Shear stress on oscillating wall in viscous incompressible fluid

Solving for C and D, we get C=1/2 and D=1/2. Therefore, the solution for T(t) is:T(t) = (1/2)e^(iλ√(μ)t)+(1/2)e^(-iλ√(μ)t)Finally, we can substitute this back into our solution for U(y,t) to get:U(y,t) = (1/2)Y(y)e^(iλ√(μ)t)+(1/2)Y(y)e^(-iλ√(μ)t)To find the shear stress at the wall, we can use the formula:Shear stress at wall = μ(∂u/
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Homework Statement


A layer of viscous incompressible fluid of thickness H lies on top of a solid wall that oscillates simple harmonically w/ angular frequency Ω. u(wall)=Acos(Ωt). Ignore the motion of air above the fluid layer and find the shear stress at the wall. (Shear stress on free surface must be zero.)


Homework Equations


Equation of motion and Navier-Stokes equations in cartesian coordinates.

Shear stress at wall = μ(∂u/∂y) for y=0


The Attempt at a Solution


Boundary conditions: For y=0, u(wall)=Acos(Ωt) and v=w=0. Solution is independent of x and z so ∂/∂x=∂/∂z=0. And where y=H you have μ(∂u/∂y).

From here, the governing equations simplify to just ∂u/∂t = μ(∂^2u/∂y^2)

Solving this PDE is where I'm running into trouble. I believe there is a way to simplify the problem by converting it to complex, but that's where I am stuck.
 
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Any help or guidance would be greatly appreciated.

First, we can convert the problem to complex by letting u=u(x,y,t)=Re{U(x,y,t)}, where U is a complex function. Then, we can rewrite the governing equation as:

∂U/∂t = μ(∂^2U/∂y^2)

Using separation of variables, we can rewrite U as U(y,t)=Y(y)T(t). Substituting this into the governing equation, we get:

T'(t)Y(y) = μ(Y''(y)T(t))

Dividing both sides by Y(y)T(t), we get:

T'(t)/T(t) = μ(Y''(y)/Y(y))

Since the left side only depends on t and the right side only depends on y, this can only be true if both sides are equal to a constant. Let's call this constant -λ^2. This gives us two ordinary differential equations:

T'(t) + λ^2μT(t) = 0

Y''(y) + λ^2Y(y) = 0

The solution to the first equation is T(t) = Ce^(iλ√(μ)t)+De^(-iλ√(μ)t), where C and D are constants. The solution to the second equation is Y(y) = C1cos(λy)+C2sin(λy), where C1 and C2 are constants.

Now, we can use the boundary conditions to determine the values of C, D, C1, and C2. At y=0, we know that u(wall)=Acos(Ωt), so Y(0)=0, which gives us C1=0. At y=H, we know that the shear stress must be zero, so μ(∂u/∂y)|y=H=0. Using the expression for Y(y), we get C2sin(λH)=0, which means that either C2=0 or λH=nπ for some integer n. If C2=0, then the solution will be trivial, so let's assume that λH=nπ. This gives us the eigenvalues λn=nπ/H.

Now, we can determine the constants C and D using the initial condition T(0)=1. This gives us:

C+D=1
C-D
 

1. What is shear stress on an oscillating wall in a viscous incompressible fluid?

Shear stress on an oscillating wall in a viscous incompressible fluid refers to the force per unit area that is exerted on the wall by the fluid as it moves back and forth, or oscillates, in a viscous and incompressible manner. It is a measure of the resistance to flow of the fluid.

2. What factors affect shear stress on an oscillating wall in a viscous incompressible fluid?

The main factors that affect shear stress on an oscillating wall in a viscous incompressible fluid are the properties of the fluid, such as its viscosity and density, the oscillation frequency and amplitude of the wall, and the distance between the wall and the fluid.

3. How is shear stress on an oscillating wall in a viscous incompressible fluid calculated?

Shear stress on an oscillating wall in a viscous incompressible fluid can be calculated using the formula τ = μ(dv/dy), where τ is the shear stress, μ is the dynamic viscosity of the fluid, and (dv/dy) is the velocity gradient perpendicular to the direction of flow at the wall.

4. Why is understanding shear stress on an oscillating wall in a viscous incompressible fluid important?

Understanding shear stress on an oscillating wall in a viscous incompressible fluid is important in many practical applications, such as in the design of pumps, turbines, and other fluid flow systems. It also plays a crucial role in the study of fluid dynamics and can help in predicting the behavior of fluids in various situations.

5. How can shear stress on an oscillating wall in a viscous incompressible fluid be controlled or reduced?

Shear stress on an oscillating wall in a viscous incompressible fluid can be controlled or reduced by changing the properties of the fluid, such as by using a less viscous fluid or adding a lubricant. It can also be reduced by minimizing the distance between the wall and the fluid, or by adjusting the oscillation frequency and amplitude of the wall.

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