How Does Critical Velocity Affect Motion in Physics?
The Velocity with which the liquid flow changes from streamlined to turbulent known as the Critical Velocity of the fluid. The fluid's streamlines are straight parallel lines when the Velocity is less for the fluid in the pipe. As the Velocity of the fluid gradually increases, the streamline continues to be straight and parallel to the pipe wall. Once the Velocity reaches the breaking point, it forms patterns. Throughout the pipe, the Critical Velocity will disperse the streamlines.
To keep the flow non-Critical, the sewer pipes are gradually sloped so gravity works on the fluid flow. The excess Velocity of flow can cause erosion of the pipe since solid particles are present in the flow, which will lead to damage to the pipe. By using the trenchless method like cured-in-place-pipe, pipe bursting and slip lining, pipe damaged by the action of high-Velocity fluid can be rectified.
The fluid's Critical Velocity can be calculated using the Reynolds number, which characterizes the flow of streamlined or turbulent air. It is a dimensionless variable which can be calculated by using the formula.
Critical Velocity Formula
The mathematical representation of Critical Velocity with the dimensional formula given below:
Critical Velocity vc = (kη/rρ)
Where,
K = Reynold’s number,
η = coefficient of viscosity of a liquid
r = radius of capillary tube and
ρ = density of the liquid.
Dimensional formula of:
Reynolds number (Re) = M0L0T0
Coefficient of viscosity (𝜂) = M1L-1T-1
Radius (r) = M0L1T0
The density of fluid (⍴) = M1L-3T0
\[V_c=\frac{\mid M^0L^0T^0\mid \mid M^1L^{-1}T^{-1}\mid }{\mid M^1L^{-3}T^0\mid \mid M^0L^1T^0\mid }\]
Critical Velocity
∴Vc=M0L1T-1
SI unit of Critical Velocity is meter/sec
Reynolds Number
The ratio between inertial forces and viscous forces is known as the Reynolds number. Reynold’s number is a pure number that helps identify the nature of the flow and Critical Velocity of a liquid through a pipe.
The number is mathematically represented as follows:
\[R_c=\frac{\rho uL}{\mu }=\frac{uL}{v}\]
Where,
⍴: density of the fluid in kg.m^-3
𝜇: dynamic viscosity of the fluid in m^2s
u: Velocity of the fluid in ms^-1
L: characteristic linear dimension in m
𝜈: kinematic viscosity of the fluid in m2s-1
By determining the value of the Reynolds number, flow type can decide as follows:
If the value of Re is between 0 to 2000, the flow is streamlined or laminar
If the value of Re is between 2000 to 3000, the flow is unstable or turbulent
If the value of Re is above 3000, the flow is highly turbulent
Reynolds number concerning laminar and turbulent flow regimes are as follows:
When the value of Reynolds number is low then the viscous forces are dominant, laminar flow transpires and are categorized as a smooth, constant fluid motion
When the value of the Reynolds number is high, then the inertial forces are dominant, turbulent flow occurs and tends to produce vortices, flow uncertainties, and disordered eddies.
Following is the derivation of Reynolds number:
\[R_c=\frac{ma}{TA}=\frac{\rho V.\frac{du}{dt}}{\mu \frac{du}{dy}.A}\alpha \frac{\rho L^3.\frac{du}{dt}}{\mu \frac{du}{dy}L^2}=\frac{\rho L\frac{dy}{dt}}{\mu }=\frac{\rho u_0L}{\mu }=\frac{u_0}{v}\]
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 = a maximum speed of the particle relative to the fluid in ms^-1
L = a characteristic linear dimension
𝜇 = fluid of dynamic viscosity in Pa.s
𝜈 = kinematic viscosity in m^2s
⍴ = density of the fluid in kg.m^-3
Critical Velocity Ratio
The idea of Critical Velocity was established that will make a channel free from silting and scouring. From long observations, a relation between Critical Velocity and full supply depth was formulated as
The values of C and n were found out as 0.546 and 0.64 respectively, thus v0=0.546 D^0.64
However, in the above formula, the Critical Velocity was affected by the grade of silt. So, another factor (m) was introduced which was known as the Critical Velocity ratio (C.V.R).
V0=0.546mD^0.64
Critical Velocity Ratio (C.V.R) is otherwise known as the ratio of mean Velocity 'V' to the Critical Velocity 'Vo' where Vo is known as the Critical Velocity ratio (CVR).
It is denoted by m i.e. CVR (m) = V/Vo
When m = 1, there will be no silting or scouring
When m>1, scouring will occur,
When m<1, silting will occur.
So, by finding the value of m, the condition of the canal can be predicted whether it will have silting or scouring.
Critical Velocity Definition
The speed at which gravity and air resistance on a falling object are equalised is defined as the speed at which the object reaches its destination Critical Velocity. The speed and direction at which a fluid will flow through a conduit without becoming turbulent is the alternative approach of elaborating Critical Velocity. Turbulent flow is defined as a fluid flow that is erratic and changes amplitude and direction continuously.
Characteristics of Turbulent Flow
At higher velocities, low viscosity, and larger associated linear dimensions, turbulent flow is more likely to occur. A turbulent flow is defined as one with a Reynolds number greater than Re > 3500.
Irregularity: The irregular motion of the fluid particles characterizes the flow. Fluid particles travel in a haphazard manner. Turbulent flow is frequently treated statistically rather than deterministically for this reason.
Diffusivity: Inflow with a relatively constant Velocity Dispersal occurs across a section of the pipe, resulting in the entire fluid flowing at a single value and rapidly dropping extremely close to the walls. Diffusivity is the property that accounts for the better mixture and exaggerated rates of mass, momentum, and energy transfers in a flow.
Rotationality:Turbulent flow is distinguished by a strong three-dimensional vortex production process. Vortex stretching is the name for this mechanism.
Dissipation: A dissipative approach is one in which viscous shear stress moulds the K.E. of flow into internal energy.
Critical Velocity-Formula, Units
With the dimensional formula, the following is a mathematical demonstration of Critical Velocity:
VC= Reηρr
Where,
Vc: Critical Velocity
Re: Regarding the Reynolds figure (ratio of mechanical phenomenon forces to viscous forces)
𝜂: coefficient of viscosity
r: radius of the tube
⍴: density of the fluid
Critical Velocity Dimensional Formula:
Vc = M0L1T-1
Unit of Critical Velocity:
SI unit of Critical Velocity is meter/sec
Types of Critical Velocity
Lower Critical Velocity:
Lower Critical Velocity is the speed at which laminar flow ceases or shifts from laminar to transition period. There is a transition time between laminar and turbulent flow. It has been discovered experimentally when a laminar flow turns into turbulence, it does not change abruptly. But there's a transition period between 2 forms of flows. This experiment was first performed by Prof. Reynolds Osborne in 1883.
Upper Critical Velocity:
The speed at which turbulence in a flow begins or ends. Greater or higher Critical Velocity refers to the Velocity at which a flow transitions from a transition period to turbulent flow.
FAQs on Critical Velocity Explained: Formula, Derivation & Applications
1. What is critical velocity in the context of fluid dynamics?
In fluid dynamics, critical velocity (Vc) is the specific speed at which the flow of a fluid transitions from a smooth, orderly state to a chaotic one. Below this velocity, the fluid exhibits a laminar or streamline flow, where layers of fluid slide smoothly past one another. Above this velocity, the flow becomes turbulent, characterised by eddies, swirls, and unpredictable changes in pressure and velocity.
2. What is the formula to calculate the critical velocity of a fluid?
The critical velocity of a fluid flowing through a pipe or tube is calculated using a formula derived from the Reynolds number. The critical velocity formula is:
Vc = (Re × η) / (ρ × r)
Where:
- Vc is the critical velocity.
- Re is the Reynolds number, a dimensionless quantity.
- η (eta) is the coefficient of viscosity of the fluid.
- ρ (rho) is the density of the fluid.
- r is the radius of the pipe or tube.
3. What is the difference between critical velocity and terminal velocity?
Critical velocity and terminal velocity are two distinct concepts in physics:
- Critical Velocity: This refers to the speed of the fluid itself. It is the threshold that determines whether the fluid's flow pattern is laminar (smooth) or turbulent (chaotic).
- Terminal Velocity: This refers to the constant speed that a body or object attains when falling through a fluid (like air or water). It occurs when the downward force of gravity is perfectly balanced by the upward force of drag or resistance from the fluid. You can learn more about this in notes on the properties of solids and liquids.
4. How is critical velocity related to the Reynolds number?
Critical velocity is fundamentally defined by the Reynolds number (Re). The Reynolds number is a dimensionless value that predicts flow patterns.
- For a fluid flowing in a pipe, if Re is less than approximately 2000, the flow is considered laminar.
- If Re is greater than approximately 4000, the flow is turbulent.
The critical velocity is the speed at which the Reynolds number crosses this transitional threshold (around 2000-4000), causing the streamline flow to become turbulent.
5. What happens to a fluid's flow when its speed exceeds the critical velocity?
When a fluid's speed surpasses its critical velocity, the nature of its flow changes dramatically from laminar to turbulent. In this turbulent state:
- The fluid particles no longer move in smooth, parallel layers.
- The motion becomes chaotic, with the formation of eddies, vortices, and swirls.
- There is a significant increase in energy loss due to internal friction.
- The pressure and velocity at any given point within the fluid begin to fluctuate unpredictably.
6. Is the critical velocity for a satellite in orbit the same as for fluid flow?
No, they are completely different concepts that happen to share the same name.
- Critical Velocity in Fluid Flow: Relates to the transition from laminar to turbulent flow within a fluid.
- Critical Velocity in Gravitation: Refers to the minimum horizontal velocity an object (like a satellite) needs to maintain a stable, circular orbit around a celestial body like Earth. It is the speed at which the gravitational force provides the exact centripetal force required for circular motion. This concept is a key part of the study of Gravitation.
7. What is the dimensional formula for critical velocity?
The dimensional formula for critical velocity is the same as that for any velocity. It is [M⁰L¹T⁻¹].
This can be confirmed by analysing the dimensions of the variables in its formula, Vc = (Re × η) / (ρ × r), where the dimensions of Reynolds number (Re) cancel out, leaving the dimensions of velocity.
8. Why is understanding critical velocity important in practical applications?
Understanding critical velocity is crucial in many engineering and medical fields for designing efficient systems. For example:
- Pipeline Design: Engineers design pipelines for transporting oil, water, or gas to ensure the flow remains laminar. Turbulent flow increases friction, requiring more energy (and higher costs) to pump the fluid.
- Aerospace Engineering: It helps in studying airflow over an aircraft's wings to maximise lift and minimise drag.
- Medical Science: It is used to analyse blood flow in arteries. Turbulent blood flow can be an indicator of blockages or other cardiovascular issues.
9. How does the density of a fluid affect its critical velocity?
The critical velocity of a fluid is inversely proportional to its density (ρ), as seen in the formula Vc ∝ 1/ρ. This means that if all other factors remain constant, a denser fluid will have a lower critical velocity. In practical terms, a denser fluid will transition from smooth (laminar) to chaotic (turbulent) flow at a much lower speed compared to a less dense fluid.
