Answer and Explanation:
The velocity u(y,t) and temperature T(y,t) fields evolve in an exact same manner under certain circumstances when a particular physical parameter is identified in two equations, rendering them similar. This physical parameter is known as the thermal diffusivity, represented by the symbol α.
The thermal diffusivity, α, is a measure of how quickly heat diffuses through a material relative to its ability to conduct heat. It is defined as the ratio of thermal conductivity (k) to the product of density (ρ) and specific heat capacity (Cp). Mathematically, it can be expressed as:
α = k / (ρ * Cp)
The unit of thermal diffusivity is square meters per second (m^2/s), obtained by dividing the unit of thermal conductivity (W/m·K) by the product of density (kg/m^3) and specific heat capacity (J/kg·K).
Under conditions where the thermal diffusivity is similar for both the fluid (governing the velocity field) and the solid material (governing the temperature field), the velocity u(y,t) and temperature T(y,t) fields will evolve in an exact same manner. This occurs when the fluid and solid material have similar thermal properties, such as similar thermal conductivities, densities, and specific heat capacities.
Moving on to the second part of the question, drawing the steady state velocity and temperature profiles in the channel and metal plate, respectively:
In a steady state, the velocity profile in a channel can be represented by a parabolic shape, with the maximum velocity occurring at the center of the channel and gradually decreasing towards the walls. The velocity is zero at the walls of the channel. This profile is known as the Hagen-Poiseuille flow profile.
On the other hand, in a metal plate, the steady state temperature profile will depend on the boundary conditions and heat transfer mechanism. One common example is the steady state temperature profile in a metal plate with one surface maintained at a constant temperature and the other surface insulated. In this case, the temperature profile will be linear, with the temperature gradient being highest near the surface with the constant temperature and gradually decreasing towards the insulated surface.
It's important to note that the specific profiles may vary depending on the specific boundary conditions and characteristics of the system. The profiles described here are simplified representations for illustration purposes.