Critical Velocity Formula

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

In terms of fluid dynamics, critical velocity is the speed and direction at which the fluid can have a laminar flow through a conduit (i.e., free from turbulence).

There is a particular formula of critical velocity along with the formula of critical velocity of liquid, critical velocity formula in open channel flow, and critical velocity formula circular motion, which we will discuss on this page.

What is Critical Velocity?

From the above text, we understand that critical velocity is the speed and direction at which the progression of a fluid through a tube changes from smooth to turbulence. Determining the critical velocity relies upon numerous factors, yet it is the Reynolds number that portrays the progression of the fluid through a tube as either turbulent or laminar. 

The Reynolds number is a dimensionless quantity, which implies that it has no units connected to it. Now, we will talk about the critical velocity formula.

Critical Velocity of Liquid Formula

The formula of critical velocity of liquid is given as;

\[v_c = \frac{k \eta}{dr} …..(1)\] (the required formula of critical velocity)

From the above formula of critical velocity, we have the description of the following quantities:

k   =  the constant of proportionality. It is Reynold’s number that helps predict the flow pattern in different fluid flow situations. 

The Value of Reynolds Number for The Different Types of Flow are as Follows:

• Laminar Flow  -  less than 2000 - It means that between 0 and 2000, the flow type is streamlined or laminar.

• Turbulent Flow  -  Between 2000 and 3500. At this moment, the flow is turbulent or unstable

• Transitional Flow  -  Greater than 3500. At this moment, the flow is highly unstable.

η  =  coefficient of viscosity. It is measured in units of poise, where one poise equals one dyne-second per square centimetre.

d  =  density of liquid/fluid, and

r  =  radius of the tube

From the above critical velocity equation for fluids (1), we see that the critical velocity ( vc ) depends on the coefficient of viscosity η, the density of the liquid d, and radius ‘r’ of the tube through which liquid/fluid flows.

Critical Velocity Dimensional Formula

The critical velocity dimensional formula is calculated as;

The formula of critical velocity is \[\frac{k \eta}{dr}\] . The dimensional formula of each quantity is:

         k  = Reynolds number = [M0L0T0]

         η  = coefficient of viscosity  = [M1L-1T-1]

         Radius (r)  =  [M0L1T0]

         Density of fluid (d)  =  [M1L-3T0]

Now, putting the dimensional formulas in the above equation (1):

\[ v_c = \frac{[M^0L^0T^0][M^1L^{-1}T^{-1}] }{[M^0L^1T^0][M^1L^{-3}T^0]} \]

Cancelling out the common terms, we get the critical velocity dimensional formula:

So, from the above dimensional formula of critical velocity, we see that the SI unit of critical velocity is meter per second, abbreviated as m/s.

Critical Velocity Formula in Open Channel Flow

The below diagram illustrates the open channel fluid flow:

(Image will be uploaded soon)

The Froude number is of interest here in light of the fact that its incentive for a specific open channel stream gives data on whether that stream is a subcritical, basic, or supercritical flow. The Froude number is characterized as follows: Fr = V/(gL)1/2, where 

Fr is the Froude Number 

V is the normal speed of the fluid in the channel (ft/sec for U.S. units or m/s for S.I.). 

g is the acceleration due to gravity, measured in 32.17 ft/sec2 for U.S. units or 9.81 m/s2 for S.I.

L is a length for the specific sort of open channel (ft for U.S. units or m for S.I.) 

For instance, with the flow in a rectangular channel, the length, L is the profundity of flow, y, making \[ F_r = \frac{V(c)}{\sqrt{gy}} \]

For a flow in a non-rectangular channel, \[F_r = \frac{v(c)}{\sqrt{g[\frac{A}{B}]}} = 1 \], on the grounds that the length, L, is A/B, where A is the cross-sectional area of fluid flow, and B is the surface width. 

As referenced above, the value of the Froude number gives data about the sort of flow. Subtleties are summed up underneath: 

In the event that Fr  > 1, it is a supercritical flow 

In the event that Fr < 1, it is a subcritical flow 

In the event that Fr = 1, it is critical flow

Rewriting the equation for Froude number in terms of critical velocity: 

                            \[v_c =( \sqrt{g}/A/B_c ). F_r \]

                            \[ v_c = \sqrt{g}/A_c/B_c . F_r \]

Since for a critical flow, the value of F is 1, so the critical velocity formula in open channel becomes:

                        \[v_c = \sqrt{g}/A_c/B_c \]

Critical Velocity Formula Circular Motion

The critical velocity is the minimum speed at a point where the normal (or tension, frictional, etc.) force is zero, and the force of gravity is the only thing that keeps the object making a circular motion.

In simple words,  the critical velocity is the minimum velocity of an object at the highest point of the rotation while having a vertical circular motion.

(Image will be uploaded soon)

Therefore, the minimum velocity required at the highest point to complete a vertical circle is expressed as the equation of critical velocity for a circular motion as;

                      \[v_c = \sqrt{rg} \]

Here,

g  = acceleration due to gravity, and

r  = the radius of the vertical circular path taken by the object

Conclusion

Critical velocity is the restricting value in which the flow becomes streamlined above which the flow becomes non-laminar or turbulent. The speed and direction in which the progression of fluid changes from smooth flow to tempestuous is known as the critical speed of the liquid.