2.1. No Slip Boundary Condition

So, when a real fluid flows past a solid, the fluid particles stick to the solid surface. So, that is one of the phenomena that happens. And the velocity of the fluid particles close to the solid boundary is actually equal to the velocity of the boundary and this phenomenon is called the no slip boundary condition.

The no slip boundary condition describes how fluids behave when they come into contact with a solid surface. When a fluid, such as water, flows over a solid surface, the particles of the fluid right at the surface do not slide or slip; instead, they adhere to the surface. This means that these fluid particles have the same velocity as the solid surface, which, if the surface is stationary, is zero.

Imagine pushing a toy car on a smooth floor. The wheels of the car are in contact with the floor and do not slip against it. If you push the car, the wheels turn but they don’t slide on the floor; they move with the same speed as the car itself. Similarly, in fluid flow, the molecules closest to the surface of a solid adhere to it and do not slip.

So, actually this is the most commonly used no slip boundary condition that we use in a viscous fluid flow. What is viscous fluid flow? A small detail which we have already talked about before but will look into more details in the upcoming lectures in coming weeks.

In viscous fluid flow, the fluid’s viscosity plays a crucial role in determining how the fluid behaves near surfaces. The no slip condition is essential because it indicates that the fluid's velocity gradually changes from zero (at the solid boundary) to the free stream velocity (the speed of the fluid further away from the surface). This change in velocity is the result of the fluid particles experiencing drag due to viscosity, as they interact with the solid surface.

Imagine stirring honey with a spoon. Near the spoon, the honey slows down and sticks to it, while further away, the honey flows more freely. Just as the honey's movement changes near the spoon due to the forces acting on it, fluid particles slow down and stick to solid surfaces, demonstrating the no slip condition.

Because of this phenomenon, what happens is there is a velocity variation and this gives a velocity gradient du / dy which exists in normal direction. So, in case of say, for example, this is the boundary because of no slip and this is stationary and suppose there has been a velocity coming in, so, we are not going to talk about let us say, in between now but we say, suppose here is a velocity u and this distance is y and, of course, because this is stationary at this point, the velocity is going to be zero.

The no slip boundary condition leads to the formation of a velocity gradient, which is a measure of how much the fluid velocity changes with distance from the solid surface. The notation du/dy represents this gradient, where 'u' is the velocity of the fluid and 'y' is the distance from the boundary. In a thin layer of flow near the surface, the velocity increases from zero at the boundary to the free stream velocity farther away. This thin layer, where the velocity changes significantly, is known as the boundary layer.

Think about how the wind feels stronger the further you get from a wall. If you stand right next to the wall, the air doesn't move at all (like the fluid at the wall, which is stationary). But as you step away, the wind speeds up. This change in wind speed as you move away from the wall is similar to the velocity gradient in a fluid near a boundary.

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.

To simplify the analysis of fluid flow around solids, Prandtl categorized the flow into two regions. The first, the boundary layer, is the layer where the effects of viscosity and rotational movements are significant. Inside this layer, the fluid experiences shear stress due to the velocity gradient defined by the no slip boundary condition. The second region is the outer flow region, where the fluid behaves more like an ideal fluid, largely unaffected by viscosity.

Consider a moving car. The air right next to the car’s surface moves slowly, affected by the car due to the no slip condition. This is similar to the boundary layer. However, the air farther away flows freely without being slowed down by the car's presence, which is like the outer flow region. This distinction helps in understanding how different forces act at various distances from the body.

--