# Laminar Couette Flow with Imposed Pressure Gradient and Heated Walls

In this application, AcuSolve is used to simulate the flow of a highly viscous fluid between a moving and a stationary plate with an imposed pressure gradient and fixed temperature on the walls. AcuSolve results are compared with analytical results described in White (1991). The close agreement of AcuSolve results with analytical results validates the ability of AcuSolve to model cases with imposed pressure gradients and viscous heating.

## Problem Description

The problem consists of a viscous fluid between two plates in a two-dimensional domain, as shown in the following image, which is not drawn to scale. The domain is 1.0 m high and 1.5 m long. The top plate moves with a constant velocity of 3.0 m/sec and the bottom plate is fixed. To simulate a fully developed flow with a relatively short length domain there is a mean-pressure gradient of -12 Pa/m applied to the fluid in the streamwise direction, defined as a body force acting on the entire fluid. The lower wall is held at a fixed temperature of 273.0 K, while the upper wall is held at a fixed temperature of 274.0 K. The problem is simulated with periodic boundaries in the streamwise direction. The Reynolds number of the flow, based on the distance between the plates is 0.3, indicating laminar flow. The flow field develops from the pressure gradient, the motion of the top plate, and the viscous shear stresses near the plates. The shear stress within the domain causes the temperature to increase within the channel.
The simulation was performed as a two dimensional problem by constructing a volume mesh that contains a single layer of elements extruded in the cross stream direction, normal to the flow plane and by imposing symmetry boundary conditions on the extruded planes. Due to the periodic natural of the flow, a single element extrusion in the streamwise direction is used to reduce the model size.

## AcuSolve Results

The AcuSolve solution converged to a steady state and the results reflect the mean flow conditions. The simulation results show a flow field with a nearly linear distribution of velocity, with the peak at the top wall. The flow develops primarily because of the shear stress acting on the fluid near both the moving plate and the stationary plate. The temperature variation between the two plates develops due to the viscous heating effects that increase the temperature within the channel, with the maximum near the central height of the domain. The following figure shows the temperature distribution between the two plates and velocity vectors indicating the velocity profile.
The following image demonstrates the accuracy of the solution compared to the analytical results as computed by the solution for viscous heating within a channel with a moving upper boundary.

## Summary

In this application, a fully developed laminar flow between two plates with fixed temperature is studied and compared against analytical data. The AcuSolve results compare well with the analytical data for the temperature distribution between the two plates. The simulation indicates that the velocity profile arises due to the combination of the viscous shearing and the velocity of the upper plate. The velocity and temperature profiles computed by AcuSolve demonstrate how highly viscous fluids tend to direct the flow regime. AcuSolve demonstrates the ability to predict the analytical solution for viscous heating between two flat plates with moving and stationary boundaries and an imposed pressure gradient.

## Simulation Settings for Laminar Couette Flow with Imposed Pressure Gradient and Heated Walls

SimLab database file: <your working directory>\couette_laminar_heat\couette_laminar_heat.slb

Global

• Problem Description
• Temperature equation - Advective Diffusive
• Turbulence equation - Laminar
• Auto Solution Strategy
• Relaxation factor - 0.2
• Material Model
• Fluid
• Density - 1.0 kg/m3
• Specific Heat - 1.0 J/kg-K
• Viscosity - 100.0 kg/m-sec
• Body Force
• DP/DL
• Gravity
• Z-component - 18.0 m/sec2

Model

• Volumes
• Fluid
• Element set
• Material model - Fluid
• Body force - DP/DL
• Viscous heating - on
• Surfaces
• Max_X
• Simple Boundary Condition
• Type - Symmetry
• Max_Y
• Simple Boundary Condition
• Type - Wall
• Wall velocity type - Cartesian
• Z-velocity - 3.0 m/s
• Temperature - 274.0 K
• Max_Z
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Min_X
• Simple Boundary Condition
• Type - Symmetry
• Min_Y
• Simple Boundary Condition
• Type - Wall
• Temperature - 273.0 K
• Min_Z
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Periodics
• Periodic 1 all
• Periodic Boundary Conditions
• Type - Periodic

## References

F. M. White. "Viscous Fluid Flow". Section 3-3.2. McGraw-Hill Book Co., Inc. New York. 1991.