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Chapter 6: Problem 6

According to Eq. 6.134 , the \(x\) -velocity in fully developed laminar flow between parallel plates is given by $$u=\frac{1}{2 \mu}\left(\frac{\partial p}{\partial x}\right)\left(y^{2}-h^{2}\right)$$ The \(y\) -velocity is \(v=0\). Determine the volumetric strain rate, the vorticity, and the rate of angular deformation. What is the shear stress at the plate surface?

Short Answer

Expert verified
The volumetric strain rate and the y-component of velocity v are zero, while the vorticity, rate of angular deformation, and shear stress are yielded by differentiating the velocity profile and can be evaluated with real number values if the other variables involved are given.

Step by step solution

01

Velocity Profile Calculation

Given the equation for velocity in x-direction, use this equation directly to assign the x-velocity. According to the problem, \(v = 0\) as the y-velocity.
02

Calculate Volumetric Strain Rate

The equation for volumetric strain rate is \(\frac {du}{dx}+\frac {dv}{dy}\), where \(du/dx\) is the rate of change of velocity in x direction and \(dv/dy\) is the rate of change of velocity in y direction. After substituting \(v=0\) into the equation, we are left with \(\frac {du}{dx}=0\), since the rate of change of \(u\) with respect to \(x\) is 0, as from the given \(u\) equation there is no \(x\) dependent term.
03

Vorticity Calculation

The vorticity is calculated by \(\frac {dv}{dx}-\frac {du}{dy}\). Substitute the given \(v = 0\) and differentiate the velocity profile \(u\) given earlier with respect to \(y\), and calculate the vorticity.
04

Rate of Angular Deformation

Similarly, the rate of angular deformation is calculated by \(\frac {1}{2}(\frac {dv}{dx}+\frac {du}{dy})\). Substituting \(v = 0\), we just need to differentiate the x-velocity with respect to \(y\) and substitute those values.
05

Shear Stress Calculation

Shear stress on the plate surface at \(y = h\) can be calculated by the expression \(τ=μ*(\frac {du}{dy}) = μ*\frac {dv}{dx}\). The value of \(du/dy\) can be obtained by differentiating the x-velocity profile with respect to \(y\).
06

Evaluate the Calculations

By using the above calculations, determine the values of volumetric strain rate, vorticity, the rate of angular deformation and shear stress.

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Most popular questions from this chapter

A liquid of constant density \(\rho\) and constant viscosity \(\mu\) flows down a wide, long inclined flat plate. The plate makes an angle \(\theta\) with the horizontal. The velocity components do not change in the direction of the plate, and the fluid depth, \(h,\) normal to the plate is constant. There is negligible shear stress by the air on the fluid. Find the velocity profile \(u(y),\) where \(u\) is the velocity parallel to the plate and \(y\) is measured perpendicular to the plate. Write an expression for the volume flow rate per unit width of the plate.

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