Question

In fully developed laminar flow inside a circular pipe, the velocities at \(r=0.5 R\) (midway between the wall surface and the centerline) are measured to be 3,6 , and \(9 \mathrm{~m} / \mathrm{s}\). (a) Determine the maximum velocity for each of the measured midway velocities. (b) By varying \(r / R\) for $-1 \leq r / R \leq 1$, plot the velocity profile for each of the measured midway velocities with \(r / R\) as the \(y\)-axis and \(V(r / R)\) as the \(x\)-axis.

Short answer

Answer: Using the Hagen-Poiseuille equation, determine the maximum velocity for each midway velocity by rearranging the equation and finding the ratios between them. Then, find the maximum velocities for each given midway velocity and plot the velocity profiles using the derived equations with a range of -1 ≤ r/R ≤ 1 for the y-axis. The resulting graphs will show the velocity profile for each given midway velocity in the laminar flow inside the circular pipe.

Step-by-step solution

Determine maximum velocity for each midway velocity

To find the maximum velocity, we can apply the Hagen-Poiseuille equation to each midway velocity. We will first rearrange the equation for maximum velocity by setting r = 0: $$ u_{max}(r=0) = \frac{dP_{m}}{4\mu L} R^2 $$ Now, we can use the given midway velocities to find the respective maximum velocities. For \(u = 3 \mathrm{~m} / \mathrm{s}\): $$ 3 = \frac{dP_{1}}{4\mu L}(R^2 - (0.5R)^2) $$ For \(u = 6 \mathrm{~m} / \mathrm{s}\): $$ 6 = \frac{dP_{2}}{4\mu L}(R^2 - (0.5R)^2) $$ For \(u = 9 \mathrm{~m} / \mathrm{s}\): $$ 9 = \frac{dP_{3}}{4\mu L}(R^2 - (0.5R)^2) $$ We can solve these equations simultaneously to find the corresponding maximum velocities.

Solve for the maximum velocities

Divide the second equation by the first equation: $$ \frac{6}{3} = \frac{dP_{2}}{dP_{1}} $$ So, \(dP_{2} = 2 dP_{1}\). Divide the third equation by the first equation: $$ \frac{9}{3} = \frac{dP_{3}}{dP_{1}} $$ So, \(dP_{3} = 3 dP_{1}\). Now we can use these ratios to find the maximum velocities for each case. For the first midway velocity: $$ u_{max_1}(r=0) = \frac{dP_{1}}{4\mu L} R^2 $$ For the second midway velocity: $$ u_{max_2}(r=0) = \frac{2 dP_{1}}{4\mu L} R^2 $$ For the third midway velocity: $$ u_{max_3}(r=0) = \frac{3 dP_{1}}{4\mu L} R^2 $$ Note that maximum velocity is directly proportional to the pressure difference, so we can find ratios for maximum velocities. Divide the second maximum velocity by the first maximum velocity: $$ \frac{u_{max_2}}{u_{max_1}} = 2 $$ So, \(u_{max_2} = 2 u_{max_1}\). Similarly, divide the third maximum velocity by the first maximum velocity: $$ \frac{u_{max_3}}{u_{max_1}} = 3 $$ So, \(u_{max_3} = 3 u_{max_1}\). Using the given midway velocity of \(3 \mathrm{~m} / \mathrm{s}\), we can find the corresponding maximum velocity at the centerline by multiplying it by 2: $$ u_{max_1} = 2 \times 3 = 6 \mathrm{~m} / \mathrm{s} $$ Now, we can find the maximum velocities for the other cases: $$ u_{max_2} = 2 u_{max_1} = 2 \times 6 = 12 \mathrm{~m} / \mathrm{s} $$ $$ u_{max_3} = 3 u_{max_1} = 3 \times 6 = 18 \mathrm{~m} / \mathrm{s} $$ So, the maximum velocities for the given midway velocities are 6, 12, and 18 m/s.

Plot the velocity profile for each midway velocity

To plot the velocity profiles, we can substitute the various values for pressure difference corresponding to the given midway velocities into the Hagen-Poiseuille equation, and use a range of \(-1 \leq r / R \leq 1\) for the y-axis. The x-axis will represent the \(V(r / R)\) values. Since the equation is quadratic, the velocity profiles will have a parabolic shape. Use a graphing tool or software to plot the equations below: 1. For \(u_{max_1} = 6 \mathrm{~m} / \mathrm{s}\): $$ V_1(r/R) = \frac{dP_{1}}{4\mu L}(1 - (r/R)^2) $$ 2. For \(u_{max_2} = 12 \mathrm{~m} / \mathrm{s}\): $$ V_2(r/R) = \frac{2 dP_{1}}{4\mu L}(1 - (r/R)^2) $$ 3. For \(u_{max_3} = 18 \mathrm{~m} / \mathrm{s}\): $$ V_3(r/R) = \frac{3 dP_{1}}{4\mu L}(1 - (r/R)^2) $$ The resulting graphs will show the velocity profile for each given midway velocity in the laminar flow inside the circular pipe.