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Chapter 6: Problem 4

The three components of velocity in a flow field are given by $$\begin{aligned}u &=x^{2}+y^{2}+z^{2} \\\v &=x y+y z+z^{2} \\\w &=-3 x z-z^{2} / 2+4\end{aligned}$$ (a) Determine the volumetric dilatation rate and interpret the results. (b) Determine an expression for the rotation vector. Is this an irrotational flow field?

Short Answer

Expert verified
The volumetric dilatation rate for the given flow field is zero, meaning there is no expansion or contraction in the fluid volume. The rotation vector for the flow field is non-zero, hence it is not an irrotational flow.

Step by step solution

01

Determine the volumetric dilatation rate

From the given, the components of velocity are \(u = x^{2} + y^{2} + z^{2}\), \(v = xy + yz + z^2\), \(w = -3xz - z^{2}/2 + 4\). The volumetric dilatation rate is given by the divergence of the velocity vector \(u, v, w\). In Cartesian coordinates, the divergence of a vector is given by \(\nabla \cdot V = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\). Substituting the given velocity components: \(\nabla \cdot V = 2x + y + z - 3z - z/2 = 2x + y - 5z/2 = 0\). Since the dilatation rate is zero, this means there is no expansion or contraction, the fluid volume does not change.
02

Determine the Rotation Vector

The rotation vector or vorticity is given by the curl of the velocity vector: \(rotV = \nabla \times V = (\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z})\hat{i} + (\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x})\hat{j} + (\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})\hat{k}\). Substituting the velocity components: \(rotV = (z - y - 2z)\hat{i} + (2z + 3x - 2x)\hat{j} + (y + x - 2y)\hat{k} = -y\hat{i} + 2z\hat{j} + x\hat{k}\).
03

Determine if this is an irrotational flow field

A flow field is considered irrotational if the rotation vector or vorticity is zero. In this case, the rotation vector \(rotV = -y\hat{i} + 2z\hat{j} + x\hat{k}\) is non-zero. Therefore, this is not an irrotational flow.

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Most popular questions from this chapter

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