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How is the expression for Reynold's number valid in steady flow?

Physics Asked by Azur Qasm on June 20, 2021

Reynold’s number is defined as the ratio of inertia force to viscous force. A common expression for this inertia force is (density)x(Area of cross section of tube)x(velocity of fluid)x(velocity of fluid) which can also be expressed as (mass flow rate)x(velocity of fluid). My question pertains to steady flow of a perfect fluid and the tube is smooth.

However force (as in inertia force) is defined as rate of change of momentum. In steady flow, change in momentum with time must be zero (any change in property of with fluid must not change with time). So how can there be a change in momentum with respect to time and thus by extension how can there be a force (inertia force) and how is the expression for Reynold’s number valid?

Inertia force is an imaginary force. It is just another way of describing what forces are driving the fluid, just with the sign reversed. For a fluid in steady flow that is not under the effect of driving forces (because force is not a requirement for motion) what is the meaning of inertia force?

I do admit that since momentum is a vector we have change in momentum possible in curved tubes without change in magnitude so I’m excluding them from the case here. Consider only straight smooth tubes.

I want to know how exactly this expression for inertia force is valid for steady flow.

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