Viscosity and Fluid Dynamics - Important Q&A (Class 11)

Physics Q&A: Viscosity and Fluid Dynamics

Q1. Define the Coefficient of Viscosity. What are its SI and CGS units?

Definition: The coefficient of viscosity \(\eta\) is defined as the tangential viscous force required per unit area to maintain a unit velocity gradient between two parallel layers of a fluid in streamline flow.

Mathematically, it is expressed as:

\( \eta = \frac{F}{A \cdot (dv/dx)} \)

Units:

• SI Unit: Newton-second per square metre \(N \cdot s \cdot m^{-2}\) or Pascal-second \(Pa \cdot s\). It is also called 1 Poiseuille (Pl).
• CGS Unit: Poise \(dyn \cdot s \cdot cm^{-2}\).

Relationship: \( 1 \text{ Pa}\cdot\text{s} = 10 \text{ poise} \).

Q2. State Newton’s Law of Viscosity and explain the concept of velocity gradient.

Newton’s Law of Viscosity: This law states that the viscous force \(F\) acting between two adjacent layers of a fluid is directly proportional to the area \(A\) of the layers and the velocity gradient \(dv/dx\) between them.

\( F = -\eta A \frac{dv}{dx} \)

(The negative sign indicates that the viscous force acts in a direction opposite to the motion of the fluid).

Velocity Gradient: When a fluid flows over a fixed surface, different layers move with different velocities. The rate of change of velocity with distance measured in the direction perpendicular to the flow is called the velocity gradient.

Q3. Explain why the viscosity of a liquid decreases with an increase in temperature, while the viscosity of a gas increases.

In Liquids: Viscosity arises primarily due to the cohesive forces between molecules. When temperature increases, the kinetic energy of the molecules increases, which overcomes the intermolecular cohesive forces. Consequently, the layers slide more easily, and viscosity decreases.

In Gases: Viscosity is caused by the transport of momentum due to random molecular collisions. As temperature increases, the random thermal motion of gas molecules increases, leading to more frequent collisions between layers. This increases the resistance to flow, thereby increasing the viscosity.

Q4. State Stokes’ Law for a small spherical body falling through a viscous medium.

Stokes’ Law: According to this law, the backward viscous force \(F\) acting on a small spherical body of radius \(r\) moving with velocity \(v\) through a fluid of coefficient of viscosity \(\eta\) is given by:

\( F = 6\pi \eta r v \)

Assumptions: The medium is infinite, the body is perfectly rigid and smooth, and there are no slip conditions at the surface.

Q5. What is Terminal Velocity? Derive an expression for the terminal velocity of a sphere falling through a viscous liquid.

Definition: Terminal velocity is the constant maximum velocity acquired by a body falling through a viscous medium when the net external force acting on it becomes zero.



Derivation: Let a sphere of radius \(r\) and density \(\rho\) fall through a fluid of density \(\sigma\) and viscosity \(\eta\). The forces acting on it are:

1. Weight (downward): \( W = \frac{4}{3}\pi r^3 \rho g \)
2. Upthrust/Buoyancy (upward): \( U = \frac{4}{3}\pi r^3 \sigma g \)
3. Viscous Force (upward): \( F = 6\pi \eta r v \)

At terminal velocity \(v_t\), the net force is zero:

\( W = U + F \) \( \frac{4}{3}\pi r^3 \rho g = \frac{4}{3}\pi r^3 \sigma g + 6\pi \eta r v_t \) \( 6\pi \eta r v_t = \frac{4}{3}\pi r^3 (\rho - \sigma) g \) \( v_t = \frac{2}{9} \frac{r^2 (\rho - \sigma) g}{\eta} \)

Q6. Distinguish between Streamline flow and Turbulent flow.

Property	Streamline (Laminar) Flow	Turbulent Flow
Nature	Orderly and steady flow; particles follow a set path.	Disorderly and chaotic flow; particles move zigzag.
Velocity	Fluid velocity is less than the critical velocity.	Fluid velocity exceeds the critical velocity.
Path	Two streamlines never cross each other.	Formation of eddies and whirlpools.

Q7. Define Critical Velocity and explain its relationship with the Reynolds Number.

Critical Velocity \(v_c\): It is that velocity of fluid flow up to which the flow is streamline and beyond which the flow becomes turbulent.

Reynolds Number \(R_e\): It is a dimensionless number that determines the nature of the flow.

\( R_e = \frac{\rho v d}{\eta} \)

Where \(\rho\) is density, \(v\) is velocity, \(d\)( is pipe diameter, and \(\eta\) is viscosity.

• If \( R_e < 2000 \), flow is Streamline.
• If \( R_e > 3000 \), flow is Turbulent.
• Between 2000 and 3000, flow is unstable.

Q8. Derive the Equation of Continuity for the steady flow of a non-viscous, incompressible fluid.

Definition: For an incompressible fluid in steady flow, the mass of fluid passing through any cross-section per unit time is constant.

Derivation: Consider a pipe of varying cross-section. Let \(A_1, v_1\) be the area and velocity at one end, and \(A_2, v_2\) at the other.

Mass entering per second = \( \rho_1 A_1 v_1 \)

Mass leaving per second = \( \rho_2 A_2 v_2 \)

Since mass is conserved: \( \rho_1 A_1 v_1 = \rho_2 A_2 v_2 \).
For an incompressible fluid, density \(\rho\) is constant \(\rho_1 = \rho_2\), therefore:

\( A_1 v_1 = A_2 v_2 \quad \text{or} \quad Av = \text{constant} \)

Q9. State and prove Bernoulli’s Theorem. Mention two of its limitations.

Statement: For an incompressible, non-viscous, irrotational fluid in a streamline flow, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline.

\( P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant} \)

Limitations:

1. It assumes zero viscosity; real fluids have internal friction (viscosity) which leads to energy loss as heat.
2. It assumes the flow is streamline; it does not hold true for turbulent flow where energy is lost in eddies.

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