Answer:
Explanation:
First we should recall how Newton's laws relates shear stress to a fluid's velocity profile:
[tex]\tau = \mu \cfrac{\partial v}{\partial y}[/tex]
where tau is the shear stress, mu is viscosity, v is the fluid's velocity and y is the direction perpendicular to flow.
Now, in this case we have a parabolic velocity profile, and also we know that the fluid's velocity is zero at the boundary (no-slip condition) and that the vertex (maximum) is at [tex]y=75 \, mm[/tex] and the velocity at that point is [tex]1.125 \, m/s[/tex]
We can put that in mathematical terms as:
[tex]v(y)= A+ By +Cy^2 \\v(0) = 0\\v(75 \, mm) = 1.125 \, m/s\\v'(75 \, mm) = 0\\[/tex]
From the no-slip condition, we can deduce that [tex]A=0[/tex] and so we are left with just two terms:
[tex]v(y) = By + C y ^2 \\[/tex]
We know that the vertex is at [tex]y= 75 \, mm[/tex] and so we can rewrite the last equation as:
[tex]v(y) = k(y-75 \, mm) ^2+h[/tex]
where k and h are constants to be determined. First we check that [tex]v( 75 \, mm) = 1.125 \, m/s[/tex] :
[tex]v( 75 \, mm) = k(75 \, mm -75 \, mm) ^2+h = h = 1.125 \, m/s\\\\h= v_{max} = 1.125 \, m/s[/tex]
So we found that h was the maximum velocity for the fluid, now we have to determine k, for that we need to make use of the no-slip condition.
[tex]v( 0) = k( -75 \, mm) ^2+ 1.125 \, m/s= 0 \quad (no \, \textendash slip) \\\\k= - \cfrac{ 1.125 \, m/s }{(75 \, mm ) ^2} = - \cfrac{ 1125 \, mm/s }{(75 \, mm ) ^2}\\\\k= - \cfrac{0.2}{mm \times s}[/tex]
And thus we find that the final expression for the fluid's velocity is:
[tex]v( y) = 1125- 0.2 ( y -75 ) ^2[/tex]
where v is in mm/s and y is in mm.
In SI units it would be:
[tex]v( y) = 1.125- 200 ( y -0.075 ) ^2[/tex]
To calculate the shear stress, we need to take the derivative of this expression and multiply by the fluid's viscosity:
[tex]\tau = \mu \cfrac{\partial v}{\partial y}[/tex]
[tex]\tau =0.048\, \cdot (-400) ( y-0.075 )[/tex]
for [tex]y= 0.050 \, m[/tex] we have:
[tex]\tau =0.048\, \cdot (-400) ( 0.050 -0.075 ) = 0.48\, Pa[/tex]
Which is our final result
