Crude oil at 20^∘ C flows over the surface of the flat plate...

Crude oil at $20^{\circ} \mathrm{C}$ flows over the surface of the flat plate that has a width of $0.7 \mathrm{~m}$. If the free-stream velocity is $U=10 \mathrm{~m} / \mathrm{s}$, plot the boundary layer thickness and the shearstress distribution along the plate. What is the friction drag on the plate?

Key Concepts

Boundary Layer

The boundary layer is a thin region adjacent to a solid surface where viscous forces dominate and the fluid velocity transitions from zero (due to the no?slip condition) to the free stream value. It is a fundamental concept in fluid dynamics that explains how the effects of viscosity are confined to a narrow region near the boundary, influencing momentum transfer between the fluid and the surface.

Laminar Flow Over a Flat Plate

Laminar flow over a flat plate refers to a flow regime in which the fluid particles move in smooth, orderly layers with little mixing between them. This behavior allows for predictable development of the boundary layer along the plate, and the flow characteristics can be analyzed using simplified mathematical models, making it a key case study in boundary layer theory.

Blasius Boundary Layer Solution

The Blasius solution is a similarity solution to the boundary layer equations for a steady, incompressible, laminar flow over a flat plate. It provides analytical expressions for the velocity profile, boundary layer thickness, and shear stress distribution along the plate, serving as a foundational result in the study of viscous flows.

Wall Shear Stress

Wall shear stress is the tangential force per unit area exerted by the fluid on the surface, resulting from the velocity gradient at the wall. It is a critical parameter in determining how viscous forces interact with boundaries and directly influences the friction drag experienced by a surface.

Friction Drag

Friction drag, often referred to as skin friction drag, is the component of drag on a body due to viscous shear stresses acting along its surface. It is obtained by integrating the wall shear stress over the entire wetted area and plays a significant role in the overall resistance experienced by objects moving through a fluid.

Reynolds Number

The Reynolds number is a dimensionless parameter defined as the ratio of inertial forces to viscous forces in a fluid flow. It is crucial in determining the flow regime, such as laminar or turbulent, and helps in scaling and predicting boundary layer behavior along surfaces.

Transcript

00:02 First, we list what we know and decide that the boundary layer is laminar or turbulent.

00:06 So we're given a plate length of 1 .5 meters, a width b of 0 .7 meters.

00:13 The free stream speed u is 10 meters per second.

00:16 The crude oil at 20 degrees celsius, i'll use the wti light as a representative crude.

00:22 So the viscosity at 20 degrees is 2 .48 times 10 to the negative 6 square meters per second, with a density of about, i'll say, 790 .2 kilogram per cubic meter.

00:41 And let's compute the reynolds number first along the plate.

00:48 So the reynolds would be, let's see, a common transition criterion is about 5 times 10 to the 5th.

00:58 So that would be 5 times 10 to the 5th times the viscosity of 2 .4.

01:11 8 times 10 to the negative 6 over 10, which is 0 .124 meters.

01:17 So it's laminar from x equals 0 to 0 .124 meters and turbulent to 1 .5.

01:41 The boundary layer thickness, now we would write the standard flat rate, the flat plate correlations.

01:50 And so for laminar, delta of x is 5x over the square end of the x times the reynolds of the negative one -fifth.

02:15 So you could do this graphically.

02:20 And at 1 .5 meters, it's about 24 .43 millimeters.

02:29 The same idea, you can convert to sheer stress...