3. Regions of Fluid Flow

This velocity variation occurs in a very thin region of flow near the solid surface. So, far away from the solid boundary the velocity is going to be the velocity with which the flow was actually coming. So, this whole phenomenon occurs in a very thin region, and this layer, this thin region is called the boundary layer.

The boundary layer is a thin region of fluid flow that develops when fluid moves past a solid object. In this area, the velocity of the fluid changes from zero at the solid surface (due to the no-slip boundary condition) to the free stream velocity away from the surface. This transition occurs because the fluid particles stick to the surface, creating a gradient of velocity.

You can think of the boundary layer like the layers in a cake. The icing on the outside represents the fluid very close to the surface, which is affected by the contact with the cake. Just like the icing sticks to the cake and behaves differently from the cake itself, the fluid close to the boundary of a solid object behaves differently than the fluid farther away.

So, Prandtl actually divided the flow of the fluid in the neighbourhood of the solid boundary into 2 regions, to have a more simplified look. So, one is called the boundary layer. So, it is in the immediate vicinity of the solid boundary where the viscous forces and rotationality cannot be ignored.

Prandtl classified the fluid flow near a solid boundary into two distinct regions: the boundary layer and the outer flow region. The boundary layer, which is close to the surface, is where the effects of viscosity and fluid rotation are significant. In contrast, in the outer flow region, the flow becomes largely inviscid, meaning viscosity has a minimal effect, and the fluid behaves in a more predictable, irrotational manner.

Imagine swimming in a pool. When you get very close to the pool wall (the boundary), the water feels different because it’s influenced by the wall - it slows down and moves differently. However, as you swim farther away from the wall, the water moves freely, and you can swim faster without any influence from the wall.

In this region there will be a velocity gradient equal to du / dy. Therefore, the fluid will exert a shear stress tau on the wall which we assume to be equal to mu du / dy, where mu is the eddy viscosity.

The presence of the velocity gradient in the boundary layer leads to the development of shear stress. This shear stress is proportional to the rate of change of velocity in the direction perpendicular to the flow (du/dy). The concept of eddy viscosity (mu) characterizes the internal friction due to the mixing of fluid layers and is a critical parameter in understanding how fluids behave near solid surfaces.

Think about spreading butter on a piece of bread. The force you apply creates a thin layer of butter (analogous to the boundary layer) right on the surface of the bread. The faster you spread it (greater du), the more resistance (shear stress) you feel. This relationship helps illustrate how shear stress is generated when different layers of the fluid move at different speeds.

The boundary layer grows with downstream distance from the leading edge x. This is the direction x.

As fluid flows along a flat plate, the boundary layer starts developing from the leading edge. As you move downstream (in the x direction), this boundary layer continuously grows thicker due to increased velocity gradients, which occur from the interaction of fluid with the surface. This growth affects the overall fluid flow and can significantly impact flow characteristics, including drag and lift on objects.

Picture a snowplow moving through a snow-covered street. At the front of the plow (the leading edge), only a small amount of snow is disturbed initially. As the plow moves forward, it pushes more snow aside (the growing boundary layer), creating a larger pile. Similarly, as fluid flows along a flat plate, the boundary layer increases its thickness with distance.

So, one of the important parameters that you have read in the fluid flow is Reynolds number. So, there is going to be a Reynolds number associated with this type of phenomenon.

The Reynolds number is a dimensionless quantity used to predict the flow regime in fluid dynamics. It is defined as the ratio of inertial forces to viscous forces, calculated as Re = U * x / ν, where U is the free stream velocity, x is the distance from the leading edge, and ν is the kinematic viscosity. Depending on the Reynolds number value, it indicates whether the flow is laminar (smooth) or turbulent (chaotic). For flow over a flat plate, a Reynolds number below 5 x 10^5 indicates a laminar flow.

Envision riding a bicycle. If you ride slowly (like a low Reynolds number), the air flows smoothly around you, and you can maintain control easily. However, if you speed up (high Reynolds number), the air creates turbulence and can make it harder to steer. The transition from laminar to turbulent flow is key in determining how a fluid interacts with surfaces.

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