This paper deals with cellular (e.g. LTE) networks that selectively offload the mobile data traffic onto WiFi (IEEE 802.11) networks to improve network performance. We propose the Seamless Internetwork Flow Mobility (SIFM) architecture that provides seamless flow-mobility support using concepts of Software Defined Networking (SDN). The SDN paradigm decouples the control and data plane, leading to a centralized network intelligence and state. The SIFM architecture utilizes this aspect of SDN and moves the mobility decisions to a centralized Flow Controller (FC). This provides a global network view while making mobility decisions and also reduces the complexity at the PGW. We implement and evaluate both basic PMIPv6 and the SIFM architectures by incorporating salient LTE and WiFi network features in the ns-3 simulator. Performance experiments validate that seamless mobility is achieved. Also, the SIFM architecture shows an improved network performance when compared to the base PMIPv6 architecture. A proof-of-concept prototype of the SIFM architecture has been implemented on an experimental testbed. The LTE network is emulated by integrating USRP B210x with the OpenLTE eNodeB and OpenLTE EPC. The WiFi network is emulated using hostapd and dnsmasq daemons running on Ubuntu 12.04. An off-the-shelf LG G2 mobile phone running Android 4.2.2 is used as the user equipment. We demonstrate seamless mobility between the LTE network and the WiFi network with the help of ICMP ping and a TCP chat application.