The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field Find where S is the upper surface of the cylindrical container.

Question

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field  Find  where S is the upper surface of the cylindrical container.

Quizplus · Accepted Answer

The normal to S is along the unit vector j. Therefore, the projection of the velocity vector on j at the upper surface of the cylinder is given by:
v . j = (10 + 2z) e^(-r^2)cos(theta)
At z=1, this becomes:
v . j = 10 e^(-r^2)cos(theta) + 2 e^(-r^2)cos(theta)
= 12 e^(-r^2)cos(theta)
Taking the gradient of this function, we get:
∇(v . j) = (-24r e^(-r^2)cos(theta), -12 e^(-r^2)sin(theta), 0)
Setting r=1, sin(theta)=0, and cos(theta)=-1 (since we want the downward vector), we get:
∇(v . j)|S = (-24 e^(-1), 0, 0)
Therefore, the correct answer is D) 11eaa595_61e1_a7cf_a696_cdaea0a2e131_TB4584_11_TB4584_11.

The Motion of a Liquid in a Cylindrical Container of Radius