# Can be used to confirm that equations are

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- can be used to confirm that equations are dimensionally correct. e.g. = (dv) / (dy) (ML -1 T -2 ) = (ML -1 T -1 )(LT -1 )(L -1 )
Example: H = f (v, , D, LHS = L V = m/s = LT -1 = kg/m 3 = ML -3 D = m = L = Pa.s = M.T -1 L -1 H = K (v a x b x D c x We now solve for “a, b, c and d” using the fact that the dimensions on the LHS must equal the dimensions on the RHS ) d )
Buckingham Theorem This theorem is one method of determining appropriate dimensionless groups, from a number of given variables. It states that: - in a physical problem involving “n” quantities / variables - in which there are “m” fundamental dimensions (M,L,T) - the quantities may be arranged in (n-m) independent dimensionless groups or parameters These are called groups. Each group is formed from the product of 3 repeating variables raised to a power and one of the remaining non-repeating variables raised to the power of unity. The choice of repeating variables is guided by the following considerations: 1. Each repeating variable must contain between them all the fundamental dimensions. 2. The repeating variables should describe a size characteristic, a fluid property characteristic and kinematic characteristic, d, and v respectively. 3. The repeating set must contain three variables that cannot themselves be formed into a dimensionless group. e.g. a) both l and d cannot be chosen as they can be formed into the dimensionless group l/d. b) p, and v cannot be used since p/ v 2 is dimensionless.
Experiment: Consider a sphere of diameter d which is towed towards the left at a velocity V through a fluid of density and dynamic viscosity . The fluid could be air, water, oil, etc. The force required to tow the sphere is simply that required to overcome the force on the sphere due to friction and pressure forces. If the sphere is brought to a rest and the fluid given a velocity V to the right then the sphere experiences the same force as the towing force, but in the opposite direction. This force is called the Drag Force D .