4.3. Velocity Gradient and Shear Stress

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. For a stationary body, the fluid velocity at the boundaries going to be zero because as we said in this point above that the velocity of the fluid particles close to the solid boundary is equal to the velocity of the boundary.

In fluid dynamics, when a fluid flows over a solid surface, it experiences a phenomenon known as the 'no slip boundary condition.' This principle states that the fluid in immediate contact with the surface has a velocity equal to that of the surface. Therefore, if the surface is stationary, the fluid's velocity at that point will be zero. This condition is crucial for accurately analyzing how fluids behave near solid boundaries, especially in viscous fluid flow, where friction between the fluid and the surface plays a significant role.

Think of it like a conveyor belt with water flowing over it. If the conveyor belt is at rest, the water right at the belt's surface does not move; it's as if it's 'stuck' to the belt. Above this layer, the water moves faster, creating a gradient in speed until it reaches the full velocity of the water stream above the belt.

So, because of this phenomenon what happens is there is a velocity variation and this gives a velocity gradient du / dy which exist in normal direction. This velocity variation occurs in a very thin region of flow near the solid surface. This layer, this thin region is called the boundary layer.

Due to the no-slip boundary condition, there is a change in fluid velocity from the solid surface to the outer moving fluid, resulting in a velocity gradient, denoted as du/dy. This gradient indicates how rapidly the fluid's velocity increases with distance from the boundary. The area where this gradient exists is referred to as the boundary layer, a crucial concept in fluid mechanics that influences how fluids interact with surfaces it flows over.

Imagine spreading peanut butter on bread. The layer of peanut butter that first touches the bread has less movement (almost like it sticks to the bread), while the top of the peanut butter spread can flow freely. The transition between the stickiness at the bottom and the free-flowing peanut butter at the top represents the velocity gradient.

So, Prandtl actually divided the flow of the fluid in the neighbourhood of the solid boundary into 2 regions. One is called the boundary layer, where the viscous forces and rotationality cannot be ignored. The second region is called the outer flow region, where the velocity is constant and is equal to the free-stream velocity.

In fluid flow near a solid boundary, there are two distinct regions identified by Prandtl. The boundary layer is where viscous forces are significant, affecting the fluid's behavior due to the interaction with the solid surface. Contrarily, in the outer flow region, the influence of viscosity diminishes, allowing the fluid to flow without significant resistance, maintaining a constant velocity equal to that of the incoming free stream.

Consider a flow of water over a riverbed. The water right by the riverbed experiences drag due to the bottom’s roughness and is less fluid compared to the water several feet above, which flows freely without much interaction with the bed. The water closer to the bed is akin to the boundary layer, while the faster-flowing water above represents the outer flow region.

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 velocity gradient in the boundary layer leads to the development of shear stress on the solid surface. This shear stress (denoted as τ) arises from the frictional forces between the fluid layers, influenced by the viscosity of the fluid (represented by μ). Higher velocity gradients lead to greater shear stress, which is vital for understanding forces acting on structures in fluid flows, like pipes and airfoils.

Imagine trying to push a stacked pile of books across a table; the harder you push (representative of high shear stress), the more friction you feel, and they slide more easily rather than sticking to the surface. Similarly, when fluid flows past an object, the interaction creates shear stress that affects fluid behavior and resistance.

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