Critical Velocity

What is critical velocity?

Critical velocity is defined as the speed at which a falling object reaches when both gravity and air resistance are equalised on the object.

The other way of defining critical velocity is the speed and direction at which the fluid can flow through a conduit without becoming turbulent. Turbulent flow is defined as the irregular flow of the fluid with continuous change in magnitude and direction. It is the opposite of laminar flow which is defined as the flow of fluid in parallel layers without disruptions of the layers.

Critical velocity formula

Following is the mathematical representation of critical velocity with the dimensional formula:

\(V_{C}=\frac{R_{e}\eta }{\rho r}\)

Where,

Vc: critical velocity

Re: Reynolds number (ratio of inertial forces to viscous forces)

𝜂: coefficient of viscosity

r: radius of the tube

⍴: density of the fluid

Dimensional formula of:

  • Reynolds number (Re): M0L0T0
  • Coefficient of viscosity (𝜂): M1L-1T-1
  • Radius (r) : M0L1T0
  • Density of fluid (⍴): M1L-3T0
  • Critical velocity: \(V_{c}=\frac{\left [ M^{0}L^{0}T^{0} \right ]\left [ M^{1}L^{-1}T^{-1} \right ]}{\left [ M^{1}L^{-3}T^{0} \right ]\left [ M^{0}L^{1}T^{0} \right ]}\)

∴\( V_{c}=M^{0}L^{1}T^{-1}\)

SI unit of critical velocity is ms-1

Reynolds number

Reynolds number is defined as the ratio of inertial forces to viscous forces. Mathematical representation is as follows:

\(R_{e}=\frac{\rho uL}{\mu }=\frac{uL}{\nu }\)

Where,

⍴: density of the fluid in kg.m-3

𝜇: dynamic viscosity of the fluid in m2s

u: velocity of the fluid in ms-1

L: characteristic linear dimension in m

𝜈: kinematic viscosity of the fluid in m2s-1

Depending upon the value of Reynolds number, flow type can be decided as follows:

  • If Re is between 0 to 2000, the flow is streamlined or laminar
  • If Re is between 2000 to 3000, the flow is unstable or turbulent
  • If Re is above 3000, the flow is highly turbulent

Reynolds number with respect to laminar and turbulent flow regimes are as follows:

  • When the Reynolds number is low that is the viscous forces are dominant, laminar flow occurs and are characterised as a smooth, constant fluid motion
  • When the Reynolds number is high that is the inertial forces are dominant, turbulent flow occurs and tends to produce vortices, flow instabilities and chaotic eddies.

Following is the derivation of Reynolds number:

\(R_{e}=\frac{ma}{\tau A}=\frac{\rho V.\frac{du}{dt}}{\mu \frac{du}{dy}.A}\propto \frac{\rho L^{3}\frac{du}{dt}}{\mu \frac{du}{dy}L^{2}}=\frac{\rho L\frac{dy}{dt}}{\mu }=\frac{\rho u_{0}L}{\mu }=\frac{u_{0}L}{\nu }\)

Where,

t: time

y: cross-sectional position

\(u=\frac{dx}{dt}\) : flow speed

τ: shear stress in Pa

A: cross-sectional area of the flow

V: volume of the fluid element

u0: maximum speed of the object relative to the fluid in ms-1

L: a characteristic linear dimension

𝜇: dynamic viscosity of the fluid in Pa.s

𝜈: kinematic viscosity in m2s

⍴: density of the fluid in kg.m-3

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