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Chapter 6: Problem 7

For a certain incompressible, two-dimensional flow field the velocity component in the \(y\) direction is given by the equation $$v=3 x y+x^{2} y$$ Determine the velocity component in the \(x\) direction so that the volumetric dilatation rate is zero.

Short Answer

Expert verified
The velocity component in the \(x\) direction such that the volumetric dilatation rate is zero is \(u = -\frac{3}{2}x^{2} - \frac{1}{3}x^{3} + f(y)\), where \(f(y)\) is an arbitrary function of \(y\).

Step by step solution

01

Understand the given

The y-component of the velocity \(v\) is given by the equation \(v=3xy+x^{2}y\). The x-component of the velocity is unknown and needs to be found. The condition given is that the volumetric dilatation rate is zero.
02

Apply the divergence of velocity field formula

For a two-dimensional incompressible flow, the divergence of the velocity field should equal zero. This yields the equation \(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\), where \(u\) is the x-component of velocity, and \(v\) is the y-component of velocity.
03

Substitute the given equation in the divergence formula

Substitute \(v = 3xy + x^{2}y\) into the divergence equation, so we have \(\frac{\partial u}{\partial x} + \frac{\partial (3xy + x^{2}y)}{\partial y} = 0\). While \(\frac{\partial (3xy + x^{2}y)}{\partial y}\) can be simplified to \(3x + x^{2}\), thus the equation becomes \(\frac{\partial u}{\partial x} + 3x + x^{2} = 0\).
04

Solve for \(u\)

Solving for \(u\), the x-component of the velocity, we get \(\frac{\partial u}{\partial x} = -3x - x^{2}\). Integrating with respect to \(x\), we obtain \(u = -\frac{3}{2}x^{2} - \frac{1}{3}x^{3} + f(y)\), where \(f(y)\) is an arbitrary function of \(y\), as a result of the integration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Velocity Components
In fluid mechanics, understanding velocity components is crucial for analyzing fluid flow. Velocity components represent the speed and direction of fluid particles in different directions of a coordinate system. For two-dimensional flow fields, we consider the x and y components of velocity, denoted by \(u\) and \(v\), respectively. The given exercise provides the y-component of the velocity as \( v = 3xy + x^{2}y \). This component reflects how the fluid moves in the y-direction based on the variables \(x\) and \(y\). The aim is often to determine the missing velocity component, here \(u\), in the x-direction to fulfill certain conditions like incompressibility.
  • Velocity components help describe the motion of fluid particles.
  • They are defined for each direction in the problem's coordinate system.
  • Incompressible flow requires special conditions for these components.
Divergence of Velocity Field
The divergence of a velocity field is a vital concept in fluid dynamics, indicating how the flow expands or contracts at a point. For incompressible flows, which are flows where density remains constant, the divergence must equal zero. Mathematically, the divergence in a two-dimensional field involving components \(u\) and \(v\) is given by \( abla \cdot \mathbf{V} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \). This equation signifies that the flow neither expands nor contracts, maintaining a constant volume at any point.
  • Divergence measures the rate of volume change within a flow.
  • In incompressible flow, the divergence is zero.
  • The divergence equation links the velocity components together.
Volumetric Dilatation Rate
Volumetric dilatation rate is another crucial concept especially in the context of incompressible fluid flows. It refers to the volume change rate of a fluid element per unit volume. For incompressible flows, as in this exercise, the volumetric dilatation rate must be zero. This means that the product of the divergence of the velocity field must also be zero, ensuring no net change in density. This constraint directly impacts how we calculate unknown velocity components, as seen in determining the x-component of velocity \(u\).
  • Measures change in volume of a fluid element.
  • Must be zero for incompressible fluids, indicating no density change.
  • Essential for solving fluid dynamics problems.
Integration in Fluid Mechanics
Integration is a mathematical process heavily utilized in fluid mechanics to deduce unknown functions from known derivatives, such as velocity components. In this problem, we derive the x-component of the velocity by integrating the result of setting the divergence of the velocity field to zero. Starting with the equation \( \frac{\partial u}{\partial x} = -3x - x^{2} \), integrating with respect to \(x\) provides \( u = -\frac{3}{2}x^{2} - \frac{1}{3}x^{3} + f(y) \). This integration requires us to understand the need for an integration constant, which is an arbitrary function of \(y\) here, \(f(y)\), due to the partial differentiation originally involved.
  • Used to find velocity from rates of change.
  • Involves finding a function based on derivatives.
  • Results in an arbitrary function due to indeterminate variables.

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Most popular questions from this chapter

The stream function for an incompressible. two-dimensional flow field is $$\psi r=3 x^{2} y+y$$ For this flow field, plot several streamlines.

The stream function for a given two-dimensional flow field is $$\psi=5 x^{2} y-(5 / 3) y^{3}$$ Determine the corresponding velocity potential.

Two immiscible, incompressible, viscous fluids having the same densities but different viscosities are contained between two infinite, horizontal, parallel plates (Fig. \(P 6.80\) ). The bottom plate is fixed, and the upper plate moves with a constant velocity \(U\) Determine the velocity at the interface. Express your answer in terms of \(U, \mu_{1},\) and \(\mu_{2}\). The mo:ion of the fluid is caused entirely by the movement of the upper plate; that is, there is no pressure gradient in the \(x\) direction. The fluid velocity and shearing stress are continuous across the interface between the two fluids. Assume laminar flow:

Air is delivered through a constant-diameter duct by a fan. The air is inviscid, so the fluid velocity profile is "flat" across each cross section. During the fan start-up, the following velocities were measured at the time \(t\) and axial positions \(x\) : $$\begin{array}{llll} & x=0 & x=10 \mathrm{m} & x=20 \mathrm{m} \\ t=0 \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} \\\t=1.0 \mathrm{s} & V=1.00 \mathrm{m} / \mathrm{s} & V=1.20 \mathrm{m} / \mathrm{s} & V=1.40 \mathrm{m} / \mathrm{s} \\\t=2.0 \mathrm{s} & V=1.70 \mathrm{m} / \mathrm{s} & V=1.80 \mathrm{m} / \mathrm{s} & V=1.90 \mathrm{m} / \mathrm{s} \\\t=3.0 \mathrm{s} & V=2.10 \mathrm{m} / \mathrm{s} & V=2.15 \mathrm{m} / \mathrm{s} & V=2.20 \mathrm{m} / \mathrm{s}\end{array}$$ Estimate the local acceleration, the convective acceleration, and the total acceleration at \(t=1.0 \mathrm{s}\) and \(x=10 \mathrm{m} .\) What is the local acceleration after the fan has reached a steady air flow rate?

Two fixed, horizontal, parallel plates are spaced 0.4 in. apart. A viscous liquid \(\left(\mu=8 \times 10^{-3} 16 \cdot \mathrm{s} / \mathrm{ft}^{2}, 3 G=0.9\right)\) flows between the plates with a mean velocity of 0.5 fls. The flow is laminar. Determine the pressure drop per unit length in the direction of flow. What is the maximum velocity in the channel?
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