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Chapter 8: Problem 2

Assume that a viscous fluid flows in a pipe with a circular cross section of radius \(a\). Choose the z-axis to be the direction of motion of the fluid. Assume that the density of the fluid is constant and that the flow is steady. Starting from the hydrodynamic equations, find the equation relating pressure gradients to velocity gradients in the fluid. Describe what is happening in the pipe. (Note: For steady flow, the velocity is independent of time, but can vary in space. At the walls the velocity is zero due to friction.)

Short Answer

Expert verified
The pressure gradient and velocity gradient relationship in the cylindrical pipe is given by: \frac{\text{d} p}{\text{d} z} = - \rho u \frac{1}{r} \frac{\text{d}}{\text{d} r} \big( r \frac{\text{d} v_z}{\text{d} r} \big).

Step by step solution

01

- Understanding Steady Flow

In steady flow, the velocity of the fluid is constant over time, meaning it does not change with respect to time. However, it can still vary at different points in the space within the pipe.
02

- Applying Navier-Stokes Equations

The Navier-Stokes equations govern the motion of viscous fluid substances. For incompressible steady flow with constant density, the Navier-Stokes equation in the z-direction is: \[ \rho \frac{\text{d} v_z}{\text{d} t} = -\frac{\text{d} p}{\text{d} z} + \rho u abla^2 v_z \] where \( \rho \) is the fluid density, \( u \) is the kinematic viscosity, and \( v_z \) is the velocity in the z-direction.
03

- Considering Steady Flow

Since the flow is steady, the term \( \frac{\text{d} v_z}{\text{d} t} \) is zero. This simplifies the Navier-Stokes equation to: \[ 0 = -\frac{\text{d} p}{\text{d} z} + \rho u abla^2 v_z \]
04

- Cylindrical Coordinates

Convert the equation to cylindrical coordinates. Assuming the velocity \( v_z \) only depends on the radial coordinate \( r \), we have: \[ abla^2 v_z = \frac{1}{r} \frac{\text{d}}{\text{d} r} \bigg( r \frac{\text{d} v_z}{\text{d} r} \bigg) \]
05

- Substituting in Simplified Navier-Stokes Equation

Now substitute \( abla^2 v_z \) into the simplified Navier-Stokes equation: \[ -\frac{\text{d} p}{\text{d} z} = \rho u \bigg( \frac{1}{r} \frac{\text{d}}{\text{d} r} \bigg( r \frac{\text{d} v_z}{\text{d} r} \bigg) \bigg) \]
06

- Rearranging for Pressure Gradient

Isolate \( \frac{\text{d} p}{\text{d} z} \): \[ \frac{\text{d} p}{\text{d} z} = -\rho u \frac{1}{r} \frac{\text{d}}{\text{d} r} \bigg( r \frac{\text{d} v_z}{\text{d} r} \bigg) \]
07

- Boundary Conditions

Consider the boundary conditions. The velocity \( v_z \) is zero at the pipe wall \( r = a \) due to friction, so: \[ v_z(a) = 0 \]
08

- Integrating the Differential Equation

Identify that the velocity \( v_z \) as a function of \( r \) helps to understand the fluid flow more deeply, but the found equation directly relates the pressure gradient to the velocity gradient.
09

- Final Equation

The derived equation connects pressure gradients to velocity gradients in the viscous fluid flow within a cylindrical pipe.

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Key Concepts

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

steady flow
Steady flow occurs when the fluid flow properties at any given point do not change over time. This means that the velocity, pressure, and density remain constant at any point as time progresses. However, this does not mean that the properties are uniform throughout the entire pipe.
In a steady flow within a pipe, the velocity of fluid particles may vary at different locations within the pipe. For instance, due to friction, the fluid velocity at the wall of the pipe is generally zero while it increases towards the center. This concept is crucial for understanding how viscous fluids behave in confined environments like pipes.
Navier-Stokes equations
The Navier-Stokes equations are fundamental in fluid dynamics, governing the motion of fluid substances. These equations describe how the velocity field of a fluid evolves over time under various forces and conditions. For viscous flow, the equation accounts for the internal friction within the fluid, also known as viscosity.
In the context of steady, incompressible flow with constant density, the Navier-Stokes equation in the z-direction simplifies. Since the density is constant and the flow is steady, the related terms can be simplified, leading to a more manageable form which aids in solving for the velocity and pressure within the fluid.
cylindrical coordinates
Cylindrical coordinates are especially useful in problems involving circular or cylindrical geometries, such as fluid flow in pipes. In cylindrical coordinates, the position is described by a radial distance, angle, and height (often denoted as \(r, \theta, z \)) instead of the traditional Cartesian coordinates (x, y, z).
When applied to our problem, the flow velocity is considered a function of the radial distance (r) and remains independent of the angular coordinate (胃). This conversion simplifies the differential equations, enabling us to solve for the velocity distribution more effectively.
pressure gradient
A pressure gradient refers to the rate of change of pressure in the direction of flow. In the context of a pipe, the pressure gradient is critical in driving the fluid from one end to the other. It essentially represents the pushing force that balances the viscous resistance encountered by the fluid.
This relationship between pressure gradient and velocity gradient is one of the key outcomes of our derived equation. Specifically, in viscuous fluid flow, the pressure gradient counteracts the resistive forces due to viscosity that hinder the fluid's motion.
velocity gradient
The velocity gradient in fluid mechanics is a measure of how the velocity changes with respect to position within the flow field. In a pipe, the velocity gradient is particularly important as it provides insights into how the fluid velocity varies radially from the pipe center to the walls.
Specifically, understanding the velocity gradient helps explain the parabolic velocity profile often observed in laminar flow. This gradient is central to the link between pressure gradients and the resulting fluid movement, as captured by the Navier-Stokes equation adapted to our problem.

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