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)
\[ \begin{aligned} u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} &= g\beta(T - T_{\infty}) \\ &\quad + \nu\frac{\partial^{2}u}{\partial y^{2}} , \end{aligned} \]
the term \(g\beta(T - T_{\infty})\) represents
GATE ME - 2022,
Answer: D
The term \(g\beta(T - T_{\infty})\) is the buoyancy force per unit mass.
Each term of the boundary-layer momentum equation is a force per unit mass (an acceleration). Where the fluid is warmer than the far field, its density is lower by about \(\rho\beta(T - T_{\infty})\), and gravity acting on that density deficit gives an upward force \(\rho g\beta(T - T_{\infty})\) per unit volume, which is \(g\beta(T - T_{\infty})\) per unit mass. This term drives natural convection.
The viscous force per unit mass is the last term, \(\nu\,\partial^{2}u/\partial y^{2}\). The ratios of forces in the first two options are dimensionless groups (the Reynolds and Grashof numbers), not individual terms of this equation.
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