8.4. Boundary Conditions

Let me introduce the no slip boundary conditions. This is the basic properties of the fluid flow. That means if a fluid is moving, the velocity of V, the fluid attached to these also will be moving the fluid particles attached to this, the fluid particles this fluid V_f and another ones moving with V_m. So, both because no slip conditions both has to move it the same velocity.

The no slip boundary condition is a fundamental concept in fluid mechanics that states that a fluid in contact with a solid boundary (like a plate) will have a velocity equal to that of the boundary at the very point of contact. This means if a plate is stationary, the fluid right next to it (the layer of fluid in contact with the plate) will also have zero velocity. Imagine running your hand over a table; the part of your hand in contact with the table does not slide, which is analogous to how fluid molecules stick to the surface of the solid due to viscosity.

Think of a row of kids holding hands on a slippery slide. The kid at the top (the solid boundary or the edge of the slide) is stationary, and the kids below are sliding down. The first kid in contact with the slide doesn’t slide down but holds onto the slide, similar to how the first layer of fluid holds onto the surface. They are stationary relative to the slide, illustrating the no slip condition.

Now if you come to very basic things as we always look at the interface between the air and waters. In that case, if I look at u_water should be equal to u_air. That means at the interfaces levels I will get it the velocity of water is equal to velocity of air at the three surface levels. Also, I will get it the shear stress working on waters that is what will be shear stress working on air that should be.

Interface boundary conditions describe the behavior of fluids at the boundary between them, such as between water and air. The velocities of the two fluids at the interface must match; otherwise, there would be discontinuity in flow. Additionally, the shear stress acting on either side of the interface should also be equal. This means that if one fluid flows faster, it would affect the other fluid's velocity to maintain equilibrium at their shared boundary.

Imagine a swimming pool where water (liquid) meets the air above it. When someone splashes into the pool, the velocity of water right at the surface (the interface) must adjust so it can slip into the air smoothly. The faster-moving water at the surface affects the layer of air just above it; both surfaces need to equilibrate to prevent turbulence. Just like how a calm lake's surface gently meets the air, showing how velocities align at the boundary.

If I have a symmetric problems, instead of solving total problems, we just solve this part subjecting the boundary conditions of v equal to 0 and the gradient of u in the y direction becomes 0.

Symmetric flow conditions allow for simplifications in fluid analysis; essentially, they enable us to solve complex problems in a simpler way. If a flow is symmetric with respect to a certain axis, the flow velocity in the direction normal to that axis (like 'v' in this context) will be zero at that axis. In such cases, we can also set the gradient of the velocity in the perpendicular direction to the axis of symmetry to zero, which can significantly reduce the complexity of the equations required to describe the flow.

Consider a seesaw where both sides are perfectly balanced. Just as the seesaw does not tilt (which would represent an imbalance something akin to asymmetric flow), the fluid behaves similarly at points of symmetry. If both fluids (or fluid flows) strike an object, like water entering a pipe, their forces distribute evenly (symmetric), making calculations simpler due to consistent behavior at both sides. Thus, we only need to analyze half of the seesaw to understand the whole system.

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