Understanding the parabolic velocity distribution is crucial for analyzing fluid flow, especially in scenarios like fully developed laminar flow between parallel, stationary plates. The question provides the velocity distribution profile and asks for the maximum and average velocities. We will derive both step-by-step.

Velocity Distribution Profile Analysis

The given velocity distribution profile for fully developed laminar flow in the x-direction between two parallel plates, separated by distance \(h\), is:

\[U = - \frac{{{h^2}}}{{8\mu }}\frac{{dp}}{{dx}}\left[ {1 - 4{{\left( {\frac{y}{h}} \right)}^2}} \right]\]

Here:

• \(U\) is the velocity at a given y-position.
• \(h\) is the distance separating the two plates.
• \(y = 0\) axis lies equidistant between the plates, at the center of the flow channel.
• \(p\) is the pressure variable.
• \(\mu\) is the dynamic viscosity term.
• \(\frac{{dp}}{{dx}}\) represents the pressure gradient in the x-direction, which drives the flow.

Maximum Velocity Determination

For a parabolic velocity profile like the one given for this laminar flow, the maximum velocity (\(U_{max}\)) always occurs at the center of the flow passage. In this specific problem, the center is defined by \(y = 0\).

To find the maximum velocity, we substitute \(y = 0\) into the given velocity expression:

\[{U_{max}} = - \frac{{{h^2}}}{{8\mu }}\frac{{dp}}{{dx}}\left[ {1 - 4{{\left( {\frac{0}{h}} \right)}^2}} \right]\]

Simplifying the expression:

\[{U_{max}} = - \frac{{{h^2}}}{{8\mu }}\frac{{dp}}{{dx}}\left[ {1 - 0} \right]\]

Thus, the maximum velocity for this parabolic velocity distribution is:

\[{U_{max}} = - \frac{{{h^2}}}{{8\mu }}\frac{{dp}}{{dx}}\]

Average Velocity Calculation

The average velocity (\(U_{avg}\)) for fully developed laminar flow between parallel plates is defined as the total volumetric flow rate divided by the cross-sectional area. Mathematically, it can be found by integrating the velocity profile over the entire flow height \(h\) and dividing by \(h\).

Since the \(y = 0\) axis is at the center, the flow extends from \(y = -h/2\) (bottom plate) to \(y = h/2\) (top plate).

The formula for average velocity is:

\[{U_{avg}} = \frac{1}{h}\int_{ - h/2}^{h/2} {Udy}\]

Substitute the expression for \(U\) into the integral:

\[{U_{avg}} = \frac{1}{h}\int_{ - h/2}^{h/2} {\left( { - \frac{{{h^2}}}{{8\mu }}\frac{{dp}}{{dx}}\left[ {1 - 4{{\left( {\frac{y}{h}} \right)}^2}} \right]} \right)dy}\]

We know that \(U_{max} = - \frac{{{h^2}}}{{8\mu }}\frac{{dp}}{{dx}}\). So, we can rewrite the velocity profile as \(U = {U_{max}}\left[ {1 - 4{{\left( {\frac{y}{h}} \right)}^2}} \right]\).

\[{U_{avg}} = \frac{1}{h}\int_{ - h/2}^{h/2} {{U_{max}}\left[ {1 - \frac{{4{y^2}}}{{{h^2}}}} \right]dy}\]

Factor out the constant \(U_{max}\):

\[{U_{avg}} = \frac{{{U_{max}}}}{h}\int_{ - h/2}^{h/2} {\left[ {1 - \frac{{4{y^2}}}{{{h^2}}}} \right]dy}\]

Since the integrand \(\left[ {1 - \frac{{4{y^2}}}{{{h^2}}}} \right]\) is an even function (symmetric about \(y=0\)), we can integrate from \(0\) to \(h/2\) and multiply by 2:

\[{U_{avg}} = \frac{{{U_{max}}}}{h} \times 2\int_0^{h/2} {\left[ {1 - \frac{{4{y^2}}}{{{h^2}}}} \right]dy}\]

Now, perform the integration:

\[{U_{avg}} = \frac{{2{U_{max}}}}{h}\left[ {y - \frac{{4{y^3}}}{{3{h^2}}}} \right]_0^{h/2}\]

Substitute the limits of integration (upper limit \(h/2\) minus lower limit \(0\)):

\[{U_{avg}} = \frac{{2{U_{max}}}}{h}\left[ {\left( {\frac{h}{2} - \frac{{4{{\left( {h/2} \right)}^3}}}{{3{h^2}}}} \right) - \left( {0 - 0} \right)} \right]\]

Simplify the term inside the bracket:

\[{U_{avg}} = \frac{{2{U_{max}}}}{h}\left[ {\frac{h}{2} - \frac{{4\left( {{h^3}/8} \right)}}{{3{h^2}}}} \right]\]

\[{U_{avg}} = \frac{{2{U_{max}}}}{h}\left[ {\frac{h}{2} - \frac{{{h^3}/2}}{{3{h^2}}}} \right]\]

\[{U_{avg}} = \frac{{2{U_{max}}}}{h}\left[ {\frac{h}{2} - \frac{h}{6}} \right]\]

Combine the terms within the bracket by finding a common denominator:

\[{U_{avg}} = \frac{{2{U_{max}}}}{h}\left[ {\frac{{3h - h}}{6}} \right]\]

\[{U_{avg}} = \frac{{2{U_{max}}}}{h}\left[ {\frac{{2h}}{6}} \right]\]

\[{U_{avg}} = \frac{{2{U_{max}}}}{h}\left[ {\frac{h}{3}} \right]\]

Finally, simplify to get the average velocity:

\[{U_{avg}} = \frac{2}{3}{U_{max}}\]

Summary of Velocities

Based on our derivations for the given parabolic velocity distribution of fully developed laminar flow between two parallel plates, the maximum and average velocities are:

• Maximum Velocity (\(U_{max}\)): \[{U_{max}} = - \frac{{{h^2}}}{{8\mu }}\frac{{dp}}{{dx}}\]
• Average Velocity (\(U_{avg}\)): \[{U_{avg}} = \frac{2}{3}{U_{max}}\]

These results highlight a key characteristic of parabolic velocity profiles in laminar flow between parallel plates: the average velocity is exactly two-thirds of the maximum velocity, which occurs at the centerline.

Comparing these derived expressions with the provided options, we find that they precisely match option 1.