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Consider two generic scalar functions y and , a vector field V, and a vector potential A, such that V = curl(A). a) Is the vector field V solenoidal? b) Use the crib formulas to show that div(grad(4) x grad(4)) = 0 c) If the additional condition A = yvo is included, then show that and are streamfunctions for the velocity field V. d) Show that vorticity is related to the streamfunctions by %3D e) Consider the volume of fluid bound by four surfaces with constant streamfunctions o=61, p=$2, 4=1, p=2. Show that the volumetric flow rate is constant through any surface cutting the volume and equal to Q = (41 – V2)($1 - $2).

Consider two generic scalar functions y and , a vector field V, and a vector potential A, such that V = curl(A). a) Is the vector field V solenoidal? b) Use the crib formulas to show that div(grad(4) x grad(4)) = 0 c) If the additional condition A = yvo is included, then show that and are streamfunctions for the velocity field V. d) Show that vorticity is related to the streamfunctions by %3D e) Consider the volume of fluid bound by four surfaces with constant streamfunctions o=61, p=$2, 4=1, p=2. Show that the volumetric flow rate is constant through any surface cutting the volume and equal to Q = (41 – V2)($1 - $2).

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Kinematics of Velocity 1. Consider two generic scalar functions y and o, a vector field V, and a vector potential A, such that V = curl(A). a) Is the vector field V solenoidal? b) Use the crib formulas to show that div(grad(y) × grad(4)) = 0 c) If the additional condition A = yvo is included, then show that and o are streamfunctions for the velocity field V. d) Show that vorticity is related to the streamfunctions by e) Consider the volume of fluid bound by four surfaces with constant streamfunctions o=0, $=02, V=1, V=p2. Show that the volumetric flow rate is constant through any surface cutting the volume and equal to Q = (41 - 42)($1- $2).
Transcribed Image Text:Kinematics of Velocity 1. Consider two generic scalar functions y and o, a vector field V, and a vector potential A, such that V = curl(A). a) Is the vector field V solenoidal? b) Use the crib formulas to show that div(grad(y) × grad(4)) = 0 c) If the additional condition A = yvo is included, then show that and o are streamfunctions for the velocity field V. d) Show that vorticity is related to the streamfunctions by e) Consider the volume of fluid bound by four surfaces with constant streamfunctions o=0, $=02, V=1, V=p2. Show that the volumetric flow rate is constant through any surface cutting the volume and equal to Q = (41 - 42)($1- $2).
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