Modified Power Law Fluids Research Articles

Investigation of Velocity Profile in Time Dependent Boundary Layer Flow of a Modified Power-Law Fluid of Fourth Grade

This paper deals with the investigation of time dependent boundary layer flow of a modified power-law fluid of fourth grade on a stretched surface with an injection or suction boundary condition. The fluid model is a mixture of fourth grade and power-law fluids in which the fluid may display shear thickening, shear thinning or normal stress textures. By using the scaling and translation transformations which is a type of Lie Group transformation, time dependent boundary layer equations are reduced into two alternative ordinary differential equations systems (ODEs) with boundary conditions. During this reduction, special Lie Group transformations are used for translation, scaling and combined transformation. Numerical solutions have been carried out for the ordinary differential equations for various fluids and boundary condition parameters. As a result of numerical analysis, it is observed that the boundary layer thickness decreases as the power-law index value increases. It was also observed that for the fourth-grade fluid parameter, as the parameter increases, the boundary layer thickness decreases while the velocity in the <i>y</i> direction increases.

Pressure drop predictions for laminar pipe flow of carreau and modified power law fluids

Fluids used in many chemical and petroleum industrial processes are often described as complex systems that exhibit non‐Newtonian character. Due to the complex nature of non‐Newtonian fluids, universally accepted flow equations are unavailable. This paper uses the Carreau and modified power law–Cross (MPL‐Cross) rheological models to derive pressure drop–flow rate relationships under laminar flow condition. The aim is to develop simple but effective equations that enable accurate estimation of pressure drop for flow in any pipe size using viscometric data. The new relationships compare favourably with experimental data. Furthermore, new Reynolds number and generalized Reynolds number expressions have been defined.

Smoothed particle hydrodynamics simulation of non-Newtonian moulding flow

Smoothed particle hydrodynamics (SPH) has been widely applied in simulating fluid flow because of its attractive properties, for example, it is fully Lagrangian and mesh free. However, this method usually uses the explicit method to solve the conservation equations and in this form it is only suitable to momentum dominated flows with low viscosity. In polymer processing, the fluid is non-Newtonian with high viscosity, O(10 3 ) to O(10 4 ) Pa-s say, and the pressure is high as O(10 6 ) to O(10 10 ) Pa. The algorithm of the standard SPH is infeasible in this case, because only very small time steps can be used for a stable simulation. We have developed an implicit SPH for non-Newtonian flow, which is completely matrix free, to solve the equation system iteratively and robustly. The artificial pressure is introduced between particles to stabilize the SPH system avoiding the tensile instability. The fluid is compressible under high pressure. Realistic state equations for polymers, such as the Tait and SSY [16] equations are adopted to describe the density/pressure relations. The method is finally applied to the simulation of moulding flow of a modified power law fluid with the zero shear rate viscosity of 1.22 × 10 4 Pa-s, Reynolds number of 3 × 10 −4 to 6 × 10 −5 and the highest pressure of O(10 8 ) to O(10 10 ) Pa.

Heat transfer in laminar flows of extended modified power law fluids in rectangular ducts

This paper reports the results of a numerical study of the Nusselt number for the H1 and T boundary conditions in laminar, fully-developed flows of pseudoplastic and dilatant fluids in rectangular ducts. Equations for the apparent viscosity that span the entire shear rate range were utilized and the Nusselt numbers were calculated as a function of a shear rate parameter that determines the flow regime where the duct is operating. Numerical results for the Nusselt number in all flow regimes are included along with correlation equations. Finally, errors associated with applying power law solutions are discussed.

Pressure drop predictions for laminar flows of extended modified power law fluids in rectangular ducts

This paper reports on a numerical study of the friction factor-modified Reynolds number product, f × Re M, for fully developed, laminar flows of pseudoplastic and dilatant fluids in rectangular ducts. Constitutive equations for the apparent viscosity that span from the low shear rate Newtonian region through the high shear rate Newtonian region were utilized, and a shear rate parameter was defined that determines the flow regime where the duct is operating. Numerical results for f × Re M in all flow regimes are included along with correlation equations. Errors associated with applying power law solutions to flows of weakly non-Newtonian fluids are discussed.

Experimental and numerical investigations of pressure drop in a rectangular duct with modified power law fluids

Numerical solutions are presented for fully developed laminar flow for a modified power law fluid (MPL) in a rectangular duct. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian behavior at low shear rates, through a transition region, to power law behavior at higher shear rates. The analysis identified a dimensionless shear rate parameter which, for a given set of operating conditions, specifies where in the shear rate range a particular system is operating, i.e. in the Newtonian, transition, or power law regions. The numerical results of the friction factor times Reynolds number for the Newtonian and power law region are compared with previously published results showing agreement within 0.05% in the Newtonian region, and 0.9% and 5.1% in the power law region. Rheological flow curves were measured for three CMC-7H4 solutions and were found to be well represented by the MPL constitutive equation. The friction factor times Reynolds number values were measured in the transition region for which previous measurements were unavailable. Good agreement was found between experiment and calculation thus confirming the validity of the analysis.

Laminar Heat Transfer With Viscous Dissipation and Fluid Axial Heat Conduction for Modified Power Law Fluids Flowing in Parallel Plates With One Plate Moving

Using the fully developed laminar velocity distributions obtained by applying the modified power-law model proposed by Irvine and Karni, the thermal-entrance-region heat transfer of non-Newtonian fluids flowing in parallel plates with one plate moving is investigated taking into account both viscous dissipation and fluid axial heat conduction for two kinds of thermal boundary conditions, namely, constant temperature and constant heat flux at the moving wall. The energy equation subject to a constant temperature at upstream infinity, fully developed temperature profile at downstream infinity and the appropriate thermal boundary conditions at the upper and lower walls is numerically solved by the finite difference method as an elliptic type problem. The effects of the moving plate velocity, rheological properties, Brinkman number and Peclet number on the temperature distribution and Nusselt numbers are discussed for both Newtonian and pseudoplastic fluids

Predictions of pressure drop for modified power law fluids in conduits of three different cross-sectional shapes

Numerical solutions are presented for fully developed laminar flow for a modified power law fluid (MPL) in conduits of arbitrary cross sections. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian behavior at low shear rates, through a transition region, to power law behavior at higher shear rates. The analysis identified a dimensionless shear rate parameter which, for a given set of operating conditions, specifies where in the shear rate range a particular system is operating, i.e. in the Newtonian, transition, or power law regions. The numerical results of the friction factor times Reynolds number for the Newtonian and power law region are compared with previously published results showing agreement within 0.05% in the Newtonian region, and 0.9% and 5.1% in the power law region.

Anomalous predictions of pressure drop and heat transfer in ducts of arbitrary cross-section with modified power law fluids

The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer solutions.

Predictions of heat transfer and pressure drop for a modified power law fluid flow in a square duct

Numerical Solutions for the Nusselt Numbers (CHF and CWT) and the Friction Factor times Reynolds Number have been obtained for fully developed laminar flow of a MPL (Modified Power Law) fluid within a square duct. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian at low shear rates through a transition region to power law behavior at higher shear rates. A shear rate parameter is identified, which allows the prediction of the shear rate range for a specified set of operating conditions. Numerical results of the Nusselt numbers (CHF and CWT) and the Friction factors times Reynolds number for the Newtonian and power law regions are compared with previous published results, showing agreement with 0.02% in Newtonian region and 4.0% in power law region.

Experimental and numerical study of friction factor for a modified power law fluid in a rectangular duct

Numerical solutions are presented for fully developed laminar flow for a modified power law fluid (MPL) in a rectangular duct. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian behavior at low shear rates, through a transition region, to power law behavior at higher shear rates. The analysis identified a dimensionless shear rate parameter that for a given set of operating conditions specifies where in the shear rate range a particular system is operating, that is, in the Newtonian, transition, or power law regions. The numerical results of the friction factor times Reynolds number for the Newtonian and power law region are compared with previously published results showing agreement within 0.05% in the Newtonian region and 0.9 and 5.1% in the power law region, respectively. Rheological flow curves were measured for three CMC-7H4 solutions and were found to be well represented by the MPL constitutive equation. The friction factor times Reynolds number values were measured in the transition region for which previous measurements were unavailable. Good agreement was found between experiment and calculation, thus confirming the validity of the analysis.

Experimental and numerical study of friction factor for a modified power law fluid in a rectangular duct

Experimental and numerical study of friction factor for a modified power law fluid in a rectangular duct

Predictions of pressure drop and heat transfer in concentric annular ducts with modified power law fluids

The following numerical analysis examines the pressure drop and heat transfer characteristics associated with flows of modified power law fluids through concentric annular ducts. The flows considered are steady, fully developed and laminar with constant properties, except for viscosity. The boundary conditions for the heat transfer problem are constant heat flux at the inner tube, and an adiabatic outer tube. Results are presented for the friction factor-Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter. These are expressed as correlation equations which give a convenient description of the results throughout the complete domain of the analysis: from low shear rates where the viscosity is Newtonian, to the higher shear rates where the behavior is purely power law, and at the intermediate shear rates where the viscosity is transitional.

Similitude considerations in laminar flow of modified power law fluids in circular ducts

A numerical solution is presented for the friction factorReynolds number relation for a fluid which is Newtonian at low shear rates, power law at high shear rates with a transition region at intermediate shear rates.