Laminar Flows Between Two Parallel Plates

In the case of Newtonian and non-Newtonian liquids the channel has a 2h = 0.01 m height and a 0.2 m length, the inlet for these liquids have standard ambient temperature (293.2 K) and a uniform inlet velocity profile of uav = 0.01 m/s (entrance disturbances are neglected). The geometry model is shown in Figure 1.The inlet pressure is not known beforehand, since it will be obtained from the calculations in accordance with the specified channel exit pressure of 1 atm. The fluids pass through the channel due to a pressure gradient.

Since the Reynolds number based on the channel height is equal to about Re2h = 100, the flow is laminar.

As for the liquids, let us consider water as a Newtonian liquid and four non-Newtonian liquids having identical density of 1000 kg/m3, identical specific heat of 4200 J/(kg·K) and identical thermal conductivity of 10 W/(m·K), but obeying different non-Newtonian liquid laws available in Flow Simulation.

The considered non-Newtonian liquids' models and their governing characteristics are presented in Table 1. These models are featured by the function connecting the flow shear stress (τ) with the flow shear rate (), i.e. , or, following Newtonian liquids, the liquid dynamic viscosity (η) with the flow shear rate (), i.e. :

1. the Herschel-Bulkley model: , where K is the consistency coefficient, n is the power-law index, and τ0 is the yield stress (a special case with n = 1 gives the Bingham model);

2. the power-law model: , i.e., , which is a special case of Herschel-Bulkley model with τ0 = 0;

3. the Carreau model: , , where η∞ is the liquid dynamic viscosity at infinite shear rate, i.e. the minimum dynamic viscosity, η0 is the liquid dynamic viscosity at zero shear rate, i.e. the maximum dynamic viscosity, K1 is the time constant, n is the power-law index (this model is a smooth version of the power-law model).

In accordance with the well-known theory presented in Ref. 21, after some entrance length, the flow profile u(y) becomes fully developed and invariable. It can be determined from the Navier-Stokes x-momentum equation corresponding to this case in the coordinate system shown in Figure 1 (y = 0 at the channel's center plane, is the longitudinal pressure gradient along the channel, in the flow under consideration).

As a result, the fully developed u(y) profile for a Newtonian fluid has the following form:

,

where η is the fluid dynamic viscosity and h is the half height of the channel,

,

where uav is the flow's mass-average velocity defined as the flow's volume flow rate divided by the area of the flow passage cross section.

For a non-Newtonian liquid described by the power-law model the fully developed u(y) profile and the corresponding pressure gradient can be determined from the following formulas:

, .

For a non-Newtonian liquid described by the Herschel-Bulkley model the fully developed u(y) profile can be determined from the following formulas:

at ,

at ,

where the unknown wall shear stress τw is determined numerically by solving the nonlinear equation

,

e.g. with the Newton method, as described in this validation. The corresponding pressure gradient is determined as .

For a non-Newtonian liquid described by the Carreau model the fully developed u(y) profile can not be determined analytically in an explicit form, so in this validation example it is obtained by solving the following parametric equation:

,

,

where p is a free parameter varied within the ±pmax range,

,

The p value is varied to satisfy .

The corresponding pressure gradient is equal to .

The geometry model for the 2D calculation is shown in Figure 2. The boundary conditions are specified as mentioned above and the initial conditions coincide with the inlet boundary conditions. The results of the calculations performed with Flow Simulation at result resolution level 5 are presented in Figure 2 – Figure 7. The channel exit u(y) profile and the channel P(x) profile were obtained along the sketches shown by green lines in Figure 2.

From Figure 4, Figure 6, and Figure 8 you can see that for all the liquids under consideration, after some entrance length of about 0.03 m, the pressure gradient governing the channel pressure loss becomes constant and nearly similar to the theoretical predictions. From Figure 3, Figure 5, and Figure 7 you can see that the fluid velocity profiles at the channel exit obtained from the calculations are close to the theoretical profiles.

In the case of compressible liquids the channel has the height of 2h = 0.001 m and the length of 0.5 m, the liquids at its inlet had standard ambient temperature (293.2 K) and a uniform inlet velocity profile corresponding to the specified mass flow rate of  = 0.01 kg/s.

The inlet pressure is not known beforehand, since it will be obtained from the calculations as providing the specified mass flow rate under the specified channel exit pressure of 1 atm. (The fluids pass through the channel due to the pressure gradient).

Let us consider two compressible liquids whose density obeys the following laws:

• the power law: ,

  where ρ0, P0, B and n are specified: ρ0 is the liquid's density under the reference pressure P0, B and n are constants,

• the logarithmic law: ,

  where ρ0, P0, B and C are specified: ρ0 is the liquid's density under the reference pressure P0, B and C are constants.

In this validation example these law's parameters values have been specified as ρ0 =103 kg/m3, P0 = 1 atm, B = 107 Pa, n = 1.4, C = 1, and these liquids have the 1 Pa·s dynamic viscosity.

Since this channel is rather long, the pressure gradient along it can be determined as ,

where η is the liquids' dynamic viscosity, is the liquid mass flow rate, S is the channel's width, ρ is the liquid density.

Therefore, by substitution the compressible liquids' ρ(P) functions, we obtain the following equations for determining P(x) along the channel:

• for the power-law liquid:

  its solution is ,

  where C1 is a constant determined from the boundary conditions;

• for the logarithmic-law liquid: ,

  this equation is solved numerically.

Both the theoretical P(x) distributions and the corresponding distributions computed within Flow Simulation on a 5x500 computational mesh are presented in Figure 9 and Figure 10. It is seen that the Flow Simulation calculations agree with the theoretical distributions.