/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Consider the flow field \(\vec{V... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 25

Consider the flow field \(\vec{V}=a x t \hat{i}+b \hat{j},\) where \(a=0.1 \mathrm{s}^{-2}\) and \(b=4 \mathrm{m} / \mathrm{s}\). Coordinates are measured in meters. For the particle that passes through the point \((x, y)=(3,1)\) at the instant \(t=0,\) plot the pathline during the interval from \(t=0\) to 3 s. Compare this pathline with the streamlines plotted through the same point at the instants \(t=1,2,\) and 3 s.

Short Answer

Expert verified
The pathline depicts the trajectory of a fluid particle over time, while the streamlines represent the flow direction at an instant in time. Since streamlines can change over time, they may not coincide with the pathline except at the initial instant. The specific plots would depend upon the constants and the flow field function.

Step by step solution

01

Solving for Pathline

The pathline is the trajectory that a fluid particle follows. It is obtained by integrating the velocity field \(\vec{V}\) from \(t=0\) to \(t=3\). This can be expressed as: \(\vec{x}(t) = \int_0^t \vec{V}(t') dt'\). Since \(\vec{V} = a x t \hat{i} + b \hat{j}\), the integral expression becomes: \(\vec{x}(t) = \int_0^t (axt' \hat{i} + b \hat{j}) dt'\). The particle initially passes through (3,1), so this information is included in the integration. By integrating, the pathline function \(\vec{x}(t)\) is obtained.
02

Plotting Pathline

Once the pathline is obtained from step 1, the next step is to plot this pathline on a graph. x coordinate (or i-component) forms the x-axis and y coordinate (or j-component) forms the y-axis. Time t ranges from 0 to 3s. By plugging in the values, the path of the particle can be visualized.
03

Solving for Streamlines

A streamline is a line that is everywhere tangent to the velocity field and shows the direction a fluid element will travel in at any point in space. It can be obtained by setting velocity field \(\vec{V}\) equal to \(dx/dt\), which leads to first order differential equations. Given field \(\vec{V} = a x t \hat{i} + b \hat{j}\), finding \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) will help solve for the streamlines, and we can repeat this process for \(t=1\), \(t=2\) and \(t=3\).
04

Plotting Streamlines

Next, plot these streamlines on the same graph as the pathline. This process includes inputting the appropriate values into the equations derived in step 3. These plots will reveal the trajectory of particles at different time intervals after they pass through the point (3,1). After successfully plotting these streamlines, we can visually compare them with the pathline plotted before.

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Most popular questions from this chapter

A block of mass \(M\) slides on a thin film of oil. The film thickness is \(h\) and the area of the block is \(A\). When released, mass \(m\) exerts tension on the cord, causing the block to accelerate. Neglect friction in the pulley and air resistance. Develop an algebraic expression for the viscous force that acts on the block when it moves at speed \(V\). Derive a differential equation for the block speed as a function of time. Obtain an expression for the block speed as a function of time. The mass \(M=5 \mathrm{kg}, m=1 \mathrm{kg}, A=25 \mathrm{cm}^{2},\) and \(h=0.5 \mathrm{mm} .\) If it takes \(1 \mathrm{s}\) for the speed to reach \(1 \mathrm{m} / \mathrm{s}\), find the oil viscosity \(\mu .\) Plot the curve for \(V(t)\).

A flow is described by velocity field \(\vec{V}=a y^{2} \hat{i}+b j\) where \(a=1 \mathrm{m}^{-1} \mathrm{s}^{-1}\) and \(b=2 \mathrm{m} / \mathrm{s}\). Coordinates are measured in meters. Obtain the equation for the streamline passing through point \((6,6) .\) At \(t=1 \mathrm{s}\), what are the coordinates of the particle that passed through point (1,4) at \(t=0 ?\) At \(t=3 \mathrm{s},\) what are the coordinates of the particle that passed through point (-3,0) 2 s earlier? Show that pathlines, streamlines, and streaklines for this flow coincide.

Tiny hydrogen bubbles are being used as tracers to visualize a flow. All the bubbles are generated at the origin \((x=0, y=0) .\) The velocity ficld is unsteady and obeys the equations: \\[\begin{array}{lll}u=1 \mathrm{m} / \mathrm{s} & v=2 \mathrm{m} / \mathrm{s} & 0 \leq t<2 \mathrm{s} \\ u=0 & v=-1 \mathrm{m} / \mathrm{s} & 0 \leq t \leq 4 \mathrm{s}\end{array}\\] Plot the pathlines of bubbles that leave the origin at \(t=0,1\) \(2,3,\) and 4 s. Mark the locations of these five bubbles at \(t=4\) s. Use a dashed line to indicate the position of a streakline at \(t=4 \mathrm{s}\).

Consider the flow ficld \(\vec{V}=a x t \hat{i}+b \tilde{j},\) where \(a=1 / 4 s^{-2}\) and \(b=1 / 3 \mathrm{m} / \mathrm{s}\). Coordinates are measured in meters. For the particle that passes through the point \((x, y)=(1,2)\) at the instant \(t=0,\) plot the pathline during the time interval from \(t=0\) to 3 s. Compare this pathline with the streakline through the same point at the instant \(t=3\) s.

A block that is \(a\) mm square slides across a flat plate on a thin film of oil. The oil has viscosity \(\mu\) and the film is \(h\) mm thick. The block of mass \(M\) moves at steady speed \(U\) under the influence of constant force \(F\). Indicate the magnitude and direction of the shear stresses on the bottom of the block and the plate. If the force is removed suddenly and the block begins to slow, sketch the resulting speed versus time curve for the block. Obtain an expression for the time required for the block to lose 95 percent of its initial speed.
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