Thermal investigation of Cu-TiO ₂ /sodium alginate Casson hybridized nanofluid flow via a stretching (shrinking) disk: Stability analysis

This study evaluates the thermal characteristics of the boundary layer flow of Casson hybrid nanofluid via a stretchable (or shrinkable) disk, taking into consideration the effects of radiative heat flux and velocity slippage. Casson liquid is often used to characterize non-Newtonian liquids that exhibit properties of a normal liquid but are influenced by the presence of two types of nanoparticles, Cu and TiO 2 . Similarity transformation is employed to derive similarity formula, which are computationally resolved utilizing the shooting technique with the assistance of Maple software. There are two solutions for both the stretchable and shrinkable states of a disk, however when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , only one solution is seen. No solution is found for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mrow> <mml:mi>λ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , whereas the obtained solutions are divided at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi>λ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> . Lobatto-III bvp4c method in MATLAB software is applied to evaluate the stability of the occurrence of dual solutions. Moreover, the analysis demonstrates that the first solution is both stable and physically possible. The Casson, shrinking, and slippage parameters decrease the fluid velocity and surface frictional profile. Furthermore, an augmentation in nanoparticle volume fractions leads to an escalation in skin friction, whereas the Nusselt number decreases for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and augments for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> .