Question 6.5: A fully developed laminar incompressible flow between two fl...
A fully developed laminar incompressible flow between two flat plates with one plate moving with a uniform velocity U with respect to other is known as Couette flow. In a Couette flow, the velocity u at a point depends on its location y (measured perpendicularly from one of the plates), the distance of separation h between the plates, the relative velocity U between the plates, the pressures gradient dp/dx imposed on the flow, and the viscosity μ of the fluid. Find a relation in dimensionless form to express u in terms of the independent variables as described above.
The Buckinghams π theorem is used for this purpose. The variables describing a Couette flow are u, U, y h, dp/dx and μ. Therefore, m (the total no. of variables) = 6.
n (the number of fundamental dimensions in which the six variables are expressed) = 3 (M, L and T)
Hence no. of independent π terms is 6 3 = 3
To determine these π terms, U, h and μ are taken as repeating variables. Then
\pi_{1}=U^{a} h^{b} \mu^{c} u\pi_{2}=U^{a} h^{b} \mu^{c} y
\pi_{3}=U^{a} h^{b} \mu^{c} d p / d x
The above three equations can be expressed in terms of the fundamental dimensions of each variable as
M ^{0} L ^{0} T ^{0}=\left( LT ^{-1}\right)^{a}( L )^{b}\left( ML ^{-1} T ^{-1}\right)^{c} LT ^{-1} (6.37)
M ^{0} L ^{0} T ^{0}=\left( LT ^{-1}\right)^{a}( L )^{b}\left( ML ^{-1} T ^{-1}\right)^{c} L (6.38)
M ^{0} L ^{0} T ^{0}=\left( LT ^{-1}\right)^{a}( L )^{b}\left( ML ^{-1} T ^{-1}\right)^{c} ML ^{-2} T ^{-2} (6.39)
Equating the exponents of M, L and T on both sides of the above equations we get the following:
From Eq. (6.37): c = 0
a + b c + 1 = 0
a c 1 = 0
which give a = 1, b = 0 and c = 0
Therefore \pi_{1}=\frac{u}{U}
From equation (6.38): c = 0
a + b c +1 = 0
c a = 0
which give a = 0, b = 1 and c = 0
Therefore \pi_{2}=\frac{y}{h}
It is known from one of the corolaries of the p theorem, as discussed earlier, that if any two physical quantities defining a problem have the same dimensions, the ratio of the quantities is a π term. Therefore, there is no need of evaluating the terms \pi_{1} \text { and } \pi_{2} through a routine application of π theorem as done here; instead they can be written straight forward as \pi_{1}=u / U \text { and } \pi_{2}=y / h.
From Eq. (6.39)
c + 1 = 0
a + b c 2 = 0
a c 2 = 0
which give a = 1, b = 2 and c = 1
Therefore \pi_{3}=\frac{h^{2}}{\mu U} \frac{ d p}{ d x}
Hence, the governing relation amongst the different variables of a couette flow in dimensionless form is
f\left(\frac{u}{U}, \frac{y}{h}, \frac{h^{2}}{\mu U} \frac{ d p}{ d x}\right)=0or \frac{u}{U}=F\left(\frac{y}{h}, \frac{h^{2}}{\mu U} \frac{ d p}{ d x}\right) (6.40)
It is interesting to note, in this context, that from the exact solution of Navier Stokes equation, the expression of velocity profile in case of a couette f low has been derived in Chapter 8 (Sec. 8.4.2) and is given by Eq. (8.39) as
\frac{u}{U}=y / h-\left(\frac{h^{2}}{2 \mu U} \frac{ d p}{ d x}\right) \frac{y}{h}\left(1-\frac{y}{h}\right)However, π theorem can never determine this explicit functional form of the relation between the variables.