This study aims to evaluate how meloxicam diffuses into the synovial fluid when delivered through a thermoresponsive hydrogel microneedle patch and determine if it is a feasible alternative to current treatments. Rheumatoid arthritis causes inflammation in the joints. Pain relief for this type of arthritis requires sustained drug delivery that targets COX-2 inhibition. To model this process, we simplified the complex 3D structure of a joint into an axisymmetric 2D geometry. We focused on the key skin layers that influence drug diffusion. Joints are enclosed in a cavity containing synovial fluid, encapsulated by the synovial membrane and further layered with skin and tissue. Our patch features microneedles that bypass the outer skin layer, allowing the drug to diffuse toward the synovial fluid. Using heat and mass transfer principles, our model evaluates how inflammation-driven temperature changes activate the hydrogel and initiate drug release. We analyzed the extent of meloxicam penetration through the anatomical layers, identifying key barriers to diffusion and assessing whether enough drug reaches the synovial membrane to provide effective pain relief. Upon modeling our transdermal patch, we determined that meloxicam diffusion stops at the hypodermis layer and fails to reach the synovial fluid. These results indicate that a passive transdermal diffusion alone is not sufficient in treating rheumatoid arthritis.