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Chapter 9: Problem 15

If the radial velocity \(v_{r}=v_{r}(r)\), the circumferential velocity \(v_{\theta}=v_{\theta}(r)\), and the axial velocity \(w=z \bar{w}(r)\) for a steady incompressible axisymmetric vortex, show that the Navier-Stokes equations become $$ v_{r} \frac{d \bar{w}}{d r}+\bar{w}^{2}-\frac{v}{r} \frac{d}{d r}\left(r \frac{d \bar{w}}{d r}\right)=C $$ and $$ \frac{\partial p}{\partial z}=c \rho z $$

Short Answer

Expert verified
The Navier-Stokes equations simplify to the given forms by applying axisymmetric vortex assumptions and integrating accordingly.

Step by step solution

01

Understanding the Problem

We are asked to transform the Navier-Stokes equations for a specific flow characterized by radial, circumferential, and axial velocities into simpler forms. This involves showing how these velocities contribute to the radial and pressure components under the axiomatic assumptions of the problem.
02

Assumptions and Set-Up

Given the problem, we assume a steady, incompressible, axisymmetric vortex flow, which means the flow properties do not change over time, density is constant, and the flow pattern does not vary in the circumferential direction. In this setup, the velocity components are expressed as: \( v_r = v_r(r) \), \( v_\theta = v_\theta(r) \), and \( w = z \bar{w}(r) \).
03

Simplifying the Navier-Stokes Equations

Start with the Navier-Stokes equation in cylindrical polar coordinates: \( \frac{D\vec{v}}{Dt} = -\frac{1}{\rho}abla p + u abla^2 \vec{v} \).For axisymmetric flow, eliminate terms associated with \( \theta \) derivatives. As \( w = z\bar{w}(r) \), substitute these expressions into the momentum equations.
04

Radial Component Derivation

From the radial momentum equation \( \rho(v_r \frac{dv_r}{dr} + v_\theta^2/r) = -\frac{dp}{dr} + \mu \left(abla^2 v_r - v_r/r^2\right) \).Given \( w = z\bar{w}(r) \), incorporate its expression in the context of an incompressible fluid to isolate the radial velocity terms and constants.
05

Derivation of Axial Pressure Equation

The axial momentum equation becomes \( \rho(v_r \frac{d(z\bar{w})}{dr}) = -\frac{\partial p}{\partial z} + \mu abla^2(z\bar{w}) \).Assume pressure gradient \( \frac{\partial p}{\partial z} = c \rho z \), resulting from integrating other terms with respect to radius while holding axial terms consistent.
06

Final Equations and Constants

For the substitution in axial direction, substitute into the simplified Navier-Stokes for differentials in the context given (steady, axisymmetric), focus on isolating pressures or velocities contributing to each equation, be it radial or axial: 1. \( v_r \frac{d \bar{w}}{d r}+\bar{w}^2-\frac{v}{r} \frac{d}{d r}(r \frac{d \bar{w}}{d r})=C \) 2. \( \frac{\partial p}{\partial z}=c \rho z \).These represent dynamics consistent with incompressible, invariant radial flows given \( \bar{w} \).

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Key Concepts

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

Axisymmetric Vortex
An axisymmetric vortex is a flow pattern where the fluid's velocity components are symmetric around a particular axis. In simpler terms, imagine swirling a straw in a glass of water鈥攅ach particle of water circles around, forming rings centered on the axis of the straw. In this setup:
  • The flow characteristics like velocity, density, and pressure are uniform in the direction around the axis, meaning they don't change as you go around the circle.
  • This simplifies the mathematics as motions or calculations concerning angular displacements can often be ignored.
Understanding this concept makes it easier to deal with complex equations like the Navier-Stokes equations by eliminating redundant terms. In engineering and fluid dynamics, exploring such symmetries helps simplify calculations and leads to more streamlined solutions.
Radial Velocity
Radial velocity is the component of velocity that points directly outwards from or inwards towards the axis of symmetry in a radial direction. It answers the question: "How fast is something moving straight out from the center?"
  • For axisymmetric flows, this velocity depends solely on the radius, denoted as \( v_r = v_r(r) \).
  • This simplification assumes there is no variation in the circumferential or axial directions.
  • It is pivotal when transforming real-world fluid flows into cylindrical or spherical models where radial components become essential in deriving solutions.
In the given problem, radial velocity contributes to how the fluid flow behaves when employed in the Navier-Stokes equations, particularly affecting pressure gradients and flow continuity.
Circumferential Velocity
Circumferential velocity, or tangential velocity, represents the velocity component around the axis - think of it as the speed at which something spins around the center. In a vortex:
  • This velocity is again dependent on the radius, expressed as \( v_\theta = v_\theta(r) \).
  • It characterizes the swirling part of the flow and is crucial in analyzing rotational fluid dynamics.
  • This component impacts the radial forces extensively, especially in circular motions driven by pressure differences.
Understanding circumferential velocity helps further break down complex flow patterns into manageable calculations and analyses, especially crucial in systems modeling phenomena like cyclones or rotating machinery.
Incompressible Flow
An incompressible flow is one wherein the fluid density remains constant as it moves through the flow field. It is an essential assumption in many fluid dynamics problems. This simplification holds true for most liquids and even gases at relatively low speeds.
  • Incompressibility implies that there aren't significant changes in volume or density as a result of pressure variations in the flow.
  • This simplifies the continuity equation immensely, reducing it primarily to the conservation of mass condition \( abla \cdot \vec{v} = 0 \).
  • Such flows are characterized by a high speed of sound within the fluid compared to fluid motion speeds, ensuring minimal compressibility effects.
Applying the incompressible condition ensures that calculations using the Navier-Stokes equations are more straightforward and often more computationally feasible, leading to quicker and more accurate predictions in many practical scenarios.

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Most popular questions from this chapter

Two different fluids flow down an inclined plane at an angle \(\theta\) to the horizontal. A fluid of depth \(d_{2}\), viscosity \(\mu_{2}\), and density \(\rho_{2}\) is below an upper fluid of depth \(d_{1}\), viscosity \(\mu_{1}\), and density \(\rho_{1}\). The upper fluid has a freesurface. Find the velocity and shear stress distribution of both fluids.

Consider a two-dimensional unsteady vortex in a viscous incompressible fluid. Using polar coordinates, transform the Navier-Stokes equations in terms of vorticity \(\zeta_{z}(r, \theta)\) and the stream function \(\psi(r, \theta)\).

A \(20^{\circ}\) roof with respect to the horizontal is to be covered with a \(0.25\) in. layer of liquid roofing asphalt. Find the minimum effective kinematic viscosity of the asphalt in a \(6 \mathrm{hr}\) period if the top surface of the layer is not to move more than \(0.00100\) in. relative to the bottom surface, and the width of the covered surface is \(30 \mathrm{ft}\).

A free vortex is characterized by two conditions: (i) a linear radial distribution of radial velocity, (ii) zero effects of viscosity. Show that from (i) this results in a linear axial distribution of the axial velocity, and from (ii) the tangential velocity \(v_{\theta}=\Gamma_{x} / 2 \pi r\).

If the radial velocity \(v_{r}\) is zero everywhere in the flow field, the vortex flow is said to rotate as a solid body at an angular speed \(\omega\), such that \(\zeta_{z}=2 \omega .\) In addition, the local radial Reynolds number \(N\) is also zero. Prove that the centerline axial velocity \(w_{o}\) cannot vary in the axial direction.
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