A quiescent fluid at temperature T₃ is confined between two vertical parallel plates. While plate 1 is kept at temperature T₁ and plate-2 is maintained at temperature T₂, A velocity profile between the plates, as shown in the figure below, is generated due to heat transfer. Which one of the following is correct relation between T₁, T₂ and T₃?

The correct option is: T₁ > T₃ and T₃ > T₂.

Let's break down the step-by-step reasoning based on the principles of fluid dynamics and heat transfer:

1. Understanding the Physical System:
Initially, a quiescent (still) fluid is at a uniform temperature of T₃.
It is confined between two vertical parallel plates: Plate-1 on the left (at temperature T₁) and Plate-2 on the right (at temperature T₂).
According to the provided image, gravity acts downwards, as indicated by the downward-pointing double arrow labeled "Gravity".

2. Analyzing the Velocity Profile and Buoyancy Forces:
When a fluid experiences a temperature difference in the presence of a gravity field, density variations occur. This drives natural convection (buoyancy-driven flow):
- Fluid rising (upward flow): Regions of the fluid that are hotter than the bulk fluid (T₃) become less dense, causing them to experience an upward buoyant force and rise.
- Fluid sinking (downward flow): Regions of the fluid that are colder than the bulk fluid (T₃) become denser, causing them to experience a downward gravitational/buoyant force and sink.

3. Applying this to the Left Plate (Plate-1):
In the image, near the label Plate-1 at T₁, the velocity vectors (represented by vertical arrows) point upwards.
This upward motion indicates that the fluid in contact with Plate-1 is warmer than the initial quiescent fluid temperature T₃, causing it to expand, lose density, and rise. Therefore, we must have:
$$T_1>T_3$$

4. Applying this to the Right Plate (Plate-2):
Near the label Plate-2 at T₂, the velocity vectors point downwards.
This downward motion indicates that the fluid in contact with Plate-2 is cooler than the initial quiescent fluid temperature T₃, causing it to contract, gain density, and sink. Therefore, we must have:
$$T_3>T_2$$

Conclusion:
Combining the two temperature relations derived from the buoyancy-driven flow direction shown in the diagram, we get:
$$T_1>T_3>T_2$$
This is equivalent to T₁ > T₃ and T₃ > T₂.