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In class 11 physics ncert Reynolds number

Re= ρA(v^2)/ (ηAv/d)

Where inertial force = ρA(v^2)

And viscous force is = ηAv/d

In the book d is described as dimensions of the pipe But they don't mention what A is at all.

As far as I understand the only way ρA(v^2) is the inertial force if A is the cross-sectional area of the pipe.

But in considering A as cross-section area it conflicts with the viscous force, as what I learnt is that

η= (F/A)/(v/l)

Where (F/A) is the shearing stress, A is the area in contact, where the force is applied. And v/l or Δx/lΔt is the rate of change of shearing strain.

Now in viscous force the area is not the cross-sectional area. From what I could gather it's the surface area of fluid in contact with the pipe.

Could someone please help me understand this discrepancy in term A correctly and also specify what exactly d is?

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I will derive the Reynolds Number denoting all the variables along the way

$A \rightarrow$ Area of cross section of tube

$v \rightarrow$ Velocity of fluid flow in tube

$\rho \rightarrow$ Density of fluid in tube

$r \rightarrow$ Radius of tube

$\eta \rightarrow$ Coefficient of viscosity

Mass of fluid flowing per second = $\mu$ = $\rho Av$

Inertial force = Force due to change in momentum = $\Delta l$= $\mu \cdot v$ = $\rho Av^2$

Viscous force = $F$ = $\eta \frac{Av}{r}$

Reynolds Number = $\frac{\Delta l}{F} = \frac{\rho A v^2}{\eta \frac{Av}{r}} = \frac{\rho A v^2 r}{\eta A v}$

Now this is of the form that was given in NCERT. which means that $A$ was cross sectional area after all.

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  • $\begingroup$ Thanks for the answer. But it would be helpful to understand how we come to this viscous force in this perticular case. That is can you also derive the viscous force for a fluid flowing in a pipe of radius r? $\endgroup$
    – Vladymort
    Commented Jan 13 at 8:17
  • $\begingroup$ The way we learnt the derivation of viscous force was by placing the fluid between to glass plates and pushing one of them by force f while keeping the other one steady and in that case area A in that viscous force was the area in contact with the glass plate. SO IT WOULD BE HELPFUL TO UNDERSTAND VISCOUS FORCE DERIVATION FOR A FLUID IN PIPE OF RADIUS R, just like in this case of Reynolds number $\endgroup$
    – Vladymort
    Commented Jan 13 at 8:24
  • $\begingroup$ Since it is not related to this particular question, I would recommend you to refer to physics.stackexchange.com/questions/218098/… or any other website that explains it well. $\endgroup$
    – Dev Not Taken
    Commented Jan 13 at 12:08
  • $\begingroup$ I was further studying regarding the Reynolds number and found about the "d" in my original question which you referred as "r". It is called the characteristic linear dimension and in case of a horizontal pipe it is actually the diameter? $\endgroup$
    – Vladymort
    Commented Jan 19 at 12:17
  • $\begingroup$ @Vladymort, both radius and diameter can be used which would return different values of Reynolds number. Just depends on the source. Wikipedia, Physics Libre all uses characteristic length whereas NCERT uses radius $\endgroup$
    – Dev Not Taken
    Commented Jan 19 at 12:58

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