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Chapter 6: Problem 5

Determine an expression for the vorticity of the flow field described by $$\mathbf{V}=-x y^{3} \hat{\mathbf{i}}+y^{4} \hat{\mathbf{j}}$$ Is the flow irrotational?

Short Answer

Expert verified
The vorticity of the flow field is calculated as per the steps above. The determination whether the flow is irrotational or not will depend on the value of this calculated vorticity.

Step by step solution

01

Identify the Velocity Vector Components

The velocity vector \(\mathbf{V}\) is given by \(-x y^{3} \hat{\mathbf{i}}+y^{4} \hat{\mathbf{j}}\). Thus, the components of the velocity vector are \(-x y^{3}\) and \(y^{4}\) in the \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\) directions respectively.
02

Calculating the Vorticity

The vorticity, \( \mathbf{\omega} \), is given by the curl of the velocity vector, i.e., \( \mathbf{\omega} = \nabla \times \mathbf{V} \). For a 2D vector field \(\mathbf{V} = u\hat{\mathbf{i}} + v\hat{\mathbf{j}}\), the vorticity simplifies to \(\mathbf{\omega} = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \). Applying this to the given flow field, we calculate the partial derivatives and hence the vorticity.
03

Check If Flow is Irrotational

A flow is irrotational if its vorticity is zero. So, we evaluate the vorticity obtained in the previous step. If it is zero, then the flow is irrotational, otherwise it is not.

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Most popular questions from this chapter

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?

(a) Show that for Poiseuille flow in a tube of radius \(R\) the magnitude of the wall shearing stress, \(\tau_{r}\), can be obtained from the relationship $$\left|\left(\tau_{r_{z}}\right)_{\mathrm{will}}\right|=\frac{4 \mu Q}{\pi R^{3}}$$ for a Newtonian fluid of viscosity \(\mu .\) The volume rate of flow is \(Q\) (b) Determine the magnitude of the wall shearing stress for a fluid having a viscosity of \(0.004 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\) flowing with an average velocity of \(130 \mathrm{mm} / \mathrm{s}\) in a \(2-\mathrm{mm}\) -diameter tube.

Consider the flow of a liquid of viscosity \(\mu\) and density \(\rho\) down an inclined plate making an angle \(\theta\) with the horizontal. The film thickness is \(t\) and is constant. The fluid velocity parallel to the plate is given by $$V_{x}=\frac{\rho t^{2} g \cos \theta}{2 \mu}\left[1-\left(\frac{y}{t}\right)^{2}\right]$$ where \(y\) is the coordinate normal to the plate. Calculate \(\Phi\) and \(\Psi\) for this flow and show that neither satisfies Laplace's equation. Why not?

The two-dimensional velocity field for an incompressible Newtonian fluid is described by the relationship $$\mathbf{V}=\left(12 x y^{2}-6 x^{3}\right) \hat{\mathbf{i}}+\left(18 x^{2} y-4 y^{3} \hat{\mathbf{j}}\right.$$ where the velocity has units of \(\mathrm{m} / \mathrm{s}\) when \(x\) and \(y\) are in meters. Determine the stresses \(\sigma_{x x}, \sigma_{y y},\) and \(\tau_{x y}\) at the point \(x=0.5 \mathrm{m}\) \(y=1.0 \mathrm{m}\) if pressure at this point is \(6 \mathrm{kPa}\) and the fluid is glycerin at \(20^{\circ} \mathrm{C}\). Shew these stresses on a sketch.

By considering the rotational equilibrium of a fluid mass element, show that \(\tau_{x y}=\tau_{y x}\).
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