Question

For a layer of liquid flowing in a laminar flow in the z direction down a vertical plate or surface, the velocity profile is: v_z = frac{ ho g delta^2}{2mu} left[ 1 - left( frac{x}{delta} ight)^2 ight] Where delta is the thickness of the layer, x is the distance from the free surface of liquid toward the plate, and v_z is the velocity at a distance x from the free surface. Consider the wall to have a depth of w in the y-direction and a length L in the z-direction. (a) Provide a sketch of the system described above with appropriate coordinates and origin point (b) What is the maximum velocity v_{z-max}? Show that the expression you get has the units of velocity. (c) Derive the expression for the average velocity v_{z-av} and relate that to v_{z-max} (d) Derive an expression for the total volumetric flow rate down the wall. Show that the expression you get has the units of m3/s. (e) Calculate the shear acting on the x-surface at x=delta. You are given the relationship between shear and velocities as: au_{xz} = -mu left[ frac{partial v_z}{partial x} + frac{partial v_x}{partial z} ight]. What are the units of shear you obtain? (f) Calculate the drag force on the wall. What is the unit of drag you obtain?

          For a layer of liquid flowing in a laminar flow in the z direction down a vertical plate or surface, the velocity profile is:
v_z = frac{
ho g delta^2}{2mu} left[ 1 - left( frac{x}{delta} 
ight)^2 
ight]
Where delta is the thickness of the layer, x is the distance from the free surface of liquid toward the plate, and v_z is the velocity at a distance x from the free surface. Consider the wall to have a depth of w in the y-direction and a length L in the z-direction.
(a) Provide a sketch of the system described above with appropriate coordinates and origin point
(b) What is the maximum velocity v_{z-max}? Show that the expression you get has the units of velocity.
(c) Derive the expression for the average velocity v_{z-av} and relate that to v_{z-max}
(d) Derive an expression for the total volumetric flow rate down the wall. Show that the expression you get has the units of m3/s.
(e) Calculate the shear acting on the x-surface at x=delta. You are given the relationship between shear and velocities as: 	au_{xz} = -mu left[ frac{partial v_z}{partial x} + frac{partial v_x}{partial z} 
ight]. What are the units of shear you obtain?
(f) Calculate the drag force on the wall. What is the unit of drag you obtain?

$For a layer of liquid flowing in a laminar flow in the z direction down a vertical plate or surface, the velocity profile is: vz = frac ho g delta^22mu left[ 1 - left( fracxdelta ight)^2 ight] Where delta is the thickness of the layer, x is the distance from the free surface of liquid toward the plate, and vz is the velocity at a distance x from the free surface. Consider the wall to have a depth of w in the y-direction and a length L in the z-direction. (a) Provide a sketch of the system described above with appropriate coordinates and origin point (b) What is the maximum velocity vz-max? Show that the expression you get has the units of velocity. (c) Derive the expression for the average velocity vz-av and relate that to vz-max (d) Derive an expression for the total volumetric flow rate down the wall. Show that the expression you get has the units of m3/s. (e) Calculate the shear acting on the x-surface at x=delta. You are given the relationship between shear and velocities as: auxz = -mu left[ fracpartial vzpartial x + fracpartial vxpartial z ight]. What are the units of shear you obtain? (f) Calculate the drag force on the wall. What is the unit of drag you obtain?$

Added by Maria O.

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