GATE ME 2022 Set 1 — Question 25
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Heat Transfer → Convection → Forced & Free Convection
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Question
In the following two-dimensional momentum equation for natural convection over a surface immersed in a quiescent fluid at temperature ( is the gravitational acceleration, is the volumetric thermal expansion coefficient, is the kinematic viscosity, and are the velocities in and directions, respectively, and is the temperature)the term represents
Correct answer
(D) Buoyancy force per unit mass.
Solution
The momentum equation for natural convection is derived from the Navier-Stokes equations using the Boussinesq approximation. In this approximation, density variations are neglected except in the buoyancy term. The equation is given by:For a quiescent fluid at infinity, the pressure gradient is balanced by the hydrostatic weight: . Substituting this into the equation:Using the definition of volumetric thermal expansion coefficient , we have . Dividing the entire equation by :Here, the term represents the net buoyancy force per unit mass acting on the fluid element.
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