# Creeping Flow in Fluids: Examples and Analysis

### Key Takeaways

• Creeping flow describes fluid flow in which inertia is insignificant.

• Creeping flow at zero Reynolds number is what we call Stokes flow.

• Compared to general fluid flow, creeping flow is easier to solve mathematically due to the absence of non-linear or advective terms.

The flow of high-viscosity fluids such as paints, heavy oils, and food-processing materials are examples of creeping flow

Do you remember learning about creepers and climbers in elementary school science class? We classify plants as creepers or climbers based on whether they grow horizontally or vertically along the soil. Creeping movements are seen in living and nonliving things, and the main characteristic of a “creeper” is gradual movement.

We can relate the gradual flow in fluids to creeping movement, provided certain conditions are met. A significant example of creeping flow is seen in the movement of heavy oils, honey, etc. These fluids flow with difficulty due to viscosity. There are so many applications in which we make use of fluids that showcase creeping flow. Let’s explore what this flow is through a few examples.

## Creeping Flow in Fluids

Creeping flow describes fluid flow in which inertia is insignificant. The viscous and pressure forces exerted on the fluid are greater than the inertia. Fluids with high viscosity have difficulty flowing and they usually travel in a creeping motion. Even though the inertia is negligible in these fluids, they are dominated by internal friction. Fluids that creep in flow are non-turbulent and never make spinning vortices. Creeping flow fluids creep around obstacles rather than become turbulent.

Creeping flow is also known as Stokes flow. In the creeping motion of fluids, viscous forces dominate over advective inertial forces. In fluids, the creeping flow is a laminar type of flow where streamlines are parallel to each other. The velocity of creeping flow is very low.

## Reynolds Number and Creeping Flow

Reynolds number is a dimensionless number that gives the relation between advective inertial forces and viscous forces. Reynolds number is directly proportional to the density of a fluid and the velocity of the fluid and is inversely proportional to the dynamic viscosity of the fluid.  It is the value of the Reynolds number that distinguishes between the laminar type and turbulent type of flow in fluids. For Reynolds numbers below 2000, the flow type is laminar. The higher the Reynolds number, the more the flow becomes chaotic. When the Reynolds number is greater than 2000, the flow type is turbulent.

For creeping flow, the Reynolds number is less than 1 (Re<<1). When Reynolds number is less than unity, inertial effects can be ignored, taking into account only the viscous resistance. The fluid flow is non-chaotic in creeping motion. Fluid flow that travels in a creeping motion is time reversible.

## Navier-Stokes Equation and Creeping Flow

To be precise, the creeping flow at zero Reynolds number is what we call Stokes flow. The Reynolds number is small in microfluidics devices and can be classified as creeping flow. The creeping flow in fluids is viscous flow and can be mathematically expressed using the Navier-Stokes equation.

In the creeping flow observed in microfluidics devices, the left side terms of the Navier-Stokes equation, which gives the rate of change of momentum of the fluid, is neglected. The momentum terms in the Navier-Stokes equation of creeping flow fluids are non-linear and neglecting these terms linearizes the equation. When the Reynolds number of a given creeping fluid flow is small, it is necessary to consider the convective terms in the Navier-Stokes equation.

## Examples of Creeping Flow

One of the applications utilizing creeping flow is hydrodynamic lubrication. Hydrodynamic lubrication utilizes the properties of highly viscous fluids and their flow through small channels to bring effective lubrication. The flow of the lubricant fluid through the gaps between bearings and races is governed by the balance between viscous friction and the pressure gradient. The heavy pressure exerted in the bearing gaps helps prevent surfaces from rubbing each other, which is effective in causing hydrodynamic lubrication.

 Fun fact: Hydrodynamic lubrication is also found in animals. In animals, synovial fluids confine in the small gap between the surfaces of the bones, preventing the contact, wear, and tear of joints.

Applications based on the creeping flow of fluids is not limited to:

• Flow of high-viscosity fluids such as paints, heavy oils, and food-processing materials
• Extrusion of melts
• Seepage in sand or rock formation
• Dust particle settling
• Any small object moving in fluids
• Locomotion of microorganisms in fluids
• Flow of groundwater or oil through small channels or cracks

Compared to general fluid flow, creeping flow is easier to solve mathematically due to the absence of non-linear or advective terms. Cadence’s suite of software can help you find solutions for creeping flow as well as complicated general fluid flows. With these tools, it is easier to run CFD simulations in complex fluid-dependent systems that facilitate fluid flow modeling.