Question

In the following two-dimensional momentum equation for natural convection over a surface immersed in a quiescent fluid at temperature \( T_\infty \) (g is the gravitational acceleration, \( \beta \) is the volumetric thermal expansion coefficient, \( \nu \) is the kinematic viscosity, \( u \) and \( v \) are the velocities in \( x \) and \( y \) directions, respectively, and \( T \) is the temperature) \[ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = g \beta (T - T_\infty) + \nu \frac{\partial^2 u}{\partial y^2}, \] the term \( g \beta (T - T_\infty) \) represents

• A. Ratio of inertial force to viscous force
• B. Ratio of buoyancy force to viscous force
• C. Viscous force per unit mass
• D. Buoyancy force per unit mass.

Hint:

In natural convection, the term \( g \beta (T - T_\infty) \) represents the buoyancy force driving the flow, and it is important to compare this to the viscous forces for understanding the flow behavior.

Answer 1

The Correct Option is B

The term \( g \beta (T - T_\infty) \) in the momentum equation represents the buoyancy force, which drives natural convection. This term accounts for the effect of temperature differences on the fluid density, causing the fluid to move due to buoyancy forces. The equation describes how the buoyancy force compares to the viscous forces in the fluid, which is the ratio of buoyancy force to viscous force. Thus, the correct answer is (B).