will be a dimensionless number. These are also called non-dimensional parameters . Dimensionless numbers are widely applied in mechanical & chemical engineering.

4. Properties A dimensionless number has no physical unit associated with it. However, it is sometimes helpful to use the same units in both the numerator and denominator, such as kg/kg, to show the quantity being measured. A dimensionless number has the same value regardless of the measurement units used to calculate it. It has the same value whether it was calculated using the metric measurement system or the imperial measurement system. However, a number may be dimensionless in one system of units (e.g., in a nonrationalized cgs system of units with the electric constant ε  = 1), and not dimensionless in another system of units (e.g., the rationalized SI system, with ε  = 8.85419×10 -12  F/m).

5. The following are most important dimensionless numbers . Reynold’s Number Froude’s Number Euler’s Number Mach’s Number Weber’s Number

6. Reynolds number Sir George Stokes  introduced Reynolds numbers. Osbome Reynolds popularised the concept. The concept was introduced by  Sir George Stokes  in 1851 ,  but the Reynolds number was named by  Arnold Sommerfeld  in 1908 after   Osborne Reynolds  (1842–1912), who popularized its use in 1883 .

7. Dimensionless Numbers Reynold’s Number,Re : It is the ratio of inertia force to the viscous force of flowing fluid. The Reynolds Number can be used to determine if flow is laminar, transient or turbulent. The flow is laminar  when  Re < 2300 transient  when  2300 < Re < 4000 turbulent  when  Re > 4000

8. Example - Calculating Reynolds Number A Newtonian fluid with a dynamic or absolute viscosity of  0.38 Ns/m 2  and a specific gravity of  0.91  flows through a  25 mm  diameter pipe with a velocity of  2.6 m/s . The density can be calculated using the specific gravity like ρ = 0.91 (1000 kg/m 3 )     =  910   kg/m 3 The Reynolds Number can then be calculated using equation (1) like Re = (910 kg/m 3 ) (2.6 m/s) (25 mm) (10 -3  m/mm) / (0.38 Ns/m 2 )     = 156 (kg m / s 2 )/N     =  156  ~ Laminar flow (1 N = 1 kg m / s 2 )

9. Froude’s number named after William Froude is a dimensionless number defined as the ratio of characteristic velocity to the gravity wave velocity.  The Froude number in terms of gravity is expressed as, It is used to determine the resistance of an body which is submerged  partially moving along with water.

10. Where, Fr is Froude number, v is velocity, g is gravity, l is characteristic length.

11. Example – Calculating frodue’s Number Question 1:  Find the Froude number if the length of the boat is 2m and velocity is 10 m/s. Solution: Given: length l = 2m, velocity v = 10 m/s The froude number is given by, Fr = v( gl )1/2v( gl )1/2 Fr = 2m/s(9.8m/s2×2m)1/22m/s(9.8m/s2×2m)1/2  Fr = 0.451 Therefore, the froude number of the boat is  0.451.

12. Question 2:  Calculate the velocity of the moving fish in the water if its froude number is 0.72 and length 0.5 m. Solution: Given: length l = 0.5m, froude number Fr = 0.72 The froude number is given by, Fr = v( gl )1/2v( gl )1/2. The velocity of the moving fish is, v = Fr ×× ( gl )1/2( gl )1/2 v = 0.72 ×× (9.8×0.5)1/2(9.8×0.5)1/2 v = 1.59 m/s.

13. Euler’s Number Euler's Number, E : It is the ratio of inertia force to the pressure force of flowing fluid.

14. Mach’s Number, M: It is the ratio of inertia force to the elastic force of flowing fluid. Mach’s number

15. Weber’s Number The Weber Number is a dimensionless value useful for analyzing fluid flows where there is an interface between two different fluids. The Weber Number is the ratio between the inertial force and the surface tension force and the Weber number indicates whether the kinetic or the surface tension energy is dominant. It can be expressed as We = ρ v 2  l / σ                                             

16. where We = Weber number (dimensionless) ρ =  density of fluid  (kg/m 3 , lb /ft 3 ) v = velocity of fluid (m/s, ft /s) l = characteristic length (m, ft ) σ =  surface tension  (N/m ) Since the Weber Number represents an index of the  inertial force  to the  surface tension force  acting on a fluid element, it can be useful analyzing thin films flows and the formation of droplets and bubbles.

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