A dimensionally consistent relation for the volume V of a liquid of coefficient of viscosity η flowing per second through a tube of radius r and length l and having a pressure difference p across its end, is

The correct answer is: V = πpr⁴/8ηl

To determine the dimensionally consistent relation for the rate of flow (volume per second, $V$) of a liquid flowing through a tube, we can use the method of dimensional analysis.

First, let us write down the dimensions of each physical quantity involved in the relation:
1. Volume flowing per second ($V$): Since it is the volume of liquid flowing per unit time,
$\left[V\right]=L^3{T}^{-1}$
2. Pressure difference ($p$): Pressure is force per unit area,
$\left[p\right]=\frac{\left[\text{Force}\right]}{\left[\text{Area}\right]}=\frac{ML{T}^{-2}}{L^2}=M{L}^{-1}{T}^{-2}$
3. Radius of the tube ($r$): It is a length,
$\left[r\right]=L$
4. Coefficient of viscosity ($\eta$):
$\left[\eta \right]=M{L}^{-1}{T}^{-1}$
5. Length of the tube ($l$): It is a length,
$\left[l\right]=L$

Let the rate of flow $V$ depend on pressure gradient (pressure difference per unit length, $\frac{p}{l}$), radius $r$, and coefficient of viscosity $\eta$ as follows:
$V\propto {\left(\frac{p}{l}\right)}^a{r}^b{\eta }^c$

Now, let us substitute the dimensions of each term into the proportionality equation:
$\left[L^3{T}^{-1}\right]={\left[\frac{M{L}^{-1}{T}^{-2}}{L}\right]}^a{\left[L\right]}^b{\left[M{L}^{-1}{T}^{-1}\right]}^c$
Simplifying the right-hand side, we get:
$M^0{L}^3{T}^{-1}={\left(M{L}^{-2}{T}^{-2}\right)}^a{L}^b{\left(M{L}^{-1}{T}^{-1}\right)}^c$
$M^0{L}^3{T}^{-1}=M^{a+c}{L}^{-2a+b-c}{T}^{-2a-c}$

Equating the exponents of $M$, $L$, and $T$ on both sides:
1. For $M$:
$a+c=0\Rightarrow c=-a$
2. For $T$:
$-2a-c=-1$
Substitute $c=-a$ into this equation:
$-2a-(-a)=-1\Rightarrow -a=-1\Rightarrow a=1$
Thus, $c=-1$.
3. For $L$:
$-2a+b-c=3$
Substitute $a=1$ and $c=-1$:
$-2\left(1\right)+b-(-1)=3\Rightarrow -2+b+1=3\Rightarrow b-1=3\Rightarrow b=4$

Substituting the values of $a$, $b$, and $c$ back into our proportionality expression:
$V\propto {\left(\frac{p}{l}\right)}^1{r}^4{\eta }^{-1}\Rightarrow V\propto \frac{p{r}^4}{\eta l}$

By experimental validation (Poiseuille's Law), the constant of proportionality is determined to be $\frac{\pi }{8}$. Therefore, the dimensionally consistent equation is:
$V=\frac{\pi p{r}^4}{8\eta l}$