13.3.1. Mass Conservation

We are talking about the boundary layer mass conservation equations. As you remember, the boundary layer equations we derive first is the mass conservation which is very simple as again I am repeating it for two-dimensional incompressible flow velocity divergence, which is equal to del u by del x plus del v by del y.

In fluid mechanics, mass conservation is a fundamental principle stating that mass cannot be created or destroyed. When we analyze fluid flow, especially in boundary layers (the thin regions near surfaces where the effects of viscosity are significant), we must ensure that the mass entering a region equals the mass exiting that region. In two-dimensional incompressible flow, this principle is mathematically represented as the sum of the rate of change of velocity in the x direction (del u by del x) and the rate of change of velocity in the y direction (del v by del y) being equal to zero. This leads to the continuity equation for mass conservation.

Consider a river flowing steadily. If water enters a narrow section of the river, the same amount of water must exit through that section, assuming there are no leaks. This reflects the principle of mass conservation: the water's mass must be conserved as it flows through different areas.

Del u by del x plus del v by del y is equal to u dy dx, this is the free stream velocity plus you have mu del square u by del y square.

The equations mentioned describe how the velocity of a fluid changes within a boundary layer. Here, 'u' represents the velocity parallel to the x-axis (the direction of flow), 'v' is the velocity perpendicular to the x-axis, and 'mu' is the dynamic viscosity of the fluid. The term 'del square u by del y square' indicates how the velocity profile (the distribution of velocity across the boundary layer's thickness) changes vertically. The combination of these terms allows us to understand how momentum and energy are transported in the flow near a surface.

Think of a river flowing over a flat surface. Near the bank, the water moves slower due to friction (viscosity) with the bank, while further out, the water flows faster. This change in speed across the width of the river mirrors the concepts found in boundary layer equations.

If I have a wall surface, like this, and you have formations of boundary layers, ... we know how the stream velocity is varying it ux also we know it.

In analyzing flow over a flat plate, specific boundary conditions are applied. At the wall (the surface of the plate), the fluid adheres to the surface, meaning both velocities u and v are zero due to the no-slip condition (the fluid does not slide along the wall). Away from the wall, in the free stream area, the velocity reaches the free stream velocity 'ux'. Understanding these conditions is crucial for solving the boundary layer equations for various fluid dynamics scenarios.

Imagine a runner on a track. When the runner is against a wall, they can't move—this represents the no-slip condition of the fluid at the wall. But as they move away from the wall, they can run freely, reaching their maximum speed, analogous to the 'free stream velocity' in fluid flow.

The x momentum equations can now be simplified as these ux components become constants.

In certain conditions, such as assuming a uniform flow with little variation (constant free stream velocity), the complexity of the equations can be reduced. The x momentum equations become simpler as the components of velocity (ux) can be treated as constants for the purpose of calculations. This allows engineers to predict and model fluid behavior more easily without intricate calculations.

Consider a straight road with constant speed traffic. The average speed (ux) can be treated as constant, allowing easier calculations for traffic flow instead of considering every vehicle's speed at all times.

The basic idea comes as the boundary layer thickness happens and if I take a just a free out streamlines, ... that deflected part is known as displacement thickness.

Displacement thickness is a concept used in boundary layer theory to account for the reduction in mass flow rate due to the boundary layer effect. When fluid flows over a surface, some of it slows down due to friction, leading to a 'displacement' in the streamline positions. The displaced flow acts as if it were being 'pushed' outward, and this push is quantified as the displacement thickness. It’s a vital property because it helps in determining the effective area through which fluid flows.

Imagine a flat plate with flowing water. The water near the plate moves slower due to friction. The 'displacement thickness' represents how much this slowed water effectively pushes the free stream of the water further away from the plate.

Momentum thickness, as defined, is nothing else but has introduced as viscous drag force on the plate per unit width, which is equal to rho u square theta.

Momentum thickness is another important property in boundary layer theory. It accounts for the momentum deficit in the boundary layer compared to the outer flow. It helps quantify the effect of the boundary layer on drag force experienced by a surface. Higher momentum thickness values indicate higher drag forces on the surface, which is crucial for designs to reduce drag in applications like aircraft wings.

Think of a boat moving through water. If the water flows smoothly around it, the drag is minimal. But if there’s a thick layer of water being dragged along with it (momentum thickness), the boat experiences more resistance. This is why boat designers aim to reduce this effect through better shapes.

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