NEET UG - Viscosity, Stokes law, terminal velocity, streamline and turbulent flow, critical velocity E-Book Study Material

Viscosity

Viscosity is a measure of a fluid's resistance to flow. It's essentially the internal friction of a fluid. Imagine trying to stir honey versus water. Honey is much harder to stir because it has higher viscosity. This resistance arises from the cohesive forces between fluid molecules and the momentum exchange between adjacent layers of the fluid moving at different speeds.

Think of a fluid as being made up of many thin layers sliding over each other. When these layers move at different velocities, there's a shearing stress between them. Viscosity quantifies how much shear stress is required to produce a certain rate of shear strain. A more viscous fluid requires a greater shear stress for the same rate of flow.

Factors Affecting Viscosity

Viscosity is primarily influenced by temperature and the nature of the fluid itself.

Temperature: For liquids, viscosity generally decreases as temperature increases. This is because increased thermal energy overcomes the intermolecular cohesive forces, allowing layers to slide more easily. For gases, viscosity increases with temperature. This is because higher temperatures lead to more frequent collisions between molecules, increasing momentum transfer and thus resistance to flow.
Intermolecular Forces: Stronger intermolecular forces lead to higher viscosity. For example, glycerol, with its strong hydrogen bonds, is much more viscous than water.
Molecular Structure: Long, chain-like molecules can become entangled, increasing resistance to flow and thus viscosity.

Coefficient of Viscosity (η)

The coefficient of viscosity, denoted by the Greek letter eta (η), is a proportionality constant that relates the shear stress to the rate of shear strain.

Consider two parallel plates separated by a distance 'd', with a fluid in between. If the lower plate is stationary and the upper plate moves with a velocity 'v', the fluid velocity varies linearly from 0 at the lower plate to 'v' at the upper plate. The velocity gradient (rate of shear strain) is given by $dv/dx$, where $x$ is the distance perpendicular to the flow.

Newton's law of viscosity states that the shear stress ($\tau$) is directly proportional to the velocity gradient: $$ \tau \propto \frac{dv}{dx} $$ Introducing the coefficient of viscosity (η) as the constant of proportionality: $$ \tau = \eta \frac{dv}{dx} $$

The coefficient of viscosity (η) can be expressed as: $$ \eta = \frac{\tau}{dv/dx} $$

Units of Viscosity

In the SI system, the unit of viscosity is Pascal-second (Pa·s) or Newton-second per square meter (N·s/m²).

In the CGS system, the unit is the poise (P). 1 poise = 0.1 Pa·s.

A more common unit used in practice is the centipoise (cP), where 1 cP = 0.01 P = 0.001 Pa·s. For reference, the viscosity of water at 20°C is approximately 1 cP.

Kinematic Viscosity

Kinematic viscosity (ν) is defined as the ratio of dynamic viscosity (η) to the density (ρ) of the fluid: $$ \nu = \frac{\eta}{\rho} $$

It represents the ratio of viscous forces to inertial forces. The SI unit for kinematic viscosity is m²/s. In the CGS system, the unit is the stokes (St), where 1 St = 1 cm²/s = 10⁻⁴ m²/s. The centistokes (cSt) is also commonly used (1 cSt = 10⁻² St = 10⁻⁶ m²/s).

Stokes Law and Terminal Velocity

When an object moves through a viscous fluid, the fluid exerts a drag force on the object. This drag force opposes the motion. Stokes' Law describes this drag force for a small, spherical object moving at a slow speed through a viscous fluid.

Stokes' Law

Stokes found that for a small sphere of radius 'r' moving with a uniform velocity 'v' through a fluid of viscosity 'η', the drag force ($F_d$) is given by: $$ F_d = 6 \pi \eta r v $$

This formula holds true under specific conditions:

The object is spherical.
The flow is slow (low Reynolds number).
The fluid is homogeneous and incompressible.
The object is small compared to the dimensions of the fluid container, so boundary effects are minimal.

The direction of this drag force is opposite to the direction of the object's velocity.

Mnemonic for Stokes' Law: Remember the formula $F_d = 6 \pi \eta r v$. Think of a Sphere (S) moving slowly in a Thick Oil (T, O) creating a Kind of Effort (K, E) i.e. a drag force, which depends on the Stickiness (S) of the oil (η), its Radius (r) and its Velocity (v). The '6 pi' is a constant factor.

Terminal Velocity

When an object is dropped into a viscous fluid, it initially accelerates due to gravity. However, as its velocity increases, the drag force also increases. Eventually, the upward forces (buoyant force and drag force) balance the downward force (gravitational force). At this point, the net force on the object becomes zero, and it stops accelerating. The constant velocity achieved is called the terminal velocity ($v_t$).

Let's analyze the forces acting on a spherical object of radius 'r', density 'ρ_s', falling through a fluid of density 'ρ_f' and viscosity 'η'.

Gravitational Force ($F_g$): This is the weight of the object. $$ F_g = m \cdot g = \left(\frac{4}{3} \pi r^3 \rho_s\right) g $$
Buoyant Force ($F_b$): This is the upward force exerted by the fluid, equal to the weight of the fluid displaced by the object. $$ F_b = \left(\frac{4}{3} \pi r^3 \rho_f\right) g $$
Drag Force ($F_d$): As per Stokes' Law, when the object reaches terminal velocity $v_t$. $$ F_d = 6 \pi \eta r v_t $$

At terminal velocity, the net force is zero: $$ F_g = F_b + F_d $$ Substituting the expressions for the forces: $$ \left(\frac{4}{3} \pi r^3 \rho_s\right) g = \left(\frac{4}{3} \pi r^3 \rho_f\right) g + 6 \pi \eta r v_t $$ Rearranging to solve for $v_t$: $$ 6 \pi \eta r v_t = \left(\frac{4}{3} \pi r^3 \rho_s\right) g - \left(\frac{4}{3} \pi r^3 \rho_f\right) g $$ $$ 6 \pi \eta r v_t = \frac{4}{3} \pi r^3 g (\rho_s - \rho_f) $$ Now, isolate $v_t$: $$ v_t = \frac{\frac{4}{3} \pi r^3 g (\rho_s - \rho_f)}{6 \pi \eta r} $$ Simplifying the expression: $$ v_t = \frac{2 r^2 g (\rho_s - \rho_f)}{9 \eta} $$

This equation shows that the terminal velocity depends on the square of the radius, the acceleration due to gravity, the difference in densities between the object and the fluid, and the viscosity of the fluid.

If the object's density ($\rho_s$) is less than the fluid's density ($\rho_f$), the term $(\rho_s - \rho_f)$ will be negative. This means the net force will initially be upwards, and the object will float or move upwards until other forces balance it. If $\rho_s > \rho_f$, the object sinks with a positive terminal velocity.

Key takeaway for Terminal Velocity: The object stops accelerating when the upward forces (buoyancy + drag) equal the downward force (gravity). The formula $v_t = \frac{2 r^2 g (\rho_s - \rho_f)}{9 \eta}$ is crucial. Notice $v_t \propto r^2$. A larger object (with the same density and falling in the same fluid) reaches a higher terminal velocity faster.

Fluid Flow: Streamline and Turbulent Flow

When a fluid flows, its motion can be described in different ways. The nature of the flow depends on factors like the fluid's velocity, viscosity, and density, as well as the dimensions of the pipe or channel it flows through. Two main types of fluid flow are streamline flow and turbulent flow.

Streamline Flow (Laminar Flow)

Streamline flow, also known as laminar flow, is characterized by smooth, orderly motion. In this type of flow, fluid particles move along well-defined paths called streamlines. Each streamline represents the path of a single particle.

Key characteristics of streamline flow:

Smooth and Orderly: Fluid layers slide smoothly past each other. There is no mixing between adjacent layers.
Constant Velocity at a Point: At any given point in the fluid, the velocity of the fluid particles remains constant over time.
Streamlines Do Not Intersect: Streamlines are continuous lines and never cross each other. The velocity vector at any point is tangent to the streamline at that point.
Viscous Forces Dominate: Streamline flow typically occurs at lower velocities where viscous forces are significant enough to damp out any irregularities.

Imagine pouring honey slowly; the flow is smooth and layered. This is an example of streamline flow.

Turbulent Flow

Turbulent flow is characterized by chaotic, irregular, and erratic motion. In this type of flow, fluid particles move in random, swirling patterns called eddies. There is significant mixing of the fluid.

Key characteristics of turbulent flow:

Chaotic and Irregular: Fluid motion is highly disorganized, with eddies and vortices.
Variable Velocity at a Point: The velocity of the fluid at any given point fluctuates randomly over time.
Mixing: Significant mixing occurs between different parts of the fluid, leading to a more uniform distribution of momentum and properties throughout the flow.
Inertial Forces Dominate: Turbulent flow typically occurs at higher velocities where inertial forces overcome viscous forces, leading to instability and chaotic motion.

Imagine a rapidly flowing river with rapids and whirlpools. This is an example of turbulent flow.

Reynolds Number (Re)

The transition between streamline and turbulent flow is often characterized by a dimensionless quantity called the Reynolds number (Re). It is defined as the ratio of inertial forces to viscous forces. For flow through a pipe of diameter 'D', the Reynolds number is given by: $$ Re = \frac{\rho v D}{\eta} $$ where:

ρ is the density of the fluid
v is the average velocity of the fluid
D is the characteristic linear dimension (e.g., diameter of the pipe)
η is the dynamic viscosity of the fluid

The value of the Reynolds number helps predict the type of flow:

Re < 2100 (approximately): The flow is generally streamline (laminar).
2100 < Re < 4000 (approximately): The flow is in a transitional region, exhibiting characteristics of both laminar and turbulent flow.
Re > 4000 (approximately): The flow is typically turbulent.

These values can vary slightly depending on the geometry of the flow situation.

Reynolds Number Shortcut: Think of it as Realistically Explaining Regimes of Everyday Flow. It compares the Rush (inertial forces) to the Effort (viscous forces). High Re means rush wins (turbulent), low Re means effort wins (streamline).

Critical Velocity

Critical velocity is a concept closely related to the transition from streamline flow to turbulent flow. It is the maximum velocity a fluid can attain in a particular situation while still maintaining streamline flow. Above this velocity, the flow becomes turbulent.

The critical velocity is not a single fixed value for a fluid but depends on the specific conditions of the flow, such as the geometry of the channel, the properties of the fluid (viscosity and density), and the presence of any disturbances.

For flow through a pipe, the critical velocity is often associated with the upper limit of the transitional Reynolds number range. If the velocity of the fluid exceeds the critical velocity, the flow will become turbulent.

Consider a fluid flowing through a pipe. As you increase the flow rate (and thus the velocity), the flow remains streamline up to a certain velocity. This velocity is the critical velocity. Beyond this point, even small disturbances can cause the flow to break down into chaotic eddies and swirls, characteristic of turbulent flow.

The concept of critical velocity is important in many engineering applications, such as the design of pipelines, aircraft wings, and hydraulic systems. Understanding when flow becomes turbulent is crucial because turbulent flow involves higher energy dissipation (due to eddies) and different drag characteristics compared to streamline flow.

Factors Influencing Critical Velocity

The critical velocity is influenced by the same factors that affect the Reynolds number:

Pipe Diameter (D): A larger diameter pipe generally allows for a higher critical velocity because the viscous forces have a larger region to act within, helping to stabilize the flow.
Fluid Viscosity (η): Higher viscosity tends to suppress turbulence, allowing for a higher critical velocity. More viscous fluids are more resistant to chaotic motion.
Fluid Density (ρ): Higher density means higher inertial forces for a given velocity. This makes the flow more prone to becoming turbulent, thus lowering the critical velocity.
Surface Roughness: Roughness of the pipe walls can trigger turbulence at lower velocities, effectively reducing the critical velocity.

While there isn't a simple universal formula for critical velocity like there is for terminal velocity, it's fundamentally linked to the Reynolds number threshold. For flow in a smooth circular pipe, the transition to turbulence typically begins around a Reynolds number of 2300. Therefore, the critical velocity ($v_c$) can be approximated using the Reynolds number formula: $$ v_c \approx \frac{Re_{critical} \cdot \eta}{\rho D} $$ Using $Re_{critical} \approx 2300$ for a pipe: $$ v_c \approx \frac{2300 \eta}{\rho D} $$

It's important to remember that this is an approximation, and the actual critical velocity can vary. The transitional range (2100 < Re < 4000) signifies that the flow is unstable and can switch between laminar and turbulent states.

Critical Velocity Essence: It's the speed limit for smooth sailing (streamline flow). Go faster, and the fluid gets chaotic (turbulent). It's directly related to the Reynolds number. Think of it as the Changeover Velocity from Calm to Vortex.