Empirical modeling and sensitivity analysis of pressure rise per wavelength and frictional forces for the peristaltic flow of Bingham plastic fluids: application of response surface methodology

Abstract

Abstract The efficiency of mixed convection peristaltic flow can be investigated through pressure rise per wavelength $$(\Delta P_{{{\uplambda }}} )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and frictional forces ( $${F}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> ). The main aim of the present study is to discover the sensitivity analysis of non-Newtonian fluids using the Bingham plastic fluid model. In order to achieve this objective, we have empirically modeled the pressure rise per wavelength $$(\Delta {P}_{\uplambda })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and frictional forces ( $${F}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> ) as a function varying with leading parameters of problem. The flow problem is governed by three coupled nonlinear partial differential equations. They are reduced to nonlinear coupled ordinary differential equations by using the long wavelength and low Reynolds number approximations. They are solved numerically using MATLAB built-in routine bvp4c to analyze the sensitivity of pressure rise per wavelength ( $$\Delta {P}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:mrow> </mml:math> ) and frictional forces ( $${F}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> ). We first derive the empirical model among each of responses $$\Delta {P}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:mrow> </mml:math> and $${F}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> and physical parameters which govern the flow using response surface methodology. The goodness of fit of empirical model is decided on the basis of coefficient of determination ( $${R}^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> ) obtained from the analysis of variance (ANOVA). The coefficients of determination ( $${R}^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> ) are 99.78% both for $$\Delta {P}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:mrow> </mml:math> and $${F}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> . The higher values of $${R}^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> determine the goodness of fit of empirical model. No correlation has been developed to optimize $$\Delta {P}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:mrow> </mml:math> and $${F}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> in peristaltic flow for Bingham plastic fluids using RSM. The results of sensitivity analysis revealed that $$\Delta {P}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:mrow> </mml:math> and $${F}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> are most sensitive to flow rate ( q ) at all levels such as low (− 1), medium (0) and high (+ 1). The sensitivity of $$\Delta {P}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:mrow> </mml:math> to Bingham number ( Bn ) shows a distinct behavior with varying levels of flow rate ( q ). At low level (− 1) of flow rate ( q ), the sensitivity is positive, and at high level (+ 1) of flow rate ( q ), the sensitivity becomes negative. Conversely, the sensitivity of $${F}_{\uplambda }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> </mml:math> to Bingham number ( Bn</jat

Topics & Concepts