In this explainer, we will learn how to model the motion of viscous liquids that can have uniform laminar flow or nonuniform turbulent flow.

The laminar flow of a fluid is modeled by considering a flowing fluid to consist of distinct, stable layers.

The simplest example of laminar flow is where the fluid in each layer moves in the same direction at the same constant speed. This is shown in the following figure.

For a flow of this simplicity, there is no obvious distinction between the motion of the fluid and the uniform motion of a solid object.

This highly simplified example is shown in the diagram with no clear boundaries to the fluid. The lack of boundaries is intentional, as the existence of boundaries to a moving fluid will affect the flow of the layers of the fluid if the fluid has nonzero viscosity.

We recall that fluid viscosity is associated with the tendency of a fluid to exert forces on objects that it is in contact with. To use a nonscientific term, viscosity corresponds to βstickiness.β

Let us see what effect the viscosity of a fluid has on the laminar flow of the fluid.

Suppose that we consider a part of a very long rectangular prism containing a flowing fluid. The fluid is modeled as consisting of layers, each of which has a particular velocity. This is shown in the following figure.

It is important to appreciate that although we can now see definite boundaries to the flow parallel to the flow direction, no such boundaries are shown perpendicular to the flow direction. This is done to avoid the need to consider points at which flow starts or stops, as such points would involve changes to the velocity of the flow. The fluid is modeled as flowing continuously without any such starting or stopping points.

If the fluid in the container has nonzero viscosity, then the layers in contact with the walls of the container (layer I and layer V) are acted on by viscous frictional forces between the fluid and the inner surface of the containerβs wall.

Like all frictional forces, viscous frictional forces act on a moving object in the opposite direction to the motion of the object. It may not be obvious to think of a layer of fluid as an object, but we can model it as an object on which frictional forces can act.

The frictional forces on layer I and layer V reduce the velocities of these layers. This is shown in the following figure.

The viscosity of the fluid does not only produce viscous friction where the fluid is in contact with the inner walls of the container.

When two adjacent layers of the fluid do not have the same velocity, this is equivalent to two objects that are in contact moving past each other. There is then viscous friction between adjacent layers of a fluid that have unequal velocities.

We see then that friction between layer I and layer II reduces the velocity of layer II and friction between layer V and layer IV reduces the velocity of layer IV.

It is important to understand that the magnitude of the frictional force on a layer of fluid flowing over some surface depends on the speed that the fluid flows with over the surface.

The friction between layer I and the inner wall of the container is due to the flow of layer I over an unmoving inner wall. This friction reduces the speed of layer I but does not reduce the speed to zero.

The friction between layer II and layer I is due to the flow of layer II over a layer that is moving in the same direction as layer II but slower. This friction reduces the speed of layer II but by less than the speed reduction of layer I. This is shown in the following figure.

In this example, we will assume that the speed at which Layer III moves past the layers adjacent to it is sufficiently small that the speed of layer III is only negligibly reduced.

The speeds of the flow of the layers of the fluid are compared to produce what is called the velocity profile of the flow. This is shown in the following figure by the red dashed curve.

Let us now look at an example that considers the layers of a flowing fluid.

### Example 1: Identifying the Flow Layer Structure for a Viscous Fluid in a Cylindrical Container

The diagrams shown represent the boundaries of the layers of a viscous liquid that move at different speeds when the liquid flows through a pipe. Which of the diagrams correctly represents the boundaries of the layers for the liquid?

### Answer

A viscous liquid experiences viscous friction when it flows past a surface. The friction between the pipe and the liquid produces such friction, reducing the speed of the layers in contact with the inside of the pipe. Layers of reduced speed produce friction with layers adjacent to them, reducing the speeds of those layers. For these reasons, the fluid flows slower near the walls of the pipe and faster farther from the walls of the pipe.

Let us first consider the option for which the layers are perpendicular to the perimeters of the ends of the pipe. This is shown in the following figure.

If this is the correct option, then two things must be true:

- The speed of the fluid in the layer shown in red, , must be unequal to the speed of the fluid in the layer shown in green, .
- The speed of the fluid flow must be constant throughout each layer.

We can see that the layer shown in red is in the widest section of the pipe. The center of this layer is therefore at a greater distance from the walls of the pipe than the center of the green layer. This difference might allow the flow speeds in the layers to differ.

However, for both the red and green layers, the distance of the fluid in the layers from the walls of the pipe changes across the layer in the direction perpendicular to the fluid flow. The fluid should flow faster farther from the walls of the pipe than at the walls of the pipe. It is not reasonable, therefore, that the flow speed be constant throughout the layer.

Let us now consider the option for which the flow layers correspond to concentric cylindrical shells.

The following figure shows possible speeds for the flow in each layer. The speed in each layer corresponds to a different color arrow.

We can see that the flow speeds become greater the greater the distance from the walls of the pipe. This is true for any diameter of the end of the pipe that is chosen.

This is then the correct option.

Let us now look at an example that considers the layers of a fluid that is made to flow by an applied force.

### Example 2: Identifying the Flow Layer Structure for a Viscous Fluid Made to Flow by an Applied Force

The diagrams shown represent the boundaries of the layers of a viscous liquid, which move at different speeds when the liquid flows through a container. Which of the diagrams correctly represents the boundaries of the layers for the liquid flowing through a container in which the top of the container is moved relative to the base of the container?

### Answer

The diagrams above show a small region in which flow occurs. The following figures show that region. No definite boundaries are shown perpendicular to the flow direction. In the figure below, the plate is not in contact with the fluid, and hence, the fluid is static.

The plate is then placed in contact with the top layer of the fluid.

The plate is then moved horizontally by the action of a constant force.

The friction between the top layer of the fluid and the plate produces a force on the plate in the opposite direction to the constant applied force on the plate.

In accordance with Newtonβs third law of motion, as the fluid exerts a force on the plate, the plate exerts a force of equal magnitude on the fluid in the opposite direction. We see then that the motion of the plate results in the top layer of the fluid moving in the same direction as the plate.

It is important to realize that only the top layer of the fluid is made to move. There is no equivalent motion induced on the sides or bottom of the fluid.

From this, we see that any motion of layers of the fluid other than the top layer must be due to the motion of the top layer. The motion of the top layer results in friction between the top layer and the layer below. This is repeated for the layer below that and so on until all the layers reach their equilibrium speeds.

When the plate moves over the top layer of the fluid, the fluid in the top layer moves slower than the plate. The friction between the top layer and the layer below therefore is less than that between the plate and the top layer. We see then that the layer below the top layer has less friction acting on it than the top layer, so it does not move as fast as the top layer.

It must then be the case that the speeds of the layers of fluid become less the greater their distance from the plate.

Let us now consider possible velocity profiles for the two flow layer arrangements shown in the question. The flow speed in each layer corresponds to a different color arrow.

We can see that for the concentric rectangular-prism arrangement, the flow speeds at the top and the bottom of the fluid are equal. This cannot be correct, as the flow speeds of layers must decrease with distance from the plate. This is consistent with the stacked rectangular-prism arrangement, which is then the correct option.

It is important to recognize that the constant force due to a moving plate does not indefinitely produce a constant acceleration of the fluid moved by the plate. The fluid accelerates until its layers each reach their equilibrium speed. The work done by the plate after the fluid stops accelerating is therefore dissipated. This should be expected, as we can recall that frictional forces are dissipative.

A plate being pushed along the surface of a very long rectangular prism of a fluid can be used to determine the viscosity of the fluid. To be more exact, the dynamic viscosity of the fluid can be so determined.

The value of the dynamic viscosity of a fluid can be calculated using the following variables:

- , the force applied to the plate
- , the area of the plate
- , the change of velocity between adjacent layers of fluid
- , the height of each layer of fluid

It is important to note that this model of fluid flow assumes that there is an equal change in velocity between adjacent fluid layers. The velocity profile of such a flow would therefore look something like the one shown in the following figure.

### Formula: The Dynamic Viscosity of a Fluid That Is Made to Move by Friction from a Rectangular Plate

The dynamic viscosity, , of a fluid is given by the formula where is the constant force applied to the plate, is the area of the plate, is the height of a fluid layer, and is the difference in flow velocity between adjacent fluid layers.

The unit of dynamic viscosity is given by

The standard value of the dynamic viscosity of water is Paβ s.

Let us now look at an example in which the dynamic viscosity of a fluid is determined.

### Example 3: Calculating the Dynamic Viscosity of a Fluid

A thin plate of mass 2.5 g is pushed by a constant force , moving at a constant 1.32 cm/s over the surface of a viscous liquid that is 2.5 mm deep, as shown in the diagram. The speeds of the layers of the liquid between the plate and the lowest layer of the liquid are shown in the diagram. The liquid in contact with the top and bottom plates moves at the same speed as the plates move. What is the dynamic viscosity of the liquid? Give your answer in scientific notation to one decimal place.

### Answer

The value of the dynamic viscosity of a fluid can be calculated using the following variables:

- , the force applied to the plate
- , the area of the plate
- , the change of velocity between adjacent layers of fluid
- , the height of each layer of fluid

The following formula is used:

The value of the force is given as

The area of the plate is the product of the lengths of its sides and is given by

There are 5 layers of fluid, and the total height of these layers is 2.5 mm. The height of a layer is therefore given by

Taking the speeds of two adjacent layers, we see that the difference between them is given by

The value of is therefore given by

This gives us

This is close to the standard value for the dynamic viscosity of water.

Up until this point, the explainer has only considered laminar fluid flow.

We can also consider turbulent flow. In turbulently flowing fluids, the layers of the fluid cross each other, as shown in the following figure.

Turbulence occurs more easily when fluids have greater flow speeds. We have seen that the frictional forces between layers depend on the speeds at which layers flow past each other. The greater the speed of the bulk flow of a fluid, the greater the difference in speeds between layers and, hence, the greater the frictional forces between layers.

We can model a layer of a fluid as consisting of parts of the layer. Let us consider one such part of a layer that has layers on either side of it, one slower and one faster. This is shown in the following figure.

We see that the faster layer pulls the part of the middle layer in the direction of the bulk flow, while the slower layer pulls the part in the opposite direction. The action of forces in opposite directions at the top and bottom of the middle layer produces a turning effect, as shown in the following figure.

The magnitude of the turning effect will generally be different for different layers, and so a layer may turn sufficiently to cross into another layer. Turbulence is then observed.

The greater the viscosity of a fluid, the more the layers of the fluid tend to turn by similar amounts. This means that the more viscous a fluid is, the more difficult it is to make the fluid flow turbulently.

Let us now look at an example in which the turbulent flow of a viscous fluid is considered.

### Example 4: Explaining Transitions between Laminar and Turbulent Flow in a Viscous Fluid

The diagram shows a cross section of the streamlines of a liquid flowing around a cylinder. Before the fluid reaches the cylinder, the flow is laminar. After the flow passes behind the cylinder, there is a region of turbulent flow. However, after the region of turbulent flow, the flow becomes laminar again. Which of the following statements most correctly explains why the turbulent flow returns to a laminar flow condition?

- In turbulent flow regions, the kinetic energy of the fluid is dissipated by viscous friction, reducing the average speed of the fluid. As the fluid slows, it becomes less turbulent.
- The flowing liquid expands in the turbulent region. The fluid compresses as it exits the turbulent region.
- The dynamic viscosity of the liquid is increased in the turbulent region. The fluid decreases in viscosity as it exits the turbulent region.

### Answer

The viscosity of a liquid does not vary with the flow speed of a fluid. The option requiring changes in viscosity can be rejected.

Liquids are effectively incompressible. The amount by which they can be compressed can be neglected for most purposes, including transitions between laminar and turbulent flow. The option requiring compression and expansion of the liquid can be rejected.

We can recall that friction is a dissipative force. For a fluid made to move by a constant force from a plate, the fluid will reach an equilibrium flow speed while the constant force acts on it rather than accelerate without limit. From this, we can see that viscous friction will dissipate the energy of flowing fluids.

In turbulently flowing fluids, the mixing of layers of fluids of different speeds results in greater viscous friction between the mixing layers than would occur in laminar flow.

The dissipation of energy by the turbulent fluid reduces the flow speeds of the layers. As layers flow past each other slower, they exert less frictional force on each other, and this reduces the turbulence produced by such forces.

We see then that the first option is the correct one.

Let us now summarize what has been learned in this explainer.

### Key Points

- A fluid in laminar flow can be modeled as consisting of parallel layers that exert viscous frictional forces on each other.
- Viscous frictional forces dissipate the energy of a flowing fluid.
- A fluid that is made to flow by a constant force applied to a plate moved across the surface of the fluid has a dynamic viscosity, , given by where is the constant force applied to the plate, is the area of the plate, is the height of a fluid layer, and is the difference in flow velocity between adjacent fluid layers.
- The unit of dynamic viscosity is pascal-seconds (Paβ s).
- The standard value of the dynamic viscosity of water is Paβ s.
- Viscous friction between layers of fluid can cause the layers to mix turbulently.
- Dissipation of the energy of a turbulently flowing fluid tends to restore the laminar flow of the fluid.