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Chapter 5: Problem 43

Observations of a fluid flow indicate that a velocity field can be represented by the velocity components \(u=8 z, v=6 x,\) and \(w=2 x,\) where gravity has a magnitude \(g\) and acts in the negative \(z\) -direction. Within the flow field, the viscosity is presumed constant. Assess whether this given velocity field satisfies the Navier-Stokes equation. If not, speculate on what may be the reason. If so, find an expression for the pressure distribution.

Short Answer

Expert verified
The velocity field does not satisfy the Navier-Stokes equation due to non-zero convective acceleration, inconsistent with the constant viscosity implicit stability assumption.

Step by step solution

01

Recall the Navier-Stokes Equation

The Navier-Stokes equations describe how the velocity field \(\vec{v} = (u, v, w)\) evolves in a fluid with constant density and viscosity. For an incompressible Newtonian fluid, the equation in vector form is given by \(- \rho \frac{D\vec{v}}{Dt} = - abla p + \mu abla^2 \vec{v} + \rho \vec{g}\), where \p\ is the pressure, \mu\ is the dynamic viscosity, \vec{g}\ is the gravitational field, \rho\ is the density, and \frac{D}{Dt}\ denotes the material derivative.
02

Calculate Material Derivative

The material derivative \(\frac{D\vec{v}}{Dt}\) for each component of the velocity field accounts for both local and convective changes. Compute this for \(u = 8z, v = 6x, w = 2x\). Each is evaluated as \(\frac{D}{Dt} = \frac{\partial}{\partial t} + u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y} + w \frac{\partial}{\partial z}\). Here, partial derivatives with respect to time and each spatial variable need to be computed separately.
03

Partial Derivatives

Calculate the necessary partial derivatives for \(u, v, w\). Like, \(\frac{\partial u}{\partial x} = 0, \frac{\partial u}{\partial y} = 0, \frac{\partial u}{\partial z} = 8\). Similarly, compute convective term contributions for each component, simplifying where components are constant or zero.
04

Viscosity Terms

Evaluate the contribution of the viscous term from \(\mu abla^2 \vec{v}\). For each velocity component, compute \(abla^2 u, abla^2 v,\) and \(abla^2 w\). Given \(abla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\), most second derivatives are zero considering constant coefficients in our problem.
05

Gravity Terms

Substitute the gravitational term \( \rho \vec{g} \) where \( \vec{g} = -g \hat{k} \) into the equation, resulting only in the \(z\)-direction component as non-zero.
06

Solve for Pressure Distribution

Set up the equations for each direction based on balance of terms. Integrate as necessary to find pressure distribution. Focus on terms exposing relationships, like \(- abla p = \rho \vec{g}\) implying pressure varies linearly with \(z\) when gravitation is significant.
07

Check Consistency

Review derived expressions and ensure they match prerequisite conditions of incompressibility or neglectful terms. Here, check that the divergence of velocity \(abla \cdot \vec{v} = 0\) to confirm incompressibility. For any inconsistencies, theorize on assumptions (e.g., neglected transient or unsteady effects).

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Key Concepts

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

Incompressible Fluid
An incompressible fluid is one where the fluid density is constant. This assumption simplifies the Navier-Stokes equations, making them more manageable to work with. In practical terms, this means that the volume of the fluid does not change significantly as it moves through space.

To confirm if a flow is incompressible, we often check the divergence of the velocity field. In mathematical terms, the velocity field abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla abla - abla abla abla abla abla abla abla abla

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Most popular questions from this chapter

The velocity components in a three-dimensional velocity field are given by \(u=\) \(a x^{2} z, v=b x z^{2},\) and \(w=c x z^{2}+d,\) where \(a, b, c,\) and \(d\) are constants. Determine the relationship between the constants that would be required for the flow to be incompressible.

Two-dimensional unsteady flow of an incompressible fluid occurs in the \(x y\) plane. When polar coordinates are used, the \(r\) component of the velocity is given by $$ v_{r}=r^{2} \sin \theta+r \cos \theta $$ Determine the required functional form of the \(\theta\) -component of the velocity.

A velocity field can be represented by the velocity components \(u=8 z, v=0,\) and \(w=2 x,\) where gravity has a magnitude \(g\) and acts in the negative \(z\) direction. Within the flow field, the viscosity is constant, and the pressure and density are equal to \(p_{0}\) and \(\rho_{0}\), respectively, at the location \(\left(x_{0}, y_{0}, z_{0}\right)\). Use the Navier-Stokes equation to determine the pressure distribution in terms of the given parameters.

Water at \(20^{\circ} \mathrm{C}\) flows around a submerged structure that has the shape of a Rankine half-body with a width of \(3 \mathrm{~m}\). The absolute pressure in the water upstream of the structure is \(130 \mathrm{kPa}\). Determine the maximum allowable approach velocity such that cavitation does not occur on the surface of the structure.

The nozzle shown in Figure 5.50 is designed such that the velocity, \(\mathbf{v}\), at any point near the centerline of the nozzle under steady- state conditions can be estimated by $$ \mathbf{v}=\frac{4}{0.8-x} \mathbf{i} \mathrm{m} / \mathrm{s} $$ where \(x\) is the distance in meters from the center of the nozzle entrance, measured along the centerline of the nozzle. The centerline velocities at the entrance and exit of the nozzle are \(4 \mathrm{~m} / \mathrm{s}\) and \(20 \mathrm{~m} / \mathrm{s}\), respectively, and the length of the nozzle is \(0.6 \mathrm{~m}\). What is the acceleration of a fluid element at the center of the midsection of the nozzle, where \(x=0.2 \mathrm{~m} ?\)
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