| Summary: | Placement of fins in enclosures has promising utilization in advanced technological processes due to their role as heat reducing/generating elements such as in conventional furnaces, economizers, gas turbines, heat exchangers, superconductive heaters and so forth. The advancement in technologies in power engineering and microelectronics requires the development of effective cooling systems. This evolution involves the utilization of fins of significantly variable geometries enclosed in cavities to increase the heat elimination from heat-generating mechanisms. Since fins are considered to play an effective role in the escalation of heat transmission, the current study is conducted to examine the transfer of heat in cavities embedding fins, as well as the effect of a range of several parameters upon the transmission of energy. The following research is supplemented with the interpretation of the thermo-physical aspects of a power-law liquid enclosed in a trapezoidal cavity embedding a U-shaped fin. The Boussinesq approximation is utilized to generate the mathematical attributes of factors describing natural convection, which are then used in the momentum equation. Furthermore, the Fourier law is applied to formulate the streaming heat inside the fluid flow region. The formulated system describing the problem is non-dimensionalized using similarity transformations. The geometry of the problem comprises a trapezoidal cavity with a non-uniformly heated U-shaped fin introduced at the center of the base of the enclosure. The boundaries of the cavity are at no-slip conditions. Non-uniform heating is provided at the walls (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>l</mi><mn>1</mn></msub><mo> </mo><mrow><mi>and</mi><mo> </mo></mrow><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, curves (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mn>1</mn></msub><mo>,</mo><msub><mi>c</mi><mn>2</mn></msub><mo> </mo><mi>and</mi><mo> </mo><msub><mi>c</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and surfaces (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>s</mi><mn>1</mn></msub><mo> </mo><mrow><mi>and</mi><mo> </mo></mrow><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of the fin; the upper wall is insulated whereas the base and sidewalls of the enclosure are kept cold. The solution of the non-dimensionalized equations is procured by the Galerkin finite element procedure. To acquire information regarding the change in displacement w.r.t time and temperature, supplementary quadratic interpolating functions are also observed. An amalgam meshing is constructed to elaborate the triangular and quadrilateral elements of the trapezoidal domain. Observation of significant variation in the flow configurations for a specified range of parameters is taken into consideration i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.5</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>1.5</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mn>10</mn></mrow><mn>4</mn></msup><mo>≤</mo><mi>R</mi><mi>a</mi><mo>≤</mo><msup><mrow><mn>10</mn></mrow><mn>6</mn></msup></mrow></semantics></math></inline-formula>. Furthermore, flow structures in the form of velocity profiles, streamlines, and temperature contours are interpreted for the parameters taken into account. It is deduced from the study that ascending magnitude of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mi>R</mi><mi>a</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> elevates level of kinetic energy and magnitude of heat flux; however, a contrary configuration is encapsulated for the power-law index. Navier–Stokes equations constituting the phenomenon are written with the help of non-dimensionalized stream function, temperature profiles, and vortices, and the solutions are acquired using the finite element method. Furthermore, the attained outcomes are accessible through velocity and temperature profiles. It is worth highlighting the fact that the following analysis enumerates the pseudo-plastic, viscous and dilatant behavior of the fluid for different values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. This study highlights that the momentum profile and the heat transportation increase by increasing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mrow><mi>R</mi><mi>a</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and decline as the viscosity of the fluid increases. Overall, it can be seen from the current study that heat transportation increases with the insertion of a fin in the cavity. The current communication signifies the phenomenon of a power-law fluid flow filling a trapezoidal cavity enclosing a U-shaped fin. Previously, researchers have studied such phenomena mostly in Newtonian fluids, hence the present effort presents novelty regarding consideration of a power-law liquid in a trapezoidal enclosure by the placement of a U-shaped fin.
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