midterm1_solution

# midterm1_solution - NAME MAE 101B – Advanced Fluid...

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Unformatted text preview: NAME: MAE 101B – Advanced Fluid Mechanics - Spring 2010 Midterm # 1- April 14, 2010 50 minutes, closed book, open notes, calculator allowed, no cell phones. Write your answers directly on the exam. 40 points total. The exam is 4 pages long. (1) (10 points) In class we saw how to non-dimensionalize the Navier-Stokes equation. Here we will do the same for a different equation. Consider the equation satisfied by the concentration of a chemical species, c , which is being transported by a flow, and also diffuses at the same time (for example, the concentration of tea in a cup). In two dimensions, and using cartesian coordinates ( x,y ), we write the concentration as c ( x,y,t ) and u ( x,y,t ) = u e x + v e y is the known velocity of the fluid. The equation satisfied by c , called the advection-diffusion equation, is then written as ∂c ∂t + u ∂c ∂x + v ∂c ∂y = D ∂ 2 c ∂x 2 + ∂ 2 c ∂y 2 · (1) In Eq. ( ?? ), c is a concentration, with dimensions of # of molecules per unit volume, so [ c ] = 1 /L 3 ; u is the ambient flow velocity field, with dimensions [ u ] = L/T ; D is the molecular diffusivity of the species considered (i.e. tea in water), with units [ D ] = L 2 /T . Use a concentration scale c , a velocity scale V and a length scale L , to non-dimensionalize Eq. ( ?? ). Derive the dimensionless groups that characterize the solution to this equation. Verify that they are dimensionless. Solution : We write u = V u * , v = V v * , x = L x * , y = L y * , t = L V t * and c = c c * where all * quantities are dimensionless. We plug this into Eq. ( ?? ) and get c V L ∂c * ∂t * + u * ∂c * ∂x * + v * ∂c * ∂y * = Dc L 2 ∂ 2 c * ∂x * 2 + ∂ 2 c * ∂y * 2 (2) and then we divide everything by c V L to get ∂c * ∂t * + u * ∂c * ∂x * + v * ∂c * ∂y * = D V L ∂ 2 c * ∂x * 2 + ∂ 2 c * ∂y * 2 (3) The dimensionless group is Π =...
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midterm1_solution - NAME MAE 101B – Advanced Fluid...

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