To determine the fluid viscosity (μ0) using the given constitutive model and the sliding plate viscometer experiment, we can start by analyzing the fluid flow between the sliding plates.
In the sliding plate viscometer, the fluid is subjected to shear stress (τyx) as the top plate moves with a velocity (V). The fluid experiences a shear rate (dvx/dy) due to the velocity gradient between the plates.
The constitutive model for the fluid is given as:
τyx = τyield - μ0 * (dy/dvx)
Here, τyield represents the yield stress of the fluid, and μ0 is the fluid viscosity that we want to determine.
In the sliding plate viscometer, the shear stress (τyx) can be related to the applied force (F) and the dimensions of the system. The shear stress can be approximated as:
τyx = F / (h * w)
Where h is the distance between the plates (plate height) and w is the width of the plate.
Now, let's consider the velocity gradient (dy/dvx) in the fluid. The top plate is moving at a constant velocity (V) to the right, and the fluid velocity near the top plate can be assumed to be approximately equal to V. Therefore, we can approximate the velocity gradient as:
(dy/dvx) ≈ (h / V)
Substituting these approximations into the constitutive model equation, we get:
F / (h * w) = τyield - μ0 * (h / V)
Rearranging the equation, we can isolate the fluid viscosity (μ0):
μ0 = (F / (h * w * h)) * V - τyield / h
So, the expression for the fluid viscosity (μ0) in terms of the experiment parameters and the yield stress (τyield) is:
μ0 = (F / (h^2 * w)) * V - τyield / h
Please note that this expression assumes a simplified approximation and may not account for all possible factors or complexities that could affect the measurement or the fluid's behavior.