Question

Shell balance in fluid mechanics is used to derive:

• A. Velocity profiles in laminar flow
• B. Turbulent flow models
• C. Critical Reynolds number
• D. Pump characteristic curves

Hint:

The shell balance method always yields the shear stress profile first (which is independent of fluid type), and then the velocity profile is derived by applying the fluid's specific rheological model (e.g., Newtonian or power-law).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the primary application of the "shell momentum balance" technique in fluid mechanics and transport phenomena.

Step 2: Key Formula or Approach:
The shell balance method involves selecting a differential control volume (a "shell") of fluid that matches the geometry of the system (e.g., cylindrical shell for pipe flow, flat plate shell for falling film).
We apply the conservation of momentum to this shell:
\[ \text{Rate of Momentum In} - \text{Rate of Momentum Out} + \text{Sum of External Forces} = 0 \]

Step 3: Detailed Explanation:

• Derivation Process:
1. A thin shell of fluid with a differential thickness (e.g., \( \Delta r \) or \( \Delta y \)) is defined.
2. Momentum enters and leaves the shell by molecular transport (viscous shear) and convective transport.
3. Gravity and pressure forces acting on the shell are accounted for.
4. Taking the limit as the shell thickness approaches zero (\( \Delta r \to 0 \)) yields a differential equation for the shear stress distribution, \( \tau(r) \).

• Laminar Flow Profiles: By substituting Newton's law of viscosity (\( \tau = -\mu \frac{dv}{dr} \)) into the shear stress equation and integrating, we obtain the velocity profile for laminar flow.
For example, this method yields the parabolic velocity profile (Hagen-Poiseuille profile) for steady laminar flow in a circular tube.

• Limitations: This analytical approach is limited to simple geometries and laminar flow regimes where the flow is steady and structured.
It cannot be directly applied to turbulent flow because of the chaotic, fluctuating nature of the fluid motion.


Step 4: Final Answer:
The shell balance method is used to derive analytical velocity profiles in laminar flow.