[Paper Review] Similarity Flow Solutions of a Non-Newtonian Power-law Fluid

Mohamed Guedda, Zakia Hammouch|ArXiv.org|Apr 2, 2009

Fluid Dynamics and Turbulent FlowsEngineering19 references18 citations

TL;DR

This paper investigates similarity solutions for steady laminar boundary layer flow of a non-Newtonian power-law fluid over a stretching plate using a generalized similarity transformation. It proves the existence of an infinite number of global unbounded solutions for dilatant fluids (n > 1) and derives their asymptotic behavior, showing that solution growth depends critically on the power-law index n and the stretching exponent m.

ABSTRACT

In this paper we present a mathematical analysis for a steady-state laminar boundary layer flow, governed by the Ostwald-de Wael power-law model of an incompressible non- Newtonian fluid past a semi-infinite power-law stretched flat plate with uniform free stream velocity. A generalization of the usual Blasius similarity transformation is used to find similarity solutions [1]. Under appropriate assumptions, partial differential equations are transformed into an autonomous third-order nonlinear degenerate ordinary differential equation with boundary conditions. Using a shooting method, we establish the existence of an infinite number of global unbounded solutions. The asymptotic behavior is also discussed. Some properties of those solutions depend on the viscosity power-law index.

Motivation & Objective

• To extend the analysis of multiple similarity solutions from Newtonian to non-Newtonian power-law fluids.
• To investigate the existence and asymptotic behavior of global unbounded solutions for dilatant fluids (n > 1).
• To examine how the power-law index n and stretching exponent m influence solution structure and decay/growth rates.
• To generalize the Blasius similarity transformation for power-law fluids under non-Newtonian rheology.

Proposed method

• A generalized similarity transformation is applied to reduce the 2D boundary layer PDEs to a third-order nonlinear ODE with boundary conditions.
• The transformation uses power-law scaling for the stream function and similarity variable, dependent on the stretching exponent m and fluid index n.
• The resulting ODE is of the form (|f''|^{n-1}f'')' + αff'' - m(f')² = 0, with f(0) = a, f'(0) = b, f'(∞) = 0.
• The shooting method is employed to establish existence of global unbounded solutions for m ∈ (−2α, 0) and n > 1.
• Asymptotic analysis is conducted via energy-type functionals and phase-plane techniques to derive long-time behavior.
• A Lyapunov-type function and integral estimates are used to prove convergence to power-law growth at infinity.

Experimental results

Research questions

• RQ1 Do multiple global unbounded similarity solutions exist for non-Newtonian power-law fluids in boundary layer flow over a stretching plate?
• RQ2 How does the power-law index n affect the existence and asymptotic behavior of these solutions?
• RQ3 What is the large-t behavior of solutions when f(t) → ∞ as t → ∞, and how does it depend on m and n?
• RQ4 Can the shooting method be used to prove existence of solutions beyond the classical Blasius case for non-Newtonian fluids?

Key findings

• For m ∈ (−2α, 0) and n > 1, the problem admits an infinite number of global unbounded similarity solutions.
• When f(t) → ∞ as t → ∞, the solution grows asymptotically as f(t) ∼ t^{α/(α−m)}(A + o(1)) for some A > 0.
• The asymptotic growth rate depends on both the power-law index n and the stretching exponent m.
• For the special case m = −α, the solution behaves like f(t) ∼ t^{1/2} at infinity, indicating slower growth.
• The existence of solutions is established via the shooting method, with rigorous analysis of the ODE’s behavior at infinity.
• The asymptotic behavior is derived using integral estimates and Lyapunov-type functionals, ensuring convergence to power-law growth.