Question

what are the steps on how to obtain a set of dimensionless groups. hint dimensional analysis

Answer 1

STEP 1: WRITE EACH VRIABLE IN DIMENSIONAL FORM. LET NUMBER OF FUNDAMENTAL (BASIC) DIMENSIONS INVOLVED ARE n.

STEP 2: IF TOTAL NUMBER OF VARIABLES INVOLVED ARE m, THEN AS PER BUCKINGHAM PI THEOREM: PROBLEM CAN BE DIVIDED IN m-n DIMENSIONLESS GROUPS DENOTED BY π_K WHERE (K=1,2,3----m-n).

STEP 3: TO FORM π GROUP WE MUST SELECT A GROUP OF REPEATING (CORE) VARIABLES WHICH SHALL APPEAR IN EACH π GROUP. EACH π GROUP SHALL ALSO CONTAIN ONE NON REPEATING VARIABLE. CRITERIA FOR SELECTING REPEATING VARIABLES IS THAT TOGETHER THEY MUST CONTAIN ALL DIMENSIONS INVOLVED IN PROBLEM.

STEP 4: NOW ONE BY ONE, FOR EACH DIMENSIONLESS π GROUP, WRITE GROUP IN THE FORM OF EACH REPEATING & ONE NON REPEATING VARIABLE RAISED TO ORBITARY POWER. SOLVE FOR EACH DIMENSION POWER = 0 (AS DIMENSIONLESS GROUP HAS NO UNIT) TO GET THE VALUE OF POWER.

BY FINDING VALUE OF THESE POWERS, DIMENSIONLESS GROUP IS FOUND.

BY FINDING REMAINING DIMENSIONLESS GROUPS, WE CAN ALSO FIND RELATION AMONG THESE VARIABLES.

ONE EXAMPLE TO CLEAR THE PROCEDURE IS ALSO ATTACHED BELOW:

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