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The equations of motion for natural convection for a certain two-dimensional Cartesian problem can be written as follows: Ôu ovo at = 0) Ou ди aาน U — a u -= gß(T-T) +V - Ov + - ax where u,v, x, y,g,B,1,1,v,p,C,,k are respectively, velocity in the x-coordinate direction (m/s), velocity in the y-coordinate direction (m/s), the x- and y-coordinate directions (m), gravity vector magnitude (m/s2), volumetric expansion coefficient (1/K), instantaneous temperature (K), refer- ence temperature (cold wall temperature, say) (K), kinematic viscosity (m²/s), density (kg/m%), specific heat at constant pressure (J/Kg.K), and thermal conductivity (W/m.K). Introduce the following dimensionless variables T-T (ur*, vt) =(4,732,63*, vt) = 6;y), 1*= 17 where Lis a reference length and T, is a second reference temperature (hot wall temperature, say) (K), and the variables in asterisks are the non-dimensional versions of those without aster- isks. (a) Obtain the non-dimensional form of the three equations using the reference scales given above. (b) Identify all the non-dimensional parameters that appear in the non-dimensional equations. (c) Identify a non-dimensional parameter that could be used to measure the relative strength of convection (left-hand side of the momentum equation above) and the buoyancy in the problem.
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