A new protective facial mask design is being trialled. Inside it is a cylindrical core W, with surface 5 having three components, 51 , 52, 53. The...
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This question employs the understanding of vector calculus and engineering mathematics

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A new protective facial mask design is being trialled. Inside it is a cylindrical core W, with surface 5 having three components, 51 , 52, 53. The core is designed to block droplets and aerosols produced from talking, coughing and sneezing [Figure 1). For convenience, we orient the mask's core as shown in Figure 1 (Left): the mouth is centred around the origin and is covered by the curved bottom surface ofthe cylinder 51. Red dashed arrows trace the trajectory of exhaled aerosol particles inside W. a) Assume the region W, oriented with an outward unit normal, lies inside the following surfaces: the paraboloid z = 1 — x2 — 3,2 (bottom surface 51}, the cylinder x2 + y2 = 1 (side surface 52) and the plane 2 = 5 (top surface 53). Use Gauss Divergence Theorem to compute the flux across 52, given the velocity vector field is F=i+ i+2(2rZ + y2)2k. b} Consider a different scenario in which the aerosol's velocity vector field F is everywhere tangent to the closed surface 5 oriented with the outward unit normal. Use Gauss' theorem to show that "IV-FdV=0. V Is this a desirable outcome given 5 is a protective shield? Briefly explain your answer.

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