This work addresses the construction of slow manifolds for chemically reactive flows. This construction relies on the same decomposition of a local eigensystem that is used in formation of what are known as Intrinsic Low Dimensional Manifolds ͑ILDMs͒. We first clarify the accuracy of the standard ILDM approximation to the set of ordinary differential equations which model spatially homogeneous reactive systems. It is shown that the ILDM is actually only an approximation of the more fundamental Slow Invariant Manifold ͑SIM͒ for the same system. Subsequently, we give an improved extension of the standard ILDM method to systems where reaction couples with convection and diffusion. Reduced model equations are obtained by equilibrating the fast dynamics of a closely coupled reaction/convection/diffusion system and resolving only the slow dynamics of the same system in order to reduce computational costs, while maintaining a desired level of accuracy. The improvement is realized through formulation of an elliptic system of partial differential equations which describe the infinite-dimensional Approximate Slow Invariant Manifold ͑ASIM͒ for the reactive flow system. This is demonstrated on a simple reaction-diffusion system, where we show that the error incurred when using the ASIM is less than that incurred by use of the Maas-Pope Projection ͑MPP͒ of the diffusion effects onto the ILDM. This comparison is further done for ozone decomposition in a premixed laminar flame where an error analysis shows a similar trend.

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Cardiovascular diseases, such as atherosclerosis and aortic valve sclerosis (AVS) are driven by inflammation induced by a variety of stimuli, including low-density lipoproteins (LDL), reactive oxygen species (ROS), infections, mechanical stress, and chemical insults. Fibrosis is the process of compensating for tissue injury caused by chronic inflammation. Fibrosis is initially beneficial and maintains extracellular homeostasis. However, in the case of AVS and atherosclerosis, persistently active resident fibroblasts, myofibroblasts, and smooth muscle cells (SMCs) perpetually remodel the extracellular matrix under the control of autocrine and paracrine signaling from the immune cells. Myofibroblasts also produce pro-fibrotic factors, such as transforming growth factor-β1 (TGF-β1), angiotensin II (Ang II), and interleukin-1 (IL-1), which allow them to assist in the activation and migration of resident immune cells. Post wound repair, these cells undergo apoptosis or become senescent; however, in the presence of unresolved inflammation and persistence signaling for myofibroblast activation, the tissue homeostasis is disturbed, leading to excessive extracellular matrix (ECM) secretion, disorganized ECM, and thickening of the affected tissue. Accumulating evidence suggests that diverse mechanisms drive fibrosis in cardiovascular pathologies, and it is crucial to understand the impact and contribution of the various mechanisms for the control of fibrosis before the onset of a severe pathological consequence.