/*! 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 77 A Bingham plastic is a fluid in ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 77

A Bingham plastic is a fluid in which the stress \(\tau\) is related to the rate of strain \(d u / d y\) by $$\tau=\tau_{0}+\mu\left(\frac{d u}{d y}\right)$$ where \(\tau_{0}\) and \(\mu\) are constants. Consider the flow of a Eingham plastic between two fixed, horizontal, infinitely wide, flat plates. For fully developed flow with \(d u / d x=0\) and \(d p / d x\) constant. find \(u(y)\).

Short Answer

Expert verified
The velocity profile \(u(y)\) for Bingham plastic flow between two plates is \(u(y) = 0\).

Step by step solution

01

Start with the constitutive equation

Start with the constitutive equation for a Bingham plastic which states \(\tau=\tau_{0}+\mu\left(\frac{d u}{d y}\right)\) where \(\tau_0\) is the yield stress and \(\mu\) is the dynamic viscosity.
02

Set the momentum balance for steady, one directional,fully developed flow.

The momentum balance equation for steady, one-dimensional and fully developed flow is given by \(\frac{d \tau}{d y}=0\). Substitute the equation of stress from step 1 into this equation, to get \(\frac{d \tau_0}{d y} + \mu \frac{d^2 u}{d y^2} = 0\). Since \(\tau_0\) is a constant, it's derivative with respect to \(y\) becomes zero. So, the simplified momentum balance equation is \(\frac{d^2 u}{d y^2} = 0\).
03

Solve the differential equation.

Solving the differential equation \(\frac{d^2 u}{d y^2} = 0\), gives, \(\frac{d u}{d y} = C_1\), where \(C_1\) is the constant of integration. Integrate further with respect to \(y\), to get \(u = C_1 y + C_2\), where \(C_1\) and \(C_2\) are integration constants to be determined from boundary conditions.
04

Evaluate boundary conditions

In the scenario of flow between two fixed plates, at \(y = 0\) and \(y = h\) the velocity \(u = 0\) (no slip condition at boundaries). Use these conditions to evaluate the constants \(C_1\) and \(C_2\). This leads to \(C_2 = 0\) and \(C_1 = 0\).
05

Formulate the result.

Substituting the constants \(C_1 = 0\) and \(C_2 = 0\) back into the equation derived in Step 3. Therefore, \(u(y)=0\) is the velocity profile for Bingham plastic flow between two plates.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Yield Stress
When studying Bingham plastics, yield stress is a crucial concept. Yield stress, denoted as \(\tau_0\), describes the minimum stress needed for a fluid to start flowing. For instance, imagine ketchup. When you tilt the bottle, it doesn鈥檛 flow until it reaches a certain angle. That angle represents the yield stress. Without reaching the yield stress, a Bingham plastic behaves like a solid.

In practical terms, this means:
  • A Bingham plastic requires an initial force to begin movement.
  • Once the yield stress is overcome, the fluid behaves more like a traditional liquid.
  • This understanding helps in designing systems where such fluids are used, ensuring that no unexpected resistance occurs unless the yield stress is surpassed.
Dynamic Viscosity
Dynamic viscosity, denoted as \(\mu\), measures a fluid's resistance to flow once it has started moving. Unlike yield stress, which is an initial threshold, dynamic viscosity describes how the fluid behaves after this point. In simpler terms, it鈥檚 how "thick" or "sticky" a fluid feels as it moves.

For Bingham plastics, once the yield stress is overcome:
  • The flow becomes proportional to the applied force minus yield stress.
  • Higher dynamic viscosity means more force is needed to maintain flow rate.
  • This property is essential in fluids like cement and drilling muds used in various industries, ensuring efficiency and effectiveness in their applications.
Momentum Balance Equation
The momentum balance equation is fundamental in fluid dynamics, describing how forces affect a fluid's motion. For the case of Bingham plastics, it's expressed as: \(\frac{d \tau}{d y} = 0\). This equation simplifies to \(\frac{d^2 u}{d y^2} = 0\) for situations like flow between two flat plates.

Here's what this equation implies:
  • The derivative condition suggests a linear relationship in stress across the flow direction.
  • When solved fully, it sets the stage for determining velocity profiles, explaining how speed changes with position.
  • In engineering, these equations are used to predict and manage flow behaviour, guaranteeing that systems behave as expected under various conditions.
Velocity Profile
The velocity profile of a fluid describes how velocity changes across its flow space 鈥� in our case, between two plates. For a Bingham plastic experiencing constant conditions, the flow profile becomes simple. Given the steps outlined in the problem, the solution yields \(u(y) = 0\).

Key points to remember are:
  • With no external pressure gradient or due to boundary conditions (like no-slip conditions), the velocity remains zero.
  • This scenario shows that without applied force overcoming yield stress, Bingham plastics behave solid-like.
  • Understanding velocity profiles helps engineers and scientists optimize systems handling Bingham plastics, ensuring they design systems where steady flow is achievable under controlled conditions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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:

The streamlines for an incompressible, inviscid, two-dimensional $$v_{\theta}=K r$$ where \(K\) is a constant. (a) For this rotational flow, determine, if possible, the stream function. (b) Can the pressure difference between the origin and any other point be determined from the Bernoulli equation? Explain.

By considering the rotational equilibrium of a fluid mass element, show that \(\tau_{x y}=\tau_{y x}\).

The velocity potential for a given two-dimensional flow field is $$\phi=\left(\frac{5}{3}\right) x^{3}-5 x y^{2}$$ Show that the continuity equation is satisfied and determine the corresponding stream function.

An infinitely long, solid, vertical cylinder of radius \(R\) is located in an infinite mass of an incompressible fluid. Start with the Navier-Stokes equation in the \(\theta\) direction and derive an expression for the velocity distribution for the steady-flow case in which the cylinder is rotating about a fixed axis with a constant angular velocity \(\omega\). You need not consider body forces. Assume that the flow is axisymmetric and the fluid is at rest at infinity.
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Astrophysics

Read Explanation

Dynamics

Read Explanation

Wave Optics

Read Explanation

Linear Momentum

Read Explanation

Kinematics Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.