1. The dimensional form of the conservation equation for total energy can be written as (p*e*)...

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1. The dimensional form of the conservation equation for total energy can be written as (p*e*) + * [(p*e* + p*)u*] = * [q + 7* u*] + p*f* u*, t* where the total internal energy, et, is defined as 1 et =e* + u* u*, 2 the energy diffusion flux, qe, is modeled using Fourier's law q = -*V*T*, and * is the thermal conductivity. Derive the dimensionless form of this equation along with the corresponding dimensionless forms of the total internal energy and energy diffusion flux. Show all steps in your formulation. 2. Consider the two-dimensional form of the Navier-Stokes equations in Cartesian coordinates. (a) Write these equations in standard vector form and define all terms (e.g., Q = [p, pu, pv, pe,], etc.). You can use either the dimensional or nondimensional form of the equations as long as you are consistent. (b) Transform the system of equations in (a) to generalized coordinates showing all steps. 1. The dimensional form of the conservation equation for total energy can be written as (p*e*) + * [(p*e* + p*)u*] = * [q + 7* u*] + p*f* u*, t* where the total internal energy, et, is defined as 1 et =e* + u* u*, 2 the energy diffusion flux, qe, is modeled using Fourier's law q = -*V*T*, and * is the thermal conductivity. Derive the dimensionless form of this equation along with the corresponding dimensionless forms of the total internal energy and energy diffusion flux. Show all steps in your formulation. 2. Consider the two-dimensional form of the Navier-Stokes equations in Cartesian coordinates. (a) Write these equations in standard vector form and define all terms (e.g., Q = [p, pu, pv, pe,], etc.). You can use either the dimensional or nondimensional form of the equations as long as you are consistent. (b) Transform the system of equations in (a) to generalized coordinates showing all steps.