# Transient Natural Convection between Two Plates at Different Temperatures

This Demonstration analyzes the transient free convection of a viscous incompressible fluid between two vertical plates at different temperatures. The fluid is initially at rest and the temperature of the fluid and the plates is the same. Convection occurs due to the Soret effect caused by a step change in the temperature of the plates .
The velocity and temperature profiles were obtained by solving the governing equations using the Mathematica function NDSolve. You can vary the temperature of the plates, the time and the Prandtl number (the ratio of the fluid momentum diffusivity to thermal diffusivity) to observe the temperature and velocity profiles.
Both the velocity and temperature are positive near the plate with positive dimensionless temperature and negative near the plate with negative dimensionless temperature; steady state conditions are reached faster with lower Prandtl numbers. Air at room temperature has a Prandtl number of 0.71, and for water at 18°C it is around 7.56; thus the thermal diffusivity is more dominant for air than for water.

### DETAILS

Consider two vertical, infinitely long parallel plates separated by a distance ; the axis is along the left plate and the axis is perpendicular to it. Initially the temperature of the fluid as well as the plates is the same ; at time the temperature of the plates is instantaneously changed to two temperatures, and . Assume that all the fluid properties are constant except the fluid density, which decreases with temperature.
The governing equations for this system using the Boussinesq approximation  are: and .
Here the dimensional variables , , and represent temperature, fluid velocity, distance between the plates and time; the parameters , , , , and are the fluid density, heat capacity, thermal diffusion coefficient, kinematic viscosity, gravity and the coefficient of thermal expansion.
It is convenient to define the following dimensionless variables: , , , ,
where , , and are the dimensionless horizontal coordinate, time, velocity and temperature, respectively.
The momentum and energy conservation equations then are , ,
where the Prandtl number is given by .
The initial and boundary conditions are , and .

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