3. A hot solid cube (having sides of equal length) of temperature Tco is put into a pot containing cold water of temperature Two. Assume that the heat exchange only occurs between the cube and water. (20%) (a) Write down the equation for determining the final equilibrium temperature. (b) Assuming both the cube and water keep spatially uniform in temperature, write down the equation for evaluating the temperature evolution for the cube and water, respectively. (c) Consider the cube is not spatially uniform in temperature but the water is. Write down the equation and boundary conditions for evaluating the temperature distribution in the cube. (d) Consider neither the cube nor the water keeps uniform in temperature. Write down the equation and boundary conditions for evaluating the temperature distribution in the water.

2. Two horizontal, infinite, parallel plates are spaced a distance b apart, please see the following figure. A viscous liquid is contained between the plates. The bottom plate is fixed (y = 0), and the upper plate (y = b) moves parallel to the bottom plate with a velocity U. Because of the no-slip boundary condition, the liquid motion is caused by the liquid being dragged along by the moving boundary. Assuming that there is no pressure gradient in the direction of flow (x-direction) and the flow is steady and incompressible without the y-component velocity. (a) Derive the governing equation of the above 2-D flow from the x-direction Navier-Stokes equation (see below). (b) Determine the velocity distribution between the plates. (c) Determine an expression for the flowrate per unit width passing between the plates. Express your answers of (b) and (c) in terms of b and U. (25%)

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