MHD stagnation point flow on a shrinking surface with hybrid nanoparticles and melting phenomenon effects

Purpose This study aims to explore the stagnation flow over a shrinking surface in a hybrid nanofluid consists of Al 2 O 3 and Cu nanoparticles. Here, the flow is subjected to the magnetohydrodynamic (MHD) and the melting phenomenon effects. Design/methodology/approach The similarity variables are used to gain the similarity equations. These equations are solved via the bvp4c solver. The effects of several physical parameters on the flow and the thermal characteristics of the hybrid nanofluid are analysed and discussed. Later, the temporal stability analysis is used to determine the stability of the dual solutions obtained as time evolves. Findings Results show that two solutions are found for the limited range of the stretching/shrinking parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>λ</m:mi></m:math> , and then these solutions are terminated at <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>λ</m:mi><m:mo>=</m:mo><m:msub><m:mrow><m:mi>λ</m:mi></m:mrow><m:mrow><m:mi>c</m:mi></m:mrow></m:msub></m:math> . The rise of the melting parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>M</m:mi><m:mi>e</m:mi></m:math> from 0 to 2 contributes to enhance 109.63% of the local Nusselt number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msubsup><m:mrow><m:mtext>Re</m:mtext></m:mrow><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msubsup><m:msub><m:mrow><m:mi>N</m:mi><m:mi>u</m:mi></m:mrow><m:mrow><m:mi>x</m:mi></m:mrow></m:msub></m:math> and 3.30% of the skin friction coefficient <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msubsup><m:mrow><m:mtext>Re</m:mtext></m:mrow><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mn>1</m:mn><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msubsup><m:msub><m:mrow><m:mi>C</m:mi></m:mrow><m:mrow><m:mi>f</m:mi></m:mrow></m:msub></m:math> . Contrarily, the values of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msubsup><m:mrow><m:mtext>Re</m:mtext></m:mrow><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msubsup><m:msub><m:mrow><m:mi>N</m:mi><m:mi>u</m:mi></m:mrow><m:mrow><m:mi>x</m:mi></m:mrow></m:msub></m:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msubsup><m:mrow><m:mtext>Re</m:mtext></m:mrow><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mn>1</m:mn><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msubsup><m:msub><m:mrow><m:mi>C</m:mi></m:mrow><m:mrow><m:mi>f</m:mi></m:mrow></m:msub></m:math> decline by 25.04% and 5.58%, respectively, as the magnetic parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>M</m:mi><m:mi>g</m:mi></m:math> increases from 0 to 0.3. Additionally, Al 2 O 3 -Cu/water has the highest values of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msubsup><m:mrow><m:mi>R</m:mi><m:mi>e</m:mi></m:mrow><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mn>1</m:mn><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msubsup><m:msub><m:mrow><m:mi>C</m:mi></m:mrow><m:mrow><m:mi>f</m:mi></m:mrow></m:msub></m:math> and the lowest values of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msubsup><m:mrow><m:mi>R</m:mi><m:mi>e</m:mi></m:mrow><m:mrow><m:mi>x</m:mi></m:mrow><m:mrow><m:mo>-</m:mo><m:mn>1</m:mn><m:mo>/</m:mo><m:mn>2</m:mn></m:mrow></m:msubsup><m:msub><m:mrow><m:mi>N</m:mi><m:mi>u</m:mi></m:mrow><m:mrow><m:mi>x</m:mi></m:mrow></m:msub></m:math> . Lastly, it is found that the first solution is physically stable as time evolves. Originality/value This paper considers the MHD stagnation point flow of a hybrid nanofluid over a shrinking surface with the melting phenomenon effects. Most importantly, it is shown that there exist dual solutions within a specific range of the physical parameters. Besides, the temporal stability of the solutions is also reported in this study. The finding can contribute to foresee the flow and thermal behaviours in industrial applications. Also, the suitable values of parameters can be determined to avoid misjudgement in flow and heat transfer analysis.