Question

The equation for the velocity V in a pipe with diameter d and length L, under laminar condition is given by the equation V=Δpdsquare/32µL, with V=Δp the pressure drop and Δp the viscosity of the fluid. Determine whether the equation is dimensionally consistent by inspecting the dimensions on both sides

Answer 1

Given that

`V=(\Delta Pd^2)/(32\mu L)`

LHS of above given equation have dimension
`[M^oL^(1)T^(-1)]`.

Now find the dimension of RHS

Dimension of P =
`[ML^(-1)T^(-2)]`.

Dimension of d=
`[M^(0)L^(1)T^(0)]`.

Dimension of μ =
`[ML^(-1)T^(-1)]`.

Dimension of L=
`[M^(0)L^(1)T^(0)]`.

So

`(\Delta Pd^2)/(32\mu L)=([ML^(-1)T^(-2)].[M^(0)L^(1)T^(0)]^2)/([ML^(-1)T^(-1)].[M^(0)L^(1)T^(0)])`

`(\Delta Pd^2)/(32\mu L)=[M^0L^(1)T^(-1)]`

It means that both sides have same dimensions.