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Chapter 6: Problem 77

A Bingham plastic is a fluid in which the stress \(\tau\) is related to the rate of strain \(d u / d y\) by $$\tau=\tau_{0}+\mu\left(\frac{d u}{d y}\right)$$ where \(\tau_{0}\) and \(\mu\) are constants. Consider the flow of a Eingham plastic between two fixed, horizontal, infinitely wide, flat plates. For fully developed flow with \(d u / d x=0\) and \(d p / d x\) constant. find \(u(y)\).

Short Answer

Expert verified
The velocity profile \(u(y)\) for Bingham plastic flow between two plates is \(u(y) = 0\).

Step by step solution

01

Start with the constitutive equation

Start with the constitutive equation for a Bingham plastic which states \(\tau=\tau_{0}+\mu\left(\frac{d u}{d y}\right)\) where \(\tau_0\) is the yield stress and \(\mu\) is the dynamic viscosity.
02

Set the momentum balance for steady, one directional,fully developed flow.

The momentum balance equation for steady, one-dimensional and fully developed flow is given by \(\frac{d \tau}{d y}=0\). Substitute the equation of stress from step 1 into this equation, to get \(\frac{d \tau_0}{d y} + \mu \frac{d^2 u}{d y^2} = 0\). Since \(\tau_0\) is a constant, it's derivative with respect to \(y\) becomes zero. So, the simplified momentum balance equation is \(\frac{d^2 u}{d y^2} = 0\).
03

Solve the differential equation.

Solving the differential equation \(\frac{d^2 u}{d y^2} = 0\), gives, \(\frac{d u}{d y} = C_1\), where \(C_1\) is the constant of integration. Integrate further with respect to \(y\), to get \(u = C_1 y + C_2\), where \(C_1\) and \(C_2\) are integration constants to be determined from boundary conditions.
04

Evaluate boundary conditions

In the scenario of flow between two fixed plates, at \(y = 0\) and \(y = h\) the velocity \(u = 0\) (no slip condition at boundaries). Use these conditions to evaluate the constants \(C_1\) and \(C_2\). This leads to \(C_2 = 0\) and \(C_1 = 0\).
05

Formulate the result.

Substituting the constants \(C_1 = 0\) and \(C_2 = 0\) back into the equation derived in Step 3. Therefore, \(u(y)=0\) is the velocity profile for Bingham plastic flow between two plates.

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