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Chapter 4: Problem 27

A certain flow field has the velocity vector $\mathbf{V}=\frac{-2 x y z}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{i}+\frac{\left(x^{2}-y^{2}\right) z}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{j}+\frac{y}{x^{2}+y^{2}} \mathbf{k} .$ Find the acceleration vector for this flow.

Short Answer

Expert verified
To compute the acceleration vector for the given velocity vector, calculate the material derivative of the velocity vector with respect to time \(t\). This involves working out the partial derivatives for each component of the velocity vector. These partial derivatives will become zeros due to the flow field being steady. Lastly, substitute these components into the formula for acceleration to determine the acceleration vector.

Step by step solution

01

Representation of Velocity Vector

Using the given information, the velocity vector \(\mathbf{V}\) is represented in Cartesian coordinates as \(\frac{-2 x y z}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{i}+\frac{\left(x^{2}-y^{2}\right) z}{\left(x^{2}+y^{2}\right)^{2}} \mathbf{j}+\frac{y}{x^{2}+y^{2}} \mathbf{k}\).
02

Partial Derivatives of the Velocity Vector Components

Compute the partial derivatives of the velocity vector with respect to time \(t\) for each component of the vector. That is \(\frac{\partial V_{i}}{\partial t}, \frac{\partial V_{j}}{\partial t}, \text{and } \frac{\partial V_{k}}{\partial t}\). Considering the flow field to be steady, the velocity doesn't change with time. These material derivatives then become zeroes.
03

Components of acceleration

Determine each of the components of acceleration \(V\frac{\partial V_{i}}{\partial x}, V\frac{\partial V_{j}}{\partial y}, V\frac{\partial V_{k}}{\partial z}\) etc. Substitute each of these into the formula for acceleration given by \(A = \frac{D \mathbf{V}}{D t}\) to get the acceleration vector.

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Most popular questions from this chapter

A fluid particle flowing along a stagnation streamline, as shown in Video \(V 4.9\) and Fig. \(P 4.38,\) slows down as it approaches the stagnation point. Measurements of the dye flow in the video indicate that the location of a particle starting on the stagnation streamline a distance \(s=0.6 \mathrm{ft}\) upstream of the stagnation point at \(t=0\) is given approximately by \(s=0.6 e^{-0.5 y},\) where \(t\) is in seconds and \(s\) is in feet. (a) Determine the speed of a fluid particle es a function of time, \(V_{\text {particle }}(t),\) as it flows along the streamline. (b) Determine the speed of the fluid as a function of position along the streamline, \(V=V(s)\) (c) Determine the fluid acceleration along the streamline as a function of position, \(a_{s}=a_{s}(s)\)

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