Question

Engine oil at \(105^{\circ} \mathrm{C}\) flows over the surface of a flat plate whose temperature is \(15^{\circ} \mathrm{C}\) with a velocity of \(1.5 \mathrm{~m} / \mathrm{s}\). The local drag force per unit surface area \(0.8 \mathrm{~m}\) from the leading edge of the plate is (a) \(21.8 \mathrm{~N} / \mathrm{m}^{2}\) (b) \(14.3 \mathrm{~N} / \mathrm{m}^{2}\) (c) \(10.9 \mathrm{~N} / \mathrm{m}^{2}\) (d) \(8.5 \mathrm{~N} / \mathrm{m}^{2}\) (e) \(5.5 \mathrm{~N} / \mathrm{m}^{2}\) (For oil, use \(\nu=8.565 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}, \rho=864 \mathrm{~kg} / \mathrm{m}^{3}\) )

Short answer

Answer: The local drag force per unit surface area is approximately 10.9 N/m².

Step-by-step solution

Calculate the Reynolds number at the given location

To calculate the Reynolds number at \(0.8\mathrm{~m}\) from the leading edge of the plate, we use the formula: \(Re=\frac{Vx}{\nu}\), where \(V=1.5\mathrm{~m} / \mathrm{s}\) is the velocity of the oil, \(x=0.8\mathrm{~m}\) is the distance from the leading edge, and \(\nu=8.565\times10^{-5}\mathrm{~m}^{2}/\mathrm{s}\) is the kinematic viscosity of the oil. $$Re=\frac{(1.5\mathrm{~m/s})(0.8\mathrm{~m})}{8.565\times10^{-5}\mathrm{~m}^{2}/\mathrm{s}}\approx 14,067$$

Use the Blasius equation to find the local drag coefficient \(C_f\)

The Blasius equation for the local drag coefficient of a flat plate in a laminar flow is: \(C_f=\frac{0.664}{Re^{1/2}}\). Using the calculated Reynolds number from Step 1: $$C_f=\frac{0.664}{\sqrt{14,067}}\approx0.005618$$

Calculate the local drag force per unit surface area

To find the local drag force per unit area, we use the drag force formula: \(F =\frac{1}{2}\rho V^2 C_f\). Here, \(\rho=864\mathrm{~kg} / \mathrm{m}^{3}\) is the density of the oil. Substituting the values, we get: $$F =\frac{1}{2}(864\mathrm{~kg/m}^{3})(1.5\mathrm{~m/s})^2(0.005618) \approx 10.89\mathrm{~N/m}^{2}$$ Thus, the local drag force per unit surface area at the given location is approximately \(10.9\mathrm{~N/m}^{2}\) which corresponds to option (c).

Key concepts

Reynolds Number

The Reynolds number is a crucial dimensionless parameter used in fluid mechanics to determine the flow characteristics of a fluid. It helps to predict the transition from laminar to turbulent flow. The formula to calculate the Reynolds number \(Re\) is given by:\[Re = \frac{Vx}{u}\]where:

• \(V\) is the velocity of the fluid (\(1.5\ \mathrm{m/s}\) in this case).


• \(x\) is the characteristic length (the specific location from the leading edge, \(0.8\ \mathrm{m}\)).


• \(u\) is the kinematic viscosity of the fluid (\(8.565\times10^{-5}\ \mathrm{m}^2/\mathrm{s}\)).

Substituting the values, the Reynolds number was found as approximately \(14067\). A value of the Reynolds number under 2300 typically indicates laminar flow, while a value over 4000 would suggest turbulent flow. However, in this scenario, since the plate is flat and it's a transitional phase, the flow can still be considered laminar despite a higher number.

Blasius Equation

The Blasius equation is an empirical relationship used to determine the local drag coefficient for laminar flow over a flat plate. The equation is derived from the boundary layer theory and is expressed as:\[C_f = \frac{0.664}{Re^{1/2}}\]Here, \(C_f\) represents the local drag coefficient. It is a measure that indicates the drag force experienced by a flat surface due to the fluid flow.
By substituting the previously calculated Reynolds number \(14067\) into the Blasius equation, we get:

• \(C_f = \frac{0.664}{\sqrt{14067}} \approx 0.005618\)

This calculates the resistance that the fluid encounters while flowing over the plate. It helps in understanding how streamlined an object needs to be to minimize the frictional effects of the fluid.

Local Drag Coefficient

The local drag coefficient \(C_f\) is a key factor in determining the drag force acting on a surface in contact with a fluid flow. It provides an indication of how much resistance the surface encounters due to the fluid's movement. The drag force per unit area \(F\) is calculated using the formula:\[F = \frac{1}{2} \rho V^2 C_f\]where:

• \(\rho\) represents the fluid density (here, \(864\ \mathrm{kg/m^3}\)).


• \(V\) is the velocity (\(1.5\ \mathrm{m/s}\)).


• \(C_f\) is the local drag coefficient (calculated as \(0.005618\)).

By substituting the appropriate values, the drag force per unit surface area was calculated to be approximately \(10.9\ \mathrm{N/m}^2\). This highlights how the choice of surface material and fluid properties can affect the drag experienced by an object.

Kinematic Viscosity

Kinematic viscosity is a measure of a fluid's internal resistance to flow. It describes how easily a fluid flows under the influence of gravity and is typically denoted by \(u\). Kinematic viscosity is calculated by dividing the dynamic viscosity by the density of the fluid. For this problem, the engine oil has a kinematic viscosity of:\[u = 8.565 \times 10^{-5} \ \mathrm{m}^2/\mathrm{s}\]This parameter plays a crucial role in influencing the Reynolds number calculation. A fluid with higher kinematic viscosity would flow more slowly and have a smaller Reynolds number under the same conditions. Conversely, a lower viscosity means easier flow and a potentially larger Reynolds number. Understanding kinematic viscosity helps engineers and scientists in designing and analyzing systems involving fluid dynamics.