Continuous nutrient delivery to cells by pressure-driven flow is desirable for cell culture in lab-on-a-chip devices. An innovative method is proposed to generate an induced pressure-driven flow by using an electrokinetically-driven pump in a H-shape microchannel. A three-dimensional numerical model is developed to study the effectiveness of the proposed mechanism. It is shown that the average velocity of the generated pressure-driven flow is linearly dependent on the applied voltage. Considering the culture of a multicellular tumour spheroid (MTS) in such a microfluidic system, numerical simulations based on EMT6/Ro tumour cells is performed to find the effects of the nutrient distribution (oxygen and glucose), bulk velocity and channel size on the cell growth. Using an empirical formula, the growth of the tumour cell is studied. For low nutrient concentrations and low speed flows, it is found that the MTS grows faster in larger channels. It is also shown that, for low nutrient concentrations, a higher bulk liquid velocity provide better environment for MTS to grow. For lower velocities, it is found that the local MTS growth along the flow direction deviates from the average growth.