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Chapter 5: Problem 29

A two-dimensional steady flow in the \(r \theta\) plane has a \(\theta\) -component of the velocity given by $$ v_{\theta}=10\left(1+\frac{1}{r^{2}}\right) \cos \theta-\frac{15}{r} $$ A boundary condition that must be satisfied by the flow field is \(v_{r}(1, \theta)=0 .\) Determine the \(r\) -component of the velocity as a function of \(r\) and \(\theta\).

Short Answer

Expert verified
The radial velocity component is \( v_r = 10 \sin \theta - \frac{10 \sin \theta}{r^2} \).

Step by step solution

01

Understand the problem

We are given the velocity component \(v_{\theta}\) in a polar coordinate system. We need to find the \(r\)-component of the velocity, \(v_{r}\), which is generally done by solving the continuity equation in polar coordinates for an incompressible flow.
02

Recall the continuity equation for incompressible flows

The continuity equation for incompressible flow in polar coordinates \((r, \theta)\) is given by: \[ \frac{1}{r} \frac{\partial (r v_{r})}{\partial r} + \frac{1}{r} \frac{\partial v_{\theta}}{\partial \theta} = 0. \] This is derived from the conservation of mass principle, assuming density is constant.
03

Substitute the given \(v_{\theta}\) into the continuity equation

We need to substitute the given expression for \(v_{\theta}\) into the continuity equation: \(v_{\theta}=10\left(1+\frac{1}{r^{2}}\right) \cos \theta-\frac{15}{r}\). Compute \(\frac{\partial v_{\theta}}{\partial \theta}\).
04

Differentiate \(v_{\theta}\) with respect to \(\theta\)

Calculate \(\frac{\partial v_{\theta}}{\partial \theta}\): \[ \frac{\partial v_{\theta}}{\partial \theta} = 10\left(1+\frac{1}{r^{2}}\right)(-\sin \theta) = -10\left(1+\frac{1}{r^{2}}\right) \sin \theta. \]This result will be used in the continuity equation.
05

Incorporate derivatives into the continuity equation

Plug \(\frac{\partial v_{\theta}}{\partial \theta}\) into the continuity equation:\[ \frac{1}{r} \frac{\partial (r v_{r})}{\partial r} + \frac{1}{r} \left(-10\left(1+\frac{1}{r^{2}}\right) \sin \theta\right) = 0. \]This simplifies to:\[ \frac{1}{r} \frac{\partial (r v_{r})}{\partial r} = \frac{10 \sin \theta}{r} + \frac{10 \sin \theta}{r^3}. \]
06

Integrate with respect to \(r\) to solve for \(v_r\)

Set up the integral to solve for \(r v_r\):\[ \frac{\partial (r v_{r})}{\partial r} = 10 \sin \theta + \frac{10 \sin \theta}{r^2}. \]Integrating both sides with respect to \(r\):\[ r v_{r} = 10 r \sin \theta + 10 \sin \theta \int \frac{1}{r^2} \; dr. \]The integration leads to:\[ r v_{r} = 10 r \sin \theta - \frac{10 \sin \theta}{r} + C, \]where \(C\) is a constant of integration.
07

Apply the boundary condition

Use the boundary condition \(v_{r}(1, \theta) = 0\) to find \(C\). Substitute \(r = 1\) into the equation:\[ 1 \cdot 0 = 10 \cdot 1 \cdot \sin \theta - 10 \sin \theta + C. \]Simplifying gives \(C = 0\).
08

Solve for \(v_r\) in terms of \(r\) and \(\theta\)

With \(C = 0\), divide by \(r\) to solve for \(v_r\):\[ v_{r} = 10 \sin \theta - \frac{10 \sin \theta}{r^2}. \]This expression gives the \(r\)-component of the velocity as a function of \(r\) and \(\theta\).

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Key Concepts

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

Fluid Mechanics
Fluid mechanics is a branch of physics that studies the behavior of fluids, both at rest and in motion. It includes the study of gases and liquids and can be applied to things from predicting weather patterns to designing aircraft. Fluid mechanics combines principles from various disciplines to understand how fluids interact with their environment. This field is essential in engineering, environmental science, and even biomedical applications. When dealing with fluid mechanics, understanding various properties like pressure, density, and velocity is crucial.
  • **Pressure** refers to the force exerted by a fluid on a surface.
  • **Density** describes how compact the mass in a fluid is.
  • **Velocity** pertains to the speed and direction in which a fluid is flowing.
Each of these properties can significantly affect fluid behavior, especially under varying conditions. For instance, in designing pipelines, engineers must account for factors such as fluid viscosity and temperature, which can change how fluids flow. Ultimately, fluid mechanics helps us understand complex systems and solve practical problems involving fluid interactions.
Continuity Equation
The continuity equation is a fundamental principle in fluid mechanics. It arises from the conservation of mass, asserting that mass cannot be created or destroyed within a closed system. When dealing with incompressible flows, the equation ensures that the volume flow rate remains constant from one cross-section to another. In mathematical terms, the continuity equation for incompressible flows in polar coordinates is expressed as:\[ \frac{1}{r} \frac{\partial (r v_{r})}{\partial r} + \frac{1}{r} \frac{\partial v_{\theta}}{\partial \theta} = 0. \]This equation emphasizes the relationship between radial and angular components of velocity in a flow, ensuring that the quantity of fluid flowing in is equal to the quantity flowing out. For practical applications, understanding this helps in calculating various flow parameters in pipelines, nozzles, or any fluid transport system. It is also critical in simulations used to predict fluid movements in engineering tasks. Applying this equation requires careful assessment of the flow characteristics and proper integration of the velocity components.
Incompressible Flow
Incompressible flow refers to fluid motion where the fluid density remains constant. This is a typical assumption in many engineering problems when dealing with liquids or gases moving at low speeds. Incompressible flow simplifies many calculations since the density term vanishes, and the focus shifts to understanding the velocity and pressure variations in the flow. This is particularly important in incompressible materials like water, where changes in pressure and temperature don't significantly alter density. However, in practice, assumptions about compressibility depend on the context of the fluid and the conditions under which it flows.
  • For liquids such as water, the incompressible assumption is valid for a wide range of applications.
  • For gases, the assumption is valid only under specific conditions, typically at low speeds where compressibility effects are negligible.
Incompressible flow models are widely used in engineering to design systems like pumps, turbines, and various fluid transport systems. By assuming constant density, engineers can focus on optimizing other performance traits like energy efficiency and material selection.
Velocity Components
Velocity components in fluid mechanics are crucial to understanding fluid flow characteristics. In polar coordinates, which are useful in systems with circular symmetry, velocity is often represented by its components: radial (\( v_r \)) and angular (\( v_\theta \)). Each plays a different role in how the fluid moves through the space. - **Radial Velocity Component** - Indicates motion towards or away from the center of rotation. - Is vital in applications like centrifugal pumps or rotating equipment. - **Angular Velocity Component** - Describes circular motion around a center point. - Important for designing systems like fans or turbines.Understanding these components allows engineers to predict how fluids behave in different conditions and to design efficient systems for fluid transport and manipulation. Computation of these components typically involves solving equations that consider the conservation laws of mass and momentum, tailored to the specifics of the flow and boundary conditions. This helps in precisely controlling how fluid moves in applications such as mixing, heating, or cooling.

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Most popular questions from this chapter

A viscous fluid is contained between two parallel plates spaced \(30 \mathrm{~mm}\) apart as shown in Figure \(5.52 .\) The bottom plate remains stationary, the top plate moves at \(2 \mathrm{~m} / \mathrm{s},\) and the velocity of the fluid varies linearly between the bottom and top plate. (a) Determine the rate of angular rotation and the vorticity of a fluid element contained between the plates and (b) determine the rate of angular deformation of the fluid element.

A particular flow field experienced by an incompressible fluid is described in polar coordinates by the stream function \(\psi(r, \theta)=-3 r \sin \theta+2 \theta \mathrm{m}^{2} / \mathrm{s},\) where \(r\) is in meters and \(\theta\) is in radians. (a) Determine the \(r\) and \(\theta\) components of the velocity vector as a function of \(r\) and \(\theta\). (b) At the point \(r=2 \mathrm{~m}\) and \(\theta=\pi / 6 \mathrm{rad}=30^{\circ}\), determine the \(r\) and \(\theta\) components of the velocity and the \(x\) and \(y\) components of the velocity, where \(x\) and \(y\) are the Cartesian coordinates.

Flow of an incompressible fluid is in the \(x y\) plane, and the \(y\) component of the velocity, \(v\), is given by \(v=2 y\). (a) Determine the required functional form of the velocity field. (b) Give a particular velocity field that satisfies the required functional form.

SAE 50 oil at \(20^{\circ} \mathrm{C}\) flows between two horizontal parallel plates spaced \(7 \mathrm{~mm}\) apart. The plates are \(3 \mathrm{~m}\) long and \(2 \mathrm{~m}\) wide, and the oil flow is driven by a pressure differential between the inflow and outflow sections. The inflow pressure is \(200 \mathrm{kPa}\) gauge, and the outflow pressure is atmospheric. (a) Estimate the volume flow rate of the oil between the plates. (b) Estimate the average flow velocity between the plates. (c) Validate your results using a Reynolds number analysis.

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.
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