SHORT NOTE Viscous dissipation pattern in incompressible Newtonian simple shear zones: an analytical model Soumyajit Mukherjee • Kieran F. Mulchrone Received: 11 October 2012 / Accepted: 23 February 2013 / Published online: 12 March 2013 Ó Springer-Verlag Berlin Heidelberg 2013 Abstract An analytical model of shear heating in an inclined simple shear zone with Newtonian rheology under a reverse shear sense and an upward resultant pressure gradient is presented. Neglecting a number of secondary factors, the shear heat is expressed as a function of the total slip rates at the boundaries, pressure gradient, dip and thickness of the shear zone, and density, viscosity, and thermal conductivity of the rock. A quartic temperature profile develops with a point of maximum temperature near the bottom part of the shear zone in general. The profile is parabolic if pressure gradient vanishes leading to a Couette flow. The profile attains a bell shape if there is no slip at the boundaries, i.e., a true Pouseille flow. The present model of shear heating is more applicable in subduction channels and some extruding salt diapirs where the rheol- ogy is Newtonian viscous and pressure gradient drives extrusion. Keywords Viscous dissipation  Newtonian fluid  Simple shear  Poiseuille flow Introduction Similar to ‘frictional heating’ in the brittle regime (Cam- acho et al. 2001; Ben-Zion and Sammis 2013), mechanical work that converts into heat during ductile shear is known as ‘shear heating’, ‘viscous dissipation’, ‘viscous heating’, or ‘strain heating’ (Brun and Cobbold 1980; Nabelek and Liu 2004; Nabelek et al. 2010). Although deciphering shear heating from field-scale observations is not well established (Brun and Cobbold 1980; Vauchez et al. 2012), the heating could affect the thermal evolution of any nearby sedimentary basins. Such evolution paths are of practical interest, especially in the petroleum geosciences (Starin et al. 2000; Souche, internet reference). Shear heating has been reported to attain significant magnitude when (1) the overthrust unit is thick (C5 km; Brewer 1981); (2) the shear stress and the strain rate are high (*1,000 MPa, 10 -11 -10 -12 s -1 ; Molnar and Eng- land 1990); (3) the slip/convergence rate is high ( [ 1 cm year -1 ; Graham and England 1976; Burg and Gerya 2005; more precisely *4 cm year -1 : Nabelek and Liu 2004) as expected in subduction zones (review by Seyfert 1987); (4) deformation that takes place at a shallower depth (Hochstein and Regenauer 1998); (5) the shear zone material is ‘cold’ and rigid (Leloup et al. 1999); and (6) the sheared rock has a high viscosity and a high magnitudes of activation energy (Regenauer-Lieb and Yuen 2003). Shear heating may (Harris et al. 2000) or may not (Nabelek et al. 2010) be dependent on the distribution pattern of radioac- tive isotopes in the shear zones. Shear zones can act as paths and source of melts by shear heating (Nabelek et al. 2010; review by Clark et al. 2011; but see Camacho et al. 2001 for counter arguments), which in collisional regimes is leucogranites (Nabelek and Liu 1999; Nabelek and Liu 2004; Nabelek et al. 2011). Had there been partial melting (‘thermal softening’: Brun and Cobbold 1980), further shear heating would have decreased, and the buoyant melt would have extruded leading to further heating. Subsequent melting could extrude the melt in a second pulse, and the cycle may S. Mukherjee (&) Department of Earth Sciences, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, Maharashtra, India e-mail: soumyajitm@gmail.com K. F. Mulchrone Department of Applied Mathematics, School of Mathematical Sciences, University College, Cork, Ireland 123 Int J Earth Sci (Geol Rundsch) (2013) 102:1165–1170 DOI 10.1007/s00531-013-0879-3