# What is the boundary condition for a moving solid boundary in a viscous fluid?

• n9e9o9
In summary, the no-slip boundary condition for viscous fluids states that the relative velocity between the fluid and the solid boundary is zero. This can be seen in the example of an air-liquid-solid system where the fluid at the boundary will have the same velocity as the solid if it is moving. However, for a fluid-fluid interface, the no-slip condition may be violated due to the deformability of the interface. The no-slip condition is important to prevent the stress tensor from diverging, but it is still an unsolved problem in fluid dynamics. Some examples where it may be violated include web printing processes and droplet migration on a car windshield.
n9e9o9
I am confused on the definition of the "no-slip" boundary condition because of two seemingly contradicting definitions.

Definition 1: The no-slip condition for viscous fluid states that at a solid boundary, the fluid will have zero velocity relative to the boundary.
Definition 2: The fluid velocity at all liquid–solid boundaries is equal to that of the solid boundary.

What is the velocity at a solid boundary if its moving? This would contradict the zero velocity definition.

Take the example of a air-liquid-solid system, with air on top, liquid in the middle, and the solid on the bottom. Suppose the bottom plate is pulled with a velocity V, at steady-state, to the right-hand side of the system. What would the boundary condition be and/or what would the velocity and shear stress profile look like? (Cartesian coordinates with y in the "north" direction and x in the "east direction")

My guess for the boundary conditions would be that the v=V at y=0 and v=0 at y=$$\delta$$.

Is this the correct logic?

The RELATIVE velocity of the fluid with respect to the solid boundary, i.e, their difference, is to be zero.

Thus, if with respect to some frame the boundary is moving with velocity V, so will the fluid at its boundary.

The no-slip boundary condition represents a 200-year old unsolved problem.

The official boundary condition is that the tangential component of velocity is continuous across a solid-liquid interface. So, if the solid is not moving, the tangential component of fluid velocity as the surface must also be zero. Because the solid-fluid interface does not deform, the normal component must also be zero. If the solid is moving, the tangential component of velocity of the fluid at the interface is the same as the velocity of the solid, and the normal component of velocity at the interface is still zero. "Stokes first problem" solves how a moving solid creates a velocity field in an initially stationary fluid.

Now, for a fluid-fluid interface, all bets are off becasue the interface can deform. There can be a velocity jump across a fluid-fluid interface.

The no-slip condition exists to ensure the stress tensor does not diverge. However, the no-slip condition is routinely violated all the time- web printing processes, droplet migration across my car windshield during a rainstorm, etc. etc. As I mentioned above, the no-slip boundary condition is a 200-year old unsolved problem.

I don't think droplet migration on your windshield is an example of no-slip violation. The droplets are actually rolling I think, so no-slip is always satisfied, like the tire of a moving car.

dameyawn said:
I don't think droplet migration on your windshield is an example of no-slip violation. The droplets are actually rolling I think, so no-slip is always satisfied, like the tire of a moving car.

Davis and Dussan did show, in a beautiful 1974 paper, that some droplets roll along the surface. But that is not a general result.

## 1. What is a "No-Slip Boundary Condition"?

A "No-Slip Boundary Condition" is a fundamental concept in fluid mechanics that describes the behavior of fluids in contact with solid surfaces. It states that the fluid particles in contact with the surface have zero velocity relative to the surface, meaning that they do not slip or move along the surface.

## 2. Why is the "No-Slip Boundary Condition" important?

The "No-Slip Boundary Condition" is important because it helps us understand the behavior of fluids in contact with solid surfaces, which is crucial in many engineering and scientific applications. It also allows us to accurately model and predict the flow of fluids in pipes, channels, and other systems.

## 3. How is the "No-Slip Boundary Condition" applied in practical situations?

The "No-Slip Boundary Condition" is applied by assuming that the fluid particles at the surface have zero velocity relative to the surface, and then solving the equations of motion for the fluid flow. This is usually done using mathematical models and computational methods.

## 4. Are there any exceptions to the "No-Slip Boundary Condition"?

While the "No-Slip Boundary Condition" is generally accepted as a fundamental principle, there are some exceptions. For example, at very small scales, such as at the molecular level, fluid particles may exhibit slip behavior at the surface. In addition, certain types of fluids, such as superfluids, may not exhibit a no-slip behavior.

## 5. How does the "No-Slip Boundary Condition" affect drag force?

The "No-Slip Boundary Condition" has a significant impact on drag force, which is the resistance force experienced by an object moving through a fluid. When the no-slip condition is applied, it results in a thin layer of fluid near the surface that has a lower velocity than the free stream, causing an increase in drag force.

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